#include <deal.II/differentiation/ad/ad_helpers.h>

Inheritance diagram for Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >:

Public Types
using	scalar_type = typename HelperBase< ADNumberTypeCode, ScalarType >::scalar_type

using	ad_type = typename HelperBase< ADNumberTypeCode, ScalarType >::ad_type

Public Types inherited from Differentiation::AD::CellLevelBase< ADNumberTypeCode, ScalarType >
using	scalar_type = typename HelperBase< ADNumberTypeCode, ScalarType >::scalar_type

using	ad_type = typename HelperBase< ADNumberTypeCode, ScalarType >::ad_type

Public Types inherited from Differentiation::AD::HelperBase< ADNumberTypeCode, ScalarType >
using	scalar_type = typename AD::NumberTraits< ScalarType, ADNumberTypeCode >::scalar_type

using	ad_type = typename AD::NumberTraits< ScalarType, ADNumberTypeCode >::ad_type

Public Member Functions
Constructor / destructor
	EnergyFunctional (const unsigned int n_independent_variables)

virtual	~EnergyFunctional ()=default

Dependent variables
void	register_energy_functional (const ad_type &energy)

scalar_type	compute_energy () const

void	compute_residual (Vector< scalar_type > &residual) const override

virtual void	compute_linearization (FullMatrix< scalar_type > &linearization) const override

Public Member Functions inherited from Differentiation::AD::CellLevelBase< ADNumberTypeCode, ScalarType >
	CellLevelBase (const unsigned int n_independent_variables, const unsigned int n_dependent_variables)

virtual	~CellLevelBase ()=default

void	register_dof_values (const std::vector< scalar_type > &dof_values)

template<typename VectorType >
void	register_dof_values (const VectorType &values, const std::vector<::types::global_dof_index > &local_dof_indices)

const std::vector< ad_type > &	get_sensitive_dof_values () const

void	set_dof_values (const std::vector< scalar_type > &dof_values)

template<typename VectorType >
void	set_dof_values (const VectorType &values, const std::vector<::types::global_dof_index > &local_dof_indices)

Public Member Functions inherited from Differentiation::AD::HelperBase< ADNumberTypeCode, ScalarType >
	HelperBase (const unsigned int n_independent_variables, const unsigned int n_dependent_variables)

virtual	~HelperBase ()=default

std::size_t	n_independent_variables () const

std::size_t	n_dependent_variables () const

void	print (std::ostream &stream) const

void	print_values (std::ostream &stream) const

void	print_tape_stats (const typename Types< ad_type >::tape_index tape_index, std::ostream &stream) const

virtual void	reset (const unsigned int n_independent_variables=::numbers::invalid_unsigned_int, const unsigned int n_dependent_variables=::numbers::invalid_unsigned_int, const bool clear_registered_tapes=true)

bool	is_recording () const

Types< ad_type >::tape_index	active_tape_index () const

bool	is_registered_tape (const typename Types< ad_type >::tape_index tape_index) const

void	set_tape_buffer_sizes (const typename Types< ad_type >::tape_buffer_sizes obufsize=64 1024 1024, const typename Types< ad_type >::tape_buffer_sizes lbufsize=64 1024 1024, const typename Types< ad_type >::tape_buffer_sizes vbufsize=64 1024 1024, const typename Types< ad_type >::tape_buffer_sizes tbufsize=64 1024 1024)

bool	start_recording_operations (const typename Types< ad_type >::tape_index tape_index, const bool overwrite_tape=false, const bool keep_independent_values=true)

void	stop_recording_operations (const bool write_tapes_to_file=false)

void	activate_recorded_tape (const typename Types< ad_type >::tape_index tape_index)

bool	recorded_tape_requires_retaping (const typename Types< ad_type >::tape_index tape_index) const

bool	active_tape_requires_retaping () const

void	clear_active_tape ()

Additional Inherited Members
Static Public Member Functions inherited from Differentiation::AD::HelperBase< ADNumberTypeCode, ScalarType >
static void	configure_tapeless_mode (const unsigned int n_independent_variables, const bool ensure_persistent_setting=true)

