In this section, we introduce the computational model in terms of its governing equations. We differentiate between the prestressing stage and the deformation stage, which, when combined, yield a mechanical state of the aortic segment under systolic blood pressure. Particular focus is placed on the prestressing stage, given that special treatment is required for the purpose of snapshot collection.

### Patient-specific computational model

Our computational model consists of an aortic segment, which fully includes the AAA as well as short segments of the iliac arteries, see [12] for a detailed description of the workflow from imaging to finite element simulation. The aortic vessel is treated as an elastic solid consisting of an intraluminal thrombus (ILT) and the aortic wall. Pressure is exerted on the luminal (i.e. inner) surface of the ILT and the aneurysm is loaded to an assumed systolic blood pressure, which is the mechanical state of interest. The proximal and distal end surfaces of the model are constrained by a zero-displacement Dirichlet condition for vessel fixation. Figure 1 exhibits an example of a patient-specific computational domain.

### Model equations

The governing equations read

$$\begin{aligned} \nabla \cdot \varvec{P}&= {\varvec{0}} \qquad \mathrm {in} \ \Omega _0 \end{aligned}$$

(1)

$$\begin{aligned} \varvec{P} \cdot \varvec{N}&= {\varvec{T}} \qquad \mathrm {on} \ \Gamma _{p,0} \end{aligned}$$

(2)

$$\begin{aligned} {\varvec{u}}&= {\varvec{0}} \qquad \mathrm {on} \ \Gamma _D \end{aligned}$$

(3)

with

$$\begin{aligned} {\varvec{T}} = {\varvec{T}}({\varvec{u}}, p)&= -p J({\varvec{u}}) \varvec{F}^{-T}({\varvec{u}}) \cdot {\varvec{N}}. \end{aligned}$$

(4)

The weak form of the governing equations is given by the principle of virtual work (PVW)

$$\begin{aligned} \begin{aligned} \delta W&= \delta W_{\mathrm {int}} - \delta W_{\mathrm {ext}} = \int _{\Omega _0} \varvec{P} : \nabla \delta \varvec{u} \ dV - \int _{\Gamma _{p, 0}} \varvec{T} \cdot \delta \varvec{u} \ dA = 0 \quad \forall \delta {\varvec{u}}. \end{aligned} \end{aligned}$$

(5)

\(\delta W, \delta W_{\mathrm {int}}\) and \(\delta W_{\mathrm {ext}}\) denote the total, internal and external virtual work, \(\varvec{P}\) denotes the first Piola-Kirchhoff stress tensor, \(\varvec{N}\) is the outward normal vector in the reference configuration and \(\Gamma _{p,0}\) denotes the reference configuration pressure load surface (i.e. the luminal ILT surface). We emphasize that the traction boundary condition \({\varvec{T}}\) depends on the displacement field \({\varvec{u}}\), see Eq. (4). Therein \(\varvec{F}({\varvec{u}}) = \varvec{I} + \frac{\partial {\varvec{u}}}{\partial {\varvec{X}}}\) is the deformation gradient with respect to the reference configuration, \({\varvec{X}} \in \Omega _0\) denotes reference configuration material coordinates, \(J({\varvec{u}})\) is the deformation gradient determinant and *p* is the pressure.

We make use of hyperelastic constitutive relations

$$\begin{aligned} \varvec{P} = \frac{\partial \Psi }{\partial \varvec{F}} \end{aligned}$$

(6)

introducing the strain-energy function \(\Psi \) and apply an isochoric-volumetric split for ILT as well as the vessel wall strain-energy

$$\begin{aligned} \Psi _{\mathrm {ILT}}(\bar{I}_1, \bar{I}_2, J)&= \Psi ^{\mathrm {ILT}}_{\mathrm {iso}}(\bar{I}_1, \bar{I}_2) + \Psi ^{\mathrm {ILT}}_{\mathrm {vol}}(J), \end{aligned}$$

