Reference documentation for deal.II version Git c3b52c4 2017-08-23 10:06:38 -0600
The step-32 tutorial program

Table of contents
  1. Introduction
  2. The commented program
  1. Results
  2. The plain program

This program was contributed by Martin Kronbichler, Wolfgang Bangerth, and Timo Heister.

This material is based upon work partly supported by the National Science Foundation under Award No. EAR-0426271 and The California Institute of Technology; and in a continuation by the National Science Foundation under Award No. EAR-0949446 and The University of California – Davis. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation, The California Institute of Technology, or of The University of California – Davis.

The work discussed here is also presented in the following publication: M. Kronbichler, T. Heister, W. Bangerth: High Accuracy Mantle Convection Simulation through Modern Numerical Methods, Geophysical Journal International, 2012, 191, 12-29. [DOI]

The continuation of development of this program has led to the much larger open source code Aspect (see http://aspect.dealii.org/ ) which is much more flexible in solving many kinds of related problems.

Introduction

This program does pretty much exactly what step-31 already does: it solves the Boussinesq equations that describe the motion of a fluid whose temperature is not in equilibrium. As such, all the equations we have described in step-31 still hold: we solve the same general partial differential equation (with only minor modifications to adjust for more realism in the problem setting), using the same finite element scheme, the same time stepping algorithm, and more or less the same stabilization method for the temperature advection-diffusion equation. As a consequence, you may first want to understand that program — and its implementation — before you work on the current one.

The difference between step-31 and the current program is that here we want to do things in parallel, using both the availability of many machines in a cluster (with parallelization based on MPI) as well as many processor cores within a single machine (with parallelization based on threads). This program's main job is therefore to introduce the changes that are necessary to utilize the availability of these parallel compute resources. In this regard, it builds on the step-40 program that first introduces the necessary classes for much of the parallel functionality, and on step-55 that shows how this is done for a vector-valued problem.

In addition to these changes, we also use a slightly different preconditioner, and we will have to make a number of changes that have to do with the fact that we want to solve a realistic problem here, not a model problem. The latter, in particular, will require that we think about scaling issues as well as what all those parameters and coefficients in the equations under consideration actually mean. We will discuss first the issues that affect changes in the mathematical formulation and solver structure, then how to parallelize things, and finally the actual testcase we will consider.

Using the "right" pressure

In step-31, we used the following Stokes model for the velocity and pressure field:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& -\rho \; \beta \; T \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0. \end{eqnarray*}

The right hand side of the first equation appears a wee bit unmotivated. Here's how things should really be. We need the external forces that act on the fluid, which we assume are given by gravity only. In the current case, we assume that the fluid does expand slightly for the purposes of this gravity force, but not enough that we need to modify the incompressibility condition (the second equation). What this means is that for the purpose of the right hand side, we can assume that \(\rho=\rho(T)\). An assumption that may not be entirely justified is that we can assume that the changes of density as a function of temperature are small, leading to an expression of the form \(\rho(T) = \rho_{\text{ref}} [1-\beta(T-T_{\text{ref}})]\), i.e. the density equals \(\rho_{\text{ref}}\) at reference temperature and decreases linearly as the temperature increases (as the material expands). The force balance equation then looks properly written like this:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& \rho_{\text{ref}} [1-\beta(T-T_{\text{ref}})] \mathbf{g}. \end{eqnarray*}

Now note that the gravity force results from a gravity potential as \(\mathbf g=-\nabla \varphi\), so that we can re-write this as follows:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& -\rho_{\text{ref}} \; \beta\; T\; \mathbf{g} -\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \nabla\varphi. \end{eqnarray*}

The second term on the right is time independent, and so we could introduce a new "dynamic" pressure \(p_{\text{dyn}}=p+\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \varphi=p_{\text{total}}-p_{\text{static}}\) with which the Stokes equations would read:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p_{\text{dyn}} &=& -\rho_{\text{ref}} \; \beta \; T \; \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0. \end{eqnarray*}

This is exactly the form we used in step-31, and it was appropriate to do so because all changes in the fluid flow are only driven by the dynamic pressure that results from temperature differences. (In other words: Any contribution to the right hand side that results from taking the gradient of a scalar field have no effect on the velocity field.)

On the other hand, we will here use the form of the Stokes equations that considers the total pressure instead:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& \rho(T)\; \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0. \end{eqnarray*}

There are several advantages to this:

Note
There is, however, a downside to this procedure. In the earth, the dynamic pressure is several orders of magnitude smaller than the total pressure. If we use the equations above and solve all variables to, say, 4 digits of accuracy, then we may be able to get the velocity and the total pressure right, but we will have no accuracy at all if we compute the dynamic pressure by subtracting from the total pressure the static part \(p_\text{static}=\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \varphi\). If, for example, the dynamic pressure is six orders of magnitude smaller than the static pressure, then we need to solve the overall pressure to at least seven digits of accuracy to get anything remotely accurate. That said, in practice this turns out not to be a limiting factor.

The scaling of discretized equations

Remember that we want to solve the following set of equations:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& \rho(T) \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0, \\ \frac{\partial T}{\partial t} + {\mathbf u} \cdot \nabla T - \nabla \cdot \kappa \nabla T &=& \gamma, \end{eqnarray*}

augmented by appropriate boundary and initial conditions. As discussed in step-31, we will solve this set of equations by solving for a Stokes problem first in each time step, and then moving the temperature equation forward by one time interval.

The problem under consideration in this current section is with the Stokes problem: if we discretize it as usual, we get a linear system

\begin{eqnarray*} M \; X = \left(\begin{array}{cc} A & B^T \\ B & 0 \end{array}\right) \left(\begin{array}{c} U \\ P \end{array}\right) = \left(\begin{array}{c} F_U \\ 0 \end{array}\right) = F \end{eqnarray*}

which in this program we will solve with a FGMRES solver. This solver iterates until the residual of these linear equations is below a certain tolerance, i.e. until

\[ \left\| \left(\begin{array}{c} F_U - A U^{(k)} - B P^{(k)} \\ B^T U^{(k)} \end{array}\right) \right\| < \text{Tol}. \]

This does not make any sense from the viewpoint of physical units: the quantities involved here have physical units so that the first part of the residual has units \(\frac{\text{Pa}}{\text{m}} \text{m}^{\text{dim}}\) (most easily established by considering the term \((\nabla \cdot \mathbf v, p)_{\Omega}\) and considering that the pressure has units \(\text{Pa}=\frac{\text{kg}}{\text{m\; s}^2}\) and the integration yields a factor of \(\text{m}^{\text{dim}}\)), whereas the second part of the residual has units \(\frac{\text{m}^{\text{dim}}}{\text{s}}\). Taking the norm of this residual vector would yield a quantity with units \(\text{m}^{\text{dim}-1} \sqrt{\left(\text{Pa}\right)^2 + \left(\frac{\text{m}}{\text{s}}\right)^2}\). This, quite obviously, does not make sense, and we should not be surprised that doing so is eventually going to come back hurting us.

So why is this an issue here, but not in step-31? The reason back there is that everything was nicely balanced: velocities were on the order of one, the pressure likewise, the viscosity was one, and the domain had a diameter of \(\sqrt{2}\). As a result, while nonsensical, nothing bad happened. On the other hand, as we will explain below, things here will not be that simply scaled: \(\eta\) will be around \(10^{21}\), velocities on the order of \(10^{-8}\), pressure around \(10^8\), and the diameter of the domain is \(10^7\). In other words, the order of magnitude for the first equation is going to be \(\eta\text{div}\varepsilon(\mathbf u) \approx 10^{21} \frac{10^{-8}}{(10^7)^2} \approx 10^{-1}\), whereas the second equation will be around \(\text{div}{\mathbf u}\approx \frac{10^{-8}}{10^7} \approx 10^{-15}\). Well, so what this will lead to is this: if the solver wants to make the residual small, it will almost entirely focus on the first set of equations because they are so much bigger, and ignore the divergence equation that describes mass conservation. That's exactly what happens: unless we set the tolerance to extremely small values, the resulting flow field is definitely not divergence free. As an auxiliary problem, it turns out that it is difficult to find a tolerance that always works; in practice, one often ends up with a tolerance that requires 30 or 40 iterations for most time steps, and 10,000 for some others.

So what's a numerical analyst to do in a case like this? The answer is to start at the root and first make sure that everything is mathematically consistent first. In our case, this means that if we want to solve the system of Stokes equations jointly, we have to scale them so that they all have the same physical dimensions. In our case, this means multiplying the second equation by something that has units \(\frac{\text{Pa\; s}}{\text{m}}\); one choice is to multiply with \(\frac{\eta}{L}\) where \(L\) is a typical lengthscale in our domain (which experiments show is best chosen to be the diameter of plumes — around 10 km — rather than the diameter of the domain). Using these numbers for \(\eta\) and \(L\), this factor is around \(10^{17}\). So, we now get this for the Stokes system:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& \rho(T) \; \mathbf{g}, \\ \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0. \end{eqnarray*}

The trouble with this is that the result is not symmetric any more (we have \(\frac{\eta}{L} \nabla \cdot\) at the bottom left, but not its transpose operator at the top right). This, however, can be cured by introducing a scaled pressure \(\hat p = \frac{L}{\eta}p\), and we get the scaled equations

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla \left(\frac{\eta}{L} \hat p\right) &=& \rho(T) \; \mathbf{g}, \\ \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0. \end{eqnarray*}

This is now symmetric. Obviously, we can easily recover the original pressure \(p\) from the scaled pressure \(\hat p\) that we compute as a result of this procedure.

In the program below, we will introduce a factor EquationData::pressure_scaling that corresponds to \(\frac{\eta}{L}\), and we will use this factor in the assembly of the system matrix and preconditioner. Because it is annoying and error prone, we will recover the unscaled pressure immediately following the solution of the linear system, i.e., the solution vector's pressure component will immediately be unscaled to retrieve the physical pressure. Since the solver uses the fact that we can use a good initial guess by extrapolating the previous solutions, we also have to scale the pressure immediately before solving.

Changes to the Stokes preconditioner and solver

In this tutorial program, we apply a variant of the preconditioner used in step-31. That preconditioner was built to operate on the system matrix M in block form such that the product matrix

\begin{eqnarray*} P^{-1} M = \left(\begin{array}{cc} A^{-1} & 0 \\ S^{-1} B A^{-1} & -S^{-1} \end{array}\right) \left(\begin{array}{cc} A & B^T \\ B & 0 \end{array}\right) \end{eqnarray*}

is of a form that Krylov-based iterative solvers like GMRES can solve in a few iterations. We then replaced the exact inverse of A by the action of an AMG preconditioner \(\tilde{A}\) based on a vector Laplace matrix, approximated the Schur complement \(S = B A^{-1} B^T\) by a mass matrix \(M_p\) on the pressure space and wrote an InverseMatrix class for implementing the action of \(M_p^{-1}\approx S^{-1}\) on vectors. In the InverseMatrix class, we used a CG solve with an incomplete Cholesky (IC) preconditioner for performing the inner solves.

An observation one can make is that we use just the action of a preconditioner for approximating the velocity inverse \(A^{-1}\) (and the outer GMRES iteration takes care of the approximate character of the inverse), whereas we use a more or less exact inverse for \(M_p^{-1}\), realized by a fully converged CG solve. This appears unbalanced, but there's system to this madness: almost all the effort goes into the upper left block to which we apply the AMG preconditioner, whereas even an exact inversion of the pressure mass matrix costs basically nothing. Consequently, if it helps us reduce the overall number of iterations somewhat, then this effort is well spent.

That said, even though the solver worked well for step-31, we have a problem here that is a bit more complicated (cells are deformed, the pressure varies by orders of magnitude, and we want to plan ahead for more complicated physics), and so we'll change a few things slightly:

As a final note, let us remark that in step-31 we computed the Schur complement \(S=B A^{-1} B^T\) by approximating \(-\text{div}(-\eta\Delta)^{-1}\nabla \approx \frac 1{\eta} \mathbf{1}\). Now, however, we have re-scaled the \(B\) and \(B^T\) operators. So \(S\) should now approximate \(-\frac{\eta}{L}\text{div}(-\eta\Delta)^{-1}\nabla \frac{\eta}{L} \approx \left(\frac{\eta}{L}\right)^2 \frac 1{\eta} \mathbf{1}\). We use the discrete form of the right hand side of this as our approximation \(\tilde S\) to \(S\).

Changes to the artificial viscosity stabilization

Similarly to step-31, we will use an artificial viscosity for stabilization based on a residual of the equation. As a difference to step-31, we will provide two slightly different definitions of the stabilization parameter. For \(\alpha=1\), we use the same definition as in step-31:

\begin{eqnarray*} \nu_\alpha(T)|_K = \nu_1(T)|_K = \beta \|\mathbf{u}\|_{L^\infty(K)} h_K \min\left\{ 1, \frac{\|R_1(T)\|_{L^\infty(K)}}{c(\mathbf{u},T)} \right\} \end{eqnarray*}

where we compute the viscosity from a residual \(\|R_1(T)\|_{L^\infty(K)}\) of the equation, limited by a diffusion proportional to the mesh size \(h_K\) in regions where the residual is large (around steep gradients). This definition has been shown to work well for the given case, \(\alpha = 1\) in step-31, but it is usually less effective as the diffusion for \(\alpha=2\). For that case, we choose a slightly more readable definition of the viscosity,

\begin{eqnarray*} \nu_2(T)|_K = \min (\nu_h^\mathrm{max}|_K,\nu_h^\mathrm{E}|_K) \end{eqnarray*}

where the first term gives again the maximum dissipation (similarly to a first order upwind scheme),

\begin{eqnarray*} \nu^\mathrm{max}_h|_K = \beta h_K \|\mathbf {u}\|_{L^\infty(K)} \end{eqnarray*}

and the entropy viscosity is defined as

\begin{eqnarray*} \nu^\mathrm{E}_h|_K = c_R \frac{h_K^2 \|R_\mathrm{2,E}(T)\|_{L^\infty(K)}} {\|E(T) - \bar{E}(T)\|_{L^\infty(\Omega)} }. \end{eqnarray*}

This formula is described in the article J.-L. Guermond, R. Pasquetti, & B. Popov, 2011. Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248–4267. Compared to the case \(\alpha = 1\), the residual is computed from the temperature entropy, \(E(T) = \frac12 (T-T_m)^2\) with \(T_m\) an average temperature (we choose the mean between the maximum and minimum temperature in the computation), which gives the following formula

\begin{eqnarray*} R_\mathrm{E}(T) = \frac{\partial E(T)}{\partial t} + (T-T_\mathrm{m}) \left(\mathbf{u} \cdot \nabla T - \kappa \nabla^2 T - \gamma\right). \end{eqnarray*}

The denominator in the formula for \(\nu^\mathrm{E}_h|_K\) is computed as the global deviation of the entropy from the space-averaged entropy \(\bar{E}(T) = \int_\Omega E(T) d\mathbf{x}/\int_\Omega d\mathbf{x}\). As in step-31, we evaluate the artificial viscosity from the temperature and velocity at two previous time levels, in order to avoid a nonlinearity in its definition.

The above definitions of the viscosity are simple, but depend on two parameters, namely \(\beta\) and \(c_R\). For the current program, we want to go about this issue a bit more systematically for both parameters in the case \(\alpha =1\), using the same line of reasoning with which we chose two other parameters in our discretization, \(c_k\) and \(\beta\), in the results section of step-31. In particular, remember that we would like to make the artificial viscosity as small as possible while keeping it as large as necessary. In the following, let us describe the general strategy one may follow. The computations shown here were done with an earlier version of the program and so the actual numerical values you get when running the program may no longer match those shown here; that said, the general approach remains valid and has been used to find the values of the parameters actually used in the program.

To see what is happening, note that below we will impose boundary conditions for the temperature between 973 and 4273 Kelvin, and initial conditions are also chosen in this range; for these considerations, we run the program without internal heat sources or sinks, and consequently the temperature should always be in this range, barring any internal oscillations. If the minimal temperature drops below 973 Kelvin, then we need to add stabilization by either increasing \(\beta\) or decreasing \(c_R\).

As we did in step-31, we first determine an optimal value of \(\beta\) by using the "traditional" formula

\begin{eqnarray*} \nu_\alpha(T)|_K = \beta \|\mathbf{u}\|_{L^\infty(K)} h_K, \end{eqnarray*}

which we know to be stable if only \(\beta\) is large enough. Doing a couple hundred time steps (on a coarser mesh than the one shown in the program, and with a different viscosity that affects transport velocities and therefore time step sizes) in 2d will produce the following graph:

As can be seen, values \(\beta \le 0.05\) are too small whereas \(\beta=0.052\) appears to work, at least to the time horizon shown here. As a remark on the side, there are at least two questions one may wonder here: First, what happens at the time when the solution becomes unstable? Looking at the graphical output, we can see that with the unreasonably coarse mesh chosen for these experiments, around time \(t=10^{15}\) seconds the plumes of hot material that have been rising towards the cold outer boundary and have then spread sideways are starting to get close to each other, squeezing out the cold material in-between. This creates a layer of cells into which fluids flows from two opposite sides and flows out toward a third, apparently a scenario that then produce these instabilities without sufficient stabilization. Second: In step-31, we used \(\beta=0.015\cdot\text{dim}\); why does this not work here? The answer to this is not entirely clear – stabilization parameters are certainly known to depend on things like the shape of cells, for which we had squares in step-31 but have trapezoids in the current program. Whatever the exact cause, we at least have a value of \(\beta\), namely 0.052 for 2d, that works for the current program. A similar set of experiments can be made in 3d where we find that \(\beta=0.078\) is a good choice — neatly leading to the formula \(\beta=0.026 \cdot \textrm{dim}\).

With this value fixed, we can go back to the original formula for the viscosity \(\nu\) and play with the constant \(c_R\), making it as large as possible in order to make \(\nu\) as small as possible. This gives us a picture like this:

Consequently, \(c_R=0.1\) would appear to be the right value here. While this graph has been obtained for an exponent \(\alpha=1\), in the program we use \(\alpha=2\) instead, and in that case one has to re-tune the parameter (and observe that \(c_R\) appears in the numerator and not in the denominator). It turns out that \(c_R=1\) works with \(\alpha=2\).

Locally conservative Stokes discretization

The standard Taylor-Hood discretization for Stokes, using the \(Q_{k+1}^d \times Q_k\) element, is globally conservative, i.e. \(\int_{\partial\Omega} \mathbf n \cdot \mathbf u_h = 0\). This can easily be seen: the weak form of the divergence equation reads \((q_h, \textrm{div}\; \mathbf u_h)=0, \forall q_h\in Q_h\). Because the pressure space does contain the function \(q_h=1\), we get

\begin{align*} 0 = (1, \textrm{div}\; \mathbf u_h)_\Omega = \int_\Omega \textrm{div}\; \mathbf u_h = \int_{\partial\Omega} \mathbf n \cdot \mathbf u_h \end{align*}

by the divergence theorem. This property is important: if we want to use the velocity field \(u_h\) to transport along other quantities (such as the temperature in the current equations, but it could also be concentrations of chemical substances or entirely artificial tracer quantities) then the conservation property guarantees that the amount of the quantity advected remains constant.

That said, there are applications where this global property is not enough. Rather, we would like that it holds locally, on every cell. This can be achieved by using the space \(Q_{k+1}^d \times DGP_k\) for discretization, where we have replaced the continuous space of tensor product polynomials of degree \(k\) for the pressure by the discontinuous space of the complete polynomials of the same degree. (Note that tensor product polynomials in 2d contain the functions \(1, x, y, xy\), whereas the complete polynomials only have the functions \(1,x,y\).) This space turns out to be stable for the Stokes equation.

Because the space is discontinuous, we can now in particular choose the test function \(q_h(\mathbf x)=\chi_K(\mathbf x)\), i.e. the characteristic function of cell \(K\). We then get in a similar fashion as above

\begin{align*} 0 = (q_h, \textrm{div}\; \mathbf u_h)_\Omega = (1, \textrm{div}\; \mathbf u_h)_K = \int_K \textrm{div}\; \mathbf u_h = \int_{\partial K} \mathbf n \cdot \mathbf u_h, \end{align*}

showing the conservation property for cell \(K\). This clearly holds for each cell individually.

There are good reasons to use this discretization. As mentioned above, this element guarantees conservation of advected quantities on each cell individually. A second advantage is that the pressure mass matrix we use as a preconditioner in place of the Schur complement becomes block diagonal and consequently very easy to invert. However, there are also downsides. For one, there are now more pressure variables, increasing the overall size of the problem, although this doesn't seem to cause much harm in practice. More importantly, though, the fact that now the divergence integrated over each cell is zero when it wasn't before does not guarantee that the divergence is pointwise smaller. In fact, as one can easily verify, the \(L_2\) norm of the divergence is larger for this than for the standard Taylor-Hood discretization. (However, both converge at the same rate to zero, since it is easy to see that \(\|\textrm{div}\; u_h\|= \|\textrm{div}\; (u-u_h)\|= \|\textrm{trace}\; \nabla (u-u_h)\|\le \|\nabla (u-u_h)\|={\cal O}(h^{k+2})\).) It is therefore not a priori clear that the error is indeed smaller just because we now have more degrees of freedom.

Given these considerations, it remains unclear which discretization one should prefer. Consequently, we leave that up to the user and make it a parameter in the input file which one to use.

Higher order mappings for curved boundaries

In the program, we will use a spherical shell as domain. This means that the inner and outer boundary of the domain are no longer "straight" (by which we usually mean that they are bilinear surfaces that can be represented by the StraightBoundary class). Rather, they are curved and it seems prudent to use a curved approximation in the program if we are already using higher order finite elements for the velocity. Consequently, we will introduce a member variable of type MappingQ that denotes such a mapping (step-10 and step-11 introduce such mappings for the first time) and that we will use in all computations on cells that are adjacent to the boundary. Since this only affects a relatively small fraction of cells, the additional effort is not very large and we will take the luxury of using a quartic mapping for these cells.

Parallelization on clusters

Running convection codes in 3d with significant Rayleigh numbers requires a lot of computations — in the case of whole earth simulations on the order of one or several hundred million unknowns. This can obviously not be done with a single machine any more (at least not in 2010 when we started writing this code). Consequently, we need to parallelize it. Parallelization of scientific codes across multiple machines in a cluster of computers is almost always done using the Message Passing Interface (MPI). This program is no exception to that, and it follows the general spirit of the step-17 and step-18 programs in this though in practice it borrows more from step-40 in which we first introduced the classes and strategies we use when we want to completely distribute all computations, and step-55 that shows how to do that for vector-valued problems: including, for example, splitting the mesh up into a number of parts so that each processor only stores its own share plus some ghost cells, and using strategies where no processor potentially has enough memory to hold the entries of the combined solution vector locally. The goal is to run this code on hundreds or maybe even thousands of processors, at reasonable scalability.

Note
Even though it has a larger number, step-40 comes logically before the current program. The same is true for step-55. You will probably want to look at these programs before you try to understand what we do here.

MPI is a rather awkward interface to program with. It is a semi-object oriented set of functions, and while one uses it to send data around a network, one needs to explicitly describe the data types because the MPI functions insist on getting the address of the data as void* objects rather than deducing the data type automatically through overloading or templates. We've already seen in step-17 and step-18 how to avoid almost all of MPI by putting all the communication necessary into either the deal.II library or, in those programs, into PETSc. We'll do something similar here: like in step-40 and step-55, deal.II and the underlying p4est library are responsible for all the communication necessary for distributing the mesh, and we will let the Trilinos library (along with the wrappers in namespace TrilinosWrappers) deal with parallelizing the linear algebra components. We have already used Trilinos in step-31, and will do so again here, with the difference that we will use its parallel capabilities.

Trilinos consists of a significant number of packages, implementing basic parallel linear algebra operations (the Epetra package), different solver and preconditioner packages, and on to things that are of less importance to deal.II (e.g., optimization, uncertainty quantification, etc). deal.II's Trilinos interfaces encapsulate many of the things Trilinos offers that are of relevance to PDE solvers, and provides wrapper classes (in namespace TrilinosWrappers) that make the Trilinos matrix, vector, solver and preconditioner classes look very much the same as deal.II's own implementations of this functionality. However, as opposed to deal.II's classes, they can be used in parallel if we give them the necessary information. As a consequence, there are two Trilinos classes that we have to deal with directly (rather than through wrappers), both of which are part of Trilinos' Epetra library of basic linear algebra and tool classes:

There are a number of other concepts relevant to distributing the mesh to a number of processors; you may want to take a look at the Parallel computing with multiple processors using distributed memory module and step-40 or step-55 before trying to understand this program. The rest of the program is almost completely agnostic about the fact that we don't store all objects completely locally. There will be a few points where we have to limit loops over all cells to those that are locally owned, or where we need to distinguish between vectors that store only locally owned elements and those that store everything that is locally relevant (see this glossary entry), but by and large the amount of heavy lifting necessary to make the program run in parallel is well hidden in the libraries upon which this program builds. In any case, we will comment on these locations as we get to them in the program code.

Parallelization within individual nodes of a cluster

The second strategy to parallelize a program is to make use of the fact that most computers today have more than one processor that all have access to the same memory. In other words, in this model, we don't explicitly have to say which pieces of data reside where – all of the data we need is directly accessible and all we have to do is split processing this data between the available processors. We will then couple this with the MPI parallelization outlined above, i.e. we will have all the processors on a machine work together to, for example, assemble the local contributions to the global matrix for the cells that this machine actually "owns" but not for those cells that are owned by other machines. We will use this strategy for four kinds of operations we frequently do in this program: assembly of the Stokes and temperature matrices, assembly of the matrix that forms the Stokes preconditioner, and assembly of the right hand side of the temperature system.

All of these operations essentially look as follows: we need to loop over all cells for which cell->subdomain_id() equals the index our machine has within the communicator object used for all communication (i.e. MPI_COMM_WORLD, as explained above). The test we are actually going to use for this, and which describes in a concise way why we test this condition, is cell->is_locally_owned(). On each such cell we need to assemble the local contributions to the global matrix or vector, and then we have to copy each cell's contribution into the global matrix or vector. Note that the first part of this (the loop) defines a range of iterators on which something has to happen. The second part, assembly of local contributions is something that takes the majority of CPU time in this sequence of steps, and is a typical example of things that can be done in parallel: each cell's contribution is entirely independent of all other cells' contributions. The third part, copying into the global matrix, must not happen in parallel since we are modifying one object and so several threads can not at the same time read an existing matrix element, add their contribution, and write the sum back into memory without danger of producing a race condition.

deal.II has a class that is made for exactly this workflow: WorkStream, first discussed in step-9 and step-13. Its use is also extensively documented in the module on Parallel computing with multiple processors accessing shared memory (in the section on the WorkStream class) and we won't repeat here the rationale and detailed instructions laid out there, though you will want to read through this module to understand the distinction between scratch space and per-cell data. Suffice it to mention that we need the following:

We will comment on a few more points in the actual code, but in general their structure should be clear from the discussion in Parallel computing with multiple processors accessing shared memory.

The underlying technology for WorkStream identifies "tasks" that need to be worked on (e.g. assembling local contributions on a cell) and schedules these tasks automatically to available processors. WorkStream creates these tasks automatically, by splitting the iterator range into suitable chunks.