Protected Member Functions inherited from Differentiation::AD::HelperBase< ADNumberTypeCode, ScalarType >
void	activate_tape (const typename Types< ad_type >::tape_index tape_index, const bool read_mode)

void	reset_registered_independent_variables ()

void	set_sensitivity_value (const unsigned int index, const scalar_type &value)

void	mark_independent_variable (const unsigned int index, ad_type &out) const

void	finalize_sensitive_independent_variables () const

void	initialize_non_sensitive_independent_variable (const unsigned int index, ad_type &out) const

unsigned int	n_registered_independent_variables () const

void	reset_registered_dependent_variables (const bool flag=false)

unsigned int	n_registered_dependent_variables () const

void	register_dependent_variable (const unsigned int index, const ad_type &func)

Protected Attributes inherited from Differentiation::AD::HelperBase< ADNumberTypeCode, ScalarType >
TapedDrivers< ad_type, scalar_type >	taped_driver

TapelessDrivers< ad_type, scalar_type >	tapeless_driver

std::vector< scalar_type >	independent_variable_values

std::vector< ad_type >	independent_variables

std::vector< bool >	registered_independent_variable_values

std::vector< bool >	registered_marked_independent_variables

std::vector< ad_type >	dependent_variables

std::vector< bool >	registered_marked_dependent_variables

Detailed Description

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double>
class Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >

A helper class that facilitates the implementation of a generic (incremental) variational formulation from which the computation of the residual vector, as well as its linearization, is automated. This class would typically be used to derive the residual vector and tangent matrix (defined on the level of a cell), or a linearized system of equations, starting from a scalar energy functional.

An example of its usage in the case of a residual and tangent computations might be as follows (in this case we'll compute the linearization of a finite-strain hyperelastic solid from a stored/strain energy density function):

