This section summarizes the formulations that are required to model the THV, which is composed of the frame, skirt, and leaflets, and the artery wall. Based on the thickness of the THV geometries relative to the overall dimensions of the system, we model the artery wall as an elastic solid and the valve leaflets and skirt as thin shell structures. The frame is fabricated with long, thin wires that are modeled using elastic beams. The following superscripts “be”, “sh”, and “so” denote the beam, shell, and solid, respectively, where
\(\mathcal {S}_y = \mathcal {S}_y^\text { be}\times \mathcal {S}_y^\text { sh}\times \mathcal {S}_y^\text { so}\) and
\(\mathcal {V}_y = \mathcal {V}_y^\text { be}\times \mathcal {V}_y^\text { sh}\times \mathcal {V}_y^\text { so}\) such that
\(\textbf{y} = \left\{ \textbf{y}^\text {be}, \textbf{y}^\text {sh}, \textbf{y}^\text {so}\right\}\) and
\(\textbf{w}_\text {s} = \left\{ \textbf{w}^\text {be}_\text {s}, \textbf{w}^\text {sh}_\text {s}, \textbf{w}^\text {so}_\text {s}\right\}\). We can then write
$$\begin{aligned} B_\text {s}(\textbf{w}_\text {s},\textbf{y})&= B^\text {be}_\text {s}(\textbf{w}^\text {be}_\text {s},\textbf{y}^\text {be}) + B^\text {sh}_\text {s}(\textbf{w}^\text {sh}_\text {s},\textbf{y}^\text {sh})\nonumber \\&~~+ B^\text {so}_\text {s}(\textbf{w}^\text {so}_\text {s},\textbf{y}^\text {so})\text { ,} \end{aligned}$$
(2)
and
$$\begin{aligned} F_\text {s}(\textbf{w}_\text {s}) = F^\text {be}_\text {s}(\textbf{w}^\text {be}_\text {s}) + F^\text {sh}_\text {s}(\textbf{w}^\text {sh}_\text {s}) + F^\text {so}_\text {s}(\textbf{w}^\text {so}_\text {s})\text { .} \end{aligned}$$
(3)
The individual structural formulations for each THV component and the formulations that account for the interaction between components are discussed in the following section.
The stent is fabricated with thin wires and is modeled using an extension of the isogeometric Bernoulli beam. Bauer et al. [
40] originally proposed the formulation for geometrically nonlinear spatial curved beams using Bernoulli kinematics. This section summarizes an extension of this beam formulation, developed by Wu et al. [
25], that accounts for more complex beam geometries.
In this section, Latin indices take on values
\(\{1,2,3\}\), and Greek indices take values
\(\{2,3\}\). The beam formulation begins by defining the continuum of the beam using a centerline and a moving local coordinate system. This coordinate system is used to describe the orientation of the cross sections. The position vector of the continuum of the beam, denoted as
\(\textbf{x}\) is defined as,
$$\begin{aligned} \textbf{x}(\xi ^1,\xi ^2,\xi ^3)=\textbf{r}(\xi ^1)+\xi ^2\,\textbf{a}_2(\xi ^1) +\xi ^3\,\textbf{a}_3(\xi ^1) \text { ,} \end{aligned}$$
(4)
where
\(\textbf{r}\) is the position vector of the centerline,
\(\xi ^1\) is the contravariant coordinate of the centerline,
\(\xi ^2\) and
\(\xi ^3\) are coordinates of the cross-sectional profile, and
\(\textbf{a}_\alpha\) are unit base vectors tangent to the cross-section and orthogonal to the tangent vector of the centerline denoted as
\(\textbf{a}_1 = \textbf{r}_{,1}\). The vectors
\(\textbf{a}_i\) directly define the moving local coordinate system for both the deformed and undeformed configurations. Variables of the latter are denoted by
\(\mathring{(\cdot )}\).