Note
Using multiple threads within each MPI process only makes sense if you have fewer MPI processes running on each node of your cluster than there are processor cores on this machine. Otherwise, MPI will already keep your processors busy and you won't get any additional speedup from using threads. For example, if your cluster nodes have 8 cores as they often have at the time of writing this, and if your batch scheduler puts 8 MPI processes on each node, then using threads doesn't make the program any faster. Consequently, you probably want to either configure your deal.II without threads, or set the number of threads in Utilities::MPI::MPI_InitFinalize to 1 (third argument), or "export DEAL_II_NUM_THREADS=1" before running. That said, at the time of writing this, we only use the WorkStream class for assembling (parts of) linear systems, while 75% or more of the run time of the program is spent in the linear solvers that are not parallelized — in other words, the best we could hope is to parallelize the remaining 25%.

The testcase

The setup for this program is mildly reminiscent of the problem we wanted to solve in the first place (see the introduction of step-31): convection in the earth mantle. As a consequence, we choose the following data, all of which appears in the program in units of meters and seconds (the SI system) even if we list them here in other units. We do note, however, that these choices are essentially still only exemplary, and not meant to result in a completely realistic description of convection in the earth mantle: for that, more and more difficult physics would have to be implemented, and several other aspects are currently missing from this program as well. We will come back to this issue in the results section again, but state for now that providing a realistic description is a goal of the Aspect code in development at the time of writing this.

As a reminder, let us again state the equations we want to solve are these:

\begin{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla \left( \frac{\eta}{L} \hat p\right) &=& \rho(T) \mathbf{g}, \\ \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0, \\ \frac{\partial T}{\partial t} + {\mathbf u} \cdot \nabla T - \nabla \cdot \kappa \nabla T &=& \gamma, \end{eqnarray*}

augmented by boundary and initial conditions. We then have to choose data for the following quantities:

All of these pieces of equation data are defined in the program in the EquationData namespace. When run, the program produces long-term maximal velocities around 10-40 centimeters per year (see the results section below), approximately the physically correct order of magnitude. We will set the end time to 1 billion years.

Note
The choice of the constants and material parameters above follows in large part the comprehensive book "Mantle Convection in the Earth and Planets, Part 1" by G. Schubert and D. L. Turcotte and P. Olson (Cambridge, 2001). It contains extensive discussion of ways to make the program more realistic.

Implementation details

Compared to step-31, this program has a number of noteworthy differences:

Outlook

This is a tutorial program. That means that at least most of its focus needs to lie on demonstrating ways of using deal.II and associated libraries, and not diluting this teaching lesson by focusing overly much on physical details. Despite the lengthy section above on the choice of physical parameters, the part of the program devoted to this is actually quite short and self contained.

That said, both step-31 and the current step-32 have not come about by chance but are certainly meant as wayposts along the path to a more comprehensive program that will simulate convection in the earth mantle. We call this code Aspect (short for Advanced Solver for Problems in Earth's ConvecTion); its development is funded by the Computational Infrastructure in Geodynamics initiative with support from the National Science Foundation. We hope to release this code not long after this tutorial program will officially be released as part of deal.II 7.1.

The commented program

Include files

The first task as usual is to include the functionality of these well-known deal.II library files and some C++ header files.

#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/function.h>
#include <deal.II/base/utilities.h>
#include <deal.II/base/conditional_ostream.h>
#include <deal.II/base/work_stream.h>
#include <deal.II/base/timer.h>
#include <deal.II/base/parameter_handler.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/solver_bicgstab.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/constraint_matrix.h>
#include <deal.II/lac/block_sparsity_pattern.h>
#include <deal.II/lac/trilinos_parallel_block_vector.h>
#include <deal.II/lac/trilinos_sparse_matrix.h>
#include <deal.II/lac/trilinos_block_sparse_matrix.h>
#include <deal.II/lac/trilinos_precondition.h>
#include <deal.II/lac/trilinos_solver.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/grid/filtered_iterator.h>
#include <deal.II/grid/manifold_lib.h>
#include <deal.II/grid/tria_boundary_lib.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/grid/grid_refinement.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/dofs/dof_renumbering.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_system.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/numerics/matrix_tools.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/error_estimator.h>
#include <deal.II/numerics/solution_transfer.h>
#include <fstream>
#include <iostream>
#include <sstream>
#include <limits>
#include <locale>
#include <string>

This is the only include file that is new: It introduces the parallel::distributed::SolutionTransfer equivalent of the SolutionTransfer class to take a solution from on mesh to the next one upon mesh refinement, but in the case of parallel distributed triangulations:

#include <deal.II/distributed/solution_transfer.h>

The following classes are used in parallel distributed computations and have all already been introduced in step-40:

#include <deal.II/base/index_set.h>
#include <deal.II/distributed/tria.h>
#include <deal.II/distributed/grid_refinement.h>

The next step is like in all previous tutorial programs: We put everything into a namespace of its own and then import the deal.II classes and functions into it:

namespace Step32
{
using namespace dealii;

Equation data

In the following namespace, we define the various pieces of equation data that describe the problem. This corresponds to the various aspects of making the problem at least slightly realistic and that were exhaustively discussed in the description of the testcase in the introduction.

We start with a few coefficients that have constant values (the comment after the value indicates its physical units):

namespace EquationData
{
const double eta = 1e21; / * Pa s * /
const double kappa = 1e-6; / * m^2 / s * /
const double reference_density = 3300; / * kg / m^3 * /
const double reference_temperature = 293; / * K * /
const double expansion_coefficient = 2e-5; / * 1/K * /
const double specific_heat = 1250; / * J / K / kg * /
const double radiogenic_heating = 7.4e-12; / * W / kg * /
const double R0 = 6371000.-2890000.; / * m * /
const double R1 = 6371000.- 35000.; / * m * /
const double T0 = 4000+273; / * K * /
const double T1 = 700+273; / * K * /

The next set of definitions are for functions that encode the density as a function of temperature, the gravity vector, and the initial values for the temperature. Again, all of these (along with the values they compute) are discussed in the introduction:

double density (const double temperature)
{
return (reference_density *
(1 - expansion_coefficient * (temperature -
reference_temperature)));
}
template <int dim>
Tensor<1,dim> gravity_vector (const Point<dim> &p)
{
const double r = p.norm();
return -(1.245e-6 * r + 7.714e13/r/r) * p / r;
}
template <int dim>
class TemperatureInitialValues : public Function<dim>
{
public:
TemperatureInitialValues () : Function<dim>(1) {}
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &value) const;
};
template <int dim>
double
TemperatureInitialValues<dim>::value (const Point<dim> &p,
const unsigned int) const
{
const double r = p.norm();
const double h = R1-R0;
const double s = (r-R0)/h;
const double q = (dim==3)?std::max(0.0,cos(numbers::PI*abs(p(2)/R1))):1.0;
const double phi = std::atan2(p(0),p(1));
const double tau = s
+
0.2 * s * (1-s) * std::sin(6*phi) * q;
return T0*(1.0-tau) + T1*tau;
}
template <int dim>
void
TemperatureInitialValues<dim>::vector_value (const Point<dim> &p,
Vector<double> &values) const
{
for (unsigned int c=0; c<this->n_components; ++c)
values(c) = TemperatureInitialValues<dim>::value (p, c);
}

As mentioned in the introduction we need to rescale the pressure to avoid the relative ill-conditioning of the momentum and mass conservation equations. The scaling factor is \(\frac{\eta}{L}\) where \(L\) was a typical length scale. By experimenting it turns out that a good length scale is the diameter of plumes, which is around 10 km:

const double pressure_scaling = eta / 10000;

The final number in this namespace is a constant that denotes the number of seconds per (average, tropical) year. We use this only when generating screen output: internally, all computations of this program happen in SI units (kilogram, meter, seconds) but writing geological times in seconds yields numbers that one can't relate to reality, and so we convert to years using the factor defined here:

const double year_in_seconds = 60*60*24*365.2425;
}

Preconditioning the Stokes system

This namespace implements the preconditioner. As discussed in the introduction, this preconditioner differs in a number of key portions from the one used in step-31. Specifically, it is a right preconditioner, implementing the matrix

\begin{align*} \left(\begin{array}{cc}A^{-1} & B^T \\0 & S^{-1} \end{array}\right) \end{align*}

where the two inverse matrix operations are approximated by linear solvers or, if the right flag is given to the constructor of this class, by a single AMG V-cycle for the velocity block. The three code blocks of the vmult function implement the multiplications with the three blocks of this preconditioner matrix and should be self explanatory if you have read through step-31 or the discussion of composing solvers in step-20.

namespace LinearSolvers
{
template <class PreconditionerTypeA, class PreconditionerTypeMp>
class BlockSchurPreconditioner : public Subscriptor
{
public:
BlockSchurPreconditioner (const TrilinosWrappers::BlockSparseMatrix &S,
const PreconditionerTypeMp &Mppreconditioner,
const PreconditionerTypeA &Apreconditioner,
const bool do_solve_A)
:
stokes_matrix (&S),
stokes_preconditioner_matrix (&Spre),
mp_preconditioner (Mppreconditioner),
a_preconditioner (Apreconditioner),
do_solve_A (do_solve_A)
{}
{
{
SolverControl solver_control(5000, 1e-6 * src.block(1).l2_norm());
solver.solve(stokes_preconditioner_matrix->block(1,1),
dst.block(1), src.block(1),
mp_preconditioner);
dst.block(1) *= -1.0;
}
{
stokes_matrix->block(0,1).vmult(utmp, dst.block(1));
utmp*=-1.0;
utmp.add(src.block(0));
}
if (do_solve_A == true)
{
SolverControl solver_control(5000, utmp.l2_norm()*1e-2);
TrilinosWrappers::SolverCG solver(solver_control);
solver.solve(stokes_matrix->block(0,0), dst.block(0), utmp,
a_preconditioner);
}
else
a_preconditioner.vmult (dst.block(0), utmp);
}
private:
const SmartPointer<const TrilinosWrappers::BlockSparseMatrix> stokes_preconditioner_matrix;
const PreconditionerTypeMp &mp_preconditioner;
const PreconditionerTypeA &a_preconditioner;
const bool do_solve_A;
};
}

Definition of assembly data structures

As described in the introduction, we will use the WorkStream mechanism discussed in the Parallel computing with multiple processors accessing shared memory module to parallelize operations among the processors of a single machine. The WorkStream class requires that data is passed around in two kinds of data structures, one for scratch data and one to pass data from the assembly function to the function that copies local contributions into global objects.

The following namespace (and the two sub-namespaces) contains a collection of data structures that serve this purpose, one pair for each of the four operations discussed in the introduction that we will want to parallelize. Each assembly routine gets two sets of data: a Scratch array that collects all the classes and arrays that are used for the calculation of the cell contribution, and a CopyData array that keeps local matrices and vectors which will be written into the global matrix. Whereas CopyData is a container for the final data that is written into the global matrices and vector (and, thus, absolutely necessary), the Scratch arrays are merely there for performance reasons — it would be much more expensive to set up a FEValues object on each cell, than creating it only once and updating some derivative data.

step-31 had four assembly routines: One for the preconditioner matrix of the Stokes system, one for the Stokes matrix and right hand side, one for the temperature matrices and one for the right hand side of the temperature equation. We here organize the scratch arrays and CopyData objects for each of those four assembly components using a struct environment (since we consider these as temporary objects we pass around, rather than classes that implement functionality of their own, though this is a more subjective point of view to distinguish between structs and classes).

Regarding the Scratch objects, each struct is equipped with a constructor that creates an FEValues object using the FiniteElement, Quadrature, Mapping (which describes the interpolation of curved boundaries), and The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues instances. Moreover, we manually implement a copy constructor (since the FEValues class is not copyable by itself), and provide some additional vector fields that are used to hold intermediate data during the computation of local contributions.

Let us start with the scratch arrays and, specifically, the one used for assembly of the Stokes preconditioner:

namespace Assembly
{
namespace Scratch
{
template <int dim>
struct StokesPreconditioner
{
StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
const Quadrature<dim> &stokes_quadrature,
const Mapping<dim> &mapping,
const UpdateFlags update_flags);
StokesPreconditioner (const StokesPreconditioner &data);
FEValues<dim> stokes_fe_values;
std::vector<Tensor<2,dim> > grad_phi_u;
std::vector<double> phi_p;
};
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
const Quadrature<dim> &stokes_quadrature,
const Mapping<dim> &mapping,
const UpdateFlags update_flags)
:
stokes_fe_values (mapping, stokes_fe, stokes_quadrature,
update_flags),
grad_phi_u (stokes_fe.dofs_per_cell),
phi_p (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const StokesPreconditioner &scratch)
:
stokes_fe_values (scratch.stokes_fe_values.get_mapping(),
scratch.stokes_fe_values.get_fe(),
scratch.stokes_fe_values.get_quadrature(),
scratch.stokes_fe_values.get_update_flags()),
grad_phi_u (scratch.grad_phi_u),
phi_p (scratch.phi_p)
{}

The next one is the scratch object used for the assembly of the full Stokes system. Observe that we derive the StokesSystem scratch class from the StokesPreconditioner class above. We do this because all the objects that are necessary for the assembly of the preconditioner are also needed for the actual matrix system and right hand side, plus some extra data. This makes the program more compact. Note also that the assembly of the Stokes system and the temperature right hand side further down requires data from temperature and velocity, respectively, so we actually need two FEValues objects for those two cases.

template <int dim>
struct StokesSystem : public StokesPreconditioner<dim>
{
StokesSystem (const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &stokes_quadrature,
const UpdateFlags stokes_update_flags,
const FiniteElement<dim> &temperature_fe,
const UpdateFlags temperature_update_flags);
StokesSystem (const StokesSystem<dim> &data);
FEValues<dim> temperature_fe_values;
std::vector<Tensor<1,dim> > phi_u;
std::vector<SymmetricTensor<2,dim> > grads_phi_u;
std::vector<double> div_phi_u;
std::vector<double> old_temperature_values;
};
template <int dim>
StokesSystem<dim>::
StokesSystem (const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &stokes_quadrature,
const UpdateFlags stokes_update_flags,
const FiniteElement<dim> &temperature_fe,
const UpdateFlags temperature_update_flags)
:
StokesPreconditioner<dim> (stokes_fe, stokes_quadrature,
mapping,
stokes_update_flags),
temperature_fe_values (mapping, temperature_fe, stokes_quadrature,
temperature_update_flags),
phi_u (stokes_fe.dofs_per_cell),
grads_phi_u (stokes_fe.dofs_per_cell),
div_phi_u (stokes_fe.dofs_per_cell),
old_temperature_values (stokes_quadrature.size())
{}
template <int dim>
StokesSystem<dim>::
StokesSystem (const StokesSystem<dim> &scratch)
:
StokesPreconditioner<dim> (scratch),
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
phi_u (scratch.phi_u),
grads_phi_u (scratch.grads_phi_u),
div_phi_u (scratch.div_phi_u),
old_temperature_values (scratch.old_temperature_values)
{}

After defining the objects used in the assembly of the Stokes system, we do the same for the assembly of the matrices necessary for the temperature system. The general structure is very similar:

template <int dim>
struct TemperatureMatrix
{
TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &temperature_quadrature);
TemperatureMatrix (const TemperatureMatrix &data);
FEValues<dim> temperature_fe_values;
std::vector<double> phi_T;
std::vector<Tensor<1,dim> > grad_phi_T;
};
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &temperature_quadrature)
:
temperature_fe_values (mapping,
temperature_fe, temperature_quadrature,
phi_T (temperature_fe.dofs_per_cell),
grad_phi_T (temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const TemperatureMatrix &scratch)
:
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
phi_T (scratch.phi_T),
grad_phi_T (scratch.grad_phi_T)
{}

The final scratch object is used in the assembly of the right hand side of the temperature system. This object is significantly larger than the ones above because a lot more quantities enter the computation of the right hand side of the temperature equation. In particular, the temperature values and gradients of the previous two time steps need to be evaluated at the quadrature points, as well as the velocities and the strain rates (i.e. the symmetric gradients of the velocity) that enter the right hand side as friction heating terms. Despite the number of terms, the following should be rather self explanatory:

template <int dim>
struct TemperatureRHS
{
TemperatureRHS (const FiniteElement<dim> &temperature_fe,
const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &quadrature);
TemperatureRHS (const TemperatureRHS &data);
FEValues<dim> temperature_fe_values;
FEValues<dim> stokes_fe_values;
std::vector<double> phi_T;
std::vector<Tensor<1,dim> > grad_phi_T;
std::vector<Tensor<1,dim> > old_velocity_values;
std::vector<Tensor<1,dim> > old_old_velocity_values;
std::vector<SymmetricTensor<2,dim> > old_strain_rates;
std::vector<SymmetricTensor<2,dim> > old_old_strain_rates;
std::vector<double> old_temperature_values;
std::vector<double> old_old_temperature_values;
std::vector<Tensor<1,dim> > old_temperature_grads;
std::vector<Tensor<1,dim> > old_old_temperature_grads;
std::vector<double> old_temperature_laplacians;
std::vector<double> old_old_temperature_laplacians;
};
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const FiniteElement<dim> &temperature_fe,
const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &quadrature)
:
temperature_fe_values (mapping,
temperature_fe, quadrature,
stokes_fe_values (mapping,
stokes_fe, quadrature,
phi_T (temperature_fe.dofs_per_cell),
grad_phi_T (temperature_fe.dofs_per_cell),
old_velocity_values (quadrature.size()),
old_old_velocity_values (quadrature.size()),
old_strain_rates (quadrature.size()),
old_old_strain_rates (quadrature.size()),
old_temperature_values (quadrature.size()),
old_old_temperature_values(quadrature.size()),
old_temperature_grads(quadrature.size()),
old_old_temperature_grads(quadrature.size()),
old_temperature_laplacians(quadrature.size()),
old_old_temperature_laplacians(quadrature.size())
{}
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const TemperatureRHS &scratch)
:
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
stokes_fe_values (scratch.stokes_fe_values.get_mapping(),
scratch.stokes_fe_values.get_fe(),
scratch.stokes_fe_values.get_quadrature(),
scratch.stokes_fe_values.get_update_flags()),
phi_T (scratch.phi_T),
grad_phi_T (scratch.grad_phi_T),
old_velocity_values (scratch.old_velocity_values),
old_old_velocity_values (scratch.old_old_velocity_values),
old_strain_rates (scratch.old_strain_rates),
old_old_strain_rates (scratch.old_old_strain_rates),
old_temperature_values (scratch.old_temperature_values),
old_old_temperature_values (scratch.old_old_temperature_values),
old_temperature_grads (scratch.old_temperature_grads),
old_old_temperature_grads (scratch.old_old_temperature_grads),
old_temperature_laplacians (scratch.old_temperature_laplacians),
old_old_temperature_laplacians (scratch.old_old_temperature_laplacians)
{}
}

The CopyData objects are even simpler than the Scratch objects as all they have to do is to store the results of local computations until they can be copied into the global matrix or vector objects. These structures therefore only need to provide a constructor, a copy operation, and some arrays for local matrix, local vectors and the relation between local and global degrees of freedom (a.k.a. local_dof_indices). Again, we have one such structure for each of the four operations we will parallelize using the WorkStream class:

namespace CopyData
{
template <int dim>
struct StokesPreconditioner
{
StokesPreconditioner (const FiniteElement<dim> &stokes_fe);
StokesPreconditioner (const StokesPreconditioner &data);
FullMatrix<double> local_matrix;
std::vector<types::global_dof_index> local_dof_indices;
};
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const FiniteElement<dim> &stokes_fe)
:
local_matrix (stokes_fe.dofs_per_cell,
stokes_fe.dofs_per_cell),
local_dof_indices (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const StokesPreconditioner &data)
:
local_matrix (data.local_matrix),
local_dof_indices (data.local_dof_indices)
{}
template <int dim>
struct StokesSystem : public StokesPreconditioner<dim>
{
StokesSystem (const FiniteElement<dim> &stokes_fe);
StokesSystem (const StokesSystem<dim> &data);
Vector<double> local_rhs;
};
template <int dim>
StokesSystem<dim>::
StokesSystem (const FiniteElement<dim> &stokes_fe)
:
StokesPreconditioner<dim> (stokes_fe),
local_rhs (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesSystem<dim>::
StokesSystem (const StokesSystem<dim> &data)
:
StokesPreconditioner<dim> (data),
local_rhs (data.local_rhs)
{}
template <int dim>
struct TemperatureMatrix
{
TemperatureMatrix (const FiniteElement<dim> &temperature_fe);
TemperatureMatrix (const TemperatureMatrix &data);
FullMatrix<double> local_mass_matrix;
FullMatrix<double> local_stiffness_matrix;
std::vector<types::global_dof_index> local_dof_indices;
};
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const FiniteElement<dim> &temperature_fe)
:
local_mass_matrix (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell),
local_stiffness_matrix (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell),
local_dof_indices (temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const TemperatureMatrix &data)
:
local_mass_matrix (data.local_mass_matrix),
local_stiffness_matrix (data.local_stiffness_matrix),
local_dof_indices (data.local_dof_indices)
{}
template <int dim>
struct TemperatureRHS
{
TemperatureRHS (const FiniteElement<dim> &temperature_fe);
TemperatureRHS (const TemperatureRHS &data);
Vector<double> local_rhs;
std::vector<types::global_dof_index> local_dof_indices;
FullMatrix<double> matrix_for_bc;
};
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const FiniteElement<dim> &temperature_fe)
:
local_rhs (temperature_fe.dofs_per_cell),
local_dof_indices (temperature_fe.dofs_per_cell),
matrix_for_bc (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const TemperatureRHS &data)
:
local_rhs (data.local_rhs),
local_dof_indices (data.local_dof_indices),
matrix_for_bc (data.matrix_for_bc)
{}
}
}

The BoussinesqFlowProblem class template

This is the declaration of the main class. It is very similar to step-31 but there are a number differences we will comment on below.

The top of the class is essentially the same as in step-31, listing the public methods and a set of private functions that do the heavy lifting. Compared to step-31 there are only two additions to this section: the function get_cfl_number() that computes the maximum CFL number over all cells which we then compute the global time step from, and the function get_entropy_variation() that is used in the computation of the entropy stabilization. It is akin to the get_extrapolated_temperature_range() we have used in step-31 for this purpose, but works on the entropy instead of the temperature instead.

template <int dim>
class BoussinesqFlowProblem
{
public:
struct Parameters;
BoussinesqFlowProblem (Parameters &parameters);
void run ();
private:
void setup_dofs ();
void assemble_stokes_preconditioner ();
void build_stokes_preconditioner ();
void assemble_stokes_system ();
void assemble_temperature_matrix ();
void assemble_temperature_system (const double maximal_velocity);
void project_temperature_field ();
double get_maximal_velocity () const;
double get_cfl_number () const;
double get_entropy_variation (const double average_temperature) const;
std::pair<double,double> get_extrapolated_temperature_range () const;
void solve ();
void output_results ();
void refine_mesh (const unsigned int max_grid_level);
double
compute_viscosity(const std::vector<double> &old_temperature,
const std::vector<double> &old_old_temperature,
const std::vector<Tensor<1,dim> > &old_temperature_grads,
const std::vector<Tensor<1,dim> > &old_old_temperature_grads,
const std::vector<double> &old_temperature_laplacians,
const std::vector<double> &old_old_temperature_laplacians,
const std::vector<Tensor<1,dim> > &old_velocity_values,
const std::vector<Tensor<1,dim> > &old_old_velocity_values,
const std::vector<SymmetricTensor<2,dim> > &old_strain_rates,
const std::vector<SymmetricTensor<2,dim> > &old_old_strain_rates,
const double global_u_infty,
const double global_T_variation,
const double average_temperature,
const double global_entropy_variation,
const double cell_diameter) const;
public:

The first significant new component is the definition of a struct for the parameters according to the discussion in the introduction. This structure is initialized by reading from a parameter file during construction of this object.

struct Parameters
{
Parameters (const std::string &parameter_filename);
static void declare_parameters (ParameterHandler &prm);
void parse_parameters (ParameterHandler &prm);
double end_time;
unsigned int initial_global_refinement;
unsigned int initial_adaptive_refinement;
bool generate_graphical_output;
unsigned int graphical_output_interval;
unsigned int adaptive_refinement_interval;
double stabilization_alpha;
double stabilization_c_R;
double stabilization_beta;
unsigned int stokes_velocity_degree;
bool use_locally_conservative_discretization;
unsigned int temperature_degree;
};
private:
Parameters &parameters;

The pcout (for parallel std::cout) object is used to simplify writing output: each MPI process can use this to generate output as usual, but since each of these processes will (hopefully) produce the same output it will just be replicated many times over; with the ConditionalOStream class, only the output generated by one MPI process will actually be printed to screen, whereas the output by all the other threads will simply be forgotten.

The following member variables will then again be similar to those in step-31 (and to other tutorial programs). As mentioned in the introduction, we fully distribute computations, so we will have to use the parallel::distributed::Triangulation class (see step-40) but the remainder of these variables is rather standard with two exceptions:

double global_Omega_diameter;
const MappingQ<dim> mapping;
const FESystem<dim> stokes_fe;
DoFHandler<dim> stokes_dof_handler;
ConstraintMatrix stokes_constraints;
TrilinosWrappers::BlockSparseMatrix stokes_preconditioner_matrix;
FE_Q<dim> temperature_fe;
DoFHandler<dim> temperature_dof_handler;
ConstraintMatrix temperature_constraints;
TrilinosWrappers::SparseMatrix temperature_mass_matrix;
TrilinosWrappers::SparseMatrix temperature_stiffness_matrix;
TrilinosWrappers::SparseMatrix temperature_matrix;
TrilinosWrappers::MPI::Vector temperature_solution;
TrilinosWrappers::MPI::Vector old_temperature_solution;
TrilinosWrappers::MPI::Vector old_old_temperature_solution;
double time_step;
double old_time_step;
unsigned int timestep_number;
std::shared_ptr<TrilinosWrappers::PreconditionAMG> Amg_preconditioner;
std::shared_ptr<TrilinosWrappers::PreconditionJacobi> Mp_preconditioner;
std::shared_ptr<TrilinosWrappers::PreconditionJacobi> T_preconditioner;
bool rebuild_stokes_matrix;
bool rebuild_stokes_preconditioner;
bool rebuild_temperature_matrices;
bool rebuild_temperature_preconditioner;

The next member variable, computing_timer is used to conveniently account for compute time spent in certain "sections" of the code that are repeatedly entered. For example, we will enter (and leave) sections for Stokes matrix assembly and would like to accumulate the run time spent in this section over all time steps. Every so many time steps as well as at the end of the program (through the destructor of the TimerOutput class) we will then produce a nice summary of the times spent in the different sections into which we categorize the run-time of this program.

