# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 2 Materials and Methods
### Multiscale Modeling Framework
If a load factor is too large and the Newton-Rhapson algorithm diverges, we appropriately scale the load factor down. To proceed with computational homogenization, we begin with (4), the weak form statement of the FE problem. The integrals in (4) are computed via numerical quadrature. At each quadrature point, the macroscopic \(\mathbf{P}\) is generally obtained from a constitutive model (for hyperelastic materials, viz. (2)). In computational homogenization, or FE\({}^{2}\), the macroscopic stress, now denoted \(\mathbf{P}^{M}\), is obtained from an embedded FE problem representative of the microstructure of the material, referred to as the representative volume element (RVE). The macroscopic deformation gradient, now denoted as \(F^{M}\), is passed from the macroscale model to the RVE (with the domain given by \(\Omega\)) as a constraint condition. Then, the energy of the RVE is minimized (i.e., an FE solution is obtained for the RVE) subject to the condition that the volume average of the pointwise RVE deformation gradient, \(\mathbf{F}^{m}\), is equal to the imposed macroscopic deformation gradient \(\mathbf{F}^{M}\). This procedure is depicted in \[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]1 The resulting stress and tangent stiffness are computed from the Hill-Mandel principle:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]2  
where, as introduced, the superscript \(M\) denotes macroscale quantities and \(m\) denotes microscale quantities. \[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]3 Computational homogenization, i.e., FE\({}^{2}\). The macro problem passes the deformation gradient \(\mathbf{F}^{M}\) to the RVE (micro problem), where (1) is solved. The homogenized stress and tangent (as given by (15) and (18)) are then computed and passed up to the macro problem. The process is typically iterated as in (7). This graphic was adapted from [PERSON] [10]. We now outline the homogenization procedure (see [PERSON] [10] for a comprehensive treatment). We begin by imposing \(\mathbf{F}^{M}\) on \(\partial\Omega\), the boundary of the representative volume element (RVE). We partition the displacements accordingly:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]4  
We refer to \(\mathbf{u}_{f}\) as the free displacements in the RVE domain and:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]5  
The free displacements are computed by satisfying (1) for the RVE, under the assumption of an appropriate material description (e.g., hyperelasticity), using a standard FE procedure as in Section 2.1. One interesting extension to the material description is a two-stage FE\({}^{2}\) procedure with a \"nanoscale\" RVE. The application of (12) leads to an expression for the homogenized stress:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]6  
We then turn to an expression for the homogenized tangent stiffness. We begin by forming the Schur complement of the partitioned tangent stiffness as given by (13)  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]7  
We define the order-4 tangent stiffness \(\mathbf{A}^{M}\) that satisfies the variational inner product:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]8  
It is not difficult to show that the application of (12) reveals:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]9  
where \(n_{d}\) is the number of boundary nodes, and thus \(\mathbf{K}_{c}^{AB}\) refers to the 3 degrees of freedom at nodes \(A,B\) leading to \(\mathbf{A}^{M}\in\mathbb{R}^{3\times 3\times 3\times 3}\). The superscript \(L\) denotes the left conjugation of a 4-tensor \((D_{ijkl})^{L}=D_{jkl}\). RVE Boundary Conditions  
The Dirichlet boundary condition specified in (14) is not unique. In fact, any boundary conditions that are compatible with (12) are feasible.