Bernoulli beam theory assumes the cross sections remain orthogonal to the centerline and the cross-sectional dimensions remain constant throughout deformation. The two components,
\(\mathring{\textbf{a}}_\alpha\) of the moving local basis at every location
i along the beam centerline in the undeformed configuration are obtained as,
$$\begin{aligned} (\mathring{\textbf{a}}_\alpha )_i = \textbf{R}\left( (\hat{\mathring{{\textbf{a}}}}_1)_i, \mathring{\theta }_i \right) \mathbf {\Lambda } \left( \textbf{A}_1,(\hat{ \mathring{\textbf{a}}}_1)_i \right) \textbf{A}_{\alpha }\text { ,} \end{aligned}$$
(5)
\(\text {for } i = 1,...,n\), where the three unit vectors
\(\textbf{A}_{1}\) and
\(\textbf{A}_{\alpha }\) define a global reference coordinate system,
\(\mathbf {\Lambda }\) is a mapping operator that maps one vector to the other,
\(\textbf{R}\) is the rotation operator,
\(\hat{(\cdot )}\) denotes the normalized vector, and
n denote the number of points along the centerline, respectively. The mapping and rotation operators used are described in full detail in Bauer et al. [
40] and derived based on the Euler–Rodrigues formula specialized to the current problem.
The same two steps of mapping and rotation can then be applied to describe the alignment of the moving local coordinate system from the undeformed configuration to the deformed configuration as follows:
$$\begin{aligned} \textbf{a}_\alpha = \textbf{R}(\hat{\textbf{a}}_1, \theta )\mathbf {\Lambda }( \hat{\mathring{\textbf{a}}}_1, \hat{\textbf{a}}_1 )\mathring{\textbf{a}}_\alpha \text { ,} \end{aligned}$$
(6)
where
\(\theta\) is a rotational degree of freedom.
Wu et al. [
25] found that the definition of the local coordinate basis described in Eq. (
5) is effective; however, for complex structures, the formulation showed some limitations. The extension proposed in Wu et al. [
25] is summarized here. Let
\((\textbf{a}_{\alpha }')_1 = \mathbf {\Lambda }(\textbf{A}_1,(\hat{ \mathring{\textbf{a}}}_1)_1)\textbf{A}_\alpha\). Then we define,
$$\begin{aligned} (\textbf{a}_{\alpha }')_i = \mathbf {\Lambda }((\hat{\textbf{a}}_1)_{i-1}, (\hat{\textbf{a}}_1)_i) (\textbf{a}_\alpha ')_{i-1}\text { ,} \end{aligned}$$
(7)
\(\text {for } i = 2,...,n\), where
n is the number of points along the centerline and
\((\textbf{a}_{\alpha }')_i\) denotes the vectors
\(\textbf{a}_{\alpha }'\) at the
\(i^{\text {th}}\) location. In this work, each
\((\textbf{a}_\alpha ')_i\) is evaluated on the quadrature points. We then combine this step with the rotation step to obtain the final formulation defined at every location along the centerline as,
$$\begin{aligned} (\mathring{\textbf{a}}_{\alpha })_i = \textbf{R}((\hat{\mathring{\textbf{a}}}_1)_i,\mathring{\theta _i})(\textbf{a}_\alpha ')_i \text { .} \end{aligned}$$
(8)
This method helps achieve the desired complex geometry by avoiding twisting at locations where the tangents of the centerline change significantly.