TimerOutput computing_timer;

After these member variables we have a number of auxiliary functions that have been broken out of the ones listed above. Specifically, there are first three functions that we call from setup_dofs and then the ones that do the assembling of linear systems:

void setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning);
void setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning);
void setup_temperature_matrices (const IndexSet &temperature_partitioning,
const IndexSet &temperature_relevant_partitioning);

Following the task-based parallelization paradigm, we split all the assembly routines into two parts: a first part that can do all the calculations on a certain cell without taking care of other threads, and a second part (which is writing the local data into the global matrices and vectors) which can be entered by only one thread at a time. In order to implement that, we provide functions for each of those two steps for all the four assembly routines that we use in this program. The following eight functions do exactly this:

void
local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesPreconditioner<dim> &scratch,
Assembly::CopyData::StokesPreconditioner<dim> &data);
void
copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data);
void
local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesSystem<dim> &scratch,
Assembly::CopyData::StokesSystem<dim> &data);
void
copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data);
void
local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::TemperatureMatrix<dim> &scratch,
Assembly::CopyData::TemperatureMatrix<dim> &data);
void
copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data);
void
local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
const double global_max_velocity,
const double global_entropy_variation,
Assembly::Scratch::TemperatureRHS<dim> &scratch,
Assembly::CopyData::TemperatureRHS<dim> &data);
void
copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data);

Finally, we forward declare a member class that we will define later on and that will be used to compute a number of quantities from our solution vectors that we'd like to put into the output files for visualization.

class Postprocessor;
};

BoussinesqFlowProblem class implementation

BoussinesqFlowProblem::Parameters

Here comes the definition of the parameters for the Stokes problem. We allow to set the end time for the simulation, the level of refinements (both global and adaptive, which in the sum specify what maximum level the cells are allowed to have), and the interval between refinements in the time stepping.

Then, we let the user specify constants for the stabilization parameters (as discussed in the introduction), the polynomial degree for the Stokes velocity space, whether to use the locally conservative discretization based on FE_DGP elements for the pressure or not (FE_Q elements for pressure), and the polynomial degree for the temperature interpolation.

The constructor checks for a valid input file (if not, a file with default parameters for the quantities is written), and eventually parses the parameters.

template <int dim>
BoussinesqFlowProblem<dim>::Parameters::Parameters (const std::string &parameter_filename)
:
end_time (1e8),
initial_global_refinement (2),
initial_adaptive_refinement (2),
adaptive_refinement_interval (10),
stabilization_alpha (2),
stabilization_c_R (0.11),
stabilization_beta (0.078),
stokes_velocity_degree (2),
use_locally_conservative_discretization (true),
temperature_degree (2)
{
BoussinesqFlowProblem<dim>::Parameters::declare_parameters (prm);
std::ifstream parameter_file (parameter_filename.c_str());
if (!parameter_file)
{
parameter_file.close ();
std::ostringstream message;
message << "Input parameter file <"
<< parameter_filename << "> not found. Creating a"
<< std::endl
<< "template file of the same name."
<< std::endl;
std::ofstream parameter_out (parameter_filename.c_str());
prm.print_parameters (parameter_out,
AssertThrow (false, ExcMessage (message.str().c_str()));
}
prm.parse_input (parameter_file);
parse_parameters (prm);
}

Next we have a function that declares the parameters that we expect in the input file, together with their data types, default values and a description:

template <int dim>
void
BoussinesqFlowProblem<dim>::Parameters::
declare_parameters (ParameterHandler &prm)
{
prm.declare_entry ("End time", "1e8",
"The end time of the simulation in years.");
prm.declare_entry ("Initial global refinement", "2",
"The number of global refinement steps performed on "
"the initial coarse mesh, before the problem is first "
"solved there.");
prm.declare_entry ("Initial adaptive refinement", "2",
"The number of adaptive refinement steps performed after "
"initial global refinement.");
prm.declare_entry ("Time steps between mesh refinement", "10",
"The number of time steps after which the mesh is to be "
"adapted based on computed error indicators.");
prm.declare_entry ("Generate graphical output", "false",
"Whether graphical output is to be generated or not. "
"You may not want to get graphical output if the number "
"of processors is large.");
prm.declare_entry ("Time steps between graphical output", "50",
"The number of time steps between each generation of "
"graphical output files.");
prm.enter_subsection ("Stabilization parameters");
{
prm.declare_entry ("alpha", "2",
"The exponent in the entropy viscosity stabilization.");
prm.declare_entry ("c_R", "0.11",
"The c_R factor in the entropy viscosity "
"stabilization.");
prm.declare_entry ("beta", "0.078",
"The beta factor in the artificial viscosity "
"stabilization. An appropriate value for 2d is 0.052 "
"and 0.078 for 3d.");
}
prm.enter_subsection ("Discretization");
{
prm.declare_entry ("Stokes velocity polynomial degree", "2",
"The polynomial degree to use for the velocity variables "
"in the Stokes system.");
prm.declare_entry ("Temperature polynomial degree", "2",
"The polynomial degree to use for the temperature variable.");
prm.declare_entry ("Use locally conservative discretization", "true",
"Whether to use a Stokes discretization that is locally "
"conservative at the expense of a larger number of degrees "
"of freedom, or to go with a cheaper discretization "
"that does not locally conserve mass (although it is "
"globally conservative.");
}
}

And then we need a function that reads the contents of the ParameterHandler object we get by reading the input file and puts the results into variables that store the values of the parameters we have previously declared:

template <int dim>
void
BoussinesqFlowProblem<dim>::Parameters::
parse_parameters (ParameterHandler &prm)
{
end_time = prm.get_double ("End time");
initial_global_refinement = prm.get_integer ("Initial global refinement");
initial_adaptive_refinement = prm.get_integer ("Initial adaptive refinement");
adaptive_refinement_interval= prm.get_integer ("Time steps between mesh refinement");
generate_graphical_output = prm.get_bool ("Generate graphical output");
graphical_output_interval = prm.get_integer ("Time steps between graphical output");
prm.enter_subsection ("Stabilization parameters");
{
stabilization_alpha = prm.get_double ("alpha");
stabilization_c_R = prm.get_double ("c_R");
stabilization_beta = prm.get_double ("beta");
}
prm.enter_subsection ("Discretization");
{
stokes_velocity_degree = prm.get_integer ("Stokes velocity polynomial degree");
temperature_degree = prm.get_integer ("Temperature polynomial degree");
use_locally_conservative_discretization
= prm.get_bool ("Use locally conservative discretization");
}
}

BoussinesqFlowProblem::BoussinesqFlowProblem

The constructor of the problem is very similar to the constructor in step-31. What is different is the parallel communication: Trilinos uses a message passing interface (MPI) for data distribution. When entering the BoussinesqFlowProblem class, we have to decide how the parallization is to be done. We choose a rather simple strategy and let all processors that are running the program work together, specified by the communicator MPI_COMM_WORLD. Next, we create the output stream (as we already did in step-18) that only generates output on the first MPI process and is completely forgetful on all others. The implementation of this idea is to check the process number when pcout gets a true argument, and it uses the std::cout stream for output. If we are one processor five, for instance, then we will give a false argument to pcout, which means that the output of that processor will not be printed. With the exception of the mapping object (for which we use polynomials of degree 4) all but the final member variable are exactly the same as in step-31.

This final object, the TimerOutput object, is then told to restrict output to the pcout stream (processor 0), and then we specify that we want to get a summary table at the end of the program which shows us wallclock times (as opposed to CPU times). We will manually also request intermediate summaries every so many time steps in the run() function below.

template <int dim>
BoussinesqFlowProblem<dim>::BoussinesqFlowProblem (Parameters &parameters_)
:
parameters (parameters_),
pcout (std::cout,
(Utilities::MPI::this_mpi_process(MPI_COMM_WORLD)
== 0)),
triangulation (MPI_COMM_WORLD,
typename Triangulation<dim>::MeshSmoothing
(Triangulation<dim>::smoothing_on_refinement |
Triangulation<dim>::smoothing_on_coarsening)),
global_Omega_diameter(0.),
mapping (4),
stokes_fe (FE_Q<dim>(parameters.stokes_velocity_degree),
dim,
(parameters.use_locally_conservative_discretization
?
static_cast<const FiniteElement<dim> &>
(FE_DGP<dim>(parameters.stokes_velocity_degree-1))
:
static_cast<const FiniteElement<dim> &>
(FE_Q<dim>(parameters.stokes_velocity_degree-1))),
1),
stokes_dof_handler (triangulation),
temperature_fe (parameters.temperature_degree),
temperature_dof_handler (triangulation),
time_step (0),
old_time_step (0),
timestep_number (0),
rebuild_stokes_matrix (true),
rebuild_stokes_preconditioner (true),
rebuild_temperature_matrices (true),
rebuild_temperature_preconditioner (true),
computing_timer (MPI_COMM_WORLD,
pcout,
TimerOutput::summary,
TimerOutput::wall_times)
{}

The BoussinesqFlowProblem helper functions

BoussinesqFlowProblem::get_maximal_velocity

Except for two small details, the function to compute the global maximum of the velocity is the same as in step-31. The first detail is actually common to all functions that implement loops over all cells in the triangulation: When operating in parallel, each processor can only work on a chunk of cells since each processor only has a certain part of the entire triangulation. This chunk of cells that we want to work on is identified via a so-called subdomain_id, as we also did in step-18. All we need to change is hence to perform the cell-related operations only on cells that are owned by the current process (as opposed to ghost or artificial cells), i.e. for which the subdomain id equals the number of the process ID. Since this is a commonly used operation, there is a shortcut for this operation: we can ask whether the cell is owned by the current processor using cell->is_locally_owned().

The second difference is the way we calculate the maximum value. Before, we could simply have a double variable that we checked against on each quadrature point for each cell. Now, we have to be a bit more careful since each processor only operates on a subset of cells. What we do is to first let each processor calculate the maximum among its cells, and then do a global communication operation Utilities::MPI::max that computes the maximum value among all the maximum values of the individual processors. MPI provides such a call, but it's even simpler to use the respective function in namespace Utilities::MPI using the MPI communicator object since that will do the right thing even if we work without MPI and on a single machine only. The call to Utilities::MPI::max needs two arguments, namely the local maximum (input) and the MPI communicator, which is MPI_COMM_WORLD in this example.

template <int dim>
double BoussinesqFlowProblem<dim>::get_maximal_velocity () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.stokes_velocity_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, stokes_fe, quadrature_formula, update_values);
std::vector<Tensor<1,dim> > velocity_values(n_q_points);
const FEValuesExtractors::Vector velocities (0);
double max_local_velocity = 0;
cell = stokes_dof_handler.begin_active(),
endc = stokes_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values[velocities].get_function_values (stokes_solution,
velocity_values);
for (unsigned int q=0; q<n_q_points; ++q)
max_local_velocity = std::max (max_local_velocity,
velocity_values[q].norm());
}
return Utilities::MPI::max (max_local_velocity, MPI_COMM_WORLD);
}

BoussinesqFlowProblem::get_cfl_number

The next function does something similar, but we now compute the CFL number, i.e., maximal velocity on a cell divided by the cell diameter. This number is necessary to determine the time step size, as we use a semi-explicit time stepping scheme for the temperature equation (see step-31 for a discussion). We compute it in the same way as above: Compute the local maximum over all locally owned cells, then exchange it via MPI to find the global maximum.

template <int dim>
double BoussinesqFlowProblem<dim>::get_cfl_number () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.stokes_velocity_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, stokes_fe, quadrature_formula, update_values);
std::vector<Tensor<1,dim> > velocity_values(n_q_points);
const FEValuesExtractors::Vector velocities (0);
double max_local_cfl = 0;
cell = stokes_dof_handler.begin_active(),
endc = stokes_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values[velocities].get_function_values (stokes_solution,
velocity_values);
double max_local_velocity = 1e-10;
for (unsigned int q=0; q<n_q_points; ++q)
max_local_velocity = std::max (max_local_velocity,
velocity_values[q].norm());
max_local_cfl = std::max(max_local_cfl,
max_local_velocity / cell->diameter());
}
return Utilities::MPI::max (max_local_cfl, MPI_COMM_WORLD);
}

BoussinesqFlowProblem::get_entropy_variation

Next comes the computation of the global entropy variation \(\|E(T)-\bar{E}(T)\|_\infty\) where the entropy \(E\) is defined as discussed in the introduction. This is needed for the evaluation of the stabilization in the temperature equation as explained in the introduction. The entropy variation is actually only needed if we use \(\alpha=2\) as a power in the residual computation. The infinity norm is computed by the maxima over quadrature points, as usual in discrete computations.

In order to compute this quantity, we first have to find the space-average \(\bar{E}(T)\) and then evaluate the maximum. However, that means that we would need to perform two loops. We can avoid the overhead by noting that \(\|E(T)-\bar{E}(T)\|_\infty = \max\big(E_{\textrm{max}}(T)-\bar{E}(T), \bar{E}(T)-E_{\textrm{min}}(T)\big)\), i.e., the maximum out of the deviation from the average entropy in positive and negative directions. The four quantities we need for the latter formula (maximum entropy, minimum entropy, average entropy, area) can all be evaluated in the same loop over all cells, so we choose this simpler variant.

template <int dim>
double
BoussinesqFlowProblem<dim>::get_entropy_variation (const double average_temperature) const
{
if (parameters.stabilization_alpha != 2)
return 1.;
const QGauss<dim> quadrature_formula (parameters.temperature_degree+1);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (temperature_fe, quadrature_formula,
std::vector<double> old_temperature_values(n_q_points);
std::vector<double> old_old_temperature_values(n_q_points);

In the two functions above we computed the maximum of numbers that were all non-negative, so we knew that zero was certainly a lower bound. On the other hand, here we need to find the maximum deviation from the average value, i.e., we will need to know the maximal and minimal values of the entropy for which we don't a priori know the sign.

To compute it, we can therefore start with the largest and smallest possible values we can store in a double precision number: The minimum is initialized with a bigger and the maximum with a smaller number than any one that is going to appear. We are then guaranteed that these numbers will be overwritten in the loop on the first cell or, if this processor does not own any cells, in the communication step at the latest. The following loop then computes the minimum and maximum local entropy as well as keeps track of the area/volume of the part of the domain we locally own and the integral over the entropy on it:

double min_entropy = std::numeric_limits<double>::max(),
max_entropy = -std::numeric_limits<double>::max(),
area = 0,
entropy_integrated = 0;
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
fe_values.get_function_values (old_old_temperature_solution,
old_old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double T = (old_temperature_values[q] +
old_old_temperature_values[q]) / 2;
const double entropy = ((T-average_temperature) *
(T-average_temperature));
min_entropy = std::min (min_entropy, entropy);
max_entropy = std::max (max_entropy, entropy);
area += fe_values.JxW(q);
entropy_integrated += fe_values.JxW(q) * entropy;
}
}

Now we only need to exchange data between processors: we need to sum the two integrals (area, entropy_integrated), and get the extrema for maximum and minimum. We could do this through four different data exchanges, but we can it with two: Utilities::MPI::sum also exists in a variant that takes an array of values that are all to be summed up. And we can also utilize the Utilities::MPI::max function by realizing that forming the minimum over the minimal entropies equals forming the negative of the maximum over the negative of the minimal entropies; this maximum can then be combined with forming the maximum over the maximal entropies.

const double local_sums[2] = { entropy_integrated, area },
local_maxima[2] = { -min_entropy, max_entropy };
double global_sums[2], global_maxima[2];
Utilities::MPI::sum (local_sums, MPI_COMM_WORLD, global_sums);
Utilities::MPI::max (local_maxima, MPI_COMM_WORLD, global_maxima);

Having computed everything this way, we can then compute the average entropy and find the \(L^\infty\) norm by taking the larger of the deviation of the maximum or minimum from the average:

const double average_entropy = global_sums[0] / global_sums[1];
const double entropy_diff = std::max(global_maxima[1] - average_entropy,
average_entropy - (-global_maxima[0]));
return entropy_diff;
}

BoussinesqFlowProblem::get_extrapolated_temperature_range

The next function computes the minimal and maximal value of the extrapolated temperature over the entire domain. Again, this is only a slightly modified version of the respective function in step-31. As in the function above, we collect local minima and maxima and then compute the global extrema using the same trick as above.

As already discussed in step-31, the function needs to distinguish between the first and all following time steps because it uses a higher order temperature extrapolation scheme when at least two previous time steps are available.

template <int dim>
std::pair<double,double>
BoussinesqFlowProblem<dim>::get_extrapolated_temperature_range () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.temperature_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, temperature_fe, quadrature_formula,
std::vector<double> old_temperature_values(n_q_points);
std::vector<double> old_old_temperature_values(n_q_points);
double min_local_temperature = std::numeric_limits<double>::max(),
max_local_temperature = -std::numeric_limits<double>::max();
if (timestep_number != 0)
{
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
fe_values.get_function_values (old_old_temperature_solution,
old_old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double temperature =
(1. + time_step/old_time_step) * old_temperature_values[q]-
time_step/old_time_step * old_old_temperature_values[q];
min_local_temperature = std::min (min_local_temperature,
temperature);
max_local_temperature = std::max (max_local_temperature,
temperature);
}
}
}
else
{
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double temperature = old_temperature_values[q];
min_local_temperature = std::min (min_local_temperature,
temperature);
max_local_temperature = std::max (max_local_temperature,
temperature);
}
}
}
double local_extrema[2] = { -min_local_temperature,
max_local_temperature
};
double global_extrema[2];
Utilities::MPI::max (local_extrema, MPI_COMM_WORLD, global_extrema);
return std::make_pair(-global_extrema[0], global_extrema[1]);
}

BoussinesqFlowProblem::compute_viscosity

The function that calculates the viscosity is purely local and so needs no communication at all. It is mostly the same as in step-31 but with an updated formulation of the viscosity if \(\alpha=2\) is chosen:

template <int dim>
double
BoussinesqFlowProblem<dim>::
compute_viscosity (const std::vector<double> &old_temperature,
const std::vector<double> &old_old_temperature,
const std::vector<Tensor<1,dim> > &old_temperature_grads,
const std::vector<Tensor<1,dim> > &old_old_temperature_grads,
const std::vector<double> &old_temperature_laplacians,
const std::vector<double> &old_old_temperature_laplacians,
const std::vector<Tensor<1,dim> > &old_velocity_values,
const std::vector<Tensor<1,dim> > &old_old_velocity_values,
const std::vector<SymmetricTensor<2,dim> > &old_strain_rates,
const std::vector<SymmetricTensor<2,dim> > &old_old_strain_rates,
const double global_u_infty,
const double global_T_variation,
const double average_temperature,
const double global_entropy_variation,
const double cell_diameter) const
{
if (global_u_infty == 0)
return 5e-3 * cell_diameter;
const unsigned int n_q_points = old_temperature.size();
double max_residual = 0;
double max_velocity = 0;
for (unsigned int q=0; q < n_q_points; ++q)
{
const Tensor<1,dim> u = (old_velocity_values[q] +
old_old_velocity_values[q]) / 2;
const SymmetricTensor<2,dim> strain_rate = (old_strain_rates[q] +
old_old_strain_rates[q]) / 2;
const double T = (old_temperature[q] + old_old_temperature[q]) / 2;
const double dT_dt = (old_temperature[q] - old_old_temperature[q])
/ old_time_step;
const double u_grad_T = u * (old_temperature_grads[q] +
old_old_temperature_grads[q]) / 2;
const double kappa_Delta_T = EquationData::kappa
* (old_temperature_laplacians[q] +
old_old_temperature_laplacians[q]) / 2;
const double gamma
= ((EquationData::radiogenic_heating * EquationData::density(T)
+
2 * EquationData::eta * strain_rate * strain_rate) /
(EquationData::density(T) * EquationData::specific_heat));
double residual
= std::abs(dT_dt + u_grad_T - kappa_Delta_T - gamma);
if (parameters.stabilization_alpha == 2)
residual *= std::abs(T - average_temperature);
max_residual = std::max (residual, max_residual);
max_velocity = std::max (std::sqrt (u*u), max_velocity);
}
const double max_viscosity = (parameters.stabilization_beta *
max_velocity * cell_diameter);
if (timestep_number == 0)
return max_viscosity;
else
{
Assert (old_time_step > 0, ExcInternalError());
double entropy_viscosity;
if (parameters.stabilization_alpha == 2)
entropy_viscosity = (parameters.stabilization_c_R *
cell_diameter * cell_diameter *
max_residual /
global_entropy_variation);
else
entropy_viscosity = (parameters.stabilization_c_R *
cell_diameter * global_Omega_diameter *
max_velocity * max_residual /
(global_u_infty * global_T_variation));
return std::min (max_viscosity, entropy_viscosity);
}
}

BoussinesqFlowProblem::project_temperature_field

This function is new compared to step-31. What is does is to re-implement the library function VectorTools::project() for an MPI-based parallelization, a function we used for generating an initial vector for temperature based on some initial function. The library function only works with shared memory but doesn't know how to utilize multiple machines coupled through MPI to compute the projected field. The details of a project() function are not very difficult. All we do is to use a mass matrix and put the evaluation of the initial value function on the right hand side. The mass matrix for temperature we can simply generate using the respective assembly function, so all we need to do here is to create the right hand side and do a CG solve. The assembly function does a loop over all cells and evaluates the function in the EquationData namespace, and does this only on cells owned by the respective processor. The implementation of this assembly differs from the assembly we do for the principal assembly functions further down (which include thread-based parallelization with the WorkStream concept). Here we chose to keep things simple (keeping in mind that this function is also only called once at the beginning of the program, not in every time step), and generating the right hand side is cheap anyway so we won't even notice that this part is not parallized by threads.

Regarding the implementation of inhomogeneous Dirichlet boundary conditions: Since we use the temperature ConstraintMatrix, we could apply the boundary conditions directly when building the respective matrix and right hand side. In this case, the boundary conditions are inhomogeneous, which makes this procedure somewhat tricky since we get the matrix from some other function that uses its own integration and assembly loop. However, the correct imposition of boundary conditions needs the matrix data we work on plus the right hand side simultaneously, since the right hand side is created by Gaussian elimination on the matrix rows. In order to not introduce the matrix assembly at this place, but still having the matrix data available, we choose to create a dummy matrix matrix_for_bc that we only fill with data when we need it for imposing boundary conditions. These positions are exactly those where we have an inhomogeneous entry in the ConstraintMatrix. There are only a few such positions (on the boundary DoFs), so it is still much cheaper to use this function than to create the full matrix here. To implement this, we ask the constraint matrix whether the DoF under consideration is inhomogeneously constrained. In that case, we generate the respective matrix column that we need for creating the correct right hand side. Note that this (manually generated) matrix entry needs to be exactly the entry that we would fill the matrix with — otherwise, this will not work.

template <int dim>
void BoussinesqFlowProblem<dim>::project_temperature_field ()
{
assemble_temperature_matrix ();
QGauss<dim> quadrature(parameters.temperature_degree+2);
FEValues<dim> fe_values (mapping, temperature_fe, quadrature, update_flags);
const unsigned int dofs_per_cell = fe_values.dofs_per_cell,
n_q_points = fe_values.n_quadrature_points;
std::vector<types::global_dof_index> local_dof_indices (dofs_per_cell);
Vector<double> cell_vector (dofs_per_cell);
FullMatrix<double> matrix_for_bc (dofs_per_cell, dofs_per_cell);
std::vector<double> rhs_values(n_q_points);
IndexSet row_temp_matrix_partitioning(temperature_mass_matrix.n());
row_temp_matrix_partitioning.add_range(temperature_mass_matrix.local_range().first,
temperature_mass_matrix.local_range().second);
TrilinosWrappers::MPI::Vector rhs (row_temp_matrix_partitioning),
solution (row_temp_matrix_partitioning);
const EquationData::TemperatureInitialValues<dim> initial_temperature;
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
cell->get_dof_indices (local_dof_indices);
fe_values.reinit (cell);
initial_temperature.value_list (fe_values.get_quadrature_points(),
rhs_values);
cell_vector = 0;
matrix_for_bc = 0;
for (unsigned int point=0; point<n_q_points; ++point)
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
cell_vector(i) += rhs_values[point] *
fe_values.shape_value(i,point) *
fe_values.JxW(point);
if (temperature_constraints.is_inhomogeneously_constrained(local_dof_indices[i]))
{
for (unsigned int j=0; j<dofs_per_cell; ++j)
matrix_for_bc(j,i) += fe_values.shape_value(i,point) *
fe_values.shape_value(j,point) *
fe_values.JxW(point);
}
}
temperature_constraints.distribute_local_to_global (cell_vector,
local_dof_indices,
rhs,
matrix_for_bc);
}

Now that we have the right linear system, we solve it using the CG method with a simple Jacobi preconditioner:

SolverControl solver_control(5*rhs.size(), 1e-12*rhs.l2_norm());
preconditioner_mass.initialize(temperature_mass_matrix, 1.3);
cg.solve (temperature_mass_matrix, solution, rhs, preconditioner_mass);
temperature_constraints.distribute (solution);

Having so computed the current temperature field, let us set the member variable that holds the temperature nodes. Strictly speaking, we really only need to set old_temperature_solution since the first thing we will do is to compute the Stokes solution that only requires the previous time step's temperature field. That said, nothing good can come from not initializing the other vectors as well (especially since it's a relatively cheap operation and we only have to do it once at the beginning of the program) if we ever want to extend our numerical method or physical model, and so we initialize temperature_solution and old_old_temperature_solution as well. As a sidenote, while the solution vector is strictly distributed (i.e. each processor only stores a mutually exclusive subset of elements), the assignment makes sure that the vectors on the left hand side (which where initialized to contain ghost elements as well) also get the correct ghost elements. In other words, the assignment here requires communication between processors:

temperature_solution = solution;
old_temperature_solution = solution;
old_old_temperature_solution = solution;
}

The BoussinesqFlowProblem setup functions

The following three functions set up the Stokes matrix, the matrix used for the Stokes preconditioner, and the temperature matrix. The code is mostly the same as in step-31, but it has been broken out into three functions of their own for simplicity.

The main functional difference between the code here and that in step-31 is that the matrices we want to set up are distributed across multiple processors. Since we still want to build up the sparsity pattern first for efficiency reasons, we could continue to build the entire sparsity pattern as a BlockDynamicSparsityPattern, as we did in step-31. However, that would be inefficient: every processor would build the same sparsity pattern, but only initialize a small part of the matrix using it. It also violates the principle that every processor should only work on those cells it owns (and, if necessary the layer of ghost cells around it).

Rather, we use an object of type TrilinosWrappers::BlockSparsityPattern, which is (obviously) a wrapper around a sparsity pattern object provided by Trilinos. The advantage is that the Trilinos sparsity pattern class can communicate across multiple processors: if this processor fills in all the nonzero entries that result from the cells it owns, and every other processor does so as well, then at the end after some MPI communication initiated by the compress() call, we will have the globally assembled sparsity pattern available with which the global matrix can be initialized.

There is one important aspect when initializing Trilinos sparsity patterns in parallel: In addition to specifying the locally owned rows and columns of the matrices via the stokes_partitioning index set, we also supply information about all the rows we are possibly going to write into when assembling on a certain processor. The set of locally relevant rows contains all such rows (possibly also a few unnecessary ones, but it is difficult to find the exact row indices before actually getting indices on all cells and resolving constraints). This additional information allows to exactly determine the structure for the off-processor data found during assembly. While Trilinos matrices are able to collect this information on the fly as well (when initializing them from some other reinit method), it is less efficient and leads to problems when assembling matrices with multiple threads. In this program, we pessimistically assume that only one processor at a time can write into the matrix while assembly (whereas the computation is parallel), which is fine for Trilinos matrices. In practice, one can do better by hinting WorkStream at cells that do not share vertices, allowing for parallelism among those cells (see the graph coloring algorithms and WorkStream with colored iterators argument). However, that only works when only one MPI processor is present because Trilinos' internal data structures for accumulating off-processor data on the fly are not thread safe. With the initialization presented here, there is no such problem and one could safely introduce graph coloring for this algorithm.

The only other change we need to make is to tell the DoFTools::make_sparsity_pattern() function that it is only supposed to work on a subset of cells, namely the ones whose subdomain_id equals the number of the current processor, and to ignore all other cells.

This strategy is replicated across all three of the following functions.