(7)

$$\begin{aligned} \Psi _{\mathrm {wall}}(\bar{I}_1, J)&= \Psi ^{\mathrm {wall}}_{\mathrm {iso}}(\bar{I}_1) + \Psi ^{\mathrm {wall}}_{\mathrm {vol}}(J), \end{aligned}$$

(8)

wherein

$$\begin{aligned} \bar{I}_1&= \mathrm {tr}(\bar{\varvec{C}}), \end{aligned}$$

(9)

$$\begin{aligned} \bar{I}_2&= \frac{1}{2} (\mathrm {tr}(\bar{\varvec{C}})^2 - \mathrm {tr}(\bar{\varvec{C}}^2)) \end{aligned}$$

(10)

are the first and second principal invariant of the modified right Cauchy Green tensor \({\bar{\varvec{C}}} = {\varvec{F}}^T_{\mathrm {iso}} {\varvec{F}}_{\mathrm {iso}}\) with \({\varvec{F}}_{\mathrm {iso}} = J^{-\frac{1}{3}} \varvec{F}\). In more detail, we model the isochoric strain-energy contribution of the ILT as given in [12, 26]

$$\begin{aligned} \Psi _{\mathrm {iso}}^{\mathrm {ILT}}(\bar{I}_1, \bar{I}_2) = c (\bar{I}_1^2 - 2 \bar{I}_2 - 3) \end{aligned}$$

(11)

and the isochoric strain-energy contribution of the vessel wall as given in [9, 12]

$$\begin{aligned} \Psi _{\mathrm {iso}}^{\mathrm {wall}}(\bar{I}_1) = \alpha (\bar{I}_1 - 3) + \beta (\bar{I}_1 - 3)^2. \end{aligned}$$

(12)

The parameter *c* is a stiffness parameter of the ILT, while \(\alpha \) (referred to as \(\alpha \)-stiffness in the following) and \(\beta \) (referred to as \(\beta \)-stiffness in the following) can be interpreted as low-strain range and high-strain range stiffness of the vessel wall, respectively. The volumetric parts \(\Psi _{\mathrm {vol}}^{\mathrm {wall}}, \Psi _{\mathrm {vol}}^{\mathrm {ILT}}\) of the strain-energies are chosen as given in [12, 27]

$$\begin{aligned} \Psi ^{\mathrm {x}}_{\mathrm {vol}}(J) = \frac{\kappa ^{\mathrm {x}}}{4} (J^2 - 2 \mathrm {ln}(J) - 1), \end{aligned}$$

(13)

with \(\mathrm {x} \in \{\mathrm {ILT}, \mathrm {wall}\}\) and \(\kappa ^{\mathrm {wall}}\), \(\kappa ^{\mathrm {ILT}}\) being sufficiently large to reflect almost incompressible material behavior.

### MULF prestressing

AAA geometries obtained from computed tomography imaging are exerted to blood pressure. From a continuum mechanics perspective, this corresponds to a non stress-free reference configuration [14, 28, 29]. Our simulations are therefore divided in two stages: The *prestressing* stage, which aims at imprinting a physiological stress-state into the imaged (i.e. fixed) geometric configuration at assumed diastolic blood pressure, is performed first. At second, the vessel is loaded to an assumed systolic blood pressure at evolving geometry in the *deformation* stage.

We apply the *Modified Updated Lagrangian Formulation* (MULF) [14] prestressing approach in the first stage. MULF is an efficient prestressing method which especially was validated for the simulation of AAAs [10, 12, 13, 30]. In the MULF prestressing approach an imprinted *prestress deformation gradient* \(\varvec{F}_p\) is built up incrementally with boundary conditions evaluated at the imaged configuration.

Snapshot collection as required for data-driven construction of a ROB (cf. section “Construction of reduced-order model components”) is not possible for MULF prestressing, given that displacement modes are not generated. To overcome this problem, we present a reformulation of MULF prestressing, shifting the wanted quantity from the prestress deformation gradient \(\varvec{F}_p\) to a virtual *prestress displacement field* \(\varvec{u}_p\).

