From 845ebe7d3dc9a54108333865ef30f67c088a68c9 Mon Sep 17 00:00:00 2001 From: Ester Comellas <29859757+ecomellas@users.noreply.github.com> Date: Tue, 7 Jul 2020 11:48:27 +0200 Subject: [PATCH] Update readme.md Fix formatting of lists. Changed from markdown to html. --- Nonlinear_PoroViscoelasticity/readme.md | 103 ++++++++++++------------ 1 file changed, 51 insertions(+), 52 deletions(-) diff --git a/Nonlinear_PoroViscoelasticity/readme.md b/Nonlinear_PoroViscoelasticity/readme.md index 372c96d..133fe56 100644 --- a/Nonlinear_PoroViscoelasticity/readme.md +++ b/Nonlinear_PoroViscoelasticity/readme.md @@ -3,37 +3,41 @@ Readme file for nonlinear-poro-viscoelasticity Overview -------- - We implemented a nonlinear poro-viscoelastic formulation with the aim of characterising brain tissue response to cyclic loading. Our model captures both experimentally observed fluid flow and conditioning aspects of brain tissue behavior in addition to its well-established nonlinear, preconditioning, hysteretic, and tension-compression asymmetric characteristics. The tissue is modelled as a biphasic material consisting of an immiscible aggregate of a nonlinear viscoelastic solid skeleton saturated with pore fluid. The governing equations are linearised using automatic differentiation and solved monolithically for the unknown solid displacements and fluid pore pressure values. A detailed description of the formulation, its verification, and the results obtained can be found in: -* 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); + In this paper we show that nonlinear poroelasticity alone can reproduce consolidation experiments, yet it is insufficient to capture stress conditioning due to cyclic loading. We also discuss how the poroelastic response exhibits preconditioning and hysteresis in the fluid flow space, with porous and viscous effects being highly interrelated. Quick facts about the code -------------------------- - -* 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 + Running the code ---------------- ### Requirements -* MPI and Trilinos (built with the Sacado library) must be enabled + ### Compiling and running Similar to the example programs, run @@ -69,22 +73,20 @@ python run-multi-calc.py The 'run-multi-calc.py' and 'runPoro.sh' files provided must both be in the main directory. This will automatically generate the required input files and run them in sequence. - Reference for this work ----------------------- - If you use this program as a basis for your own work, please consider citing the paper referenced in the introduction. The initial version of this work was contributed to the deal.II project by E. Comellas and J-P. Pelteret. - Recommended literature ---------------------- - -* 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); + Results @@ -92,29 +94,25 @@ Results The results shown here are a selection of those presented and discussed in the paper referenced in the introduction. ### Consolidation experiments -We reproduce the uniaxial consolidation experiments of brain tissue from the seminal paper by Franceschini et al. (2006) using a reduction of our formulation to nonlinear poroelasticity without the viscous component. - -The following geometry (320 cells and 9544 degrees of freedom), boundary conditions, and loading are used: +We reproduce the uniaxial consolidation experiments of brain tissue from the seminal paper by Franceschini et al. (2006) using a reduction of our formulation to nonlinear poroelasticity without the viscous component. The following geometry (320 cells and 9544 degrees of freedom), boundary conditions, and loading are used: ![Consolidation geometry](./doc/images-results/consolidation-mesh.png) The material properties are: -* First Lamé parameter: 334kPa -* Ogden parameters: μ1=1.044kPa, μ2=1.183kPa, α1=4.309, and α2=7.736 -* Initial solid volume fraction: 0.80 -* Isotropic initial intrinsic permeability: 8.0e-11mm2 -* Deformation-dependency control parameter for the permeability: 40 -* Fluid viscosity: 0.89mPa·s - -We consider the effect of gravity in our formulation, with the simplifying assumption that the fluid and solid density are both 0.997mg/mm3. + -Using these parameters and simulation conditions, we obtain a reasonable fit to the experimental curve: +We consider the effect of gravity in our formulation, with the simplifying assumption that the fluid and solid density are both 0.997mg/mm3. Using these parameters and simulation conditions, we obtain a reasonable fit to the experimental curve: ![Consolidation results](./doc/images-results/Franceschini-consolidation.png) -What is interesting about these results is that they show that nonlinear poroelasticity alone, i.e. with a purely hyperelastic solid component, can capture the consolidation behavior of brain tissue in response to an [oedometric test](https://en.wikipedia.org/wiki/Oedometer_test). - -However, as we will see in the next section, the viscous component is necessary to also capture stress conditioning due to cyclic loading. +What is interesting about these results is that they show that nonlinear poroelasticity alone, i.e. with a purely hyperelastic solid component, can capture the consolidation behavior of brain tissue in response to an [oedometric test](https://en.wikipedia.org/wiki/Oedometer_test). However, as we will see in the next section, the viscous component is necessary to also capture stress conditioning due to cyclic loading. ### Cyclic experiments under multiple loading modes @@ -125,14 +123,16 @@ As an example, we show here the results for cyclic compressive loading. The foll ![Cyclic loading geometry](./doc/images-results/cyclic-loading-mesh.png) The material properties are: -* First Lamé parameter: 24.5kPa -* Ogden hyperelastic parameters: μ∞,1=-83.9Pa, and α∞,1=-11.92 -* Ogden viscous parameters: μ1=-2100Pa, and α1=-2.2 -* Deformation-independent viscosity of the solid: 14kPa·s -* Initial solid volume fraction: 0.75 -* Isotropic initial intrinsic permeability: 1.0e-8mm2 -* Deformation-dependency control parameter for the permeability: 50 -* Fluid viscosity: 0.89mPa·s + To simplify the problem, we neglect the effect of gravity for this example. Here is an animation of our results, visualised with Paraview: @@ -144,11 +144,10 @@ To compare with the experimental results, our code computes the nominal stress o ![Cyclic loading poroelastic results](./doc/images-results/brain_cube_stress-stretch-poro-COMP.png) ![Cyclic loading poro-viscoelastic results](./doc/images-results/brain_cube_stress-stretch-poro-visco-COMP.png) -We see that viscosity is required in the solid component to reproduce the preconditioning and hysteretic response seen in the experiments. - -We were somewhat surprised by these results, because we were expecting to see preconditioning and hysteresis with poroelasticity alone. We discussed this in a conference talk: - -* 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); +We see that viscosity is required in the solid component to reproduce the preconditioning and hysteretic response seen in the experiments. We were somewhat surprised by these results, because we were expecting to see preconditioning and hysteresis with poroelasticity alone. We discussed this in a conference talk: + So, we set out to explore why this is not the case. For that, we studied the porous and viscous dissipation, which are computed in the code and also provided in the "data-for-gnuplot.sol" output file. We determined that there is dissipation occurring in the poroelastic case, but we were barely seeing it in the stress-stretch plot. Even if we played around with the material parameters to increase the amount of dissipation, the slight hysteresis in our plot barely changed. *Where is all that energy going?* -- 2.39.5