Note that Trilinos matrices store the information contained in the sparsity patterns, so we can safely release the sp variable once the matrix has been given the sparsity structure.

template <int dim>
void BoussinesqFlowProblem<dim>::
setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning)
{
stokes_matrix.clear ();
TrilinosWrappers::BlockSparsityPattern sp(stokes_partitioning, stokes_partitioning,
stokes_relevant_partitioning,
MPI_COMM_WORLD);
Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
for (unsigned int c=0; c<dim+1; ++c)
for (unsigned int d=0; d<dim+1; ++d)
if (! ((c==dim) && (d==dim)))
coupling[c][d] = DoFTools::always;
else
coupling[c][d] = DoFTools::none;
DoFTools::make_sparsity_pattern (stokes_dof_handler,
coupling, sp,
stokes_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
stokes_matrix.reinit (sp);
}
template <int dim>
void BoussinesqFlowProblem<dim>::
setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning)
{
Amg_preconditioner.reset ();
Mp_preconditioner.reset ();
stokes_preconditioner_matrix.clear ();
TrilinosWrappers::BlockSparsityPattern sp(stokes_partitioning, stokes_partitioning,
stokes_relevant_partitioning,
MPI_COMM_WORLD);
Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
for (unsigned int c=0; c<dim+1; ++c)
for (unsigned int d=0; d<dim+1; ++d)
if (c == d)
coupling[c][d] = DoFTools::always;
else
coupling[c][d] = DoFTools::none;
DoFTools::make_sparsity_pattern (stokes_dof_handler,
coupling, sp,
stokes_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
stokes_preconditioner_matrix.reinit (sp);
}
template <int dim>
void BoussinesqFlowProblem<dim>::
setup_temperature_matrices (const IndexSet &temperature_partitioner,
const IndexSet &temperature_relevant_partitioner)
{
T_preconditioner.reset ();
temperature_mass_matrix.clear ();
temperature_stiffness_matrix.clear ();
temperature_matrix.clear ();
TrilinosWrappers::SparsityPattern sp(temperature_partitioner,
temperature_partitioner,
temperature_relevant_partitioner,
MPI_COMM_WORLD);
DoFTools::make_sparsity_pattern (temperature_dof_handler, sp,
temperature_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
temperature_matrix.reinit (sp);
temperature_mass_matrix.reinit (sp);
temperature_stiffness_matrix.reinit (sp);
}

The remainder of the setup function (after splitting out the three functions above) mostly has to deal with the things we need to do for parallelization across processors. Because setting all of this up is a significant compute time expense of the program, we put everything we do here into a timer group so that we can get summary information about the fraction of time spent in this part of the program at its end.

At the top as usual we enumerate degrees of freedom and sort them by component/block, followed by writing their numbers to the screen from processor zero. The DoFHandler::distributed_dofs() function, when applied to a parallel::distributed::Triangulation object, sorts degrees of freedom in such a way that all degrees of freedom associated with subdomain zero come before all those associated with subdomain one, etc. For the Stokes part, this entails, however, that velocities and pressures become intermixed, but this is trivially solved by sorting again by blocks; it is worth noting that this latter operation leaves the relative ordering of all velocities and pressures alone, i.e. within the velocity block we will still have all those associated with subdomain zero before all velocities associated with subdomain one, etc. This is important since we store each of the blocks of this matrix distributed across all processors and want this to be done in such a way that each processor stores that part of the matrix that is roughly equal to the degrees of freedom located on those cells that it will actually work on.

When printing the numbers of degrees of freedom, note that these numbers are going to be large if we use many processors. Consequently, we let the stream put a comma separator in between every three digits. The state of the stream, using the locale, is saved from before to after this operation. While slightly opaque, the code works because the default locale (which we get using the constructor call std::locale("")) implies printing numbers with a comma separator for every third digit (i.e., thousands, millions, billions).

template <int dim>
void BoussinesqFlowProblem<dim>::setup_dofs ()
{
computing_timer.enter_section("Setup dof systems");
std::vector<unsigned int> stokes_sub_blocks (dim+1,0);
stokes_sub_blocks[dim] = 1;
stokes_dof_handler.distribute_dofs (stokes_fe);
DoFRenumbering::component_wise (stokes_dof_handler, stokes_sub_blocks);
temperature_dof_handler.distribute_dofs (temperature_fe);
std::vector<types::global_dof_index> stokes_dofs_per_block (2);
DoFTools::count_dofs_per_block (stokes_dof_handler, stokes_dofs_per_block,
stokes_sub_blocks);
const unsigned int n_u = stokes_dofs_per_block[0],
n_p = stokes_dofs_per_block[1],
n_T = temperature_dof_handler.n_dofs();
std::locale s = pcout.get_stream().getloc();
pcout.get_stream().imbue(std::locale(""));
pcout << "Number of active cells: "
<< triangulation.n_global_active_cells()
<< " (on "
<< triangulation.n_levels()
<< " levels)"
<< std::endl
<< "Number of degrees of freedom: "
<< n_u + n_p + n_T
<< " (" << n_u << '+' << n_p << '+'<< n_T <<')'
<< std::endl
<< std::endl;
pcout.get_stream().imbue(s);

After this, we have to set up the various partitioners (of type IndexSet, see the introduction) that describe which parts of each matrix or vector will be stored where, then call the functions that actually set up the matrices, and at the end also resize the various vectors we keep around in this program.

std::vector<IndexSet> stokes_partitioning, stokes_relevant_partitioning;
IndexSet temperature_partitioning (n_T), temperature_relevant_partitioning (n_T);
IndexSet stokes_relevant_set;
{
IndexSet stokes_index_set = stokes_dof_handler.locally_owned_dofs();
stokes_partitioning.push_back(stokes_index_set.get_view(0,n_u));
stokes_partitioning.push_back(stokes_index_set.get_view(n_u,n_u+n_p));
stokes_relevant_set);
stokes_relevant_partitioning.push_back(stokes_relevant_set.get_view(0,n_u));
stokes_relevant_partitioning.push_back(stokes_relevant_set.get_view(n_u,n_u+n_p));
temperature_partitioning = temperature_dof_handler.locally_owned_dofs();
DoFTools::extract_locally_relevant_dofs (temperature_dof_handler,
temperature_relevant_partitioning);
}

Following this, we can compute constraints for the solution vectors, including hanging node constraints and homogeneous and inhomogeneous boundary values for the Stokes and temperature fields. Note that as for everything else, the constraint objects can not hold all constraints on every processor. Rather, each processor needs to store only those that are actually necessary for correctness given that it only assembles linear systems on cells it owns. As discussed in the this paper, the set of constraints we need to know about is exactly the set of constraints on all locally relevant degrees of freedom, so this is what we use to initialize the constraint objects.

{
stokes_constraints.clear ();
stokes_constraints.reinit (stokes_relevant_set);
stokes_constraints);
FEValuesExtractors::Vector velocity_components(0);
0,
stokes_constraints,
stokes_fe.component_mask(velocity_components));
std::set<types::boundary_id> no_normal_flux_boundaries;
no_normal_flux_boundaries.insert (1);
no_normal_flux_boundaries,
stokes_constraints,
mapping);
stokes_constraints.close ();
}
{
temperature_constraints.clear ();
temperature_constraints.reinit (temperature_relevant_partitioning);
DoFTools::make_hanging_node_constraints (temperature_dof_handler,
temperature_constraints);
VectorTools::interpolate_boundary_values (temperature_dof_handler,
0,
EquationData::TemperatureInitialValues<dim>(),
temperature_constraints);
VectorTools::interpolate_boundary_values (temperature_dof_handler,
1,
EquationData::TemperatureInitialValues<dim>(),
temperature_constraints);
temperature_constraints.close ();
}

All this done, we can then initialize the various matrix and vector objects to their proper sizes. At the end, we also record that all matrices and preconditioners have to be re-computed at the beginning of the next time step. Note how we initialize the vectors for the Stokes and temperature right hand sides: These are writable vectors (last boolean argument set to true) that have the correct one-to-one partitioning of locally owned elements but are still given the relevant partitioning for means of figuring out the vector entries that are going to be set right away. As for matrices, this allows for writing local contributions into the vector with multiple threads (always assuming that the same vector entry is not accessed by multiple threads at the same time). The other vectors only allow for read access of individual elements, including ghosts, but are not suitable for solvers.

setup_stokes_matrix (stokes_partitioning, stokes_relevant_partitioning);
setup_stokes_preconditioner (stokes_partitioning,
stokes_relevant_partitioning);
setup_temperature_matrices (temperature_partitioning,
temperature_relevant_partitioning);
stokes_rhs.reinit (stokes_partitioning, stokes_relevant_partitioning,
MPI_COMM_WORLD, true);
stokes_solution.reinit (stokes_relevant_partitioning, MPI_COMM_WORLD);
old_stokes_solution.reinit (stokes_solution);
temperature_rhs.reinit (temperature_partitioning,
temperature_relevant_partitioning,
MPI_COMM_WORLD, true);
temperature_solution.reinit (temperature_relevant_partitioning, MPI_COMM_WORLD);
old_temperature_solution.reinit (temperature_solution);
old_old_temperature_solution.reinit (temperature_solution);
rebuild_stokes_matrix = true;
rebuild_stokes_preconditioner = true;
rebuild_temperature_matrices = true;
rebuild_temperature_preconditioner = true;
computing_timer.exit_section();
}

The BoussinesqFlowProblem assembly functions

Following the discussion in the introduction and in the Parallel computing with multiple processors accessing shared memory module, we split the assembly functions into different parts:

Stokes preconditioner assembly

Let us start with the functions that builds the Stokes preconditioner. The first two of these are pretty trivial, given the discussion above. Note in particular that the main point in using the scratch data object is that we want to avoid allocating any objects on the free space each time we visit a new cell. As a consequence, the assembly function below only has automatic local variables, and everything else is accessed through the scratch data object, which is allocated only once before we start the loop over all cells:

template <int dim>
void
BoussinesqFlowProblem<dim>::
local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesPreconditioner<dim> &scratch,
Assembly::CopyData::StokesPreconditioner<dim> &data)
{
const unsigned int dofs_per_cell = stokes_fe.dofs_per_cell;
const unsigned int n_q_points = scratch.stokes_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
const FEValuesExtractors::Scalar pressure (dim);
scratch.stokes_fe_values.reinit (cell);
cell->get_dof_indices (data.local_dof_indices);
data.local_matrix = 0;
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.grad_phi_u[k] = scratch.stokes_fe_values[velocities].gradient(k,q);
scratch.phi_p[k] = scratch.stokes_fe_values[pressure].value (k, q);
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.local_matrix(i,j) += (EquationData::eta *
scalar_product (scratch.grad_phi_u[i],
scratch.grad_phi_u[j])
+
(1./EquationData::eta) *
EquationData::pressure_scaling *
EquationData::pressure_scaling *
(scratch.phi_p[i] * scratch.phi_p[j]))
* scratch.stokes_fe_values.JxW(q);
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data)
{
stokes_constraints.distribute_local_to_global (data.local_matrix,
data.local_dof_indices,
stokes_preconditioner_matrix);
}

Now for the function that actually puts things together, using the WorkStream functions. WorkStream::run needs a start and end iterator to enumerate the cells it is supposed to work on. Typically, one would use DoFHandler::begin_active() and DoFHandler::end() for that but here we actually only want the subset of cells that in fact are owned by the current processor. This is where the FilteredIterator class comes into play: you give it a range of cells and it provides an iterator that only iterates over that subset of cells that satisfy a certain predicate (a predicate is a function of one argument that either returns true or false). The predicate we use here is IteratorFilters::LocallyOwnedCell, i.e., it returns true exactly if the cell is owned by the current processor. The resulting iterator range is then exactly what we need.

With this obstacle out of the way, we call the WorkStream::run function with this set of cells, scratch and copy objects, and with pointers to two functions: the local assembly and copy-local-to-global function. These functions need to have very specific signatures: three arguments in the first and one argument in the latter case (see the documentation of the WorkStream::run function for the meaning of these arguments). Note how we use the construct std::bind to create a function object that satisfies this requirement. It uses placeholders std::placeholders::_1, std::placeholders::_2, std::placeholders::_3 for the local assembly function that specify cell, scratch data, and copy data, as well as the placeholder std::placeholders::_1 for the copy function that expects the data to be written into the global matrix (for placeholder arguments, also see the discussion in step-13's assemble_linear_system() function). On the other hand, the implicit zeroth argument of member functions (namely the this pointer of the object on which that member function is to operate on) is bound to the this pointer of the current function. The WorkStream::run function, as a consequence, does not need to know anything about the object these functions work on.

When the WorkStream is executed, it will create several local assembly routines of the first kind for several cells and let some available processors work on them. The function that needs to be synchronized, i.e., the write operation into the global matrix, however, is executed by only one thread at a time in the prescribed order. Of course, this only holds for the parallelization on a single MPI process. Different MPI processes will have their own WorkStream objects and do that work completely independently (and in different memory spaces). In a distributed calculation, some data will accumulate at degrees of freedom that are not owned by the respective processor. It would be inefficient to send data around every time we encounter such a dof. What happens instead is that the Trilinos sparse matrix will keep that data and send it to the owner at the end of assembly, by calling the compress() command.

template <int dim>
void
BoussinesqFlowProblem<dim>::assemble_stokes_preconditioner ()
{
stokes_preconditioner_matrix = 0;
const QGauss<dim> quadrature_formula(parameters.stokes_velocity_degree+1);
typedef
CellFilter;
stokes_dof_handler.begin_active()),
stokes_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_stokes_preconditioner,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_preconditioner,
this,
std::placeholders::_1),
Assembly::Scratch::
StokesPreconditioner<dim> (stokes_fe, quadrature_formula,
mapping,
Assembly::CopyData::
StokesPreconditioner<dim> (stokes_fe));
stokes_preconditioner_matrix.compress(VectorOperation::add);
}

The final function in this block initiates assembly of the Stokes preconditioner matrix and then in fact builds the Stokes preconditioner. It is mostly the same as in the serial case. The only difference to step-31 is that we use a Jacobi preconditioner for the pressure mass matrix instead of IC, as discussed in the introduction.

template <int dim>
void
BoussinesqFlowProblem<dim>::build_stokes_preconditioner ()
{
if (rebuild_stokes_preconditioner == false)
return;
computing_timer.enter_section (" Build Stokes preconditioner");
pcout << " Rebuilding Stokes preconditioner..." << std::flush;
assemble_stokes_preconditioner ();
std::vector<std::vector<bool> > constant_modes;
FEValuesExtractors::Vector velocity_components(0);
DoFTools::extract_constant_modes (stokes_dof_handler,
stokes_fe.component_mask(velocity_components),
constant_modes);
Mp_preconditioner.reset (new TrilinosWrappers::PreconditionJacobi());
Amg_preconditioner.reset (new TrilinosWrappers::PreconditionAMG());
Amg_data.constant_modes = constant_modes;
Amg_data.elliptic = true;
Amg_data.higher_order_elements = true;
Amg_data.smoother_sweeps = 2;
Amg_data.aggregation_threshold = 0.02;
Mp_preconditioner->initialize (stokes_preconditioner_matrix.block(1,1));
Amg_preconditioner->initialize (stokes_preconditioner_matrix.block(0,0),
Amg_data);
rebuild_stokes_preconditioner = false;
pcout << std::endl;
computing_timer.exit_section();
}

Stokes system assembly

The next three functions implement the assembly of the Stokes system, again split up into a part performing local calculations, one for writing the local data into the global matrix and vector, and one for actually running the loop over all cells with the help of the WorkStream class. Note that the assembly of the Stokes matrix needs only to be done in case we have changed the mesh. Otherwise, just the (temperature-dependent) right hand side needs to be calculated here. Since we are working with distributed matrices and vectors, we have to call the respective compress() functions in the end of the assembly in order to send non-local data to the owner process.

template <int dim>
void
BoussinesqFlowProblem<dim>::
local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesSystem<dim> &scratch,
Assembly::CopyData::StokesSystem<dim> &data)
{
const unsigned int dofs_per_cell = scratch.stokes_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.stokes_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
const FEValuesExtractors::Scalar pressure (dim);
scratch.stokes_fe_values.reinit (cell);
temperature_cell (&triangulation,
cell->level(),
cell->index(),
&temperature_dof_handler);
scratch.temperature_fe_values.reinit (temperature_cell);
if (rebuild_stokes_matrix)
data.local_matrix = 0;
data.local_rhs = 0;
scratch.temperature_fe_values.get_function_values (old_temperature_solution,
scratch.old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double old_temperature = scratch.old_temperature_values[q];
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.phi_u[k] = scratch.stokes_fe_values[velocities].value (k,q);
if (rebuild_stokes_matrix)
{
scratch.grads_phi_u[k] = scratch.stokes_fe_values[velocities].symmetric_gradient(k,q);
scratch.div_phi_u[k] = scratch.stokes_fe_values[velocities].divergence (k, q);
scratch.phi_p[k] = scratch.stokes_fe_values[pressure].value (k, q);
}
}
if (rebuild_stokes_matrix == true)
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.local_matrix(i,j) += (EquationData::eta * 2 *
(scratch.grads_phi_u[i] * scratch.grads_phi_u[j])
- (EquationData::pressure_scaling *
scratch.div_phi_u[i] * scratch.phi_p[j])
- (EquationData::pressure_scaling *
scratch.phi_p[i] * scratch.div_phi_u[j]))
* scratch.stokes_fe_values.JxW(q);
gravity = EquationData::gravity_vector (scratch.stokes_fe_values
.quadrature_point(q));
for (unsigned int i=0; i<dofs_per_cell; ++i)
data.local_rhs(i) += (EquationData::density(old_temperature) *
gravity *
scratch.phi_u[i]) *
scratch.stokes_fe_values.JxW(q);
}
cell->get_dof_indices (data.local_dof_indices);
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data)
{
if (rebuild_stokes_matrix == true)
stokes_constraints.distribute_local_to_global (data.local_matrix,
data.local_rhs,
data.local_dof_indices,
stokes_matrix,
stokes_rhs);
else
stokes_constraints.distribute_local_to_global (data.local_rhs,
data.local_dof_indices,
stokes_rhs);
}
template <int dim>
void BoussinesqFlowProblem<dim>::assemble_stokes_system ()
{
computing_timer.enter_section (" Assemble Stokes system");
if (rebuild_stokes_matrix == true)
stokes_matrix=0;
stokes_rhs=0;
const QGauss<dim> quadrature_formula(parameters.stokes_velocity_degree+1);
typedef
CellFilter;
stokes_dof_handler.begin_active()),
stokes_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_stokes_system,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_system,
this,
std::placeholders::_1),
Assembly::Scratch::
StokesSystem<dim> (stokes_fe, mapping, quadrature_formula,
(rebuild_stokes_matrix == true
?
:
temperature_fe,
Assembly::CopyData::
StokesSystem<dim> (stokes_fe));
if (rebuild_stokes_matrix == true)
stokes_matrix.compress(VectorOperation::add);
stokes_rhs.compress(VectorOperation::add);
rebuild_stokes_matrix = false;
pcout << std::endl;
computing_timer.exit_section();
}

Temperature matrix assembly

The task to be performed by the next three functions is to calculate a mass matrix and a Laplace matrix on the temperature system. These will be combined in order to yield the semi-implicit time stepping matrix that consists of the mass matrix plus a time step-dependent weight factor times the Laplace matrix. This function is again essentially the body of the loop over all cells from step-31.

The two following functions perform similar services as the ones above.

template <int dim>
void BoussinesqFlowProblem<dim>::
local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::TemperatureMatrix<dim> &scratch,
Assembly::CopyData::TemperatureMatrix<dim> &data)
{
const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.temperature_fe_values.n_quadrature_points;
scratch.temperature_fe_values.reinit (cell);
cell->get_dof_indices (data.local_dof_indices);
data.local_mass_matrix = 0;
data.local_stiffness_matrix = 0;
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
scratch.phi_T[k] = scratch.temperature_fe_values.shape_value (k, q);
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
{
data.local_mass_matrix(i,j)
+= (scratch.phi_T[i] * scratch.phi_T[j]
*
scratch.temperature_fe_values.JxW(q));
data.local_stiffness_matrix(i,j)
+= (EquationData::kappa * scratch.grad_phi_T[i] * scratch.grad_phi_T[j]
*
scratch.temperature_fe_values.JxW(q));
}
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data)
{
temperature_constraints.distribute_local_to_global (data.local_mass_matrix,
data.local_dof_indices,
temperature_mass_matrix);
temperature_constraints.distribute_local_to_global (data.local_stiffness_matrix,
data.local_dof_indices,
temperature_stiffness_matrix);
}
template <int dim>
void BoussinesqFlowProblem<dim>::assemble_temperature_matrix ()
{
if (rebuild_temperature_matrices == false)
return;
computing_timer.enter_section (" Assemble temperature matrices");
temperature_mass_matrix = 0;
temperature_stiffness_matrix = 0;
const QGauss<dim> quadrature_formula(parameters.temperature_degree+2);
typedef
CellFilter;
temperature_dof_handler.begin_active()),
temperature_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_temperature_matrix,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_matrix,
this,
std::placeholders::_1),
Assembly::Scratch::
TemperatureMatrix<dim> (temperature_fe, mapping, quadrature_formula),
Assembly::CopyData::
TemperatureMatrix<dim> (temperature_fe));
temperature_mass_matrix.compress(VectorOperation::add);
temperature_stiffness_matrix.compress(VectorOperation::add);
rebuild_temperature_matrices = false;
rebuild_temperature_preconditioner = true;
computing_timer.exit_section();
}

Temperature right hand side assembly

This is the last assembly function. It calculates the right hand side of the temperature system, which includes the convection and the stabilization terms. It includes a lot of evaluations of old solutions at the quadrature points (which are necessary for calculating the artificial viscosity of stabilization), but is otherwise similar to the other assembly functions. Notice, once again, how we resolve the dilemma of having inhomogeneous boundary conditions, by just making a right hand side at this point (compare the comments for the project() function above): We create some matrix columns with exactly the values that would be entered for the temperature stiffness matrix, in case we have inhomogeneously constrained dofs. That will account for the correct balance of the right hand side vector with the matrix system of temperature.

template <int dim>
void BoussinesqFlowProblem<dim>::
local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
const double global_max_velocity,
const double global_entropy_variation,
Assembly::Scratch::TemperatureRHS<dim> &scratch,
Assembly::CopyData::TemperatureRHS<dim> &data)
{
const bool use_bdf2_scheme = (timestep_number != 0);
const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.temperature_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
data.local_rhs = 0;
data.matrix_for_bc = 0;
cell->get_dof_indices (data.local_dof_indices);
scratch.temperature_fe_values.reinit (cell);
stokes_cell (&triangulation,
cell->level(),
cell->index(),
&stokes_dof_handler);
scratch.stokes_fe_values.reinit (stokes_cell);
scratch.temperature_fe_values.get_function_values (old_temperature_solution,
scratch.old_temperature_values);
scratch.temperature_fe_values.get_function_values (old_old_temperature_solution,
scratch.old_old_temperature_values);
scratch.temperature_fe_values.get_function_gradients (old_temperature_solution,
scratch.old_temperature_grads);
scratch.temperature_fe_values.get_function_gradients (old_old_temperature_solution,
scratch.old_old_temperature_grads);
scratch.temperature_fe_values.get_function_laplacians (old_temperature_solution,
scratch.old_temperature_laplacians);
scratch.temperature_fe_values.get_function_laplacians (old_old_temperature_solution,
scratch.old_old_temperature_laplacians);
scratch.stokes_fe_values[velocities].get_function_values (stokes_solution,
scratch.old_velocity_values);
scratch.stokes_fe_values[velocities].get_function_values (old_stokes_solution,
scratch.old_old_velocity_values);
scratch.stokes_fe_values[velocities].get_function_symmetric_gradients (stokes_solution,
scratch.old_strain_rates);
scratch.stokes_fe_values[velocities].get_function_symmetric_gradients (old_stokes_solution,
scratch.old_old_strain_rates);
const double nu
= compute_viscosity (scratch.old_temperature_values,
scratch.old_old_temperature_values,
scratch.old_temperature_grads,
scratch.old_old_temperature_grads,
scratch.old_temperature_laplacians,
scratch.old_old_temperature_laplacians,
scratch.old_velocity_values,
scratch.old_old_velocity_values,
scratch.old_strain_rates,
scratch.old_old_strain_rates,
global_max_velocity,
global_T_range.second - global_T_range.first,
0.5 * (global_T_range.second + global_T_range.first),
global_entropy_variation,
cell->diameter());
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.phi_T[k] = scratch.temperature_fe_values.shape_value (k, q);
scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
}
const double T_term_for_rhs
= (use_bdf2_scheme ?
(scratch.old_temperature_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_values[q] *
(time_step * time_step) /
(old_time_step * (time_step + old_time_step)))
:
scratch.old_temperature_values[q]);
const double ext_T
= (use_bdf2_scheme ?
(scratch.old_temperature_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_values[q] *
time_step/old_time_step)
:
scratch.old_temperature_values[q]);
const Tensor<1,dim> ext_grad_T
= (use_bdf2_scheme ?
(scratch.old_temperature_grads[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_grads[q] *
time_step/old_time_step)
:
scratch.old_temperature_grads[q]);
const Tensor<1,dim> extrapolated_u
= (use_bdf2_scheme ?
(scratch.old_velocity_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_velocity_values[q] *
time_step/old_time_step)
:
scratch.old_velocity_values[q]);
const SymmetricTensor<2,dim> extrapolated_strain_rate
= (use_bdf2_scheme ?
(scratch.old_strain_rates[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_strain_rates[q] *
time_step/old_time_step)
:
scratch.old_strain_rates[q]);
const double gamma
= ((EquationData::radiogenic_heating * EquationData::density(ext_T)
+
2 * EquationData::eta * extrapolated_strain_rate * extrapolated_strain_rate) /
(EquationData::density(ext_T) * EquationData::specific_heat));
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
data.local_rhs(i) += (T_term_for_rhs * scratch.phi_T[i]
-
time_step *
extrapolated_u * ext_grad_T * scratch.phi_T[i]
-
time_step *
nu * ext_grad_T * scratch.grad_phi_T[i]
+
time_step *
gamma * scratch.phi_T[i])
*
scratch.temperature_fe_values.JxW(q);
if (temperature_constraints.is_inhomogeneously_constrained(data.local_dof_indices[i]))
{
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.matrix_for_bc(j,i) += (scratch.phi_T[i] * scratch.phi_T[j] *
(use_bdf2_scheme ?
((2*time_step + old_time_step) /
(time_step + old_time_step)) : 1.)
+
scratch.grad_phi_T[i] *
scratch.grad_phi_T[j] *
EquationData::kappa *
time_step)
*
scratch.temperature_fe_values.JxW(q);
}
}
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data)
{
temperature_constraints.distribute_local_to_global (data.local_rhs,
data.local_dof_indices,
temperature_rhs,
data.matrix_for_bc);
}

In the function that runs the WorkStream for actually calculating the right hand side, we also generate the final matrix. As mentioned above, it is a sum of the mass matrix and the Laplace matrix, times some time step-dependent weight. This weight is specified by the BDF-2 time integration scheme, see the introduction in step-31. What is new in this tutorial program (in addition to the use of MPI parallelization and the WorkStream class), is that we now precompute the temperature preconditioner as well. The reason is that the setup of the Jacobi preconditioner takes a noticeable time compared to the solver because we usually only need between 10 and 20 iterations for solving the temperature system (this might sound strange, as Jacobi really only consists of a diagonal, but in Trilinos it is derived from more general framework for point relaxation preconditioners which is a bit inefficient). Hence, it is more efficient to precompute the preconditioner, even though the matrix entries may slightly change because the time step might change. This is not too big a problem because we remesh every few time steps (and regenerate the preconditioner then).

template <int dim>
void BoussinesqFlowProblem<dim>::assemble_temperature_system (const double maximal_velocity)
{
const bool use_bdf2_scheme = (timestep_number != 0);
if (use_bdf2_scheme == true)
{
temperature_matrix.copy_from (temperature_mass_matrix);
temperature_matrix *= (2*time_step + old_time_step) /
(time_step + old_time_step);
temperature_matrix.add (time_step, temperature_stiffness_matrix);
}
else
{
temperature_matrix.copy_from (temperature_mass_matrix);
temperature_matrix.add (time_step, temperature_stiffness_matrix);
}
if (rebuild_temperature_preconditioner == true)
{
T_preconditioner.reset (new TrilinosWrappers::PreconditionJacobi());
T_preconditioner->initialize (temperature_matrix);
rebuild_temperature_preconditioner = false;
}

The next part is computing the right hand side vectors. To do so, we first compute the average temperature \(T_m\) that we use for evaluating the artificial viscosity stabilization through the residual \(E(T) = (T-T_m)^2\). We do this by defining the midpoint between maximum and minimum temperature as average temperature in the definition of the entropy viscosity. An alternative would be to use the integral average, but the results are not very sensitive to this choice. The rest then only requires calling WorkStream::run again, binding the arguments to the local_assemble_temperature_rhs function that are the same in every call to the correct values:

temperature_rhs = 0;
const QGauss<dim> quadrature_formula(parameters.temperature_degree+2);
const std::pair<double,double>
global_T_range = get_extrapolated_temperature_range();
const double average_temperature = 0.5 * (global_T_range.first +
global_T_range.second);
const double global_entropy_variation =
get_entropy_variation (average_temperature);
typedef
CellFilter;
temperature_dof_handler.begin_active()),
temperature_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_temperature_rhs,
this,
global_T_range,
maximal_velocity,
global_entropy_variation,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_rhs,
this,
std::placeholders::_1),
Assembly::Scratch::
TemperatureRHS<dim> (temperature_fe, stokes_fe, mapping,
quadrature_formula),
Assembly::CopyData::
TemperatureRHS<dim> (temperature_fe));
temperature_rhs.compress(VectorOperation::add);
}

BoussinesqFlowProblem::solve

This function solves the linear systems in each time step of the Boussinesq problem. First, we work on the Stokes system and then on the temperature system. In essence, it does the same things as the respective function in step-31. However, there are a few changes here.

