Geometry-Preserving Reduced-Order Modeling via Immersed Tensor Decomposition (ITD)

2026-06-25

AI Summary

Traditional body-fitted finite element methods rely on high-quality meshes, rendering them computationally expensive for CAD models with geometric imperfections, image data, or voxelized representations. Existing immersed methods, while avoiding meshing, typically disrupt the tensor-product structure of the underlying discretization, hindering compatibility with efficient reduced-order modeling. This work proposes an Immersed Tensor Decomposition (ITD) framework that integrates a meshfree geometric representation with a separable C-HiDeNN-TD reduced-order solver on a regular Cartesian voxel grid. Complex geometries are encoded through a three-step process involving a signed distance function, a boundary-enforcing function Φ, and Tucker decomposition. Dirichlet boundary conditions are strongly enforced by multiplying trial functions with Φ—eliminating the need for penalty terms or boundary integrals. The approach preserves the tensor structure of the background grid, enabling unified high-order accuracy and model order reduction, and demonstrates optimal convergence and robustness for non-Cartesian geometries in both 2D and 3D benchmark domains.

Abstract

Body-fitted finite-element methods deliver high-order accuracy but hinge on a clean, watertight, conforming mesh, a requirement that breaks down for the geometrically imperfect CAD assemblies, image-based volumetric data, and voxel-native designs that pervade biomedical engineering and additive manufacturing, where mesh generation has become the dominant cost of the analysis cycle. Immersed methods on regular background Cartesian grids sidestep body-fitted meshing, but classical implementations integrate over irregular cut subdomains, destroying the tensor-product structure that enables separable, reduced-order methods such as tensor decomposition. In this paper we propose the \emph{Immersed Tensor Decomposition} (ITD) framework, which couples a mesh-free geometric representation via body-fitted function with the separable C-HiDeNN-TD reduced-order solver to enable large-scale simulation directly on regular background voxel meshes. The geometry is encoded in three steps: a signed-distance function represents the boundary, a body-fitted function $Φ$ approximates it with controllable error, and a low-rank Tucker decomposition provides model-order reduction; for a fixed grid spacing $h$, accuracy is improved by raising the approximation order of C-HiDeNN interpolation up to degree $p$ with a linear background mesh. The central contribution is an exact Dirichlet formulation that enforces the boundary condition strongly by multiplying the trial function with $Φ$, so that $u=g$ holds by construction without any variational penalty or interface quadrature. We establish an a priori error estimate for the formulation and assess it on canonical 2D/3D domains, demonstrating optimal convergence and robustness on non-Cartesian geometries discretized by regular voxel meshes.

Problem

• immersed methods
• geometry representation
• tensor decomposition
• mesh generation
• reduced-order modeling

Innovation

• Immersed Tensor Decomposition
• geometry-preserving
• reduced-order modeling
• strong Dirichlet enforcement
• voxel-based simulation