The weak form of the beam subproblem can then be defined as
$$\begin{aligned} B^\text {be}_\text {s}(\textbf{w}_\text {s},\textbf{y}) - F^\text {be}_\text {s}(\textbf{w}_\text {s}) =&\int _{(\mathcal {L}^\text {be})_0}\textbf{w}_\text {s}\cdot \rho ^\text {be}_\text {s} A\left. \frac{\partial ^2\textbf{y}}{\partial t^2}\right| _{\mathring{\textbf{x}}}\text {d}\mathcal {L} \nonumber \\&+ \int _{(\mathcal {L}^\text {be})_0}\int _{A}\delta \textbf{E}:\textbf{S}~\text {d}A\text {d}\mathcal {L} \nonumber \\&-\int _{(\mathcal {L}^\text {be})_0}\textbf{w}_\text {s}\cdot \rho ^\text {be}_\text {s} A\textbf{f}_\text {s} ~\text {d}\mathcal {L} \nonumber \\ {}&-\int _{(\mathcal {L}^\text {be})_t}\textbf{w}_\text {s}\cdot \textbf{h}_\text {s}^{\text {net}} ~\text {d}\mathcal {L} \text { ,} \end{aligned}$$
(9)
where
\((\mathcal {L}^\text {be})_t\) and
\((\mathcal {L}^\text {be})_0\) are the centerline of the beam in the deformed and undeformed configurations, respectively,
\(\textbf{y}\) is the displacement of the centerline,
\(\partial /\partial t |_{\mathring{\textbf{x}}}\) is the time derivative holding material coordinates
\(\mathring{\textbf{x}}\) fixed,
\(\rho ^\text {be}_\text {s}\) is the beam density,
\(\textbf{S}\) is the second Piola–Kirchhoff stress,
\(\delta \textbf{E}\) is the variation of the Green–Lagrange strain corresponding to a displacement variation
\(\textbf{w}_\text {s}\),
\(\textbf{f}_\text {s}\) is a prescribed body force,
\(\textbf{h}^\text {net}\) is the total traction from two sides of the beam, and
A is the cross-sectional area of the beam.
In this work, the stent is modeled using this extension for the IGA Bernoulli beam. Once the centerline curves are defined as discussed in Sect.
2.1, the cross-sectional profiles must also be described. These profiles are defined over the entire NURBS curve using two local unit vectors,
\(\textbf{v}_2\), and,
\(\textbf{v}_3\). In this work, the following approach is proposed to determine the orientation of these orthogonal base vectors that define the cross-sectional profiles. Let
\(\textbf{v}_1\) be the local unit vector that defines the tangent of the curve. Then, define the local unit vector,
\(\textbf{v}_2\), to be perpendicular to
\(\textbf{v}_1\) and the cylindrical unit vector,
$$\begin{aligned} \hat{\pmb {\phi }}_c = \begin{pmatrix} -\sin {\phi _c} \\ \cos {\phi _c} \\ 0 \end{pmatrix}\text {,} \end{aligned}$$
(10)
and
\(\textbf{v}_3\) to be perpendicular to
\(\textbf{v}_1\) and
\(\textbf{v}_2\). Hence, these two unit vectors representing the cross-sectional profile can be defined as
$$\begin{aligned} \textbf{v}_2&= \textbf{v}_1\times \hat{\pmb {\phi }}_c \end{aligned}$$
(11)
$$\begin{aligned} \textbf{v}_3&= \textbf{v}_1\times \textbf{v}_2 \text {.} \end{aligned}$$
(12)
In this work, we define a rectangular cross-section with a constant width of 0.54 mm along
\(\textbf{v}_2\) and a constant width of 0.21 mm along
\(\textbf{v}_3\) based on the information in Hopf [
41]. For the formulation of the Bernoulli beam, we choose
\(\mathbf {\mathring{a}}_2\) and
\(\mathbf {\mathring{a}}_3\) to be
\(\textbf{v}_2\) and
\(\textbf{v}_3\), respectively. The final stent geometry can be constructed from the curves defined in Sect.
2.1 and the cross-sectional profiles.
The beam structures are discretized using IGA and the Galerkin method. The beam centerlines are defined using cubic NURBS curves and at least \(C^1\)-continuous NURBS patches to represent the full beam geometry. These basis functions are utilized for the discretization to provide the needed C1-continuity for the Bernoulli beam formulation.