The first change is related to the way we store our solution: we keep the vectors with locally owned degrees of freedom plus ghost nodes on each MPI node. When we enter a solver which is supposed to perform matrix-vector products with a distributed matrix, this is not the appropriate form, though. There, we will want to have the solution vector to be distributed in the same way as the matrix, i.e. without any ghosts. So what we do first is to generate a distributed vector called distributed_stokes_solution and put only the locally owned dofs into that, which is neatly done by the operator= of the Trilinos vector.

Next, we scale the pressure solution (or rather, the initial guess) for the solver so that it matches with the length scales in the matrices, as discussed in the introduction. We also immediately scale the pressure solution back to the correct units after the solution is completed. We also need to set the pressure values at hanging nodes to zero. This we also did in step-31 in order not to disturb the Schur complement by some vector entries that actually are irrelevant during the solve stage. As a difference to step-31, here we do it only for the locally owned pressure dofs. After solving for the Stokes solution, each processor copies the distributed solution back into the solution vector that also includes ghost elements.

The third and most obvious change is that we have two variants for the Stokes solver: A fast solver that sometimes breaks down, and a robust solver that is slower. This is what we already discussed in the introduction. Here is how we realize it: First, we perform 30 iterations with the fast solver based on the simple preconditioner based on the AMG V-cycle instead of an approximate solve (this is indicated by the false argument to the LinearSolvers::BlockSchurPreconditioner object). If we converge, everything is fine. If we do not converge, the solver control object will throw an exception SolverControl::NoConvergence. Usually, this would abort the program because we don't catch them in our usual solve() functions. This is certainly not what we want to happen here. Rather, we want to switch to the strong solver and continue the solution process with whatever vector we got so far. Hence, we catch the exception with the C++ try/catch mechanism. We then simply go through the same solver sequence again in the catch clause, this time passing the true flag to the preconditioner for the strong solver, signaling an approximate CG solve.

template <int dim>
void BoussinesqFlowProblem<dim>::solve ()
{
computing_timer.enter_section (" Solve Stokes system");
{
pcout << " Solving Stokes system... " << std::flush;
distributed_stokes_solution (stokes_rhs);
distributed_stokes_solution = stokes_solution;
distributed_stokes_solution.block(1) /= EquationData::pressure_scaling;
const unsigned int
start = (distributed_stokes_solution.block(0).size() +
distributed_stokes_solution.block(1).local_range().first),
end = (distributed_stokes_solution.block(0).size() +
distributed_stokes_solution.block(1).local_range().second);
for (unsigned int i=start; i<end; ++i)
if (stokes_constraints.is_constrained (i))
distributed_stokes_solution(i) = 0;
unsigned int n_iterations = 0;
const double solver_tolerance = 1e-8 * stokes_rhs.l2_norm();
SolverControl solver_control (30, solver_tolerance);
try
{
const LinearSolvers::BlockSchurPreconditioner<TrilinosWrappers::PreconditionAMG,
preconditioner (stokes_matrix, stokes_preconditioner_matrix,
*Mp_preconditioner, *Amg_preconditioner,
false);
solver(solver_control, mem,
AdditionalData(30, true));
solver.solve(stokes_matrix, distributed_stokes_solution, stokes_rhs,
preconditioner);
n_iterations = solver_control.last_step();
}
{
const LinearSolvers::BlockSchurPreconditioner<TrilinosWrappers::PreconditionAMG,
preconditioner (stokes_matrix, stokes_preconditioner_matrix,
*Mp_preconditioner, *Amg_preconditioner,
true);
SolverControl solver_control_refined (stokes_matrix.m(), solver_tolerance);
solver(solver_control_refined, mem,
AdditionalData(50, true));
solver.solve(stokes_matrix, distributed_stokes_solution, stokes_rhs,
preconditioner);
n_iterations = (solver_control.last_step() +
solver_control_refined.last_step());
}
stokes_constraints.distribute (distributed_stokes_solution);
distributed_stokes_solution.block(1) *= EquationData::pressure_scaling;
stokes_solution = distributed_stokes_solution;
pcout << n_iterations << " iterations."
<< std::endl;
}
computing_timer.exit_section();

Now let's turn to the temperature part: First, we compute the time step size. We found that we need smaller time steps for 3D than for 2D for the shell geometry. This is because the cells are more distorted in that case (it is the smallest edge length that determines the CFL number). Instead of computing the time step from maximum velocity and minimal mesh size as in step-31, we compute local CFL numbers, i.e., on each cell we compute the maximum velocity times the mesh size, and compute the maximum of them. Hence, we need to choose the factor in front of the time step slightly smaller.

After temperature right hand side assembly, we solve the linear system for temperature (with fully distributed vectors without any ghosts), apply constraints and copy the vector back to one with ghosts.

In the end, we extract the temperature range similarly to step-31 to produce some output (for example in order to help us choose the stabilization constants, as discussed in the introduction). The only difference is that we need to exchange maxima over all processors.

computing_timer.enter_section (" Assemble temperature rhs");
{
old_time_step = time_step;
const double scaling = (dim==3 ? 0.25 : 1.0);
time_step = (scaling/(2.1*dim*std::sqrt(1.*dim)) /
(parameters.temperature_degree *
get_cfl_number()));
const double maximal_velocity = get_maximal_velocity();
pcout << " Maximal velocity: "
<< maximal_velocity *EquationData::year_in_seconds * 100
<< " cm/year"
<< std::endl;
pcout << " " << "Time step: "
<< time_step/EquationData::year_in_seconds
<< " years"
<< std::endl;
temperature_solution = old_temperature_solution;
assemble_temperature_system (maximal_velocity);
}
computing_timer.exit_section ();
computing_timer.enter_section (" Solve temperature system");
{
SolverControl solver_control (temperature_matrix.m(),
1e-12*temperature_rhs.l2_norm());
distributed_temperature_solution (temperature_rhs);
distributed_temperature_solution = temperature_solution;
cg.solve (temperature_matrix, distributed_temperature_solution,
temperature_rhs, *T_preconditioner);
temperature_constraints.distribute (distributed_temperature_solution);
temperature_solution = distributed_temperature_solution;
pcout << " "
<< solver_control.last_step()
<< " CG iterations for temperature" << std::endl;
computing_timer.exit_section();
double temperature[2] = { std::numeric_limits<double>::max(),
-std::numeric_limits<double>::max()
};
double global_temperature[2];
for (unsigned int i=distributed_temperature_solution.local_range().first;
i < distributed_temperature_solution.local_range().second; ++i)
{
temperature[0] = std::min<double> (temperature[0],
distributed_temperature_solution(i));
temperature[1] = std::max<double> (temperature[1],
distributed_temperature_solution(i));
}
temperature[0] *= -1.0;
Utilities::MPI::max (temperature, MPI_COMM_WORLD, global_temperature);
global_temperature[0] *= -1.0;
pcout << " Temperature range: "
<< global_temperature[0] << ' ' << global_temperature[1]
<< std::endl;
}
}

BoussinesqFlowProblem::output_results

Next comes the function that generates the output. The quantities to output could be introduced manually like we did in step-31. An alternative is to hand this task over to a class PostProcessor that inherits from the class DataPostprocessor, which can be attached to DataOut. This allows us to output derived quantities from the solution, like the friction heating included in this example. It overloads the virtual function DataPostprocessor::evaluate_vector_field(), which is then internally called from DataOut::build_patches(). We have to give it values of the numerical solution, its derivatives, normals to the cell, the actual evaluation points and any additional quantities. This follows the same procedure as discussed in step-29 and other programs.

template <int dim>
class BoussinesqFlowProblem<dim>::Postprocessor : public DataPostprocessor<dim>
{
public:
Postprocessor (const unsigned int partition,
const double minimal_pressure);
virtual
void
std::vector<Vector<double> > &computed_quantities) const;
virtual std::vector<std::string> get_names () const;
virtual
std::vector<DataComponentInterpretation::DataComponentInterpretation>
private:
const unsigned int partition;
const double minimal_pressure;
};
template <int dim>
BoussinesqFlowProblem<dim>::Postprocessor::
Postprocessor (const unsigned int partition,
const double minimal_pressure)
:
partition (partition),
minimal_pressure (minimal_pressure)
{}

Here we define the names for the variables we want to output. These are the actual solution values for velocity, pressure, and temperature, as well as the friction heating and to each cell the number of the processor that owns it. This allows us to visualize the partitioning of the domain among the processors. Except for the velocity, which is vector-valued, all other quantities are scalar.

template <int dim>
std::vector<std::string>
BoussinesqFlowProblem<dim>::Postprocessor::get_names() const
{
std::vector<std::string> solution_names (dim, "velocity");
solution_names.push_back ("p");
solution_names.push_back ("T");
solution_names.push_back ("friction_heating");
solution_names.push_back ("partition");
return solution_names;
}
template <int dim>
std::vector<DataComponentInterpretation::DataComponentInterpretation>
BoussinesqFlowProblem<dim>::Postprocessor::
get_data_component_interpretation () const
{
std::vector<DataComponentInterpretation::DataComponentInterpretation>
interpretation (dim,
return interpretation;
}
template <int dim>
BoussinesqFlowProblem<dim>::Postprocessor::get_needed_update_flags() const
{
}

Now we implement the function that computes the derived quantities. As we also did for the output, we rescale the velocity from its SI units to something more readable, namely cm/year. Next, the pressure is scaled to be between 0 and the maximum pressure. This makes it more easily comparable – in essence making all pressure variables positive or zero. Temperature is taken as is, and the friction heating is computed as \(2 \eta \varepsilon(\mathbf{u}) \cdot \varepsilon(\mathbf{u})\).

The quantities we output here are more for illustration, rather than for actual scientific value. We come back to this briefly in the results section of this program and explain what one may in fact be interested in.

template <int dim>
void
BoussinesqFlowProblem<dim>::Postprocessor::
evaluate_vector_field
std::vector<Vector<double> > &computed_quantities) const
{
const unsigned int n_quadrature_points = inputs.solution_values.size();
Assert (inputs.solution_gradients.size() == n_quadrature_points,
Assert (computed_quantities.size() == n_quadrature_points,
Assert (inputs.solution_values[0].size() == dim+2,
for (unsigned int q=0; q<n_quadrature_points; ++q)
{
for (unsigned int d=0; d<dim; ++d)
computed_quantities[q](d)
= (inputs.solution_values[q](d) * EquationData::year_in_seconds * 100);
const double pressure = (inputs.solution_values[q](dim)-minimal_pressure);
computed_quantities[q](dim) = pressure;
const double temperature = inputs.solution_values[q](dim+1);
computed_quantities[q](dim+1) = temperature;
Tensor<2,dim> grad_u;
for (unsigned int d=0; d<dim; ++d)
grad_u[d] = inputs.solution_gradients[q][d];
const SymmetricTensor<2,dim> strain_rate = symmetrize (grad_u);
computed_quantities[q](dim+2) = 2 * EquationData::eta *
strain_rate * strain_rate;
computed_quantities[q](dim+3) = partition;
}
}

The output_results() function has a similar task to the one in step-31. However, here we are going to demonstrate a different technique on how to merge output from different DoFHandler objects. The way we're going to achieve this recombination is to create a joint DoFHandler that collects both components, the Stokes solution and the temperature solution. This can be nicely done by combining the finite elements from the two systems to form one FESystem, and let this collective system define a new DoFHandler object. To be sure that everything was done correctly, we perform a sanity check that ensures that we got all the dofs from both Stokes and temperature even in the combined system. We then combine the data vectors. Unfortunately, there is no straight-forward relation that tells us how to sort Stokes and temperature vector into the joint vector. The way we can get around this trouble is to rely on the information collected in the FESystem. For each dof on a cell, the joint finite element knows to which equation component (velocity component, pressure, or temperature) it belongs – that's the information we need! So we step through all cells (with iterators into all three DoFHandlers moving in sync), and for each joint cell dof, we read out that component using the FiniteElement::system_to_base_index function (see there for a description of what the various parts of its return value contain). We also need to keep track whether we're on a Stokes dof or a temperature dof, which is contained in joint_fe.system_to_base_index(i).first.first. Eventually, the dof_indices data structures on either of the three systems tell us how the relation between global vector and local dofs looks like on the present cell, which concludes this tedious work. We make sure that each processor only works on the subdomain it owns locally (and not on ghost or artificial cells) when building the joint solution vector. The same will then have to be done in DataOut::build_patches(), but that function does so automatically.

What we end up with is a set of patches that we can write using the functions in DataOutBase in a variety of output formats. Here, we then have to pay attention that what each processor writes is really only its own part of the domain, i.e. we will want to write each processor's contribution into a separate file. This we do by adding an additional number to the filename when we write the solution. This is not really new, we did it similarly in step-40. Note that we write in the compressed format .vtu instead of plain vtk files, which saves quite some storage.

All the rest of the work is done in the PostProcessor class.

template <int dim>
void BoussinesqFlowProblem<dim>::output_results ()
{
computing_timer.enter_section ("Postprocessing");
const FESystem<dim> joint_fe (stokes_fe, 1,
temperature_fe, 1);
DoFHandler<dim> joint_dof_handler (triangulation);
joint_dof_handler.distribute_dofs (joint_fe);
Assert (joint_dof_handler.n_dofs() ==
stokes_dof_handler.n_dofs() + temperature_dof_handler.n_dofs(),
joint_solution.reinit (joint_dof_handler.locally_owned_dofs(), MPI_COMM_WORLD);
{
std::vector<types::global_dof_index> local_joint_dof_indices (joint_fe.dofs_per_cell);
std::vector<types::global_dof_index> local_stokes_dof_indices (stokes_fe.dofs_per_cell);
std::vector<types::global_dof_index> local_temperature_dof_indices (temperature_fe.dofs_per_cell);
joint_cell = joint_dof_handler.begin_active(),
joint_endc = joint_dof_handler.end(),
stokes_cell = stokes_dof_handler.begin_active(),
temperature_cell = temperature_dof_handler.begin_active();
for (; joint_cell!=joint_endc;
++joint_cell, ++stokes_cell, ++temperature_cell)
if (joint_cell->is_locally_owned())
{
joint_cell->get_dof_indices (local_joint_dof_indices);
stokes_cell->get_dof_indices (local_stokes_dof_indices);
temperature_cell->get_dof_indices (local_temperature_dof_indices);
for (unsigned int i=0; i<joint_fe.dofs_per_cell; ++i)
if (joint_fe.system_to_base_index(i).first.first == 0)
{
Assert (joint_fe.system_to_base_index(i).second
<
local_stokes_dof_indices.size(),
joint_solution(local_joint_dof_indices[i])
= stokes_solution(local_stokes_dof_indices
[joint_fe.system_to_base_index(i).second]);
}
else
{
Assert (joint_fe.system_to_base_index(i).first.first == 1,
Assert (joint_fe.system_to_base_index(i).second
<
local_temperature_dof_indices.size(),
joint_solution(local_joint_dof_indices[i])
= temperature_solution(local_temperature_dof_indices
[joint_fe.system_to_base_index(i).second]);
}
}
}
joint_solution.compress(VectorOperation::insert);
IndexSet locally_relevant_joint_dofs(joint_dof_handler.n_dofs());
DoFTools::extract_locally_relevant_dofs (joint_dof_handler, locally_relevant_joint_dofs);
TrilinosWrappers::MPI::Vector locally_relevant_joint_solution;
locally_relevant_joint_solution.reinit (locally_relevant_joint_dofs, MPI_COMM_WORLD);
locally_relevant_joint_solution = joint_solution;
Postprocessor postprocessor (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD),
stokes_solution.block(1).min());
DataOut<dim> data_out;
data_out.attach_dof_handler (joint_dof_handler);
data_out.add_data_vector (locally_relevant_joint_solution, postprocessor);
data_out.build_patches ();
static int out_index=0;
const std::string filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
"." +
(triangulation.locally_owned_subdomain(), 4) +
".vtu");
std::ofstream output (filename.c_str());
data_out.write_vtu (output);

At this point, all processors have written their own files to disk. We could visualize them individually in Visit or Paraview, but in reality we of course want to visualize the whole set of files at once. To this end, we create a master file in each of the formats understood by Visit (.visit) and Paraview (.pvtu) on the zeroth processor that describes how the individual files are defining the global data set.

if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
{
std::vector<std::string> filenames;
for (unsigned int i=0; i<Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD); ++i)
filenames.push_back (std::string("solution-") +
Utilities::int_to_string (out_index, 5) +
"." +
".vtu");
const std::string
pvtu_master_filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
".pvtu");
std::ofstream pvtu_master (pvtu_master_filename.c_str());
data_out.write_pvtu_record (pvtu_master, filenames);
const std::string
visit_master_filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
".visit");
std::ofstream visit_master (visit_master_filename.c_str());
DataOutBase::write_visit_record (visit_master, filenames);
}
computing_timer.exit_section ();
out_index++;
}

BoussinesqFlowProblem::refine_mesh

This function isn't really new either. Since the setup_dofs function that we call in the middle has its own timer section, we split timing this function into two sections. It will also allow us to easily identify which of the two is more expensive.

One thing of note, however, is that we only want to compute error indicators on the locally owned subdomain. In order to achieve this, we pass one additional argument to the KellyErrorEstimator::estimate function. Note that the vector for error estimates is resized to the number of active cells present on the current process, which is less than the total number of active cells on all processors (but more than the number of locally owned active cells); each processor only has a few coarse cells around the locally owned ones, as also explained in step-40.

The local error estimates are then handed to a parallel version of GridRefinement (in namespace parallel::distributed::GridRefinement, see also step-40) which looks at the errors and finds the cells that need refinement by comparing the error values across processors. As in step-31, we want to limit the maximum grid level. So in case some cells have been marked that are already at the finest level, we simply clear the refine flags.

template <int dim>
void BoussinesqFlowProblem<dim>::refine_mesh (const unsigned int max_grid_level)
{
computing_timer.enter_section ("Refine mesh structure, part 1");
Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
KellyErrorEstimator<dim>::estimate (temperature_dof_handler,
QGauss<dim-1>(parameters.temperature_degree+1),
temperature_solution,
estimated_error_per_cell,
nullptr,
0,
triangulation.locally_owned_subdomain());
estimated_error_per_cell,
0.3, 0.1);
if (triangulation.n_levels() > max_grid_level)
cell = triangulation.begin_active(max_grid_level);
cell != triangulation.end(); ++cell)
cell->clear_refine_flag ();

With all flags marked as necessary, we set up the parallel::distributed::SolutionTransfer object to transfer the solutions for the current time level and the next older one. The syntax is similar to the non-parallel solution transfer (with the exception that here a pointer to the vector entries is enough). The remainder of the function is concerned with setting up the data structures again after mesh refinement and restoring the solution vectors on the new mesh.

std::vector<const TrilinosWrappers::MPI::Vector *> x_temperature (2);
x_temperature[0] = &temperature_solution;
x_temperature[1] = &old_temperature_solution;
std::vector<const TrilinosWrappers::MPI::BlockVector *> x_stokes (2);
x_stokes[0] = &stokes_solution;
x_stokes[1] = &old_stokes_solution;
temperature_trans(temperature_dof_handler);
stokes_trans(stokes_dof_handler);
triangulation.prepare_coarsening_and_refinement();
temperature_trans.prepare_for_coarsening_and_refinement(x_temperature);
stokes_trans.prepare_for_coarsening_and_refinement(x_stokes);
triangulation.execute_coarsening_and_refinement ();
computing_timer.exit_section();
setup_dofs ();
computing_timer.enter_section ("Refine mesh structure, part 2");
{
TrilinosWrappers::MPI::Vector distributed_temp1 (temperature_rhs);
TrilinosWrappers::MPI::Vector distributed_temp2 (temperature_rhs);
std::vector<TrilinosWrappers::MPI::Vector *> tmp (2);
tmp[0] = &(distributed_temp1);
tmp[1] = &(distributed_temp2);
temperature_trans.interpolate(tmp);

enforce constraints to make the interpolated solution conforming on the new mesh:

temperature_constraints.distribute(distributed_temp1);
temperature_constraints.distribute(distributed_temp2);
temperature_solution = distributed_temp1;
old_temperature_solution = distributed_temp2;
}
{
TrilinosWrappers::MPI::BlockVector distributed_stokes (stokes_rhs);
TrilinosWrappers::MPI::BlockVector old_distributed_stokes (stokes_rhs);
std::vector<TrilinosWrappers::MPI::BlockVector *> stokes_tmp (2);
stokes_tmp[0] = &(distributed_stokes);
stokes_tmp[1] = &(old_distributed_stokes);
stokes_trans.interpolate (stokes_tmp);

enforce constraints to make the interpolated solution conforming on the new mesh:

stokes_constraints.distribute(distributed_stokes);
stokes_constraints.distribute(old_distributed_stokes);
stokes_solution = distributed_stokes;
old_stokes_solution = old_distributed_stokes;
}
computing_timer.exit_section();
}

BoussinesqFlowProblem::run

This is the final and controlling function in this class. It, in fact, runs the entire rest of the program and is, once more, very similar to step-31. We use a different mesh now (a GridGenerator::hyper_shell instead of a simple cube geometry), and use the project_temperature_field() function instead of the library function VectorTools::project. In this example, however, we define both a SphericalManifold() and a HyperShellBoundary() object to describe the geometry of the domain. The reason we do so here, is because we want to impose no normal flux boundary conditions, and they require knowledge of the normals to a boundary, which a SphericalManifold() alone cannot compute. Consequently, we set all manifold indicators of cells and adjacent edges to zero, then overwrite the manifold indicators of all boundary objects by one. We then associate a SphericalManifold object with the former, and the HyperShellBoundary object that can also provide normal vectors with the latter.

template <int dim>
void BoussinesqFlowProblem<dim>::run ()
{
EquationData::R0,
EquationData::R1,
(dim==3) ? 96 : 12,
true);
triangulation.set_all_manifold_ids(0);
triangulation.set_all_manifold_ids_on_boundary(1);
static SphericalManifold<dim> manifold;
static HyperShellBoundary<dim> boundary;
triangulation.set_manifold (0, manifold);
triangulation.set_manifold (1, boundary);
global_Omega_diameter = GridTools::diameter (triangulation);
triangulation.refine_global (parameters.initial_global_refinement);
setup_dofs();
unsigned int pre_refinement_step = 0;
start_time_iteration:
project_temperature_field ();
timestep_number = 0;
time_step = old_time_step = 0;
double time = 0;
do
{
pcout << "Timestep " << timestep_number
<< ": t=" << time/EquationData::year_in_seconds
<< " years"
<< std::endl;
assemble_stokes_system ();
build_stokes_preconditioner ();
assemble_temperature_matrix ();
solve ();
pcout << std::endl;
if ((timestep_number == 0) &&
(pre_refinement_step < parameters.initial_adaptive_refinement))
{
refine_mesh (parameters.initial_global_refinement +
parameters.initial_adaptive_refinement);
++pre_refinement_step;
goto start_time_iteration;
}
else if ((timestep_number > 0)
&&
(timestep_number % parameters.adaptive_refinement_interval == 0))
refine_mesh (parameters.initial_global_refinement +
parameters.initial_adaptive_refinement);
if ((parameters.generate_graphical_output == true)
&&
(timestep_number % parameters.graphical_output_interval == 0))
output_results ();

In order to speed up linear solvers, we extrapolate the solutions from the old time levels to the new one. This gives a very good initial guess, cutting the number of iterations needed in solvers by more than one half. We do not need to extrapolate in the last iteration, so if we reached the final time, we stop here.

As the last thing during a time step (before actually bumping up the number of the time step), we check whether the current time step number is divisible by 100, and if so we let the computing timer print a summary of CPU times spent so far.

if (time > parameters.end_time * EquationData::year_in_seconds)
break;
TrilinosWrappers::MPI::BlockVector old_old_stokes_solution;
old_old_stokes_solution = old_stokes_solution;
old_stokes_solution = stokes_solution;
old_old_temperature_solution = old_temperature_solution;
old_temperature_solution = temperature_solution;
if (old_time_step > 0)
{

Trilinos sadd does not like ghost vectors even as input. Copy into distributed vectors for now:

{
TrilinosWrappers::MPI::BlockVector distr_solution (stokes_rhs);
distr_solution = stokes_solution;
TrilinosWrappers::MPI::BlockVector distr_old_solution (stokes_rhs);
distr_old_solution = old_old_stokes_solution;
distr_solution .sadd (1.+time_step/old_time_step, -time_step/old_time_step,
distr_old_solution);
stokes_solution = distr_solution;
}
{
TrilinosWrappers::MPI::Vector distr_solution (temperature_rhs);
distr_solution = temperature_solution;
TrilinosWrappers::MPI::Vector distr_old_solution (temperature_rhs);
distr_old_solution = old_old_temperature_solution;
distr_solution .sadd (1.+time_step/old_time_step, -time_step/old_time_step,
distr_old_solution);
temperature_solution = distr_solution;
}
}
if ((timestep_number > 0) && (timestep_number % 100 == 0))
computing_timer.print_summary ();
time += time_step;
++timestep_number;
}
while (true);

If we are generating graphical output, do so also for the last time step unless we had just done so before we left the do-while loop

if ((parameters.generate_graphical_output == true)
&&
!((timestep_number-1) % parameters.graphical_output_interval == 0))
output_results ();
}
}

The main function

The main function is short as usual and very similar to the one in step-31. Since we use a parameter file which is specified as an argument in the command line, we have to read it in here and pass it on to the Parameters class for parsing. If no filename is given in the command line, we simply use the step-32.prm file which is distributed together with the program.

