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Merge pull request #46 from divyaadil23/Documentation_update
Add material model documentation to svMultiPhysics user guide
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‎documentation/multi_physics.html‎

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@@ -258,6 +258,10 @@ <h4 class="skipTo user-guide"> User Guide </h4>
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<p class="skipTo user_guide_computational_hemodynamics"> Computational Hemodynamics </p>
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<p class="skipTo user_guide_fluid_solid_interaction"> Fluid Solid Interaction </p>
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<p class="skipTo user_guide_material_models"> Material Models </p>
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<div>
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<p class="skipTo user_guide_material_models"> List of Material Models </p>
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<p class="skipTo references "> References </p>
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</div>
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<p class="skipTo user_guide_nonlinear_solid_dynamics"> Nonlinear Solid Dynamics </p>
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</div>
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<zero-md src="multi_physics/user-guide/material_models/introduction/readme.md" no-shadow></zero-md>
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</span>
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<span id="user_guide_material_models">
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<zero-md src="multi_physics/user-guide/material_models/material_models_list/readme.md" no-shadow></zero-md>
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</span>
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<span id="user_guide_material_models">
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<zero-md src="multi_physics/user-guide/material_models/references/readme.md" no-shadow></zero-md>
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</span>
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<!-- -------------------- nonlinear solid dynamics -------------------- -->
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<span id="user_guide_nonlinear_solid_dynamics">

‎documentation/multi_physics/user-guide/material_models/introduction/readme.md‎

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<h2 id ="user_guide_material_models"> Material Models </h2>
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svMultiPhysics supports a number of material models that can be used for simulations of complex solid mechanics.
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The ALE formulation that svMultiPhysics uses (see section on FSI) solves for displacements and stresses in the solid domain. Different materials have different relationships between the stress and the displacement (or strain) which are represented by material models (also known as constitutive models). These material models are required to solve the system of equations for solids.
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One commonly known material model is the linear elastic model, where the Cauchy stress tensor is related to the small strain tensor through a fourth order stiffness tensor. In 1D, the stress and strain are proportional and the constant of proportionality is known as the Young’s modulus.
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$$\boldsymbol{\sigma} = \mathbb{C}\boldsymbol{\epsilon}$$
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where
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$$\boldsymbol{\epsilon} = \frac{1}{2} [\nabla \mathbf{u} + (\nabla \mathbf{u})^T]$$
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This relationship holds well for metals which experience small deformations. Soft biological tissues on the other hand undergo large nonlinear deformations and are better represented by a class of material models called hyperelastic models. The passive material behavior for hyperelastic materials can be described through the strain energy function $\Psi$. Various stress measures can be obtained from the strain energy function by taking a tensor derivative <a href="#ref-2_derive_stress_elasticity">[2]</a>. svMultiPhysics uses the 2nd Piola Kirchhoff Stress $\mathbf{S}$.
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$$ \mathbf{S} = 2\frac{\partial \Psi}{\partial \mathbf{C}} $$
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where $\mathbf{C}$ is the right Cauchy-green tensor. The general constitutive relation for these materials can be written as
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$$ \mathbf{S} = \mathbb{C}:\mathbf{E}$$
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where $\mathbf{E}$ is the Green-Lagrange strain tensor. The small strain tensor used for linear elasticity is a linearized form of this strain tensor.
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The strain energy function consists of two parts - the isochoric (volume preserving) and the volumetric parts.
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$$ \Psi = \Psi_{vol} (J) + \Psi_{iso} (\bar{\mathbf{C}})$$
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Biological materials are rubber-like and are modelled as incompressible or nearly incompressible, and this decomposition is helpful for such cases.
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Tissues have extremely complex microstructure due to which we rely on phenomenological material models. svMultiPhysics has a number of options available for material models suitable for different applications.
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‎documentation/multi_physics/user-guide/material_models/material_models_list/readme.md‎

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<h3 id ="user_guide_material_models"> List of Available Hyperelastic Models </h3>
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Volumetric constitutive models for struct/ustruct equations:
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<table class="table table-bordered" style="width:100%">
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<tr>
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<th> Volumetric Model </th>
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<th> Input Keyword </th>
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</tr>
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<tr>
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<td> Quadratic model </td>
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<td> "quad", "Quad", "quadratic", "Quadratic" </td>
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</tr>
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<tr>
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<td> Simo-Taylor91 model </td>
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<td> "ST91", "Simo-Taylor91" </td>
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</tr>
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<tr>
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<td> Miehe94 model </td>
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<td> "M94", "Miehe94" </td>
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</tr>
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</table>
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Isochoric constitutive models for struct/ustruct equations.
