# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 2 Materials and Methods
### Multiscale Modeling Framework
Equation (1) is solved by satisfying the weak form statement:  
Find:  
\[\mathbf{\varphi}\in\mathcal{S}:=\{\mathbf{\varphi}\mid\mathbf{\varphi}=\bar{\mathbf{\varphi}} \text{ on }\partial\mathcal{B}_{u}\},\]  
such that:  
\[\int_{\mathcal{B}_{0}}\mathbf{P}\cdot\
abla(\delta\mathbf{\varphi})\;dV=\int_{ \mathcal{B}_{0}^{-}}\!\!\mathbf{B}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\(g\) denote discretized parameters, and \(N^{A}(\mathbf{x})\) denote interpolating shape functions at each node, we construct a Galerkin discretization as  
\[\mathbf{B}_{0}^{h}=\mathbf{\sum_{e=1}^{n_{el}}}\mathbf{B}_{0}^{e},\quad\mathbf{u}^{g}=\sum_{A=1} ^{n_{n}}N^{A}\mathbf{u}_{A},\quad\delta\mathbf{u}^{g}=\sum_{A=1}^{n_{n}}N^{A}\delta\mathbf{u} _{A}, \tag{5}\]  
where \(e\) indexes the \(n_{el}\) elements in the domain, \(\mathbf{u}\) denotes displacements, \(\delta\mathbf{u}\) denotes variational displacements, \(\mathbf{u}_{A}\) denotes nodal displacements indexed by \(A\) over \(n_{n}\) nodes in the tessellation, and \(\mathbf{\sum_{e=1}^{n_{el}}}\) is the assembly operator [33]. Substituting (5) into (4) leads to the nonlinear (static) equilibrium equations:  
\[\mathbf{R}(\mathbf{u}_{t})=\mathbf{f}_{t}-\mathbf{\sum_{e=1}^{n_{el}}}\int_{\mathbf{B}_{0}^{e}} \
abla\mathbf{N}_{e}^{T}\mathbf{P}_{e}\ dV_{e}=0, \tag{6}\]  
where \(\mathbf{R}\) is the residual for a state of displacements \(\mathbf{u}_{t}\) at time \(t\), which must be in equilibrium with the applied nodal forces \(\mathbf{f}_{t}\) at time \(t\), and \(\
abla\mathbf{N}\) is the matrix formed from derivatives of the shape functions \(N^{A}(\mathbf{X})\) with respect to \(\mathbf{X}\). The reader is referred to [PERSON] and [PERSON] [33] for a comprehensive treatment of the FE procedure. We use an iterative Newton-Rhapson approach to solve (6). Given an initial state \(\mathbf{u}_{t}^{0}\), the updated equations are:  
\[\mathbf{u}_{t}^{k+1}\leftarrow\mathbf{u}_{t}^{k}-K_{T}^{-1}(\mathbf{u}_{t}^{k})\mathbf{f}_{t}, \tag{7}\]  
where:  
\[\mathbf{K}_{T}=\frac{\partial\mathbf{R}}{\partial\mathbf{u}}=\mathbf{\sum_{e=1}^{n_{el}}}(k_{ e,\text{mat}}+\mathbf{k}_{e,\text{gen}}), \tag{8}\]  
is the linearized tangent stiffness. The element material stiffness is:  
\[\mathbf{k}_{e,\text{mat}}=\int_{\mathbf{B}_{e}}\bar{\
abla}\mathbf{N}_{e}^{T}\mathbf{c}\bar{ \
abla}\mathbf{N}_{e}\ dV, \tag{9}\]  
where \(\bar{\
abla}\) is the gradient operator with respect to the spatial manifold, and \(\mathbf{c}\) is the spatial material tangent, defined as  
\[c_{ijkl}=\frac{1}{J}F_{iA}F_{jB}F_{kC}F_{lD}\mathbb{C}_{ABCD}, \tag{10}\]  
the push-forward of the material tangent \(\mathbb{C}=2\partial\mathbf{S}/\partial\mathbb{C}\). The element geometric stiffness is:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]0  
The summation convention is implied throughout, with lower-case subscripts indicating the spatial coordinates, subscript commas indicating partial differentiation, upper-case subscripts indicating the reference coordinates, and upper-case _super_scripts indicating nodal numbers. Note that the integrals for the stiffnesses are taken over the deformed element. The iterations are carried out until a stopping criterion, such as the satisfaction of (6), is achieved within some tolerance. For the Newton-Rhapson strategy to converge, the initial guess must be in the neighborhood of the solution. This requirement poses an issue for the highly nonlinear AV tissue, particularly in the low stiffness regime. To address this problem, we incrementally and adaptively apply the load. We start with a small load factor \(\alpha_{t}\) (\(\mathbf{f}_{t}=\alpha_{t}\mathbf{f}_{0}\)) and adjust the factor heuristically based on the number of iterations (\(n_{i}\)) it takes for (7) to converge (\(\alpha_{t}\propto n_{i}^{-1}\)). In this manner, we are able to circumvent the use of unreasonably small load factors (i.e., excessive computational time) during the entire load path.