The leaflets and skirt are thin tissue structures and are modeled as hyperelastic isogeometric Kirchhoff–Love shells [
42]. The Kirchhoff–Love hypothesis of straight and normal cross-sections implies that a point
\(\textbf{x}\) in the shell continuum can be described by a point
\(\textbf{r}\) on the midsurface and a vector
\(\textbf{a}_3\) normal to the midsurface:
\(\textbf{x}(\xi ^1,\xi ^2,\xi ^3)=\textbf{r}(\xi ^1,\xi ^2)+\xi ^3\,\textbf{a}_3(\xi ^1,\xi ^2)\), where
\(\xi ^1,\xi ^2\) are the contravariant coordinates of the midsurface,
\(\xi ^3 \in [-H/2,H/2]\) is the through-thickness coordinate, and
\(H\) is the shell thickness.
The covariant base vectors and metric coefficients of the convective curvilinear coordinate system are defined by \(\textbf{g}_{i}=\textbf{x}_{,i}\) and \(g_{ij}=\textbf{g}_{i}\cdot \textbf{g}_{j}\), respectively, where \((\cdot )_{,i}=\partial (\cdot )/\partial \xi ^i\). For the shell formulation, we adopt the convention that Latin indices take on values \(\{1,2,3\}\), and Greek indices take on values \(\{1,2\}\) to denote the in-plane components. The contravariant base vectors \(\textbf{g}^{i}\) are defined by \(\textbf{g}^{i}\cdot \textbf{g}_{j}=\delta ^i_j\) and contravariant metric coefficients are given by \([g^{ij}]=[g_{ij}]^{-1}\). For the Kirchhoff–Love shell theory, both normal and transverse shear strains are neglected; only the in-plane strain components are considered. The theory assumes a linear strain distribution through the thickness and defines \(g_{\alpha \beta } = a_{\alpha \beta } -2\, \xi ^3b_{\alpha \beta }\), where \(a_{\alpha \beta }=\textbf{a}_{\alpha }\cdot \textbf{a}_{\beta },\) \(b_{\alpha \beta }=\textbf{a}_{\alpha ,\beta }\cdot \textbf{a}_{3},\) \(\textbf{a}_{\alpha }=\textbf{r}_{,\alpha },\) \(\textbf{a}_3 = (\textbf{a}_1\times \textbf{a}_2)/||\textbf{a}_1\times \textbf{a}_2||\), and \(\Vert \cdot \Vert\) is the Euclidean norm. These definitions hold for both deformed and undeformed configurations where variables of the latter are indicated by \(\mathring{(\cdot )}\). The Jacobian determinant of the structure’s motion is \(J=\sqrt{\vert g_{ij}\vert /\vert \mathring{g}_{ij}\vert }\) and the in-plane Jacobian determinant is \(J_o=\sqrt{\vert g_{\alpha \beta }\vert /\vert \mathring{g}_{\alpha \beta } \vert }\).
