The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training (Zhang et al., 2024). We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers (Lee et al., 2025a), which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner, which can reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers on complex curved domains, performing comparably to state-of-the-art algebraic multigrid preconditioners, and substantially improving on other widely used standard preconditioners, such as Jacobi or SOR, in both number of iterations and computational time. We expect that the findings in this study lead to more efficient hybrid preconditioners that complement the convergence properties of current gold-standard preconditioning methods for solid mechanics PDEs in three dimensions. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold-standard preconditioning methods when applied to solid mechanics PDEs.
Acute ascending aortic dissection (AAD) is a life-threatening cardiovascular emergency characterized by the delamination of the aortic wall and the subsequent formation of a false lumen. A hallmark histopathological feature of AAD is the localized accumulation of glycosaminoglycans (GAGs) and proteoglycans within the middle layer, a phenomenon known as translamellar mucoid extracellular matrix accumulation (T-MEMA). It is hypothesized that T-MEMA induces local swelling via Donnan osmotic pressure, generating highly localized radial stresses that predispose the tissue to delamination. In this study, we developed a multiscale computational framework to investigate the mechanistic implications of T-MEMA in the ascending aorta. By integrating finite element simulations with a Donnan-based swelling formulation, we quantified how GAG-induced swelling alters macro-scale stress distributions and exacerbates micro-scale stresses within interlamellar components, specifically the radial elastin struts. Our analysis establishes a quantitative relationship between histopathological observations and the biomechanical triggers of AAD, demonstrating that the transmural location of T-MEMA significantly determines the propensity for wall delamination.
The design of high-performance structures capable of enduring severe thermoelastic environments is of paramount importance in aerospace and mechanical engineering. However, multi-material topology optimization (TO) in such strongly coupled scenarios remains challenging due to mesh distortion-induced numerical singularities under large deformations and the intricate treatment of temperature-dependent material properties (TDMPs). To address these issues, this work proposes a strongly coupled thermoelastic TO framework based on a novel unified nonlinear multi-material (UNM) interpolation scheme. In contrast to conventional approaches, the UNM method employs a kinematic scaling technique, which effectively mitigates numerical instabilities in low-stiffness regions and allows the use of the considered constitutive models, including modified Saint Venant–Kirchhoff, neo-Hookean, and Ogden hyperelastic materials. A strongly coupled thermoelastic formulation is developed based on the multiplicative decomposition of the deformation gradient, thereby incorporating TDMPs and the influence of large deformation on heat conduction paths. Fully coupled adjoint sensitivities for end-compliance and stored-strain–energy objectives are derived. In the numerical cases investigated, kinematic scaling is shown to improve convergence under large deformation. For the prescribed material laws and loading conditions, accounting for deformation-dependent heat conduction and TDMPs leads to differences in the computed responses and optimized layouts. The numerical examples also exhibit structural bifurcation, boundary detachment, and load-direction-dependent material distributions. An additional exponential-integral thermal-expansion example demonstrates the applicability of the proposed framework to an alternative prescribed constitutive law. Overall, the proposed framework provides a computational basis for investigating material-specific thermoelastic designs.
We present a computational formulation for the coupled thermo-viscoelastic analysis of geometrically exact shape-memory polymer beam systems. The proposed formulation is based on a primal isogeometric collocation approach in which the temperature field is obtained from a transient thermal balance accounting for heat conduction, convection, and thermal inertia. To reproduce the shape-memory effect of polymeric structures, the model exploits a temperature-dependent generalized Maxwell viscoelastic model, which relies on the Time–Temperature Superposition Principle. Spatial discretization of the strong-form governing equations through isogeometric collocation preserves the exact geometric representation and high-order accuracy of isogeometric analysis, while avoiding costly element-wise numerical integration. A consistent thermomechanical linearization on the SO(3) manifold, combined with a trapezoidal time-integration scheme, ensures robustness for large rotations and strongly temperature-dependent viscoelastic responses. Numerical examples demonstrate the accuracy and efficiency of the proposed formulation, ranging from simple thermomechanical beam problems to complex programmed morphing cycles. Particular attention is devoted to stent-like shape-memory polymer structures, highlighting the role of heat transfer and material behavior in determining deployment time.