For consistency, we briefly review the original MULF prestressing formulation from a continuum mechanics perspective (details on implementation in the realm of the finite element method can be found in [14]) and state the mentioned reformulation in direct comparison with the original.

As starting point we recall the following kinematic relations. Given a virtual displacement field \({\tilde{\varvec{u}}}\), from a virtual configuration \({\tilde{\Omega }} \ni {\tilde{\varvec{X}}}\) to the current configuration \(\Omega \ni \varvec{x}\), a displacement field \(\varvec{u}\) from \(\Omega _0 \ni \varvec{X}\) to \(\Omega \), a deformation gradient \(\varvec{F} = \frac{\partial {\varvec{x}}}{\partial {\varvec{X}}}\) and a virtual deformation gradient \({\tilde{\varvec{F}}} = \frac{\partial {\tilde{\varvec{X}}}}{\partial {\varvec{X}}}\), we state

$$\begin{aligned} {\varvec{x}}&= {\tilde{\varvec{X}}} + {\tilde{\varvec{u}}} = {\varvec{X}} + {\varvec{u}}, \end{aligned}$$

(14)

$$\begin{aligned} \varvec{F}&= \varvec{I} + \frac{\partial \varvec{u}}{\partial \varvec{X}} = \frac{\partial ({\varvec{X}} + {\varvec{u}})}{\partial {\varvec{X}}} = \frac{\partial ({\tilde{\varvec{X}}} + {\tilde{\varvec{u}}})}{\partial {\varvec{X}}} \nonumber \\&= \frac{\partial ({\tilde{\varvec{X}}} + {\tilde{\varvec{u}}}) }{\partial {\tilde{\varvec{X}}}} \frac{\partial {\tilde{\varvec{X}}}}{\partial \varvec{X}} = \left( \varvec{I} + \frac{\partial {\tilde{\varvec{u}}}}{\partial {\tilde{\varvec{X}}}} \right) \cdot {\tilde{\varvec{F}}}. \end{aligned}$$

(15)

As a result, the identical first Piola-Kirchhoff stress field \(\varvec{P}\) can be expressed as

$$\begin{aligned} \varvec{P}&= \varvec{P}_F(\varvec{F}), \end{aligned}$$

(16)

$$\begin{aligned} \varvec{P}&= \varvec{P}_u(\varvec{u}), \end{aligned}$$

(17)

$$\begin{aligned} \varvec{P}&= \varvec{P}_{u,F}({\tilde{\varvec{u}}}, {\tilde{\varvec{F}}}), \end{aligned}$$

(18)

defining

$$\begin{aligned} \varvec{P}_{F}&: \varvec{F}\mapsto \frac{\partial \Psi }{\partial \varvec{F}}(\varvec{F}), \end{aligned}$$

(19)

$$\begin{aligned} \varvec{P}_u&: \varvec{u} \mapsto \frac{\partial \Psi }{\partial \varvec{F}} \left( \varvec{I} + \frac{\partial \varvec{u}}{\partial \varvec{X}}\right) , \end{aligned}$$

(20)

$$\begin{aligned} \varvec{P}_{u,F}&: ({\tilde{\varvec{u}}}, {\tilde{\varvec{F}}}) \mapsto \frac{\partial \Psi }{\partial \varvec{F}} \left( \left( \varvec{I} + \frac{\partial {\tilde{\varvec{u}}}}{\partial {\tilde{\varvec{X}}}}\right) \cdot {\tilde{\varvec{F}}}\right) . \end{aligned}$$

(21)

Applying the introduced notation into the PVW, we review the original MULF prestressing and subsequent deformation stage as

$$\begin{aligned}&\mathrm {In \ prestressing \ stage, \ find} \ \varvec{F}_p \ \mathrm {such \ that}: \nonumber \\&\int _{\Omega _0} \varvec{P}_{u,F}(\varvec{0}, \varvec{F}_p) : \nabla \delta \varvec{u} \ dV - \int _{\Gamma _{p, 0}} \varvec{T}(\varvec{0}, p_{\mathrm {dia}}) \cdot \delta \varvec{u} \ dA = 0 \quad \forall \delta {\varvec{u}}, \end{aligned}$$