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<table class="table table-bordered" style="width:100%">
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<tr>
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<th> Isochoric Model </th>
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<th> Input Keyword </th>
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</tr>
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<tr>
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<td> Saint Venant-Kirchhoff &dagger; </td>
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<td> "stVK", "stVenantKirchhoff" </td>
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</tr>
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<tr>
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<td> Neo-Hookean model </td>
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<td> "nHK", "nHK91", "neoHookean", "neoHookeanSimo91" </td>
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</tr>
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<tr>
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<td> Holzapfel-Gasser-Ogden model <a href="#ref-3_hgo">[3]</a></td>
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<td> "HGO" </td>
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</tr>
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<tr>
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<td> Guccione model <a href="#ref-4_guccione">[4]</a></td>
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<td> "Guccione", "Gucci" </td>
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</tr>
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<tr>
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<td> Holzapfel-Ogden model <a href="#ref-5_ho">[5]</a></td>
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<td> "HO", "HolzapfelOgden" </td>
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</tr>
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<tr>
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<td> Holzapfel-Ogden Modified Anisotropy model<a href="#ref-6_ho-ma">[6]</a> </td>
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<td> “HO_ma”, “HolzapfelOgden-ModifiedAnisotropy” </td>
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</tr>
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</table>
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&dagger; : These models are not available for ustruct.
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svMultiPhysics has two options for solving the solid equations - struct and ustruct. “Struct” uses a displacement based formulation i.e. the unknowns that we are solving for in each element are displacements. “Ustruct” uses a mixed formulation where the unknowns are displacements and pressures.<a href="#ref-1_ustruct_formulation">[1]</a>
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Add_equation</strong> type="<i>struct</i>"&gt; <span style="color: #888">// or "ustruct"</span><br>
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&nbsp;&nbsp;&lt;<strong>Coupled</strong>&gt; <i>true</i> &lt;/<strong>Coupled</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>Min_iterations</strong>&gt; <i>1</i> &lt;/<strong>Min_iterations</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>Max_iterations</strong>&gt; <i>3</i> &lt;/<strong>Max_iterations</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>Tolerance</strong>&gt; <i>1e-9</i> &lt;/<strong>Tolerance</strong>&gt;<br><br>
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<span style="color: #888">/*<br>
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&nbsp;&nbsp;Add constitutive model, output type, solver type, boundary conditions<br>
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*/</span><br><br>
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&lt;/<strong>Add_equation</strong>&gt;
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</div>
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Volumetric Models: These models set the volumetric part of the strain energy function. There is only one material parameter needed in the input file to define this term.
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For a displacement based formulation (“struct”), the volumetric part of the strain energy function is a penalty to allow for small amounts of compressibility (models the material as nearly incompressible).
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$$ \Psi_{vol} = K_p G(J) $$
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where $ K_p$ can be interpreted as the bulk modulus. $G(J)$ is the penalty function and takes different forms depending on the type of model. Two parameters are p and pl are defined internally to add to the stresses and elasticity tensors. “Struct” , the displacement based formulation calculates these as:
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$$ p = \frac{\partial \Psi_{vol}}{\partial J}$$
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$$ pl = p + J\frac{dp}{dJ}$$
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The mixed displacement-pressure formulation does not calculate for p and pl this way. Instead, they are solved along with the displacements.
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**Quadratic Model:**
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$$ G(J) = \frac{1}{2} (J-1)^2 $$
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$$p = K_p (J -1) $$
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$$ pl = K_p (2J - 1) $$
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**Simo-Taylor91 Model:**
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$$ G(J) = \frac{1}{4}(J^2 - 2 ln(J)) $$
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$$p = \frac{1}{2} K_p (J -\frac{1}{J}) $$
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$$ pl = K_p J $$
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**Miehe94 Model:**
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$$ G(J) = J - ln(J) $$
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$$p = K_p (1 - \frac{1}{J}) $$
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$$ pl = K_p$$
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So, if using “struct”, this is how you would input the volumetric model:
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Dilational_penalty_model</strong>&gt; <i>ST91</i> &lt;/<strong>Dilational_penalty_model</strong>&gt;<br>
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&lt;<strong>Penalty_parameter</strong>&gt; <i>4.0E9</i> &lt;/<strong>Penalty_parameter</strong>&gt;
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</div>
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For “ustruct”:
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Dilational_penalty_model</strong>&gt; <i>ST91</i> &lt;/<strong>Dilational_penalty_model</strong>&gt;
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</div>
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Isochoric Models:
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**Saint Venant-Kirchhoff**
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This model is an extension of the linear elastic model with the strain energy postulated as a quadratic function of the Green-Lagrange strain tensor. It is an isotropic material model.