Because 3d computations are simply very slow unless you throw a lot of processors at them, the program defaults to 2d. You can get the 3d version by changing the constant dimension below to 3.

int main (int argc, char *argv[])
{
using namespace Step32;
using namespace dealii;
Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv,
try
{
std::string parameter_filename;
if (argc>=2)
parameter_filename = argv[1];
else
parameter_filename = "step-32.prm";
const int dim = 2;
BoussinesqFlowProblem<dim>::Parameters parameters(parameter_filename);
BoussinesqFlowProblem<dim> flow_problem (parameters);
flow_problem.run ();
}
catch (std::exception &exc)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Exception on processing: " << std::endl
<< exc.what() << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
catch (...)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Unknown exception!" << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
return 0;
}

Results

When run, the program simulates convection in 3d in much the same way as step-31 did, though with an entirely different testcase.

Comparison of results with step-31

Before we go to this testcase, however, let us show a few results from a slightly earlier version of this program that was solving exactly the testcase we used in step-31, just that we now solve it in parallel and with much higher resolution. We show these results mainly for comparison.

Here are two images that show this higher resolution if we choose a 3d computation in main() and if we set initial_refinement=3 and n_pre_refinement_steps=4. At the time steps shown, the meshes had around 72,000 and 236,000 cells, for a total of 2,680,000 and 8,250,000 degrees of freedom, respectively, more than an order of magnitude more than we had available in step-31:

The computation was done on a subset of 50 processors of the Brazos cluster at Texas A&M University.

Results for a 2d circular shell testcase

Next, we will run step-32 with the parameter file in the directory. Here we are using 50 processors. The command to launch is (note that step-32.prm is the default):

$ mpirun -np 50 ./step-32

Note that running a job on a cluster typically requires going through a job scheduler, which we won't discuss here. The output will look roughly like this:

Number of active cells: 12,288 (on 6 levels)
Number of degrees of freedom: 186,624 (99,840+36,864+49,920)

Timestep 0:  t=0 years

   Rebuilding Stokes preconditioner...
   Solving Stokes system... 40 iterations.
   Maximal velocity: 60.4935 cm/year
   Time step: 18166.9 years
   17 CG iterations for temperature
   Temperature range: 973 4273.16

Number of active cells: 16,389 (on 7 levels)
Number of degrees of freedom: 259,617 (140,300+49,167+70,150)

Timestep 0:  t=0 years

   Rebuilding Stokes preconditioner...
   Solving Stokes system... 49 iterations.
   Maximal velocity: 60.5202 cm/year
   Time step: 10619.2 years
   19 CG iterations for temperature
   Temperature range: 973 4273.16

Number of active cells: 20,124 (on 8 levels)
Number of degrees of freedom: 323,145 (175,182+60,372+87,591)

Timestep 0:  t=0 years

   Rebuilding Stokes preconditioner...
   Solving Stokes system... 51 iterations.
   Maximal velocity: 59.9101 cm/year
   Time step: 5334.14 years
   18 CG iterations for temperature
   Temperature range: 973 4273.43

Timestep 1:  t=5334.14 years

   Solving Stokes system... 49 iterations.
   Maximal velocity: 61.0837 cm/year
   Time step: 5232.24 years
   18 CG iterations for temperature
   Temperature range: 973 4273.43

Timestep 2:  t=10566.4 years

   Solving Stokes system... 24 iterations.
   Maximal velocity: 62.265 cm/year
   Time step: 5133.56 years
   18 CG iterations for temperature
   Temperature range: 973 4273.43

[...]

Timestep 100:  t=270918 years

   Solving Stokes system... 22 iterations.
   Maximal velocity: 161.391 cm/year
   Time step: 1683.55 years
   17 CG iterations for temperature
   Temperature range: 973 4282.45

Number of active cells: 54,717 (on 8 levels)
Number of degrees of freedom: 880,626 (477,650+164,151+238,825)

+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |       170s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| Assemble Stokes system          |       103 |      2.25s |       1.3% |
| Assemble temperature matrices   |        12 |     0.296s |      0.17% |
| Assemble temperature rhs        |       103 |      9.66s |       5.7% |
| Build Stokes preconditioner     |        12 |      1.78s |         1% |
| Solve Stokes system             |       103 |       149s |        88% |
| Solve temperature system        |       103 |      1.84s |       1.1% |
| Postprocessing                  |         3 |     0.398s |      0.23% |
| Refine mesh structure, part 1   |        12 |     0.599s |      0.35% |
| Refine mesh structure, part 2   |        12 |     0.216s |      0.13% |
| Setup dof systems               |        13 |      2.15s |       1.3% |
+---------------------------------+-----------+------------+------------+

[...]

Timestep 6946:  t=1.00001e+08 years

   Solving Stokes system... 12 iterations.
   Maximal velocity: 8.94391 cm/year
   Time step: 31505.4 years
   18 CG iterations for temperature
   Temperature range: 973 4273.04

+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |  8.86e+03s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| Assemble Stokes system          |      6949 |       153s |       1.7% |
| Assemble temperature matrices   |       697 |      18.2s |       0.2% |
| Assemble temperature rhs        |      6949 |       600s |       6.8% |
| Build Stokes preconditioner     |       697 |       115s |       1.3% |
| Solve Stokes system             |      6949 |  7.56e+03s |        85% |
| Solve temperature system        |      6949 |       123s |       1.4% |
| Postprocessing                  |       140 |      18.1s |       0.2% |
| Refine mesh structure, part 1   |       696 |      34.6s |      0.39% |
| Refine mesh structure, part 2   |       696 |      12.6s |      0.14% |
| Setup dof systems               |       697 |       130s |       1.5% |
+---------------------------------+-----------+------------+------------+

The simulation terminates when the time reaches the 100 million years selected in the input file. You can extrapolate from this how long a simulation would take for a different final time (the time step size ultimately settles on somewhere around 20,000 years, so computing for one billion years will take 50,000 time steps, give or take 20%). As can be seen here, we spend most of the compute time in assembling linear systems and — above all — in solving Stokes systems.

We can clearly not show all output files produced by this program when run for a longer period of time, so let us only show the output from every 2500th time step here (these figures are from a version of this tutorial program before the introduction of manifolds, when it was not yet possible to arrange vertices along circles; the figures may therefore look slightly different if you run the current version):

The last two images show the grid as well as the partitioning of the mesh for a computation with 10 subdomains on 10 processors. The full dynamics of this simulation are really only visible by looking at an animation, for example the one shown on this site. Beware that this animation is about 20MB large, though it is well worth watching due to its almost artistic quality.

If you watch the movie, you'll see that the convection pattern goes through several stages: First, it gets rid of the instable temperature layering with the hot material overlain by the dense cold material. After this great driver is removed and we have a sort of stable situation, a few blobs start to separate from the hot boundary layer at the inner ring and rise up, with a few cold fingers also dropping down from the outer boundary layer. During this phase, the solution remains mostly symmetric, reflecting the 12-fold symmetry of the original mesh. In a final phase, the fluid enters vigorous chaotic stirring in which all symmetries are lost. This is a pattern that then continues to dominate flow.

These different phases can also be identified if we look at the maximal velocity as a function of time in the simulation:

Here, the velocity (shown in centimeters per year) becomes very large, to the order of several meters per year) at the beginning when the temperature layering is instable. It then calms down to relatively small values before picking up again in the chaotic stirring regime. There, it remains in the range of 10-40 centimeters per year, quite within the physically expected region.

Results for a 3d spherical shell testcase

3d computations are very expensive computationally. Furthermore, as seen above, interesting behavior only starts after quite a long time requiring more CPU hours than is available on a typical cluster. Consequently, rather than showing a complete simulation here, let us simply show a couple of pictures we have obtained using the successor to this program, called Aspect (short for Advanced Solver for Problems in Earth's ConvecTion), that is being developed independently of deal.II and that already incorporates some of the extensions discussed below. The following two pictures show isocontours of the temperature and the partition of the domain (along with the mesh) onto 512 processors:

Possibilities for extensions

There are many directions in which this program could be extended. As mentioned at the end of the introduction, most of these are under active development in the Aspect (short for Advanced Solver for Problems in Earth's ConvecTion) code at the time this tutorial program is being finished. Specifically, the following are certainly topics that one should address to make the program more useful:

There are many other ways to extend the current program. However, rather than discussing them here, let us point to the much larger open source code Aspect (see http://aspect.dealii.org/ ) that constitutes the further development of step-32 and that already includes many such possible extensions.