(22)

$$\begin{aligned}&\mathrm {In \ deformation \ stage, \ find} \ \varvec{u}_d \ (\mathrm {with\ given} \ \varvec{F}_p) \ \mathrm {such \ that}: \nonumber \\&\int _{\Omega _0} \varvec{P}_{u,F}(\varvec{u}_d, \varvec{F}_p) : \nabla \delta \varvec{u} \ dV - \int _{\Gamma _{p, 0}} \varvec{T}(\varvec{u}_d, p_{\mathrm {sys}}) \cdot \delta \varvec{u} \ dA = 0 \quad \forall \delta {\varvec{u}}. \end{aligned}$$

(23)

Equation (22) implicitly defines the prestress deformation gradient \(\varvec{F}_p\), which is evaluated applying an assumed diastolic blood pressure load \(\varvec{T}({\varvec{0}}, p_{\mathrm {dia}})\) at the known imaged geometry. Equation (23) utilizes the precomputed deformation gradient \(\varvec{F}_p\) in order to evaluate the deformation stage displacement field \(\varvec{u}_d\) applying an assumed systolic blood pressure load \(\varvec{T}({\varvec{u}}_d, p_{\mathrm {sys}})\) at the deformed geometry.

Recalling Eqs. (20) and (21), we can equivalently state the prestressing and deformation stage PVW as

$$\begin{aligned}&\mathrm {In \ prestressing \ stage, \ find} \ \varvec{u}_p \ \mathrm {such \ that}: \nonumber \\&\int _{\Omega _0} \varvec{P}_{u}(\varvec{u}_p) : \nabla \delta \varvec{u} \ dV - \int _{\Gamma _{p, 0}} \varvec{T}(\varvec{0}, p_{\mathrm {dia}}) \cdot \delta \varvec{u} \ dA = 0 \quad \forall \delta {\varvec{u}}, \end{aligned}$$

(24)

$$\begin{aligned}&\mathrm {In \ deformation \ stage, \ find} \ \varvec{u}_d \ (\mathrm {with\ given}\ \varvec{u}_p) \ \mathrm {such \ that}: \nonumber \\&\int _{\Omega _0} \varvec{P}_{u}(\varvec{u}_d + \varvec{u}_p) : \nabla \delta \varvec{u} \ dV - \int _{\Gamma _{p, 0}} \varvec{T}(\varvec{u}_d, p_{\mathrm {sys}}) \cdot \delta \varvec{u} \ dA = 0 \quad \forall \delta {\varvec{u}}. \end{aligned}$$

(25)

A comparison of (22), (23) with (24), (25) reveals the following. Instead of seeking a prestress deformation gradient \(\varvec{F}_p\), we solve for a virtual prestress displacement field \(\varvec{u}_p\) fulfilling the PVW at a diastolic blood pressure load of the imaged geometry \(\varvec{T}({\varvec{0}}, p_{\mathrm {dia}})\). \(\varvec{u}_p\) is then used in the deformation stage to account for the stress in the imaged configuration at a systolic blood pressure load of the deformed configuration \(\varvec{T}(\varvec{u}_d, p_{\mathrm {sys}})\).

We emphasize that the reformulation from (22), (23) to (24), (25) corresponds to a mathematical transformation of variables, physics remains unchanged. We also emphasize that both formulations are a well-posed approximation to the ill-posed inverse design problem as further detailed in [14]. From the perspective of projection-based MOR however, formulation (24), (25) enables a collection of virtual prestress displacement mode snapshots, an essential step in the data-driven construction of the ROB.