// Existing data structures:
Vector<double> solution (...); // Or another vector type
std::vector<types::global_dof_index> local_dof_indices (...);
const FEValuesExtractors::Vector u_fe (...);
FEValues<dim> fe_values (...);
const unsigned int n_q_points (...);
FullMatrix<double> cell_matrix (...);
Vector<double> cell_rhs (...);
// Assembly loop:
for (auto & cell : ...)
{
  cell->get_dof_indices(local_dof_indices);
  const unsigned int n_independent_variables =
    local_dof_indices.size();
  // Create some aliases for the AD helper.
  // In the example, the AD_typecode used for the template argument can
  // be refer to either a taped or tapeless type.
  using ADHelper = AD::EnergyFunctional<...>;
  using ADNumberType = typename ADHelper::ad_type;
  // Create and initialize an instance of the helper class.
  ADHelper ad_helper(n_independent_variables);
  // Initialize the local data structures for assembly.
  // This is also taken care of by the ADHelper, so this step could
  // be skipped.
  cell_rhs.reinit(n_independent_variables);
  cell_matrix.reinit(n_independent_variables,n_independent_variables);
  // An optional call to set the amount of memory to be allocated to
  // storing taped data.
  // If using a taped AD number then we would likely want to increase
  // the buffer size from the default values as the expression for each
  // residual component will likely be lengthy, and therefore memory
  // intensive.
  ad_helper.set_tape_buffer_sizes(...);
  // If using a taped AD number, then at this point we would initiate
  // taping of the expression for the energy for this FE type and
  // material combination:
  // Select a tape number to record to
  const typename Types<ad_type>::tape_index tape_index = ...;
  // Indicate that we are about to start tracing the operations for
  // function evaluation on the tape. If this tape has already been
  // used (i.e. the operations are already recorded) then we
  // (optionally) load the tape and reuse this data.
  const bool is_recording
    = ad_helper.start_recording_operations(tape_index);
  // The steps that follow in the recording phase are required for
  // tapeless methods as well.
  if (is_recording == true)
  {
    // This is the "recording" phase of the operations.
    // First, we set the values for all DoFs.
    ad_helper.register_dof_values(solution, local_dof_indices);
    // Then we get the complete set of degree of freedom values as
    // represented by auto-differentiable numbers. The operations
    // performed with these variables are tracked by the AD library
    // from this point until stop_recording_operations() is called.
    const std::vector<ADNumberType> dof_values_ad
      = ad_helper.get_sensitive_dof_values();
    // Then we do some problem specific tasks, the first being to
    // compute all values, gradients, etc. based on sensitive AD DoF
    // values. Here we are fetching the displacement gradients at each
    // quadrature point.
    std::vector<Tensor<2, dim, ADNumberType>> Grad_u(
      n_q_points, Tensor<2, dim, ADNumberType>());
    fe_values[u_fe].get_function_gradients_from_local_dof_values(
      dof_values_ad, Grad_u);
    // This variable stores the cell total energy.
    // IMPORTANT: Note that it is hand-initialized with a value of
    // zero. This is a highly recommended practise, as some AD numbers
    // appear not to safely initialize their internal data structures.
    ADNumberType energy_ad = ADNumberType(0.0);
    // Compute the cell total energy = (internal + external) energies
    for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
    {
      // Calculate the deformation gradient at this quadrature point
      const Tensor<2, dim, ADNumberType> F =
        unit_symmetric_tensor<dim>() + Grad_u[q_point];
      Assert(numbers::value_is_greater_than(determinant(F), 0.0),
             ExcMessage("Negative determinant of the deformation "
                        "gradient detected!"));
      // Add contribution of the internal energy:
      // Integrate the stored energy density function with the current
      // solution.
      energy_ad += get_Psi(F) * fe_values.JxW(q_point);
    }
    // Add contribution from external energy:
    // Loop over faces and accumulate external energy into cell
    // total energy.
    for (unsigned int face : ...)
      if (cell->face(face)->at_boundary())
        energy_ad += ...
    // Register the definition of the total cell energy
    ad_helper.register_energy_functional(energy_ad);
    // Indicate that we have completed tracing the operations onto
    // the tape.
    ad_helper.stop_recording_operations(false); // write_tapes_to_file
  }
  else
  {
    // This is the "tape reuse" phase of the operations.
    // Here we will leverage the already traced operations that reside
    // on a tape, and simply re-evaluate the tape at a different point
    // to get the function values and their derivatives.
    // Load the existing tape to be reused
    ad_helper.activate_recorded_tape(tape_index);
    // Set the new values of the independent variables where the
    // recorded dependent functions are to be evaluated (and
    // differentiated around).
    ad_helper.set_dof_values(solution, local_dof_indices);
  }
  // Compute the residual values and their Jacobian at the
  // evaluation point
  ad_helper.compute_residual(cell_rhs);
  cell_rhs *= -1.0; // RHS = - residual
  ad_helper.compute_linearization(cell_matrix);
}

In most use cases, and in particular in the code example shown above, the number of independent variables equals the number of dofs_per_cell for the used finite element.

Warning: ADOL-C does not support the standard threading models used by deal.II, so this class should not be embedded within a multithreaded function when using ADOL-C number types. It is, however, suitable for use in both serial and MPI routines.

Author: Jean-Paul Pelteret, 2016, 2017, 2018

Definition at line 1224 of file ad_helpers.h.

Member Typedef Documentation

◆ scalar_type

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double>

using Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::scalar_type = typename HelperBase<ADNumberTypeCode, ScalarType>::scalar_type

Type definition for the floating point number type that is used in, and results from, all computations.

Definition at line 1232 of file ad_helpers.h.

◆ ad_type

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double>

using Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::ad_type = typename HelperBase<ADNumberTypeCode, ScalarType>::ad_type

Type definition for the auto-differentiation number type that is used in all computations.

Definition at line 1239 of file ad_helpers.h.

Constructor & Destructor Documentation

◆ EnergyFunctional()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType >

Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::EnergyFunctional ( const unsigned int n_independent_variables )

The constructor for the class.

Parameters

[in] n_independent_variables The number of independent variables that will be used in the definition of the functions that it is desired to compute the sensitivities of. In the computation of \(\Psi(\mathbf{X})\), this will be the number of inputs \(\mathbf{X}\), i.e., the dimension of the domain space.