The plain program

/* ---------------------------------------------------------------------
*
* Copyright (C) 2008 - 2017 by the deal.II authors
*
* This file is part of the deal.II library.
*
* The deal.II library is free software; you can use it, redistribute
* it, and/or modify it under the terms of the GNU Lesser General
* Public License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
* The full text of the license can be found in the file LICENSE at
* the top level of the deal.II distribution.
*
* ---------------------------------------------------------------------
*
* Authors: Martin Kronbichler, Uppsala University,
* Wolfgang Bangerth, Texas A&M University,
* Timo Heister, University of Goettingen, 2008-2011
*/
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/function.h>
#include <deal.II/base/utilities.h>
#include <deal.II/base/conditional_ostream.h>
#include <deal.II/base/work_stream.h>
#include <deal.II/base/timer.h>
#include <deal.II/base/parameter_handler.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/solver_bicgstab.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/constraint_matrix.h>
#include <deal.II/lac/block_sparsity_pattern.h>
#include <deal.II/lac/trilinos_parallel_block_vector.h>
#include <deal.II/lac/trilinos_sparse_matrix.h>
#include <deal.II/lac/trilinos_block_sparse_matrix.h>
#include <deal.II/lac/trilinos_precondition.h>
#include <deal.II/lac/trilinos_solver.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/grid/filtered_iterator.h>
#include <deal.II/grid/manifold_lib.h>
#include <deal.II/grid/tria_boundary_lib.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/grid/grid_refinement.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/dofs/dof_renumbering.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_system.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/numerics/matrix_tools.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/error_estimator.h>
#include <deal.II/numerics/solution_transfer.h>
#include <fstream>
#include <iostream>
#include <sstream>
#include <limits>
#include <locale>
#include <string>
#include <deal.II/distributed/solution_transfer.h>
#include <deal.II/base/index_set.h>
#include <deal.II/distributed/tria.h>
#include <deal.II/distributed/grid_refinement.h>
namespace Step32
{
using namespace dealii;
namespace EquationData
{
const double eta = 1e21; /* Pa s */
const double kappa = 1e-6; /* m^2 / s */
const double reference_density = 3300; /* kg / m^3 */
const double reference_temperature = 293; /* K */
const double expansion_coefficient = 2e-5; /* 1/K */
const double specific_heat = 1250; /* J / K / kg */
const double radiogenic_heating = 7.4e-12; /* W / kg */
const double R0 = 6371000.-2890000.; /* m */
const double R1 = 6371000.- 35000.; /* m */
const double T0 = 4000+273; /* K */
const double T1 = 700+273; /* K */
double density (const double temperature)
{
return (reference_density *
(1 - expansion_coefficient * (temperature -
reference_temperature)));
}
template <int dim>
Tensor<1,dim> gravity_vector (const Point<dim> &p)
{
const double r = p.norm();
return -(1.245e-6 * r + 7.714e13/r/r) * p / r;
}
template <int dim>
class TemperatureInitialValues : public Function<dim>
{
public:
TemperatureInitialValues () : Function<dim>(1) {}
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &value) const;
};
template <int dim>
double
TemperatureInitialValues<dim>::value (const Point<dim> &p,
const unsigned int) const
{
const double r = p.norm();
const double h = R1-R0;
const double s = (r-R0)/h;
const double q = (dim==3)?std::max(0.0,cos(numbers::PI*abs(p(2)/R1))):1.0;
const double phi = std::atan2(p(0),p(1));
const double tau = s
+
0.2 * s * (1-s) * std::sin(6*phi) * q;
return T0*(1.0-tau) + T1*tau;
}
template <int dim>
void
TemperatureInitialValues<dim>::vector_value (const Point<dim> &p,
Vector<double> &values) const
{
for (unsigned int c=0; c<this->n_components; ++c)
values(c) = TemperatureInitialValues<dim>::value (p, c);
}
const double pressure_scaling = eta / 10000;
const double year_in_seconds = 60*60*24*365.2425;
}
namespace LinearSolvers
{
template <class PreconditionerTypeA, class PreconditionerTypeMp>
class BlockSchurPreconditioner : public Subscriptor
{
public:
BlockSchurPreconditioner (const TrilinosWrappers::BlockSparseMatrix &S,
const PreconditionerTypeMp &Mppreconditioner,
const PreconditionerTypeA &Apreconditioner,
const bool do_solve_A)
:
stokes_matrix (&S),
stokes_preconditioner_matrix (&Spre),
mp_preconditioner (Mppreconditioner),
a_preconditioner (Apreconditioner),
do_solve_A (do_solve_A)
{}
{
{
SolverControl solver_control(5000, 1e-6 * src.block(1).l2_norm());
solver.solve(stokes_preconditioner_matrix->block(1,1),
dst.block(1), src.block(1),
mp_preconditioner);
dst.block(1) *= -1.0;
}
{
stokes_matrix->block(0,1).vmult(utmp, dst.block(1));
utmp*=-1.0;
utmp.add(src.block(0));
}
if (do_solve_A == true)
{
SolverControl solver_control(5000, utmp.l2_norm()*1e-2);
TrilinosWrappers::SolverCG solver(solver_control);
solver.solve(stokes_matrix->block(0,0), dst.block(0), utmp,
a_preconditioner);
}
else
a_preconditioner.vmult (dst.block(0), utmp);
}
private:
const SmartPointer<const TrilinosWrappers::BlockSparseMatrix> stokes_preconditioner_matrix;
const PreconditionerTypeMp &mp_preconditioner;
const PreconditionerTypeA &a_preconditioner;
const bool do_solve_A;
};
}
namespace Assembly
{
namespace Scratch
{
template <int dim>
struct StokesPreconditioner
{
StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
const Quadrature<dim> &stokes_quadrature,
const Mapping<dim> &mapping,
const UpdateFlags update_flags);
StokesPreconditioner (const StokesPreconditioner &data);
FEValues<dim> stokes_fe_values;
std::vector<Tensor<2,dim> > grad_phi_u;
std::vector<double> phi_p;
};
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
const Quadrature<dim> &stokes_quadrature,
const Mapping<dim> &mapping,
const UpdateFlags update_flags)
:
stokes_fe_values (mapping, stokes_fe, stokes_quadrature,
update_flags),
grad_phi_u (stokes_fe.dofs_per_cell),
phi_p (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const StokesPreconditioner &scratch)
:
stokes_fe_values (scratch.stokes_fe_values.get_mapping(),
scratch.stokes_fe_values.get_fe(),
scratch.stokes_fe_values.get_quadrature(),
scratch.stokes_fe_values.get_update_flags()),
grad_phi_u (scratch.grad_phi_u),
phi_p (scratch.phi_p)
{}
template <int dim>
struct StokesSystem : public StokesPreconditioner<dim>
{
StokesSystem (const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &stokes_quadrature,
const UpdateFlags stokes_update_flags,
const FiniteElement<dim> &temperature_fe,
const UpdateFlags temperature_update_flags);
StokesSystem (const StokesSystem<dim> &data);
FEValues<dim> temperature_fe_values;
std::vector<Tensor<1,dim> > phi_u;
std::vector<SymmetricTensor<2,dim> > grads_phi_u;
std::vector<double> div_phi_u;
std::vector<double> old_temperature_values;
};
template <int dim>
StokesSystem<dim>::
StokesSystem (const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &stokes_quadrature,
const UpdateFlags stokes_update_flags,
const FiniteElement<dim> &temperature_fe,
const UpdateFlags temperature_update_flags)
:
StokesPreconditioner<dim> (stokes_fe, stokes_quadrature,
mapping,
stokes_update_flags),
temperature_fe_values (mapping, temperature_fe, stokes_quadrature,
temperature_update_flags),
phi_u (stokes_fe.dofs_per_cell),
grads_phi_u (stokes_fe.dofs_per_cell),
div_phi_u (stokes_fe.dofs_per_cell),
old_temperature_values (stokes_quadrature.size())
{}
template <int dim>
StokesSystem<dim>::
StokesSystem (const StokesSystem<dim> &scratch)
:
StokesPreconditioner<dim> (scratch),
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
phi_u (scratch.phi_u),
grads_phi_u (scratch.grads_phi_u),
div_phi_u (scratch.div_phi_u),
old_temperature_values (scratch.old_temperature_values)
{}
template <int dim>
struct TemperatureMatrix
{
TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &temperature_quadrature);
TemperatureMatrix (const TemperatureMatrix &data);
FEValues<dim> temperature_fe_values;
std::vector<double> phi_T;
std::vector<Tensor<1,dim> > grad_phi_T;
};
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &temperature_quadrature)
:
temperature_fe_values (mapping,
temperature_fe, temperature_quadrature,
phi_T (temperature_fe.dofs_per_cell),
grad_phi_T (temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const TemperatureMatrix &scratch)
:
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
phi_T (scratch.phi_T),
grad_phi_T (scratch.grad_phi_T)
{}
template <int dim>
struct TemperatureRHS
{
TemperatureRHS (const FiniteElement<dim> &temperature_fe,
const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &quadrature);
TemperatureRHS (const TemperatureRHS &data);
FEValues<dim> temperature_fe_values;
FEValues<dim> stokes_fe_values;
std::vector<double> phi_T;
std::vector<Tensor<1,dim> > grad_phi_T;
std::vector<Tensor<1,dim> > old_velocity_values;
std::vector<Tensor<1,dim> > old_old_velocity_values;
std::vector<SymmetricTensor<2,dim> > old_strain_rates;
std::vector<SymmetricTensor<2,dim> > old_old_strain_rates;
std::vector<double> old_temperature_values;
std::vector<double> old_old_temperature_values;
std::vector<Tensor<1,dim> > old_temperature_grads;
std::vector<Tensor<1,dim> > old_old_temperature_grads;
std::vector<double> old_temperature_laplacians;
std::vector<double> old_old_temperature_laplacians;
};
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const FiniteElement<dim> &temperature_fe,
const FiniteElement<dim> &stokes_fe,
const Mapping<dim> &mapping,
const Quadrature<dim> &quadrature)
:
temperature_fe_values (mapping,
temperature_fe, quadrature,
stokes_fe_values (mapping,
stokes_fe, quadrature,
phi_T (temperature_fe.dofs_per_cell),
grad_phi_T (temperature_fe.dofs_per_cell),
old_velocity_values (quadrature.size()),
old_old_velocity_values (quadrature.size()),
old_strain_rates (quadrature.size()),
old_old_strain_rates (quadrature.size()),
old_temperature_values (quadrature.size()),
old_old_temperature_values(quadrature.size()),
old_temperature_grads(quadrature.size()),
old_old_temperature_grads(quadrature.size()),
old_temperature_laplacians(quadrature.size()),
old_old_temperature_laplacians(quadrature.size())
{}
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const TemperatureRHS &scratch)
:
temperature_fe_values (scratch.temperature_fe_values.get_mapping(),
scratch.temperature_fe_values.get_fe(),
scratch.temperature_fe_values.get_quadrature(),
scratch.temperature_fe_values.get_update_flags()),
stokes_fe_values (scratch.stokes_fe_values.get_mapping(),
scratch.stokes_fe_values.get_fe(),
scratch.stokes_fe_values.get_quadrature(),
scratch.stokes_fe_values.get_update_flags()),
phi_T (scratch.phi_T),
grad_phi_T (scratch.grad_phi_T),
old_velocity_values (scratch.old_velocity_values),
old_old_velocity_values (scratch.old_old_velocity_values),
old_strain_rates (scratch.old_strain_rates),
old_old_strain_rates (scratch.old_old_strain_rates),
old_temperature_values (scratch.old_temperature_values),
old_old_temperature_values (scratch.old_old_temperature_values),
old_temperature_grads (scratch.old_temperature_grads),
old_old_temperature_grads (scratch.old_old_temperature_grads),
old_temperature_laplacians (scratch.old_temperature_laplacians),
old_old_temperature_laplacians (scratch.old_old_temperature_laplacians)
{}
}
namespace CopyData
{
template <int dim>
struct StokesPreconditioner
{
StokesPreconditioner (const FiniteElement<dim> &stokes_fe);
StokesPreconditioner (const StokesPreconditioner &data);
FullMatrix<double> local_matrix;
std::vector<types::global_dof_index> local_dof_indices;
};
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const FiniteElement<dim> &stokes_fe)
:
local_matrix (stokes_fe.dofs_per_cell,
stokes_fe.dofs_per_cell),
local_dof_indices (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesPreconditioner<dim>::
StokesPreconditioner (const StokesPreconditioner &data)
:
local_matrix (data.local_matrix),
local_dof_indices (data.local_dof_indices)
{}
template <int dim>
struct StokesSystem : public StokesPreconditioner<dim>
{
StokesSystem (const FiniteElement<dim> &stokes_fe);
StokesSystem (const StokesSystem<dim> &data);
Vector<double> local_rhs;
};
template <int dim>
StokesSystem<dim>::
StokesSystem (const FiniteElement<dim> &stokes_fe)
:
StokesPreconditioner<dim> (stokes_fe),
local_rhs (stokes_fe.dofs_per_cell)
{}
template <int dim>
StokesSystem<dim>::
StokesSystem (const StokesSystem<dim> &data)
:
StokesPreconditioner<dim> (data),
local_rhs (data.local_rhs)
{}
template <int dim>
struct TemperatureMatrix
{
TemperatureMatrix (const FiniteElement<dim> &temperature_fe);
TemperatureMatrix (const TemperatureMatrix &data);
FullMatrix<double> local_mass_matrix;
FullMatrix<double> local_stiffness_matrix;
std::vector<types::global_dof_index> local_dof_indices;
};
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const FiniteElement<dim> &temperature_fe)
:
local_mass_matrix (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell),
local_stiffness_matrix (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell),
local_dof_indices (temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureMatrix<dim>::
TemperatureMatrix (const TemperatureMatrix &data)
:
local_mass_matrix (data.local_mass_matrix),
local_stiffness_matrix (data.local_stiffness_matrix),
local_dof_indices (data.local_dof_indices)
{}
template <int dim>
struct TemperatureRHS
{
TemperatureRHS (const FiniteElement<dim> &temperature_fe);
TemperatureRHS (const TemperatureRHS &data);
Vector<double> local_rhs;
std::vector<types::global_dof_index> local_dof_indices;
FullMatrix<double> matrix_for_bc;
};
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const FiniteElement<dim> &temperature_fe)
:
local_rhs (temperature_fe.dofs_per_cell),
local_dof_indices (temperature_fe.dofs_per_cell),
matrix_for_bc (temperature_fe.dofs_per_cell,
temperature_fe.dofs_per_cell)
{}
template <int dim>
TemperatureRHS<dim>::
TemperatureRHS (const TemperatureRHS &data)
:
local_rhs (data.local_rhs),
local_dof_indices (data.local_dof_indices),
matrix_for_bc (data.matrix_for_bc)
{}
}
}
template <int dim>
class BoussinesqFlowProblem
{
public:
struct Parameters;
BoussinesqFlowProblem (Parameters &parameters);
void run ();
private:
void setup_dofs ();
void assemble_stokes_preconditioner ();
void build_stokes_preconditioner ();
void assemble_stokes_system ();
void assemble_temperature_matrix ();
void assemble_temperature_system (const double maximal_velocity);
void project_temperature_field ();
double get_maximal_velocity () const;
double get_cfl_number () const;
double get_entropy_variation (const double average_temperature) const;
std::pair<double,double> get_extrapolated_temperature_range () const;
void solve ();
void output_results ();
void refine_mesh (const unsigned int max_grid_level);
double
compute_viscosity(const std::vector<double> &old_temperature,
const std::vector<double> &old_old_temperature,
const std::vector<Tensor<1,dim> > &old_temperature_grads,
const std::vector<Tensor<1,dim> > &old_old_temperature_grads,
const std::vector<double> &old_temperature_laplacians,
const std::vector<double> &old_old_temperature_laplacians,
const std::vector<Tensor<1,dim> > &old_velocity_values,
const std::vector<Tensor<1,dim> > &old_old_velocity_values,
const std::vector<SymmetricTensor<2,dim> > &old_strain_rates,
const std::vector<SymmetricTensor<2,dim> > &old_old_strain_rates,
const double global_u_infty,
const double global_T_variation,
const double average_temperature,
const double global_entropy_variation,
const double cell_diameter) const;
public:
struct Parameters
{
Parameters (const std::string &parameter_filename);
static void declare_parameters (ParameterHandler &prm);
void parse_parameters (ParameterHandler &prm);
double end_time;
unsigned int initial_global_refinement;
unsigned int initial_adaptive_refinement;
bool generate_graphical_output;
unsigned int graphical_output_interval;
unsigned int adaptive_refinement_interval;
double stabilization_alpha;
double stabilization_c_R;
double stabilization_beta;
unsigned int stokes_velocity_degree;
bool use_locally_conservative_discretization;
unsigned int temperature_degree;
};
private:
Parameters &parameters;
double global_Omega_diameter;
const MappingQ<dim> mapping;
const FESystem<dim> stokes_fe;
DoFHandler<dim> stokes_dof_handler;
ConstraintMatrix stokes_constraints;
TrilinosWrappers::BlockSparseMatrix stokes_preconditioner_matrix;
FE_Q<dim> temperature_fe;
DoFHandler<dim> temperature_dof_handler;
ConstraintMatrix temperature_constraints;
TrilinosWrappers::SparseMatrix temperature_mass_matrix;
TrilinosWrappers::SparseMatrix temperature_stiffness_matrix;
TrilinosWrappers::SparseMatrix temperature_matrix;
TrilinosWrappers::MPI::Vector temperature_solution;
TrilinosWrappers::MPI::Vector old_temperature_solution;
TrilinosWrappers::MPI::Vector old_old_temperature_solution;
double time_step;
double old_time_step;
unsigned int timestep_number;
std::shared_ptr<TrilinosWrappers::PreconditionAMG> Amg_preconditioner;
std::shared_ptr<TrilinosWrappers::PreconditionJacobi> Mp_preconditioner;
std::shared_ptr<TrilinosWrappers::PreconditionJacobi> T_preconditioner;
bool rebuild_stokes_matrix;
bool rebuild_stokes_preconditioner;
bool rebuild_temperature_matrices;
bool rebuild_temperature_preconditioner;
TimerOutput computing_timer;
void setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning);
void setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning);
void setup_temperature_matrices (const IndexSet &temperature_partitioning,
const IndexSet &temperature_relevant_partitioning);
void
local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesPreconditioner<dim> &scratch,
Assembly::CopyData::StokesPreconditioner<dim> &data);
void
copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data);
void
local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesSystem<dim> &scratch,
Assembly::CopyData::StokesSystem<dim> &data);
void
copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data);
void
local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::TemperatureMatrix<dim> &scratch,
Assembly::CopyData::TemperatureMatrix<dim> &data);
void
copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data);
void
local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
const double global_max_velocity,
const double global_entropy_variation,
Assembly::Scratch::TemperatureRHS<dim> &scratch,
Assembly::CopyData::TemperatureRHS<dim> &data);
void
copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data);
class Postprocessor;
};
template <int dim>
BoussinesqFlowProblem<dim>::Parameters::Parameters (const std::string &parameter_filename)
:
end_time (1e8),
initial_global_refinement (2),
initial_adaptive_refinement (2),
adaptive_refinement_interval (10),
stabilization_alpha (2),
stabilization_c_R (0.11),
stabilization_beta (0.078),
stokes_velocity_degree (2),
use_locally_conservative_discretization (true),
temperature_degree (2)
{
BoussinesqFlowProblem<dim>::Parameters::declare_parameters (prm);
std::ifstream parameter_file (parameter_filename.c_str());
if (!parameter_file)
{
parameter_file.close ();
std::ostringstream message;
message << "Input parameter file <"
<< parameter_filename << "> not found. Creating a"
<< std::endl
<< "template file of the same name."
<< std::endl;
std::ofstream parameter_out (parameter_filename.c_str());
prm.print_parameters (parameter_out,
AssertThrow (false, ExcMessage (message.str().c_str()));
}
prm.parse_input (parameter_file);
parse_parameters (prm);
}
template <int dim>
void
BoussinesqFlowProblem<dim>::Parameters::
declare_parameters (ParameterHandler &prm)
{
prm.declare_entry ("End time", "1e8",
"The end time of the simulation in years.");
prm.declare_entry ("Initial global refinement", "2",
"The number of global refinement steps performed on "
"the initial coarse mesh, before the problem is first "
"solved there.");
prm.declare_entry ("Initial adaptive refinement", "2",
"The number of adaptive refinement steps performed after "
"initial global refinement.");
prm.declare_entry ("Time steps between mesh refinement", "10",
"The number of time steps after which the mesh is to be "
"adapted based on computed error indicators.");
prm.declare_entry ("Generate graphical output", "false",
"Whether graphical output is to be generated or not. "
"You may not want to get graphical output if the number "
"of processors is large.");
prm.declare_entry ("Time steps between graphical output", "50",
"The number of time steps between each generation of "
"graphical output files.");
prm.enter_subsection ("Stabilization parameters");
{
prm.declare_entry ("alpha", "2",
"The exponent in the entropy viscosity stabilization.");
prm.declare_entry ("c_R", "0.11",
"The c_R factor in the entropy viscosity "
"stabilization.");
prm.declare_entry ("beta", "0.078",
"The beta factor in the artificial viscosity "
"stabilization. An appropriate value for 2d is 0.052 "
"and 0.078 for 3d.");
}
prm.enter_subsection ("Discretization");
{
prm.declare_entry ("Stokes velocity polynomial degree", "2",
"The polynomial degree to use for the velocity variables "
"in the Stokes system.");
prm.declare_entry ("Temperature polynomial degree", "2",
"The polynomial degree to use for the temperature variable.");
prm.declare_entry ("Use locally conservative discretization", "true",
"Whether to use a Stokes discretization that is locally "
"conservative at the expense of a larger number of degrees "
"of freedom, or to go with a cheaper discretization "
"that does not locally conserve mass (although it is "
"globally conservative.");
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::Parameters::
parse_parameters (ParameterHandler &prm)
{
end_time = prm.get_double ("End time");
initial_global_refinement = prm.get_integer ("Initial global refinement");
initial_adaptive_refinement = prm.get_integer ("Initial adaptive refinement");
adaptive_refinement_interval= prm.get_integer ("Time steps between mesh refinement");
generate_graphical_output = prm.get_bool ("Generate graphical output");
graphical_output_interval = prm.get_integer ("Time steps between graphical output");
prm.enter_subsection ("Stabilization parameters");
{
stabilization_alpha = prm.get_double ("alpha");
stabilization_c_R = prm.get_double ("c_R");
stabilization_beta = prm.get_double ("beta");
}
prm.enter_subsection ("Discretization");
{
stokes_velocity_degree = prm.get_integer ("Stokes velocity polynomial degree");
temperature_degree = prm.get_integer ("Temperature polynomial degree");
use_locally_conservative_discretization
= prm.get_bool ("Use locally conservative discretization");
}
}
template <int dim>
BoussinesqFlowProblem<dim>::BoussinesqFlowProblem (Parameters &parameters_)
:
parameters (parameters_),
pcout (std::cout,
(Utilities::MPI::this_mpi_process(MPI_COMM_WORLD)
== 0)),
triangulation (MPI_COMM_WORLD,
typename Triangulation<dim>::MeshSmoothing
(Triangulation<dim>::smoothing_on_refinement |
Triangulation<dim>::smoothing_on_coarsening)),
global_Omega_diameter(0.),
mapping (4),
stokes_fe (FE_Q<dim>(parameters.stokes_velocity_degree),
dim,
(parameters.use_locally_conservative_discretization
?
static_cast<const FiniteElement<dim> &>
(FE_DGP<dim>(parameters.stokes_velocity_degree-1))
:
static_cast<const FiniteElement<dim> &>
(FE_Q<dim>(parameters.stokes_velocity_degree-1))),
1),
stokes_dof_handler (triangulation),
temperature_fe (parameters.temperature_degree),
temperature_dof_handler (triangulation),
time_step (0),
old_time_step (0),
timestep_number (0),
rebuild_stokes_matrix (true),
rebuild_stokes_preconditioner (true),
rebuild_temperature_matrices (true),
rebuild_temperature_preconditioner (true),
computing_timer (MPI_COMM_WORLD,
pcout,
TimerOutput::summary,
TimerOutput::wall_times)
{}
template <int dim>
double BoussinesqFlowProblem<dim>::get_maximal_velocity () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.stokes_velocity_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, stokes_fe, quadrature_formula, update_values);
std::vector<Tensor<1,dim> > velocity_values(n_q_points);
const FEValuesExtractors::Vector velocities (0);
double max_local_velocity = 0;
cell = stokes_dof_handler.begin_active(),
endc = stokes_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values[velocities].get_function_values (stokes_solution,
velocity_values);
for (unsigned int q=0; q<n_q_points; ++q)
max_local_velocity = std::max (max_local_velocity,
velocity_values[q].norm());
}
return Utilities::MPI::max (max_local_velocity, MPI_COMM_WORLD);
}
template <int dim>
double BoussinesqFlowProblem<dim>::get_cfl_number () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.stokes_velocity_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, stokes_fe, quadrature_formula, update_values);
std::vector<Tensor<1,dim> > velocity_values(n_q_points);
const FEValuesExtractors::Vector velocities (0);
double max_local_cfl = 0;
cell = stokes_dof_handler.begin_active(),
endc = stokes_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values[velocities].get_function_values (stokes_solution,
velocity_values);
double max_local_velocity = 1e-10;
for (unsigned int q=0; q<n_q_points; ++q)
max_local_velocity = std::max (max_local_velocity,
velocity_values[q].norm());
max_local_cfl = std::max(max_local_cfl,
max_local_velocity / cell->diameter());
}
return Utilities::MPI::max (max_local_cfl, MPI_COMM_WORLD);
}
template <int dim>
double
BoussinesqFlowProblem<dim>::get_entropy_variation (const double average_temperature) const
{
if (parameters.stabilization_alpha != 2)
return 1.;
const QGauss<dim> quadrature_formula (parameters.temperature_degree+1);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (temperature_fe, quadrature_formula,
std::vector<double> old_temperature_values(n_q_points);
std::vector<double> old_old_temperature_values(n_q_points);
double min_entropy = std::numeric_limits<double>::max(),
max_entropy = -std::numeric_limits<double>::max(),
area = 0,
entropy_integrated = 0;
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
fe_values.get_function_values (old_old_temperature_solution,
old_old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double T = (old_temperature_values[q] +
old_old_temperature_values[q]) / 2;
const double entropy = ((T-average_temperature) *
(T-average_temperature));
min_entropy = std::min (min_entropy, entropy);
max_entropy = std::max (max_entropy, entropy);
area += fe_values.JxW(q);
entropy_integrated += fe_values.JxW(q) * entropy;
}
}
const double local_sums[2] = { entropy_integrated, area },
local_maxima[2] = { -min_entropy, max_entropy };
double global_sums[2], global_maxima[2];
Utilities::MPI::sum (local_sums, MPI_COMM_WORLD, global_sums);
Utilities::MPI::max (local_maxima, MPI_COMM_WORLD, global_maxima);
const double average_entropy = global_sums[0] / global_sums[1];
const double entropy_diff = std::max(global_maxima[1] - average_entropy,
average_entropy - (-global_maxima[0]));
return entropy_diff;
}
template <int dim>
std::pair<double,double>
BoussinesqFlowProblem<dim>::get_extrapolated_temperature_range () const
{
const QIterated<dim> quadrature_formula (QTrapez<1>(),
parameters.temperature_degree);
const unsigned int n_q_points = quadrature_formula.size();
FEValues<dim> fe_values (mapping, temperature_fe, quadrature_formula,
std::vector<double> old_temperature_values(n_q_points);
std::vector<double> old_old_temperature_values(n_q_points);
double min_local_temperature = std::numeric_limits<double>::max(),
max_local_temperature = -std::numeric_limits<double>::max();
if (timestep_number != 0)
{
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
fe_values.get_function_values (old_old_temperature_solution,
old_old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double temperature =
(1. + time_step/old_time_step) * old_temperature_values[q]-
time_step/old_time_step * old_old_temperature_values[q];
min_local_temperature = std::min (min_local_temperature,
temperature);
max_local_temperature = std::max (max_local_temperature,
temperature);
}
}
}
else
{
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
fe_values.get_function_values (old_temperature_solution,
old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double temperature = old_temperature_values[q];
min_local_temperature = std::min (min_local_temperature,
temperature);
max_local_temperature = std::max (max_local_temperature,
temperature);
}
}
}
double local_extrema[2] = { -min_local_temperature,
max_local_temperature
};
double global_extrema[2];
Utilities::MPI::max (local_extrema, MPI_COMM_WORLD, global_extrema);
return std::make_pair(-global_extrema[0], global_extrema[1]);
}
template <int dim>
double
BoussinesqFlowProblem<dim>::
compute_viscosity (const std::vector<double> &old_temperature,
const std::vector<double> &old_old_temperature,
const std::vector<Tensor<1,dim> > &old_temperature_grads,
const std::vector<Tensor<1,dim> > &old_old_temperature_grads,
const std::vector<double> &old_temperature_laplacians,
const std::vector<double> &old_old_temperature_laplacians,
const std::vector<Tensor<1,dim> > &old_velocity_values,
const std::vector<Tensor<1,dim> > &old_old_velocity_values,
const std::vector<SymmetricTensor<2,dim> > &old_strain_rates,
const std::vector<SymmetricTensor<2,dim> > &old_old_strain_rates,
const double global_u_infty,
const double global_T_variation,
const double average_temperature,
const double global_entropy_variation,
const double cell_diameter) const
{
if (global_u_infty == 0)
return 5e-3 * cell_diameter;
const unsigned int n_q_points = old_temperature.size();
double max_residual = 0;
double max_velocity = 0;
for (unsigned int q=0; q < n_q_points; ++q)
{
const Tensor<1,dim> u = (old_velocity_values[q] +
old_old_velocity_values[q]) / 2;
const SymmetricTensor<2,dim> strain_rate = (old_strain_rates[q] +
old_old_strain_rates[q]) / 2;
const double T = (old_temperature[q] + old_old_temperature[q]) / 2;
const double dT_dt = (old_temperature[q] - old_old_temperature[q])
/ old_time_step;
const double u_grad_T = u * (old_temperature_grads[q] +
old_old_temperature_grads[q]) / 2;
const double kappa_Delta_T = EquationData::kappa
* (old_temperature_laplacians[q] +
old_old_temperature_laplacians[q]) / 2;
const double gamma
= ((EquationData::radiogenic_heating * EquationData::density(T)
+
2 * EquationData::eta * strain_rate * strain_rate) /
(EquationData::density(T) * EquationData::specific_heat));
double residual
= std::abs(dT_dt + u_grad_T - kappa_Delta_T - gamma);
if (parameters.stabilization_alpha == 2)
residual *= std::abs(T - average_temperature);
max_residual = std::max (residual, max_residual);
max_velocity = std::max (std::sqrt (u*u), max_velocity);
}
const double max_viscosity = (parameters.stabilization_beta *
max_velocity * cell_diameter);
if (timestep_number == 0)
return max_viscosity;
else
{
Assert (old_time_step > 0, ExcInternalError());
double entropy_viscosity;
if (parameters.stabilization_alpha == 2)
entropy_viscosity = (parameters.stabilization_c_R *
cell_diameter * cell_diameter *
max_residual /
global_entropy_variation);
else
entropy_viscosity = (parameters.stabilization_c_R *
cell_diameter * global_Omega_diameter *
max_velocity * max_residual /
(global_u_infty * global_T_variation));
return std::min (max_viscosity, entropy_viscosity);
}
}
template <int dim>
void BoussinesqFlowProblem<dim>::project_temperature_field ()
{
assemble_temperature_matrix ();
QGauss<dim> quadrature(parameters.temperature_degree+2);
FEValues<dim> fe_values (mapping, temperature_fe, quadrature, update_flags);
const unsigned int dofs_per_cell = fe_values.dofs_per_cell,
n_q_points = fe_values.n_quadrature_points;
std::vector<types::global_dof_index> local_dof_indices (dofs_per_cell);
Vector<double> cell_vector (dofs_per_cell);
FullMatrix<double> matrix_for_bc (dofs_per_cell, dofs_per_cell);
std::vector<double> rhs_values(n_q_points);
IndexSet row_temp_matrix_partitioning(temperature_mass_matrix.n());
row_temp_matrix_partitioning.add_range(temperature_mass_matrix.local_range().first,
temperature_mass_matrix.local_range().second);
TrilinosWrappers::MPI::Vector rhs (row_temp_matrix_partitioning),
solution (row_temp_matrix_partitioning);
const EquationData::TemperatureInitialValues<dim> initial_temperature;
cell = temperature_dof_handler.begin_active(),
endc = temperature_dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
cell->get_dof_indices (local_dof_indices);
fe_values.reinit (cell);
initial_temperature.value_list (fe_values.get_quadrature_points(),
rhs_values);
cell_vector = 0;
matrix_for_bc = 0;
for (unsigned int point=0; point<n_q_points; ++point)
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
cell_vector(i) += rhs_values[point] *
fe_values.shape_value(i,point) *
fe_values.JxW(point);
if (temperature_constraints.is_inhomogeneously_constrained(local_dof_indices[i]))
{
for (unsigned int j=0; j<dofs_per_cell; ++j)
matrix_for_bc(j,i) += fe_values.shape_value(i,point) *
fe_values.shape_value(j,point) *
fe_values.JxW(point);
}
}
temperature_constraints.distribute_local_to_global (cell_vector,
local_dof_indices,
rhs,
matrix_for_bc);
}
SolverControl solver_control(5*rhs.size(), 1e-12*rhs.l2_norm());
preconditioner_mass.initialize(temperature_mass_matrix, 1.3);
cg.solve (temperature_mass_matrix, solution, rhs, preconditioner_mass);
temperature_constraints.distribute (solution);
temperature_solution = solution;
old_temperature_solution = solution;
old_old_temperature_solution = solution;
}
template <int dim>
void BoussinesqFlowProblem<dim>::
setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning)
{
stokes_matrix.clear ();
TrilinosWrappers::BlockSparsityPattern sp(stokes_partitioning, stokes_partitioning,
stokes_relevant_partitioning,
MPI_COMM_WORLD);
Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
for (unsigned int c=0; c<dim+1; ++c)
for (unsigned int d=0; d<dim+1; ++d)
if (! ((c==dim) && (d==dim)))
coupling[c][d] = DoFTools::always;
else
coupling[c][d] = DoFTools::none;
DoFTools::make_sparsity_pattern (stokes_dof_handler,
coupling, sp,
stokes_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
stokes_matrix.reinit (sp);
}
template <int dim>
void BoussinesqFlowProblem<dim>::
setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning,
const std::vector<IndexSet> &stokes_relevant_partitioning)
{
Amg_preconditioner.reset ();
Mp_preconditioner.reset ();
stokes_preconditioner_matrix.clear ();
TrilinosWrappers::BlockSparsityPattern sp(stokes_partitioning, stokes_partitioning,
stokes_relevant_partitioning,
MPI_COMM_WORLD);
Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
for (unsigned int c=0; c<dim+1; ++c)
for (unsigned int d=0; d<dim+1; ++d)
if (c == d)
coupling[c][d] = DoFTools::always;
else
coupling[c][d] = DoFTools::none;
DoFTools::make_sparsity_pattern (stokes_dof_handler,
coupling, sp,
stokes_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
stokes_preconditioner_matrix.reinit (sp);
}
template <int dim>
void BoussinesqFlowProblem<dim>::
setup_temperature_matrices (const IndexSet &temperature_partitioner,
const IndexSet &temperature_relevant_partitioner)
{
T_preconditioner.reset ();
temperature_mass_matrix.clear ();