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$$\Psi_{iso} = \frac{\lambda}{2} tr(\mathbf{E})^2 + \mu tr(\mathbf{E}^2) $$
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where $\lambda$ and $\mu$ are Lamé constants.
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In the code (file set_material_props.h),
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$$ C_{10} = \lambda $$
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$$ C_{01} = \mu $$
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Since these parameters are set automatically, we only need to specify the constitutive model type.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="stVK"</i>&gt; &lt;/<strong>Constitutive_model</strong>&gt;
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</div>
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The 2nd Piola-Kirchoff stress is given by
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$$ \mathbf{S} = \lambda tr(\mathbf{E}) \mathbf{I} + 2\mu \mathbf{E}$$
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**NOTE:** To modify the Lamé constants for any model that uses default parameters, we do it through specifying the elasticity modulus $E$ and poisson’s ratio $\nu$.
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$$ \mu = \frac{E}{2(1+\nu)} $$
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$$ \lambda = \frac{E \nu}{(1+\nu)(1-2\nu)} $$
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The bulk modulus $\kappa$ is given by
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$$ \kappa = \frac{E}{3(1-2\nu)} $$
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$\lambda$ and $\kappa$ are set to zero if the material is incompressible, i.e. $\nu=0.5$.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Elasticity_modulus</strong>&gt; <i>240.56596e6</i> &lt;/<strong>Elasticity_modulus</strong>&gt;<br>
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&lt;<strong>Poisson_ratio</strong>&gt; <i>0.4999999</i> &lt;/<strong>Poisson_ratio</strong>&gt;
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</div>
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**Neo-Hookean model**
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$$ \Psi_{iso} = C_{10} (\bar{I}_1 - 3) $$
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<div class="nhk">
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&lt;<strong>Constitutive_model</strong> <i>type="neoHookean"</i> &gt; &lt;/<strong>Constitutive_model</strong>&gt;
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</div>
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The parameter $ C_{10}$ is automatically set (file set_material_props.h):
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$$ C_{10} = \frac{\mu}{2} $$
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**Holzapfel-Gasser-Ogden model**
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$$
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\Psi_{aniso} = \frac{a_4}{b_4} \left( \exp\left( b_4\left( \kappa \bar{I}_1 + (1-3\kappa)\bar{I}_4 - 1\right)^2 \right) - 1 \right) + \frac{a_6}{b_6} \left( \exp\left( b_6\left( \kappa \bar{I}_1 + (1-3\kappa)\bar{I}_6 - 1 \right)^2 \right) - 1 \right)
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$$
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="HGO"</i>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a4</strong>&gt; <i>9.966e5</i> &lt;/<strong>a4</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b4</strong>&gt; <i>524.6</i> &lt;/<strong>b4</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a6</strong>&gt; <i>9.966e5</i> &lt;/<strong>a6</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b6</strong>&gt; <i>524.6</i> &lt;/<strong>b6</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>kappa</strong>&gt; <i>0.1</i> &lt;/<strong>kappa</strong>&gt;<br>
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&lt;/<strong>Constitutive_model</strong>&gt;
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</div>
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The isotropic part is the same as neoHookean - the parameters are automatically assigned from elasticity modulus and poisson’s ratio.
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Apart from this, need to add fiber direction file path under Add_mesh for the solid domain:
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Fiber_direction_file_path</strong>&gt; <i>mesh/fibersLong1.vtu</i> &lt;/<strong>Fiber_direction_file_path</strong>&gt;<br>
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&lt;<strong>Fiber_direction_file_path</strong>&gt; <i>mesh/fibersLong2.vtu</i> &lt;/<strong>Fiber_direction_file_path</strong>&gt;
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</div>
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**Guccione model**
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$$ \Psi = \frac{c}{2} \left( \exp\left( Q(\bar{\mathbf{E}}) \right) - 1 \right) $$
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where $\bar{\mathbf{E}}$ is the local Green-Lagrange strain tensor, and
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<p>
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$$
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Q(\bar{\mathbf{E}}) = b_{ff} \left( \bar{E}_{ff} \right)^2 + b_{ss} \left( \bar{E}_{ss}^2 + \bar{E}_{nn}^2 + \bar{E}_{sn}^2 \right) + 2b_{fs} \left( \bar{E}_{fs}^2 + \bar{E}_{fn}^2 \right)
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$$
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</p>
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In the code, $b_f = b_{ff}$ and $b_t = b_{ss}$.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="Gucci"</i>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>c</strong>&gt; <i>880</i> &lt;/<strong>c</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>bf</strong>&gt; <i>8</i> &lt;/<strong>bf</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>bt</strong>&gt; <i>6</i> &lt;/<strong>bt</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>bfs</strong>&gt; <i>12</i> &lt;/<strong>bfs</strong>&gt;<br>