The weak form of the shell structural formulation is defined as
$$\begin{aligned} B^\text {sh}_\text {s}(\textbf{w}_\text {s},\textbf{y}) - F^\text {sh}_\text {s}(\textbf{w}_\text {s})&=\int _{(\mathcal {S}^\text { sh})_0}\textbf{w}_\text {s}\cdot \rho ^\text {sh}_\text {s} H\left. \frac{\partial ^2\textbf{y}}{\partial t^2}\right| _{\mathring{\textbf{x}}}\text {d}\mathcal {S} \nonumber \\&~~+\int _{(\mathcal {S}^\text { sh})_0}\int _{\frac{-H}{2}}^{\frac{H}{2}}\delta \textbf{E}:\textbf{S}~\text {d}\xi ^3\text {d}\mathcal {S} \nonumber \\&~~-\int _{(\mathcal {S}^\text { sh})_0}\textbf{w}_\text {s}\cdot \rho ^\text {sh}_\text {s} H\textbf{f}_\text {s} ~\text {d}\mathcal {S} \nonumber \\&~~- \int _{(\mathcal {S}^\text { sh})_t}\textbf{w}_\text {s}\cdot \textbf{h}_\text {s}^{\text {net}} ~\text {d}\mathcal {S} \text { ,} \end{aligned}$$
(13)
where
\((\mathcal {S}^\text { sh})_0\) and
\((\mathcal {S}^\text { sh})_t\) are the shell midsurfaces in the reference and deformed configurations,
\(\textbf{y}\) is the midsurface displacement,
\(\rho ^\text {sh}_\text {s}\) is the shell density,
\(\textbf{S}\) is the second Piola–Kirchhoff stress tensor obtained from a hyperelastic strain energy density functional
\(\psi\):
\(\textbf{S} =\partial _{\textbf{E}}\psi\),
\(\textbf{E}= \frac{1}{2}(\textbf{C}-\textbf{I})\) is the Green–Lagrange strain tensor,
\(\textbf{C}\) is the right Cauchy–Green deformation tensor,
\(\textbf{I}\) is the identity tensor,
\(\delta \textbf{E}\) is the variation of
\(\textbf{E}\) corresponding to displacement variation
\(\textbf{w}_\text {s}\),
\(\textbf{f}_\text {s}\) is a prescribed body force, and
\(\textbf{h}^{\text {net}}_\text {s}\) is the total traction from the two sides of the shell. In this work, the material is assumed to be incompressible. The elastic strain energy functional
\(\psi _{el}\) is augmented by a constraint term enforcing
\(J=\sqrt{\text {det}\,\textbf{C}}=1\), via a Lagrange multiplier
p:
\(\psi = \psi _{el}-p(J-1)\). Readers are referred to Wu et al. [
25] for detailed descriptions of the stress and strain tensors that are used in this work.
3.2.3 Artery wall modeling
The artery wall is modeled as a hyperelastic solid subject to damping forces. We define
$$\begin{aligned} B^\text {so}_\text {s}(\textbf{w}_\text {s},\textbf{y}) - F^\text {so}_\text {s}(\textbf{w}_\text {s})&= \int _{(\Omega ^\text {so}_\text {s})_0}\textbf{w}_\text {s}\cdot \rho ^\text {so}_\text {s}\left. \frac{\partial ^2\textbf{y}}{\partial t^2}\right| _{\mathring{\textbf{x}}}~\text {d}\Omega \nonumber \\&~~+ \int _{(\Omega ^\text {so}_\text {s})_0} \pmb {\nabla }_{\mathring{\textbf{x}}} \textbf{w}_\text {s} : \textbf{F}(\textbf{S} + \textbf{S}_0)~\text {d}\Omega \nonumber \\&~~-\int _{(\Omega ^\text {so}_\text {s})_0}\textbf{w}_\text {s}\cdot \rho ^\text {so}_\text {s}\textbf{f}_\text {s}~\text {d}\Omega \nonumber \\&~~- \int _{(\Gamma _\text {s}^{\text {so},\text {h}})_t}\textbf{w}_\text {s}\cdot \textbf{h}_\text {s}~\text {d}\Gamma \text { ,} \end{aligned}$$
(14)
where
\((\Omega ^\text {so}_\text {s})_0\) is the portion of
\(\Omega _\text {s}\) corresponding to the artery wall in the reference configuration,
\(\rho ^\text {so}_\text {s}\) is the solid mass density,
\(\partial (\cdot )/\partial t|_{\mathring{\textbf{x}}}\) is the time derivative holding the material coordinates
\(\mathring{\textbf{x}}\) fixed,
\(\pmb {\nabla }_{\mathring{\textbf{x}}}\) is the gradient operator on
\((\Omega ^\text {so}_\text {s})_0\),
\(\textbf{F}\) is the deformation gradient associated with displacement
\(\textbf{y}\),
\(\textbf{S}\) is the hyperelastic contribution to the second Piola–Kirchhoff stress tensor,
\(\textbf{S}_0\) is a prescribed prestress tensor [
43,
44],
\(\textbf{f}_\text {s}\) is a prescribed body force, and
\(\textbf{h}_\text {s}\) is a prescribed traction on the Neumann boundary
\(\Gamma _\text {s}^{\text {so},\text {h}}\). In this work, the elastic contribution to the second Piola–Kirchhoff stress in Eq. (
14) is derived from a compressible neo-Hookean model with dilatational penalty [
43,
45], which is shown to be appropriate for arterial wall modeling in FSI simulations. The additional pressure
\(\textbf{S}_0\) in Eq. (
14) is required because the initial aorta configuration is subject to blood pressure and viscous traction and is therefore not stress-free. We follow the procedure from Xu et al. [
29] to determine
\(\textbf{S}_0\). Additional details can also be found in Wu et al. [
25].
