A ROM-based BDDC solver for unfitted p-FEM level-set-based lattice structures

TL;DR

This paper introduces a fast, scalable domain decomposition solver for large lattice structures modeled with unfitted p-FEM and level set functions, utilizing ROM techniques for efficiency.

Contribution

It develops a ROM-accelerated BDDC solver for unfitted p-FEM lattice structures that handles complex geometries without homogenization assumptions.

Findings

Solver efficiently handles 17,000+ cells with high accuracy.

Scalability is maintained as the number of subdomains increases.

ROM reduces assembly time, enabling rapid simulations.

Abstract

We present a domain decomposition method for the fast simulation of large lattice structures described by level set functions. The method does not rely on homogenization or multiscale techniques, and therefore avoids their underlying assumptions such as scale separation and periodicity. Individual cells are defined through level set functions and mapped into physical space using arbitrary order mappings, allowing the creation of complex graded designs with varying geometries and topologies. The discretization is based on unfitted p-FEM, where each cell is approximated by a single high order element. This choice naturally handles the implicit geometric description and provides high accuracy with a moderate number of degrees of freedom. The solver is built on the Balanced Domain Decomposition by Constraints (BDDC) method, where each cell corresponds to one subdomain. To accelerate the assembly of the cell stiffness matrices, we combine a fast assembly technique that separates the contributions of the geometric mapping from the trimmed domain with a reduced order model (ROM) based on the matrix discrete empirical interpolation method (MDEIM). The ROM surrogate is trained offline and reused for any geometric mapping, restricting the expensive quadrature on cut elements to the training stage. A stabilization term ensures the scalability of the solver when using the ROM approximation, at the cost of a small and controllable error. We validate the method through numerical experiments and demonstrate its performance on a complex 2D problem with more than 17,000 cells of varying geometry, solved in approximately 30 seconds on a standard laptop. The number of solver iterations remains bounded as the number of subdomains grows, provided the ratio between subdomain and mesh sizes is kept constant, in agreement with classical BDDC scalability properties.