temperature_stiffness_matrix.clear ();
temperature_matrix.clear ();
TrilinosWrappers::SparsityPattern sp(temperature_partitioner,
temperature_partitioner,
temperature_relevant_partitioner,
MPI_COMM_WORLD);
DoFTools::make_sparsity_pattern (temperature_dof_handler, sp,
temperature_constraints, false,
this_mpi_process(MPI_COMM_WORLD));
sp.compress();
temperature_matrix.reinit (sp);
temperature_mass_matrix.reinit (sp);
temperature_stiffness_matrix.reinit (sp);
}
template <int dim>
void BoussinesqFlowProblem<dim>::setup_dofs ()
{
computing_timer.enter_section("Setup dof systems");
std::vector<unsigned int> stokes_sub_blocks (dim+1,0);
stokes_sub_blocks[dim] = 1;
stokes_dof_handler.distribute_dofs (stokes_fe);
DoFRenumbering::component_wise (stokes_dof_handler, stokes_sub_blocks);
temperature_dof_handler.distribute_dofs (temperature_fe);
std::vector<types::global_dof_index> stokes_dofs_per_block (2);
DoFTools::count_dofs_per_block (stokes_dof_handler, stokes_dofs_per_block,
stokes_sub_blocks);
const unsigned int n_u = stokes_dofs_per_block[0],
n_p = stokes_dofs_per_block[1],
n_T = temperature_dof_handler.n_dofs();
std::locale s = pcout.get_stream().getloc();
pcout.get_stream().imbue(std::locale(""));
pcout << "Number of active cells: "
<< triangulation.n_global_active_cells()
<< " (on "
<< triangulation.n_levels()
<< " levels)"
<< std::endl
<< "Number of degrees of freedom: "
<< n_u + n_p + n_T
<< " (" << n_u << '+' << n_p << '+'<< n_T <<')'
<< std::endl
<< std::endl;
pcout.get_stream().imbue(s);
std::vector<IndexSet> stokes_partitioning, stokes_relevant_partitioning;
IndexSet temperature_partitioning (n_T), temperature_relevant_partitioning (n_T);
IndexSet stokes_relevant_set;
{
IndexSet stokes_index_set = stokes_dof_handler.locally_owned_dofs();
stokes_partitioning.push_back(stokes_index_set.get_view(0,n_u));
stokes_partitioning.push_back(stokes_index_set.get_view(n_u,n_u+n_p));
stokes_relevant_set);
stokes_relevant_partitioning.push_back(stokes_relevant_set.get_view(0,n_u));
stokes_relevant_partitioning.push_back(stokes_relevant_set.get_view(n_u,n_u+n_p));
temperature_partitioning = temperature_dof_handler.locally_owned_dofs();
DoFTools::extract_locally_relevant_dofs (temperature_dof_handler,
temperature_relevant_partitioning);
}
{
stokes_constraints.clear ();
stokes_constraints.reinit (stokes_relevant_set);
stokes_constraints);
FEValuesExtractors::Vector velocity_components(0);
0,
stokes_constraints,
stokes_fe.component_mask(velocity_components));
std::set<types::boundary_id> no_normal_flux_boundaries;
no_normal_flux_boundaries.insert (1);
no_normal_flux_boundaries,
stokes_constraints,
mapping);
stokes_constraints.close ();
}
{
temperature_constraints.clear ();
temperature_constraints.reinit (temperature_relevant_partitioning);
DoFTools::make_hanging_node_constraints (temperature_dof_handler,
temperature_constraints);
VectorTools::interpolate_boundary_values (temperature_dof_handler,
0,
EquationData::TemperatureInitialValues<dim>(),
temperature_constraints);
VectorTools::interpolate_boundary_values (temperature_dof_handler,
1,
EquationData::TemperatureInitialValues<dim>(),
temperature_constraints);
temperature_constraints.close ();
}
setup_stokes_matrix (stokes_partitioning, stokes_relevant_partitioning);
setup_stokes_preconditioner (stokes_partitioning,
stokes_relevant_partitioning);
setup_temperature_matrices (temperature_partitioning,
temperature_relevant_partitioning);
stokes_rhs.reinit (stokes_partitioning, stokes_relevant_partitioning,
MPI_COMM_WORLD, true);
stokes_solution.reinit (stokes_relevant_partitioning, MPI_COMM_WORLD);
old_stokes_solution.reinit (stokes_solution);
temperature_rhs.reinit (temperature_partitioning,
temperature_relevant_partitioning,
MPI_COMM_WORLD, true);
temperature_solution.reinit (temperature_relevant_partitioning, MPI_COMM_WORLD);
old_temperature_solution.reinit (temperature_solution);
old_old_temperature_solution.reinit (temperature_solution);
rebuild_stokes_matrix = true;
rebuild_stokes_preconditioner = true;
rebuild_temperature_matrices = true;
rebuild_temperature_preconditioner = true;
computing_timer.exit_section();
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesPreconditioner<dim> &scratch,
Assembly::CopyData::StokesPreconditioner<dim> &data)
{
const unsigned int dofs_per_cell = stokes_fe.dofs_per_cell;
const unsigned int n_q_points = scratch.stokes_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
const FEValuesExtractors::Scalar pressure (dim);
scratch.stokes_fe_values.reinit (cell);
cell->get_dof_indices (data.local_dof_indices);
data.local_matrix = 0;
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.grad_phi_u[k] = scratch.stokes_fe_values[velocities].gradient(k,q);
scratch.phi_p[k] = scratch.stokes_fe_values[pressure].value (k, q);
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.local_matrix(i,j) += (EquationData::eta *
scalar_product (scratch.grad_phi_u[i],
scratch.grad_phi_u[j])
+
(1./EquationData::eta) *
EquationData::pressure_scaling *
EquationData::pressure_scaling *
(scratch.phi_p[i] * scratch.phi_p[j]))
* scratch.stokes_fe_values.JxW(q);
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data)
{
stokes_constraints.distribute_local_to_global (data.local_matrix,
data.local_dof_indices,
stokes_preconditioner_matrix);
}
template <int dim>
void
BoussinesqFlowProblem<dim>::assemble_stokes_preconditioner ()
{
stokes_preconditioner_matrix = 0;
const QGauss<dim> quadrature_formula(parameters.stokes_velocity_degree+1);
typedef
CellFilter;
stokes_dof_handler.begin_active()),
stokes_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_stokes_preconditioner,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_preconditioner,
this,
std::placeholders::_1),
Assembly::Scratch::
StokesPreconditioner<dim> (stokes_fe, quadrature_formula,
mapping,
Assembly::CopyData::
StokesPreconditioner<dim> (stokes_fe));
stokes_preconditioner_matrix.compress(VectorOperation::add);
}
template <int dim>
void
BoussinesqFlowProblem<dim>::build_stokes_preconditioner ()
{
if (rebuild_stokes_preconditioner == false)
return;
computing_timer.enter_section (" Build Stokes preconditioner");
pcout << " Rebuilding Stokes preconditioner..." << std::flush;
assemble_stokes_preconditioner ();
std::vector<std::vector<bool> > constant_modes;
FEValuesExtractors::Vector velocity_components(0);
DoFTools::extract_constant_modes (stokes_dof_handler,
stokes_fe.component_mask(velocity_components),
constant_modes);
Mp_preconditioner.reset (new TrilinosWrappers::PreconditionJacobi());
Amg_preconditioner.reset (new TrilinosWrappers::PreconditionAMG());
Amg_data.constant_modes = constant_modes;
Amg_data.elliptic = true;
Amg_data.higher_order_elements = true;
Amg_data.smoother_sweeps = 2;
Amg_data.aggregation_threshold = 0.02;
Mp_preconditioner->initialize (stokes_preconditioner_matrix.block(1,1));
Amg_preconditioner->initialize (stokes_preconditioner_matrix.block(0,0),
Amg_data);
rebuild_stokes_preconditioner = false;
pcout << std::endl;
computing_timer.exit_section();
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::StokesSystem<dim> &scratch,
Assembly::CopyData::StokesSystem<dim> &data)
{
const unsigned int dofs_per_cell = scratch.stokes_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.stokes_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
const FEValuesExtractors::Scalar pressure (dim);
scratch.stokes_fe_values.reinit (cell);
temperature_cell (&triangulation,
cell->level(),
cell->index(),
&temperature_dof_handler);
scratch.temperature_fe_values.reinit (temperature_cell);
if (rebuild_stokes_matrix)
data.local_matrix = 0;
data.local_rhs = 0;
scratch.temperature_fe_values.get_function_values (old_temperature_solution,
scratch.old_temperature_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
const double old_temperature = scratch.old_temperature_values[q];
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.phi_u[k] = scratch.stokes_fe_values[velocities].value (k,q);
if (rebuild_stokes_matrix)
{
scratch.grads_phi_u[k] = scratch.stokes_fe_values[velocities].symmetric_gradient(k,q);
scratch.div_phi_u[k] = scratch.stokes_fe_values[velocities].divergence (k, q);
scratch.phi_p[k] = scratch.stokes_fe_values[pressure].value (k, q);
}
}
if (rebuild_stokes_matrix == true)
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.local_matrix(i,j) += (EquationData::eta * 2 *
(scratch.grads_phi_u[i] * scratch.grads_phi_u[j])
- (EquationData::pressure_scaling *
scratch.div_phi_u[i] * scratch.phi_p[j])
- (EquationData::pressure_scaling *
scratch.phi_p[i] * scratch.div_phi_u[j]))
* scratch.stokes_fe_values.JxW(q);
gravity = EquationData::gravity_vector (scratch.stokes_fe_values
.quadrature_point(q));
for (unsigned int i=0; i<dofs_per_cell; ++i)
data.local_rhs(i) += (EquationData::density(old_temperature) *
gravity *
scratch.phi_u[i]) *
scratch.stokes_fe_values.JxW(q);
}
cell->get_dof_indices (data.local_dof_indices);
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data)
{
if (rebuild_stokes_matrix == true)
stokes_constraints.distribute_local_to_global (data.local_matrix,
data.local_rhs,
data.local_dof_indices,
stokes_matrix,
stokes_rhs);
else
stokes_constraints.distribute_local_to_global (data.local_rhs,
data.local_dof_indices,
stokes_rhs);
}
template <int dim>
void BoussinesqFlowProblem<dim>::assemble_stokes_system ()
{
computing_timer.enter_section (" Assemble Stokes system");
if (rebuild_stokes_matrix == true)
stokes_matrix=0;
stokes_rhs=0;
const QGauss<dim> quadrature_formula(parameters.stokes_velocity_degree+1);
typedef
CellFilter;
stokes_dof_handler.begin_active()),
stokes_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_stokes_system,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_stokes_system,
this,
std::placeholders::_1),
Assembly::Scratch::
StokesSystem<dim> (stokes_fe, mapping, quadrature_formula,
(rebuild_stokes_matrix == true
?
:
temperature_fe,
Assembly::CopyData::
StokesSystem<dim> (stokes_fe));
if (rebuild_stokes_matrix == true)
stokes_matrix.compress(VectorOperation::add);
stokes_rhs.compress(VectorOperation::add);
rebuild_stokes_matrix = false;
pcout << std::endl;
computing_timer.exit_section();
}
template <int dim>
void BoussinesqFlowProblem<dim>::
local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
Assembly::Scratch::TemperatureMatrix<dim> &scratch,
Assembly::CopyData::TemperatureMatrix<dim> &data)
{
const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.temperature_fe_values.n_quadrature_points;
scratch.temperature_fe_values.reinit (cell);
cell->get_dof_indices (data.local_dof_indices);
data.local_mass_matrix = 0;
data.local_stiffness_matrix = 0;
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
scratch.phi_T[k] = scratch.temperature_fe_values.shape_value (k, q);
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=0; j<dofs_per_cell; ++j)
{
data.local_mass_matrix(i,j)
+= (scratch.phi_T[i] * scratch.phi_T[j]
*
scratch.temperature_fe_values.JxW(q));
data.local_stiffness_matrix(i,j)
+= (EquationData::kappa * scratch.grad_phi_T[i] * scratch.grad_phi_T[j]
*
scratch.temperature_fe_values.JxW(q));
}
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data)
{
temperature_constraints.distribute_local_to_global (data.local_mass_matrix,
data.local_dof_indices,
temperature_mass_matrix);
temperature_constraints.distribute_local_to_global (data.local_stiffness_matrix,
data.local_dof_indices,
temperature_stiffness_matrix);
}
template <int dim>
void BoussinesqFlowProblem<dim>::assemble_temperature_matrix ()
{
if (rebuild_temperature_matrices == false)
return;
computing_timer.enter_section (" Assemble temperature matrices");
temperature_mass_matrix = 0;
temperature_stiffness_matrix = 0;
const QGauss<dim> quadrature_formula(parameters.temperature_degree+2);
typedef
CellFilter;
temperature_dof_handler.begin_active()),
temperature_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_temperature_matrix,
this,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_matrix,
this,
std::placeholders::_1),
Assembly::Scratch::
TemperatureMatrix<dim> (temperature_fe, mapping, quadrature_formula),
Assembly::CopyData::
TemperatureMatrix<dim> (temperature_fe));
temperature_mass_matrix.compress(VectorOperation::add);
temperature_stiffness_matrix.compress(VectorOperation::add);
rebuild_temperature_matrices = false;
rebuild_temperature_preconditioner = true;
computing_timer.exit_section();
}
template <int dim>
void BoussinesqFlowProblem<dim>::
local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
const double global_max_velocity,
const double global_entropy_variation,
Assembly::Scratch::TemperatureRHS<dim> &scratch,
Assembly::CopyData::TemperatureRHS<dim> &data)
{
const bool use_bdf2_scheme = (timestep_number != 0);
const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
const unsigned int n_q_points = scratch.temperature_fe_values.n_quadrature_points;
const FEValuesExtractors::Vector velocities (0);
data.local_rhs = 0;
data.matrix_for_bc = 0;
cell->get_dof_indices (data.local_dof_indices);
scratch.temperature_fe_values.reinit (cell);
stokes_cell (&triangulation,
cell->level(),
cell->index(),
&stokes_dof_handler);
scratch.stokes_fe_values.reinit (stokes_cell);
scratch.temperature_fe_values.get_function_values (old_temperature_solution,
scratch.old_temperature_values);
scratch.temperature_fe_values.get_function_values (old_old_temperature_solution,
scratch.old_old_temperature_values);
scratch.temperature_fe_values.get_function_gradients (old_temperature_solution,
scratch.old_temperature_grads);
scratch.temperature_fe_values.get_function_gradients (old_old_temperature_solution,
scratch.old_old_temperature_grads);
scratch.temperature_fe_values.get_function_laplacians (old_temperature_solution,
scratch.old_temperature_laplacians);
scratch.temperature_fe_values.get_function_laplacians (old_old_temperature_solution,
scratch.old_old_temperature_laplacians);
scratch.stokes_fe_values[velocities].get_function_values (stokes_solution,
scratch.old_velocity_values);
scratch.stokes_fe_values[velocities].get_function_values (old_stokes_solution,
scratch.old_old_velocity_values);
scratch.stokes_fe_values[velocities].get_function_symmetric_gradients (stokes_solution,
scratch.old_strain_rates);
scratch.stokes_fe_values[velocities].get_function_symmetric_gradients (old_stokes_solution,
scratch.old_old_strain_rates);
const double nu
= compute_viscosity (scratch.old_temperature_values,
scratch.old_old_temperature_values,
scratch.old_temperature_grads,
scratch.old_old_temperature_grads,
scratch.old_temperature_laplacians,
scratch.old_old_temperature_laplacians,
scratch.old_velocity_values,
scratch.old_old_velocity_values,
scratch.old_strain_rates,
scratch.old_old_strain_rates,
global_max_velocity,
global_T_range.second - global_T_range.first,
0.5 * (global_T_range.second + global_T_range.first),
global_entropy_variation,
cell->diameter());
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
scratch.phi_T[k] = scratch.temperature_fe_values.shape_value (k, q);
scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
}
const double T_term_for_rhs
= (use_bdf2_scheme ?
(scratch.old_temperature_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_values[q] *
(time_step * time_step) /
(old_time_step * (time_step + old_time_step)))
:
scratch.old_temperature_values[q]);
const double ext_T
= (use_bdf2_scheme ?
(scratch.old_temperature_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_values[q] *
time_step/old_time_step)
:
scratch.old_temperature_values[q]);
const Tensor<1,dim> ext_grad_T
= (use_bdf2_scheme ?
(scratch.old_temperature_grads[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_temperature_grads[q] *
time_step/old_time_step)
:
scratch.old_temperature_grads[q]);
const Tensor<1,dim> extrapolated_u
= (use_bdf2_scheme ?
(scratch.old_velocity_values[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_velocity_values[q] *
time_step/old_time_step)
:
scratch.old_velocity_values[q]);
const SymmetricTensor<2,dim> extrapolated_strain_rate
= (use_bdf2_scheme ?
(scratch.old_strain_rates[q] *
(1 + time_step/old_time_step)
-
scratch.old_old_strain_rates[q] *
time_step/old_time_step)
:
scratch.old_strain_rates[q]);
const double gamma
= ((EquationData::radiogenic_heating * EquationData::density(ext_T)
+
2 * EquationData::eta * extrapolated_strain_rate * extrapolated_strain_rate) /
(EquationData::density(ext_T) * EquationData::specific_heat));
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
data.local_rhs(i) += (T_term_for_rhs * scratch.phi_T[i]
-
time_step *
extrapolated_u * ext_grad_T * scratch.phi_T[i]
-
time_step *
nu * ext_grad_T * scratch.grad_phi_T[i]
+
time_step *
gamma * scratch.phi_T[i])
*
scratch.temperature_fe_values.JxW(q);
if (temperature_constraints.is_inhomogeneously_constrained(data.local_dof_indices[i]))
{
for (unsigned int j=0; j<dofs_per_cell; ++j)
data.matrix_for_bc(j,i) += (scratch.phi_T[i] * scratch.phi_T[j] *
(use_bdf2_scheme ?
((2*time_step + old_time_step) /
(time_step + old_time_step)) : 1.)
+
scratch.grad_phi_T[i] *
scratch.grad_phi_T[j] *
EquationData::kappa *
time_step)
*
scratch.temperature_fe_values.JxW(q);
}
}
}
}
template <int dim>
void
BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data)
{
temperature_constraints.distribute_local_to_global (data.local_rhs,
data.local_dof_indices,
temperature_rhs,
data.matrix_for_bc);
}
template <int dim>
void BoussinesqFlowProblem<dim>::assemble_temperature_system (const double maximal_velocity)
{
const bool use_bdf2_scheme = (timestep_number != 0);
if (use_bdf2_scheme == true)
{
temperature_matrix.copy_from (temperature_mass_matrix);
temperature_matrix *= (2*time_step + old_time_step) /
(time_step + old_time_step);
temperature_matrix.add (time_step, temperature_stiffness_matrix);
}
else
{
temperature_matrix.copy_from (temperature_mass_matrix);
temperature_matrix.add (time_step, temperature_stiffness_matrix);
}
if (rebuild_temperature_preconditioner == true)
{
T_preconditioner.reset (new TrilinosWrappers::PreconditionJacobi());
T_preconditioner->initialize (temperature_matrix);
rebuild_temperature_preconditioner = false;
}
temperature_rhs = 0;
const QGauss<dim> quadrature_formula(parameters.temperature_degree+2);
const std::pair<double,double>
global_T_range = get_extrapolated_temperature_range();
const double average_temperature = 0.5 * (global_T_range.first +
global_T_range.second);
const double global_entropy_variation =
get_entropy_variation (average_temperature);
typedef
CellFilter;
temperature_dof_handler.begin_active()),
temperature_dof_handler.end()),
std::bind (&BoussinesqFlowProblem<dim>::
local_assemble_temperature_rhs,
this,
global_T_range,
maximal_velocity,
global_entropy_variation,
std::placeholders::_1,
std::placeholders::_2,
std::placeholders::_3),
std::bind (&BoussinesqFlowProblem<dim>::
copy_local_to_global_temperature_rhs,
this,
std::placeholders::_1),
Assembly::Scratch::
TemperatureRHS<dim> (temperature_fe, stokes_fe, mapping,
quadrature_formula),
Assembly::CopyData::
TemperatureRHS<dim> (temperature_fe));
temperature_rhs.compress(VectorOperation::add);
}
template <int dim>
void BoussinesqFlowProblem<dim>::solve ()
{
computing_timer.enter_section (" Solve Stokes system");
{
pcout << " Solving Stokes system... " << std::flush;
distributed_stokes_solution (stokes_rhs);
distributed_stokes_solution = stokes_solution;
distributed_stokes_solution.block(1) /= EquationData::pressure_scaling;
const unsigned int
start = (distributed_stokes_solution.block(0).size() +
distributed_stokes_solution.block(1).local_range().first),
end = (distributed_stokes_solution.block(0).size() +
distributed_stokes_solution.block(1).local_range().second);
for (unsigned int i=start; i<end; ++i)
if (stokes_constraints.is_constrained (i))
distributed_stokes_solution(i) = 0;
unsigned int n_iterations = 0;
const double solver_tolerance = 1e-8 * stokes_rhs.l2_norm();
SolverControl solver_control (30, solver_tolerance);
try
{
const LinearSolvers::BlockSchurPreconditioner<TrilinosWrappers::PreconditionAMG,
preconditioner (stokes_matrix, stokes_preconditioner_matrix,
*Mp_preconditioner, *Amg_preconditioner,
false);
solver(solver_control, mem,
AdditionalData(30, true));
solver.solve(stokes_matrix, distributed_stokes_solution, stokes_rhs,
preconditioner);
n_iterations = solver_control.last_step();
}
{
const LinearSolvers::BlockSchurPreconditioner<TrilinosWrappers::PreconditionAMG,
preconditioner (stokes_matrix, stokes_preconditioner_matrix,
*Mp_preconditioner, *Amg_preconditioner,
true);
SolverControl solver_control_refined (stokes_matrix.m(), solver_tolerance);
solver(solver_control_refined, mem,
AdditionalData(50, true));
solver.solve(stokes_matrix, distributed_stokes_solution, stokes_rhs,
preconditioner);
n_iterations = (solver_control.last_step() +
solver_control_refined.last_step());
}
stokes_constraints.distribute (distributed_stokes_solution);
distributed_stokes_solution.block(1) *= EquationData::pressure_scaling;
stokes_solution = distributed_stokes_solution;
pcout << n_iterations << " iterations."
<< std::endl;
}
computing_timer.exit_section();
computing_timer.enter_section (" Assemble temperature rhs");
{
old_time_step = time_step;
const double scaling = (dim==3 ? 0.25 : 1.0);
time_step = (scaling/(2.1*dim*std::sqrt(1.*dim)) /
(parameters.temperature_degree *
get_cfl_number()));
const double maximal_velocity = get_maximal_velocity();
pcout << " Maximal velocity: "
<< maximal_velocity *EquationData::year_in_seconds * 100
<< " cm/year"
<< std::endl;
pcout << " " << "Time step: "
<< time_step/EquationData::year_in_seconds
<< " years"
<< std::endl;
temperature_solution = old_temperature_solution;
assemble_temperature_system (maximal_velocity);
}
computing_timer.exit_section ();
computing_timer.enter_section (" Solve temperature system");
{
SolverControl solver_control (temperature_matrix.m(),
1e-12*temperature_rhs.l2_norm());
distributed_temperature_solution (temperature_rhs);
distributed_temperature_solution = temperature_solution;
cg.solve (temperature_matrix, distributed_temperature_solution,
temperature_rhs, *T_preconditioner);
temperature_constraints.distribute (distributed_temperature_solution);
temperature_solution = distributed_temperature_solution;
pcout << " "
<< solver_control.last_step()
<< " CG iterations for temperature" << std::endl;
computing_timer.exit_section();
double temperature[2] = { std::numeric_limits<double>::max(),
-std::numeric_limits<double>::max()
};
double global_temperature[2];
for (unsigned int i=distributed_temperature_solution.local_range().first;
i < distributed_temperature_solution.local_range().second; ++i)
{
temperature[0] = std::min<double> (temperature[0],
distributed_temperature_solution(i));
temperature[1] = std::max<double> (temperature[1],
distributed_temperature_solution(i));
}
temperature[0] *= -1.0;
Utilities::MPI::max (temperature, MPI_COMM_WORLD, global_temperature);
global_temperature[0] *= -1.0;
pcout << " Temperature range: "
<< global_temperature[0] << ' ' << global_temperature[1]
<< std::endl;
}
}
template <int dim>
class BoussinesqFlowProblem<dim>::Postprocessor : public DataPostprocessor<dim>
{
public:
Postprocessor (const unsigned int partition,
const double minimal_pressure);
virtual
void
evaluate_vector_field
std::vector<Vector<double> > &computed_quantities) const;
virtual std::vector<std::string> get_names () const;
virtual
std::vector<DataComponentInterpretation::DataComponentInterpretation>
get_data_component_interpretation () const;
virtual UpdateFlags get_needed_update_flags () const;
private:
const unsigned int partition;
const double minimal_pressure;
};
template <int dim>
BoussinesqFlowProblem<dim>::Postprocessor::
Postprocessor (const unsigned int partition,
const double minimal_pressure)
:
partition (partition),
minimal_pressure (minimal_pressure)
{}
template <int dim>
std::vector<std::string>
BoussinesqFlowProblem<dim>::Postprocessor::get_names() const
{
std::vector<std::string> solution_names (dim, "velocity");
solution_names.push_back ("p");
solution_names.push_back ("T");
solution_names.push_back ("friction_heating");
solution_names.push_back ("partition");
return solution_names;
}
template <int dim>
std::vector<DataComponentInterpretation::DataComponentInterpretation>
BoussinesqFlowProblem<dim>::Postprocessor::
get_data_component_interpretation () const
{
std::vector<DataComponentInterpretation::DataComponentInterpretation>
interpretation (dim,
return interpretation;
}
template <int dim>
BoussinesqFlowProblem<dim>::Postprocessor::get_needed_update_flags() const
{
}
template <int dim>
void
BoussinesqFlowProblem<dim>::Postprocessor::
evaluate_vector_field
std::vector<Vector<double> > &computed_quantities) const
{
const unsigned int n_quadrature_points = inputs.solution_values.size();
Assert (inputs.solution_gradients.size() == n_quadrature_points,
Assert (computed_quantities.size() == n_quadrature_points,
Assert (inputs.solution_values[0].size() == dim+2,
for (unsigned int q=0; q<n_quadrature_points; ++q)
{
for (unsigned int d=0; d<dim; ++d)
computed_quantities[q](d)
= (inputs.solution_values[q](d) * EquationData::year_in_seconds * 100);
const double pressure = (inputs.solution_values[q](dim)-minimal_pressure);
computed_quantities[q](dim) = pressure;
const double temperature = inputs.solution_values[q](dim+1);
computed_quantities[q](dim+1) = temperature;
Tensor<2,dim> grad_u;
for (unsigned int d=0; d<dim; ++d)
grad_u[d] = inputs.solution_gradients[q][d];
const SymmetricTensor<2,dim> strain_rate = symmetrize (grad_u);
computed_quantities[q](dim+2) = 2 * EquationData::eta *
strain_rate * strain_rate;
computed_quantities[q](dim+3) = partition;
}
}
template <int dim>
void BoussinesqFlowProblem<dim>::output_results ()
{
computing_timer.enter_section ("Postprocessing");
const FESystem<dim> joint_fe (stokes_fe, 1,
temperature_fe, 1);
DoFHandler<dim> joint_dof_handler (triangulation);
joint_dof_handler.distribute_dofs (joint_fe);
Assert (joint_dof_handler.n_dofs() ==
stokes_dof_handler.n_dofs() + temperature_dof_handler.n_dofs(),
joint_solution.reinit (joint_dof_handler.locally_owned_dofs(), MPI_COMM_WORLD);
{
std::vector<types::global_dof_index> local_joint_dof_indices (joint_fe.dofs_per_cell);
std::vector<types::global_dof_index> local_stokes_dof_indices (stokes_fe.dofs_per_cell);
std::vector<types::global_dof_index> local_temperature_dof_indices (temperature_fe.dofs_per_cell);
joint_cell = joint_dof_handler.begin_active(),
joint_endc = joint_dof_handler.end(),
stokes_cell = stokes_dof_handler.begin_active(),
temperature_cell = temperature_dof_handler.begin_active();
for (; joint_cell!=joint_endc;
++joint_cell, ++stokes_cell, ++temperature_cell)
if (joint_cell->is_locally_owned())
{
joint_cell->get_dof_indices (local_joint_dof_indices);
stokes_cell->get_dof_indices (local_stokes_dof_indices);
temperature_cell->get_dof_indices (local_temperature_dof_indices);
for (unsigned int i=0; i<joint_fe.dofs_per_cell; ++i)
if (joint_fe.system_to_base_index(i).first.first == 0)
{
Assert (joint_fe.system_to_base_index(i).second
<
local_stokes_dof_indices.size(),
joint_solution(local_joint_dof_indices[i])
= stokes_solution(local_stokes_dof_indices
[joint_fe.system_to_base_index(i).second]);
}
else
{
Assert (joint_fe.system_to_base_index(i).first.first == 1,
Assert (joint_fe.system_to_base_index(i).second
<
local_temperature_dof_indices.size(),
joint_solution(local_joint_dof_indices[i])
= temperature_solution(local_temperature_dof_indices
[joint_fe.system_to_base_index(i).second]);
}
}
}
joint_solution.compress(VectorOperation::insert);
IndexSet locally_relevant_joint_dofs(joint_dof_handler.n_dofs());
DoFTools::extract_locally_relevant_dofs (joint_dof_handler, locally_relevant_joint_dofs);
TrilinosWrappers::MPI::Vector locally_relevant_joint_solution;
locally_relevant_joint_solution.reinit (locally_relevant_joint_dofs, MPI_COMM_WORLD);
locally_relevant_joint_solution = joint_solution;
Postprocessor postprocessor (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD),
stokes_solution.block(1).min());
DataOut<dim> data_out;
data_out.attach_dof_handler (joint_dof_handler);
data_out.add_data_vector (locally_relevant_joint_solution, postprocessor);
data_out.build_patches ();
static int out_index=0;
const std::string filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
"." +
(triangulation.locally_owned_subdomain(), 4) +
".vtu");
std::ofstream output (filename.c_str());
data_out.write_vtu (output);
if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
{
std::vector<std::string> filenames;
for (unsigned int i=0; i<Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD); ++i)
filenames.push_back (std::string("solution-") +
Utilities::int_to_string (out_index, 5) +
"." +
".vtu");
const std::string
pvtu_master_filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
".pvtu");
std::ofstream pvtu_master (pvtu_master_filename.c_str());
data_out.write_pvtu_record (pvtu_master, filenames);
const std::string
visit_master_filename = ("solution-" +
Utilities::int_to_string (out_index, 5) +
".visit");
std::ofstream visit_master (visit_master_filename.c_str());
DataOutBase::write_visit_record (visit_master, filenames);
}
computing_timer.exit_section ();
out_index++;
}
template <int dim>
void BoussinesqFlowProblem<dim>::refine_mesh (const unsigned int max_grid_level)
{
computing_timer.enter_section ("Refine mesh structure, part 1");
Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
KellyErrorEstimator<dim>::estimate (temperature_dof_handler,
QGauss<dim-1>(parameters.temperature_degree+1),
temperature_solution,
estimated_error_per_cell,
nullptr,
0,
triangulation.locally_owned_subdomain());
estimated_error_per_cell,
0.3, 0.1);
if (triangulation.n_levels() > max_grid_level)
cell = triangulation.begin_active(max_grid_level);
cell != triangulation.end(); ++cell)
cell->clear_refine_flag ();
std::vector<const TrilinosWrappers::MPI::Vector *> x_temperature (2);
x_temperature[0] = &temperature_solution;
x_temperature[1] = &old_temperature_solution;
std::vector<const TrilinosWrappers::MPI::BlockVector *> x_stokes (2);
x_stokes[0] = &stokes_solution;
x_stokes[1] = &old_stokes_solution;
temperature_trans(temperature_dof_handler);
stokes_trans(stokes_dof_handler);
triangulation.prepare_coarsening_and_refinement();
temperature_trans.prepare_for_coarsening_and_refinement(x_temperature);
stokes_trans.prepare_for_coarsening_and_refinement(x_stokes);
triangulation.execute_coarsening_and_refinement ();
computing_timer.exit_section();
setup_dofs ();
computing_timer.enter_section ("Refine mesh structure, part 2");
{
TrilinosWrappers::MPI::Vector distributed_temp1 (temperature_rhs);
TrilinosWrappers::MPI::Vector distributed_temp2 (temperature_rhs);
std::vector<TrilinosWrappers::MPI::Vector *> tmp (2);
tmp[0] = &(distributed_temp1);
tmp[1] = &(distributed_temp2);
temperature_trans.interpolate(tmp);
temperature_constraints.distribute(distributed_temp1);
temperature_constraints.distribute(distributed_temp2);
temperature_solution = distributed_temp1;
old_temperature_solution = distributed_temp2;
}
{
TrilinosWrappers::MPI::BlockVector distributed_stokes (stokes_rhs);
TrilinosWrappers::MPI::BlockVector old_distributed_stokes (stokes_rhs);
std::vector<TrilinosWrappers::MPI::BlockVector *> stokes_tmp (2);
stokes_tmp[0] = &(distributed_stokes);
stokes_tmp[1] = &(old_distributed_stokes);
stokes_trans.interpolate (stokes_tmp);
stokes_constraints.distribute(distributed_stokes);
stokes_constraints.distribute(old_distributed_stokes);
stokes_solution = distributed_stokes;
old_stokes_solution = old_distributed_stokes;
}
computing_timer.exit_section();
}
template <int dim>
void BoussinesqFlowProblem<dim>::run ()
{
EquationData::R0,
EquationData::R1,
(dim==3) ? 96 : 12,
true);
triangulation.set_all_manifold_ids(0);
triangulation.set_all_manifold_ids_on_boundary(1);
static SphericalManifold<dim> manifold;
static HyperShellBoundary<dim> boundary;
triangulation.set_manifold (0, manifold);
triangulation.set_manifold (1, boundary);
global_Omega_diameter = GridTools::diameter (triangulation);
triangulation.refine_global (parameters.initial_global_refinement);
setup_dofs();
unsigned int pre_refinement_step = 0;
start_time_iteration:
project_temperature_field ();
timestep_number = 0;
time_step = old_time_step = 0;
double time = 0;
do
{
pcout << "Timestep " << timestep_number
<< ": t=" << time/EquationData::year_in_seconds
<< " years"
<< std::endl;
assemble_stokes_system ();
build_stokes_preconditioner ();
assemble_temperature_matrix ();
solve ();
pcout << std::endl;
if ((timestep_number == 0) &&
(pre_refinement_step < parameters.initial_adaptive_refinement))
{
refine_mesh (parameters.initial_global_refinement +
parameters.initial_adaptive_refinement);
++pre_refinement_step;
goto start_time_iteration;
}
else if ((timestep_number > 0)
&&
(timestep_number % parameters.adaptive_refinement_interval == 0))
refine_mesh (parameters.initial_global_refinement +
parameters.initial_adaptive_refinement);
if ((parameters.generate_graphical_output == true)
&&
(timestep_number % parameters.graphical_output_interval == 0))
output_results ();
if (time > parameters.end_time * EquationData::year_in_seconds)
break;
TrilinosWrappers::MPI::BlockVector old_old_stokes_solution;
old_old_stokes_solution = old_stokes_solution;
old_stokes_solution = stokes_solution;
old_old_temperature_solution = old_temperature_solution;
old_temperature_solution = temperature_solution;
if (old_time_step > 0)
{
{
TrilinosWrappers::MPI::BlockVector distr_solution (stokes_rhs);
distr_solution = stokes_solution;
TrilinosWrappers::MPI::BlockVector distr_old_solution (stokes_rhs);
distr_old_solution = old_old_stokes_solution;
distr_solution .sadd (1.+time_step/old_time_step, -time_step/old_time_step,
distr_old_solution);
stokes_solution = distr_solution;
}
{
TrilinosWrappers::MPI::Vector distr_solution (temperature_rhs);
distr_solution = temperature_solution;
TrilinosWrappers::MPI::Vector distr_old_solution (temperature_rhs);
distr_old_solution = old_old_temperature_solution;
distr_solution .sadd (1.+time_step/old_time_step, -time_step/old_time_step,
distr_old_solution);
temperature_solution = distr_solution;
}
}
if ((timestep_number > 0) && (timestep_number % 100 == 0))
computing_timer.print_summary ();
time += time_step;
++timestep_number;
}
while (true);
if ((parameters.generate_graphical_output == true)
&&
!((timestep_number-1) % parameters.graphical_output_interval == 0))
output_results ();
}
}
int main (int argc, char *argv[])
{
using namespace Step32;
using namespace dealii;
Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv,
try
{
std::string parameter_filename;
if (argc>=2)
parameter_filename = argv[1];
else
parameter_filename = "step-32.prm";
const int dim = 2;
BoussinesqFlowProblem<dim>::Parameters parameters(parameter_filename);
BoussinesqFlowProblem<dim> flow_problem (parameters);
flow_problem.run ();
}
catch (std::exception &exc)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Exception on processing: " << std::endl
<< exc.what() << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
catch (...)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Unknown exception!" << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
return 0;
}