Note: There is only one dependent variable associated with the total energy attributed to the local finite element. That is to say, this class assumes that the (local) right hand side and matrix contribution is computed from the first and second derivatives of a scalar function \(\Psi(\mathbf{X})\).

Definition at line 802 of file ad_helpers.cc.

◆ ~EnergyFunctional()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double>

virtual Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::~EnergyFunctional ( )

virtualdefault

Destructor

Member Function Documentation

◆ register_energy_functional()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType >

void Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::register_energy_functional ( const ad_type & energy )

Register the definition of the total cell energy \(\Psi(\mathbf{X})\).

Parameters

[in] energy A recorded function that defines the total cell energy. This represents the single dependent variable from which both the residual and its linearization are to be computed.

Note: For this class that expects only a single scalar dependent variable, this function must only be called once per tape.; For taped AD numbers, this operation is only valid in recording mode.

Definition at line 811 of file ad_helpers.cc.

◆ compute_energy()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType >

EnergyFunctional< ADNumberTypeCode, ScalarType >::scalar_type Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::compute_energy ( ) const

Evaluation of the total scalar energy functional for a chosen set of degree of freedom values, i.e.

\[ \Psi(\mathbf{X}) \vert_{\mathbf{X}} \]

The values at the evaluation point \(\mathbf{X}\) are obtained by calling CellLevelBase::set_dof_values().

Returns: The value of the energy functional at the evaluation point corresponding to a chosen set of local degree of freedom values.

Definition at line 823 of file ad_helpers.cc.

◆ compute_residual()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType >

void Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::compute_residual ( Vector< scalar_type > & residual ) const

overridevirtual

Evaluation of the residual for a chosen set of degree of freedom values. Underlying this is the computation of the gradient (first derivative) of the scalar function \(\Psi\) with respect to all independent variables, i.e.

\[ \mathbf{r}(\mathbf{X}) = \frac{\partial\Psi(\mathbf{X})}{\partial\mathbf{X}} \Big\vert_{\mathbf{X}} \]

The values at the evaluation point \(\mathbf{X}\) are obtained by calling CellLevelBase::set_dof_values().

Parameters

[out] residual A Vector object, for which the value for each entry represents the residual value for the corresponding local degree of freedom. The output residual vector has a length corresponding to n_independent_variables.

Implements Differentiation::AD::CellLevelBase< ADNumberTypeCode, ScalarType >.

Definition at line 877 of file ad_helpers.cc.

◆ compute_linearization()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType >

void Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >::compute_linearization ( FullMatrix< scalar_type > & linearization ) const

overridevirtual

Compute the linearization of the residual vector around a chosen set of degree of freedom values. Underlying this is the computation of the Hessian (second derivative) of the scalar function \(\Psi\) with respect to all independent variables, i.e.

\[ \frac{\partial\mathbf{r}(\mathbf{X})}{\partial\mathbf{X}} = \frac{\partial^{2}\Psi(\mathbf{X})}{\partial\mathbf{X} \otimes \partial\mathbf{X}} \Big\vert_{\mathbf{X}} \]

The values at the evaluation point \(\mathbf{X}\) are obtained by calling CellLevelBase::set_dof_values().

Parameters

[out] linearization A FullMatrix representing the linearization of the residual vector. The output linearization matrix has dimensions corresponding to n_independent_variables \(\times\)n_independent_variables.

Implements Differentiation::AD::CellLevelBase< ADNumberTypeCode, ScalarType >.

Definition at line 942 of file ad_helpers.cc.

The documentation for this class was generated from the following files:

deal.II/differentiation/ad/ad_helpers.h
source/differentiation/ad/ad_helpers.cc

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double> class Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >

Member Typedef Documentation

◆ scalar_type

◆ ad_type

Constructor & Destructor Documentation

◆ EnergyFunctional()

◆ ~EnergyFunctional()

Member Function Documentation

◆ register_energy_functional()

◆ compute_energy()

◆ compute_residual()

◆ compute_linearization()

template<enum AD::NumberTypes ADNumberTypeCode, typename ScalarType = double>
class Differentiation::AD::EnergyFunctional< ADNumberTypeCode, ScalarType >