### Finite element discretization

Applying the usual finite element discretization to the PVW for the MULF prestressing and deformation stage gives

$$\begin{aligned}&\mathrm {In \ prestressing \ stage, \ find} \ \varvec{u}_p = \sum \nolimits _{e \in \mathcal {E}} {\varvec{u}}^{(e)}_p \ \mathrm {such \ that}: \nonumber \\&\sum _{e \in \mathcal {E}} \int _{\Omega _0^{(e)}} \varvec{P}_{u}(\varvec{u}_p^{(e)}) : \nabla \delta \varvec{u}^{(e)} \ dV \nonumber \\&- \sum _{e \in \mathcal {F}} \int _{\Gamma _{p, 0}^{(e)}} \varvec{T}(\varvec{0}, p_{\mathrm {dia}}) \cdot \delta \varvec{u}^{(e)} \ dA = 0 \quad \forall \delta {\varvec{u}}^{(e)}, \end{aligned}$$

(26)

$$\begin{aligned}&\mathrm {In \ deformation \ stage, \ find} \ \varvec{u}_d = \sum \nolimits _{e \in \mathcal {E}} {\varvec{u}}^{(e)}_d \ (\mathrm {with\ given}\ \varvec{u}_p^{(e)}) \ \mathrm {such \ that}: \nonumber \\&\sum _{e \in \mathcal {E}} \int _{\Omega _0^{(e)}} \varvec{P}_{u}(\varvec{u}_d^{(e)} + \varvec{u}_p^{(e)}) : \nabla \delta \varvec{u}^{(e)} \ dV \nonumber \\&- \sum _{e \in \mathcal {F}} \int _{\Gamma _{p, 0}^{(e)}} \varvec{T}(\varvec{u}_d^{(e)}, p_{\mathrm {sys}}) \cdot \delta \varvec{u}^{(e)} \ dA = 0 \quad \forall \delta {\varvec{u}}^{(e)}, \end{aligned}$$

(27)

wherein \(\varvec{u}^{(e)} = \varvec{\Phi }^{(e)} \varvec{d}^{(e)}, \delta \varvec{u}^{(e)} = \varvec{\Phi }^{(e)} \delta \varvec{d}^{(e)}\) are the continuous element-wise displacement field and weighting function, which are interpolated by finite element shape functions contained in \({\varvec{\Phi }}^{(e)}\) and the element-wise displacement and weighting degree of freedom (DOF) vectors \(\varvec{d}^{(e)}, \delta \varvec{d}^{(e)}\), respectively. Furthermore, we introduced the computational domain mesh element set \(\mathcal {E}\) as well as the set \(\mathcal {F}\) of elements loaded by the pressure load boundary condition.

Given element-wise internal and external force vectors such that

$$\begin{aligned} \begin{aligned}&{\varvec{f}}_{\mathrm {int}}^{(e)}(\varvec{d}^{(e)}) \cdot \delta \varvec{d}^{(e)} =\int _{\Omega _0^{(e)}} \varvec{P}_{u}({\varvec{\Phi }}^{(e)} \varvec{d}^{(e)}) : \nabla ({\varvec{\Phi }}^{(e)} \delta \varvec{d}^{(e)}) \ dV \quad \forall \delta \varvec{d}^{(e)}, \end{aligned} \end{aligned}$$

(28)

$$\begin{aligned} \begin{aligned}&{\varvec{f}}_{\mathrm {ext}}^{(e)}(\varvec{d}^{(e)}, p) \cdot \delta \varvec{d}^{(e)} = \int _{\Gamma _{p, 0}^{(e)}} \varvec{T}({\varvec{\Phi }}^{(e)} \varvec{d}^{(e)}, p) \cdot ({\varvec{\Phi }}^{(e)} \delta \varvec{d}^{(e)}) \ dA \quad \forall \delta \varvec{d}^{(e)}, \end{aligned} \end{aligned}$$

(29)