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&lt;/<strong>Constitutive_model</strong>&gt;
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</div>
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**Holzapfel-Ogden model**
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<p>
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\(
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\Psi_{\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_1 - 3) \right)
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+ \sum_{i \in \{f,s\}} \frac{a_i}{2b_i} \, \chi(\bar{I}_{4i})
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\left( \exp\left( b_i (\bar{I}_{4i} - 1)^2 \right) - 1 \right)
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+ \frac{a_{fs}}{2b_{fs}} \left( \exp\left( b_{fs} \bar{I}_{8fs}^2 \right) - 1 \right)
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\)
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</p>
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where $\chi (\eta)$ is the smoother heaviside function defined as
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$$ \chi(\eta) = \frac{1}{1 + exp\{ -k_{\chi} (\eta -1)\} } $$
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The heaviside function is multiplied as a switching function to turn off the fibers during contraction. This is useful for modeling collagen in cardiac mechanics for example which does not support contraction.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="HolzapfelOgden"</i>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a</strong>&gt; <i>590.0</i> &lt;/<strong>a</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b</strong>&gt; <i>8.023</i> &lt;/<strong>b</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a4f</strong>&gt; <i>184720.0</i> &lt;/<strong>a4f</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b4f</strong>&gt; <i>16.026</i> &lt;/<strong>b4f</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a4s</strong>&gt; <i>24810.0</i> &lt;/<strong>a4s</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b4s</strong>&gt; <i>11.12</i> &lt;/<strong>b4s</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>afs</strong>&gt; <i>2160.0</i> &lt;/<strong>afs</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>bfs</strong>&gt; <i>11.436</i> &lt;/<strong>bfs</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>k</strong>&gt; <i>100.0</i> &lt;/<strong>k</strong>&gt;<br>
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&lt;/<strong>Constitutive_model</strong>&gt;
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</div>
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**Holzapfel-Ogden Modified Anisotropy model**
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This model is very similar to the Holzapfel Ogden model - the only difference is the use of full invariants instead of isochoric.
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<p>
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\[
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\Psi_{\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_1 - 3) \right)
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+ \sum_{i \in \{f,s\}} \frac{a_i}{2b_i} \, \chi(I_{4i})
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\left( \exp\left( b_i (I_{4i} - 1)^2 \right) - 1 \right)
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+ \frac{a_{fs}}{2b_{fs}} \left( \exp\left( b_{fs} I_{8fs}^2 \right) - 1 \right)
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\]
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</p>
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where f and s are the fiber and sheet directions and the smoothed heaviside function is:
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$$ \chi(\eta) = \frac{1}{1 + exp\{ -k_{\chi} (\eta -1)\} } $$
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="HolzapfelOgden-ModifiedAnisotropy"</i>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a</strong>&gt; <i>590.0</i> &lt;/<strong>a</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b</strong>&gt; <i>8.023</i> &lt;/<strong>b</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a4f</strong>&gt; <i>184720.0</i> &lt;/<strong>a4f</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b4f</strong>&gt; <i>16.026</i> &lt;/<strong>b4f</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>a4s</strong>&gt; <i>24810.0</i> &lt;/<strong>a4s</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>b4s</strong>&gt; <i>11.12</i> &lt;/<strong>b4s</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>afs</strong>&gt; <i>2160.0</i> &lt;/<strong>afs</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>bfs</strong>&gt; <i>11.436</i> &lt;/<strong>bfs</strong>&gt;<br>
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&nbsp;&nbsp;&lt;<strong>k</strong>&gt; <i>100.0</i> &lt;/<strong>k</strong>&gt;<br>
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&lt;/<strong>Constitutive_model</strong>&gt;
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</div>

‎documentation/multi_physics/user-guide/material_models/references/readme.md‎

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## References
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<p><a id="ref-1_ustruct_formulation"> <a href="https://doi.org/10.1016/j.cma.2018.03.045">
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[1] Ju Liu, Alison L. Marsden. <strong>A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction.</strong> Computer Methods in Applied Mechanics and Engineering; 1 August 2018. </a></p>
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<p><a id="ref-2_derive_stress_elasticity"> <a href="https://doi.org/10.1142/S0219876218500287">
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[2] Jie Cheng and Lucy T. Zhang. <strong>A General Approach to Derive Stress and Elasticity Tensors for Hyperelastic Isotropic and Anisotropic Biomaterials</strong>. International Journal of Computational Methods; Vol. 15, No. 04, 1850028 (2018).</a></a></p>
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