3.2.4 Structural component interaction
To model the connections between different components of the THV, which would be sutured together in the actual device, the penalty coupling approach proposed by Herrema et al. [
46] is adapted for the THV structures. To enforce the shell–shell displacement coupling of the skirt and leaflets at a patch interface
\(\mathcal {L}_\text {I}^\text {ss}\) between two shell surfaces
\(\mathcal {S}^\text { sh,A}\) and
\(\mathcal {S}^\text { sh,B}\), the following displacement penalty term is added to
\(B^\text {sh}_\text {s}(\textbf{w}_\text {s},\textbf{y})\):
$$\begin{aligned} +~\int _{\mathcal {L}_\text {I}^\text {ss}} \alpha _\text {d}^\text {ss} \left( \textbf{w}_\text {s}^\text {sh,A} - \textbf{w}_\text {s}^\text {sh,B}\right) \cdot \left( \textbf{y}^\text {sh,A} - \textbf{y}^\text {sh,B}\right) \text {d}\mathcal {L} \text{, } \end{aligned}$$
(15)
where
\(\textbf{y}^\text {sh,A}\) and
\(\textbf{y}^\text {sh,B}\) are the displacements on surface patches
\(\mathcal {S}^\text { sh,A}\) and
\(\mathcal {S}^\text { sh,B}\), respectively, along the penalty curve
\(\mathcal {L}_\text {I}^\text {ss}\),
\(\textbf{w}_\text {s}^\text {sh,A}\) and
\(\textbf{w}_\text {s}^\text {sh,B}\) are their respective weighting functions, and
\(\alpha _\text {d}^\text {ss}\) is a variable penalty parameter of sufficiently large magnitude. For shell–beam coupling of the skirt and the frame, the same displacement penalty approach is used, and the following displacement penalty term is added to
\(B_\text {s}(\textbf{w}_\text {s}, \textbf{y})\):
$$\begin{aligned} +~\int _{\mathcal {L}_\text {I}^\text {sb}} \alpha _\text {d}^\text {sb} \left( \textbf{w}_\text {s}^\text {sh} - \textbf{w}_\text {s}^\text {be}\right) \cdot \left( \textbf{y}^\text {sh} - \textbf{y}^\text {be}\right) \text {d}\mathcal {L} \text{, } \end{aligned}$$
(16)
where
\(\alpha _\text {d}^\text {sb}\) is an adjustable penalty parameter, and
\(\mathcal {L}_\text {I}^\text {sb}\) is a penalty curve along the shell–beam interface. For the shell–beam coupling, in practice, we choose the penalty curve
\(\mathcal {L}_\text {I}^\text {sb}\) to be
\(\mathcal {L}^\text {be}\).