Eqs. (26) and (27) in assembled form read

$$\begin{aligned}&\mathrm {In \ prestressing \ stage, \ find} \ {\varvec{d}}_p \ \mathrm {such \ that}: \nonumber \\&{\varvec{f}}_{\mathrm {int}}({\varvec{d}}_p) \cdot \delta {\varvec{d}} - {\varvec{f}}_{\mathrm {ext}}({\varvec{0}}, p_{\mathrm {dia}}) \cdot \delta {\varvec{d}} = 0 \quad \forall \delta \varvec{d}\end{aligned}$$

(30)

$$\begin{aligned}&\quad \Rightarrow {\varvec{f}}_{\mathrm {int}}({\varvec{d}}_p) - {\varvec{f}}_{\mathrm {ext}}({\varvec{0}}, p_{\mathrm {dia}}) = {\varvec{0}}, \end{aligned}$$

(31)

$$\begin{aligned}&\mathrm {In \ deformation \ stage, \ find} \ \varvec{d}_d \ (\mathrm {with\ given}\ \varvec{d}_p)\ \mathrm {such \ that}: \nonumber \\&{\varvec{f}}_{\mathrm {int}}({\varvec{d}}_d + {\varvec{d}}_p) \cdot \delta {\varvec{d}} - {\varvec{f}}_{\mathrm {ext}}({\varvec{d}}_d, p_{\mathrm {sys}}) \cdot \delta {\varvec{d}} = 0 \quad \forall \delta \varvec{d}\end{aligned}$$

(32)

$$\begin{aligned}&\quad \Rightarrow {\varvec{f}}_{\mathrm {int}}({\varvec{d}}_d + {\varvec{d}}_p) - {\varvec{f}}_{\mathrm {ext}}({\varvec{d}}_d, p_{\mathrm {sys}}) = {\varvec{0}}. \end{aligned}$$

(33)

Thereby the global internal force vector \(\varvec{f}_{\mathrm {int}} = \sum _{e \in \mathcal {E}} \varvec{L}^{(e)} \varvec{f}_{\mathrm {int}}^{(e)} \), global external force vector \(\varvec{f}_{\mathrm {ext}} = \sum _{e \in \mathcal {F}} \varvec{L}^{(e)} \varvec{f}_{\mathrm {ext}}^{(e)} \), global displacement DOF vector \(\varvec{d}= \sum _{e \in \mathcal {E}} \varvec{L}^{(e)} \varvec{d}^{(e)}\) as well as global weighting DOF vector \(\delta \varvec{d}= \sum _{e \in \mathcal {E}} \varvec{L}^{(e)} \delta \varvec{d}^{(e)}\) result from an assembly of the corresponding element-wise vectors, while \(\varvec{L}^{(e)}\) is the usual finite element assembly operator towards the global system.

Summarizing, we denote the high-fidelity finite element model residual as

$$\begin{aligned} \varvec{r}: \left\{ \begin{array}{ll} \mathbb {R}^{N} \rightarrow \mathbb {R}^{N} \\ \\ \mathrm {for \ prestressing \ stage}: \\ \varvec{d}_p \mapsto \varvec{f}_{\mathrm {int}} (\varvec{d}_p) - {\varvec{f}}_{\mathrm {ext}}({\varvec{0}}, p_{\mathrm {dia}}) \\ \mathrm {for \ deformation \ stage\ (with\ given}\ \varvec{d}_p): \\ \varvec{d}_d \mapsto \varvec{f}_{\mathrm {int}} (\varvec{d}_d + \varvec{d}_p) - {\varvec{f}}_{\mathrm {ext}}({\varvec{d}}_d, p_{\mathrm {sys}}) \end{array} \right. , \end{aligned}$$

(34)

wherein the deformation stage only can be evaluated after the prestressing stage, which yields the virtual prestress displacement field \(\varvec{d}_p\) as a solution. The nonlinear finite element system of equations in residual form reads

$$\begin{aligned} \varvec{r}(\varvec{d}) = \varvec{0} \end{aligned}$$

(35)

and is solved applying Newton-Raphson iterations.