- E. Comellas, S. Budday, J.-P. Pelteret, G. A. Holzapfel and P. Steinmann (2020), Modeling the porous and viscous responses of human brain tissue behavior, Computer Methods in Applied Mechanics and Engineering, 113128. DOI: [10.1016/j.cma.2020.113128](https://doi.org/10.1016/j.cma.2020.113128).

- Biphasic material following the Theory of Porous Media
- Nonlinear finite viscoelasticity built on Ogden hyperelasticity
- Darcy-like fluid flow
- Spatial discretisation with continuous Q2-P1 Lagrangian finite elements
- Temporal discretisation with a stable implicit one-step backward differentiation method
- Newton-Raphson scheme to solve the nonlinear system of governing equations
- Forward mode automatic differentiation with the number of derivative components chosen at run-time (Sacado library within Trilinos package) to linearise the governing equations (and, implicitly, the constitutive laws)
- Trilinos direct solver for the (non-symmetric) linear system of equations using a monolithic scheme
- Parallelization through Threading Building Blocks and across nodes via MPI (using Trilinos linear algebra)
- Based on step-44 and the code gallery contributions 'Quasi-Static Finite-Strain Compressible Elasticity' and 'Quasi-Static Finite-Strain Quasi-incompressible Visco-elasticity'
- Only works in 3D

- MPI and Trilinos (built with the Sacado library) must be enabled

- W. Ehlers and G. Eipper (1999), Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids, Transport in Porous Media 34(1/3):179-191. DOI: [10.1023/A:1006565509095](https://doi.org/10.1023/A:1006565509095).
- S. Reese and S. Govindjee (2001), A theory of finite viscoelasticity and numerical aspects, International Journal of Solids and Structures 35(26-27):3455-3482. DOI: [10.1016/S0020-7683(97)00217-5](https://doi.org/10.1016/S0020-7683(97)00217-5).
- G. Franceschini, D. Bigoni, P. Regitnig and G. A. Holzapfel (2006), Brain tissue deforms similarly to filled elastomers and follows consolidation theory, Journal of the Mechanics and Physics of Solids 54(12):2592-2620. DOI: [10.1016/j.jmps.2006.05.004](https://doi.org/10.1016/j.jmps.2006.05.004).
- S. Budday, G. Sommer, J. Haybaeck, P. Steinmann, G. A. Holzapfel and E. Kuhl (2017), Rheological characterization of human brain tissue, Acta Biomaterialia 60:315-329. DOI: [10.1016/j.actbio.2017.06.024](https://doi.org/10.1016/j.actbio.2017.06.024).
- G.A. Holzapfel (2001), Nonlinear Solid Mechanics. A Continuum Approach for Engineering, John Wiley & Sons. ISBN: [978-0-471-82319-3](http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471823198.html).

- First Lamé parameter: 334kPa
- Ogden parameters: μ
_{1}=1.044kPa, μ_{2}=1.183kPa, α_{1}=4.309, and α_{2}=7.736 - Initial solid volume fraction: 0.80
- Isotropic initial intrinsic permeability: 8.0e-11mm
^{2} - Deformation-dependency control parameter for the permeability: 40
- Fluid viscosity: 0.89mPa·s

- First Lamé parameter: 24.5kPa
- Ogden hyperelastic parameters: μ
_{∞,1}=-83.9Pa, and α_{∞,1}=-11.92 - Ogden viscous parameters: μ
_{1}=-2100Pa, and α_{1}=-2.2 - Deformation-independent viscosity of the solid: 14kPa·s
- Initial solid volume fraction: 0.75
- Isotropic initial intrinsic permeability: 1.0e-8mm
^{2} - Deformation-dependency control parameter for the permeability: 50
- Fluid viscosity: 0.89mPa·s

- E. Comellas, J.-P. Pelteret, S. Budday and P. Steinmann (2018). Unsolved issues in the numerical modelling of experimentally-observed porous effects in brain tissue behaviour, 6th European Conference on Computational Mechanics and 7th European Conference on Computational Fluid Dynamics (ECCM-ECFD 2018), Glasgow (UK), 11th–15th June 2018. DOI: [10.13140/RG.2.2.18553.29283](https://doi.org/10.13140/RG.2.2.18553.29283).