To enforce the rotational continuity between two shell surfaces, the penalty approach is also used to maintain the angle formed at the coupled patch interface. The following rotational penalty term [
46] is added to
\(B^\text {sh}_\text {s}(\textbf{w}_\text {s},\textbf{y})\):
$$\begin{aligned} +~\int _{\mathcal {L}_\text {I}^\text {ss}}&\alpha _\text {r}^\text {ss}\left( \left( \delta \cos {\phi }-\delta \cos {\mathring{\phi }} \right) \left( \cos {\phi }-\cos {\mathring{\phi }}\right) \right. \nonumber \\ {}&~+ \left. \left( \delta \sin {\phi } - \delta \sin {\mathring{\phi }}\right) \left( \sin {\phi } - \sin {\mathring{\phi }}\right) \right) \text {d}\mathcal {L} \text{, } \end{aligned}$$
(17)
where
\(\mathring{\phi }\) and
\(\phi\) are the angles between the surfaces before and after deformation, respectively, and
\(\delta\) denotes the variation. The structure of the THV frame geometry restrains local twisting, so the effect of the beam rotation on the shell is assumed to be negligible, and a rotational penalty is not imposed for the shell–beam coupling.
The penalty parameters for shell–shell coupling are given as
$$\begin{aligned} \alpha _\text {d}^\text {ss}&= \alpha ^\text {ss}\frac{E^\text {sh} \, H}{h^\text {sh,I} \, (1-(\nu ^\text {sh})^2)} \text { ,} \end{aligned}$$
(18)
$$\begin{aligned} \alpha _\text {r}^\text {ss}&= \alpha ^\text {ss}\frac{E^\text {sh} \, H^3}{12 \, h^\text {sh,I} \, (1-(\nu ^\text {sh})^2)} \text { ,} \end{aligned}$$
(19)
where
\(\alpha ^\text {ss}\) is a dimensionless penalty coefficient,
\(E^\text {sh}\) is some effective material stiffness with units of stress (e.g., the Young’s modulus in a linear isotropic material),
\(h^\text {sh,I} = (h^\text {sh,A} + h^\text {sh,B})/2\), where
\(h^\text {sh,A}\) and
\(h^\text {sh,B}\) are the lengths of the local elements in the direction most parallel to the penalty curve, and
\(\nu ^\text {sh}\) is the Poisson’s ratio. For shell–beam coupling, the following penalty parameter is employed:
$$\begin{aligned} \alpha _\text {d}^\text {sb} = \alpha ^\text {sb}\min \left\{ \frac{E^\text {sh} \, H}{h^\text {sh} \, (1-(\nu ^\text {sh})^2)},\frac{E^\text {be} \sqrt{A}}{h^\text {be} \, (1-(\nu ^\text {be})^2)}\right\} \text { ,} \end{aligned}$$
(20)
where
\(\alpha ^\text {sb}\) is a dimensionless penalty coefficient,
\(h^\text {sh}\) is defined to be an effective shell element length [
25], and
\(h^\text {be}\) is defined to be the local beam element length. The selection of
\(\alpha _\text {d}^{\text {sb}}\) from the minimum parameter value between coupled materials produces a sufficiently high penalty value that is not excessive for the lower stiffness material. The penalty parameter is also high enough to ensure constraint satisfaction without creating excessive ill-conditioning.
