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

Abstract

We present a domain decomposition method for the fast simulation of large two-dimensional 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, which allows 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 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 based on the matrix discrete empirical interpolation method. The ROM surrogate is trained offline and can be reused for any geometric mapping, restricting the expensive quadrature on cut elements to the training stage. A stabilization term is introduced to ensure the scalability of the solver when using the ROM approximation, at the cost of a small and controllable error. We validate the method through a series of numerical experiments and demonstrate its performance on a 2D problem with more than 17,000 cells of varying geometry, which is solved in approximately 30 seconds on a standard laptop. The number of solver iterations grows only mildly as the number of subdomains increases, provided the ratio between subdomain and mesh sizes is kept constant, consistent with the scalability properties of BDDC methods.

44 pages, 21 figures, 5 algorithms