The THV presents multiple contact problems, including leaflet-to-leaflet, leaflet-to-frame, and frame-to-artery wall contact that can occur during crimping, deployment, and during the cardiac cycle. To model the multiple contact problems, a nonlocal contact formulation with a linear contact kernel is imposed [
47]. For contact between two points,
\(\textbf{x}^\text {A}\) and
\(\textbf{x}^{\text {B}}\) in a single body with reference configuration
\(\Omega _0\), the following contact term is added to
\(B^\text {sh}_\text {s}(\textbf{w}_\text {s},\textbf{y})\):
$$\begin{aligned} +\int _{\Omega _0\backslash B_R(\mathring{\textbf{x}}^\text {A})} \int _{\Omega _0 } \delta \textbf{r}^\text {AB}\cdot \phi '_\text {c}(r^\text {AB}) \frac{\textbf{r}^\text {AB}}{r^\text {AB}} \text {d} \mathring{\textbf{x}}^\text {A} \text {d} \mathring{\textbf{x}}^\text {B} \text { ,} \end{aligned}$$
(21)
where
\(B_R(\mathring{\textbf{x}}^\text {A})\) is the Euclidean ball of radius R centered at
\(\mathring{\textbf{x}}^\text {A}\) in
\(\Omega _0\),
\(\textbf{r}^\text {AB} = \textbf{x}^\text {B}-\textbf{x}^\text {A}\),
\(r^\text {AB} = \Vert \textbf{r}^\text {AB}\Vert\), and
\(\phi '_\text {c}(r^\text {AB})\) is the linear contact kernel defined as,
$$\begin{aligned} \phi '_\text {c}(r^\text {AB}) = -\max (k_\text {c}(r_{\text {max}} -r^\text {AB} ) ,0) \text{, } \end{aligned}$$
(22)
where
\(k_\text {c}\) is a penalty parameter that needs to be sufficiently large and
\(r_{\text {max}}\) is a cutoff distance within which the contact force is active.
The role of friction is also introduced in the THV system as the deployed device expands and anchors to the aortic wall. In order to model this effect, Wu et al. [
25] proposed a simple static friction model to estimate the friction force. The friction model utilizes the following penalty term in locations where contact occurs,
$$\begin{aligned} +\int _{\Omega _0^\text {be}}\int _{\Omega _0^\text {so}}\mathcal {F}~ \text {d} \mathring{\textbf{x}}^\text {so} \text {d} \mathring{\textbf{x}}^\text {be} \text { , for } \Vert {\textbf {r}} \Vert < r_\text {max}\text { ,} \end{aligned}$$
(23)
where
$$\begin{aligned} \mathcal {F} = \delta (\textbf{r})_{\tau }\cdot \alpha _\text {f}~\vert \phi '_\text {c}(\Vert \textbf{r}\Vert )\vert \left( \Vert (\textbf{r})_{\tau }\Vert - \Vert (\textbf{r}_\text {d})_{\tau } \Vert \right) \textstyle \frac{(\textbf{r})_{\tau } }{ \Vert (\textbf{r})_{\tau } \Vert }\text { ,} \end{aligned}$$
(24)
\(\Omega _0^\text {be}\) and
\(\Omega _0^\text {so}\) are the reference configurations of the beam and solid, respectively,
\(\mathring{\textbf{x}}^\text {be}\in \Omega _0^\text {be}\),
\(\mathring{\textbf{x}}^\text {so}\in \Omega _0^\text {so}\), subscript “d” indicates the deployed configuration, subscript “
\(\tau\)” indicates the tangential component,
\(\textbf{r} = \textbf{x}^\text {be}-\textbf{x}^\text {so}\),
\(\textbf{r}_{\text {d}} = \textbf{x}^\text {be}_{\text {d}}-\textbf{x}^\text {so}_{\text {d}}\),
$$\begin{aligned} (\textbf{r})_{\tau }&= \textbf{r} - (\textbf{r}\cdot \textbf{n})\textbf{n} \text { ,} \end{aligned}$$
(25)
$$\begin{aligned} (\textbf{r}_\text {d})_{\tau }&= \textbf{r}_\text {d} - (\textbf{r}_\text {d}\cdot \textbf{n}_\text {d})\textbf{n}_\text {d} \text { ,} \end{aligned}$$
(26)
\(\textbf{n}\) and
\(\textbf{n}_\text {d}\) are the outward normal vectors on the artery wall in the current and deployed configurations, respectively, and
\(\alpha _\text {f}\) is a penalty parameter that is set to
\(\alpha _\text {f} = 10^{11}\) in this work.