Enriched Immersed Finite Element and Isogeometric Analysis -- Algorithms and Data Structures

TL;DR

The paper addresses the challenge of performing reliable immersed finite element analysis on geometrically complex, multi-material domains by introducing a robust preprocessing framework that generates conforming foreground meshes, topology information, and enriched basis functions. It combines a Nitsche-based weak form with generalized Heaviside enrichment and face-oriented ghost stabilization, organized around a tessellation-driven pipeline that yields clusters with custom quadrature rules suitable for standard FE assembly. Key contributions include the background ancestry concept, regular and templated subdivision for conformal mesh generation, flood-fill-based enrichment, and a parallelizable workflow with demonstrated robustness in edge cases and strong scalability results. The framework enables efficient, stable analysis of topology-optimized and multi-material microstructures, with practical impact in computational mechanics where complex interfaces and evolving geometries are common.

Abstract

Immersed finite element methods provide a convenient analysis framework for problems involving geometrically complex domains, such as those found in topology optimization and microstructures for engineered materials. However, their implementation remains a major challenge due to, among other things, the need to apply nontrivial stabilization schemes and generate custom quadrature rules. This article introduces the robust and computationally efficient algorithms and data structures comprising an immersed finite element preprocessing framework. The input to the preprocessor consists of a background mesh and one or more geometries defined on its domain. The output is structured into groups of elements with custom quadrature rules formatted such that common finite element assembly routines may be used without or with only minimal modifications. The key to the preprocessing framework is the construction of material topology information, concurrently with the generation of a quadrature rule, which is then used to perform enrichment and generate stabilization rules. While the algorithmic framework applies to a wide range of immersed finite element methods using different types of meshes, integration, and stabilization schemes, the preprocessor is presented within the context of the extended isogeometric analysis. This method utilizes a structured B-spline mesh, a generalized Heaviside enrichment strategy considering the material layout within individual basis functions$'$ supports, and face-oriented ghost stabilization. Using a set of examples, the effectiveness of the enrichment and stabilization strategies is demonstrated alongside the preprocessor$'$s robustness in geometric edge cases. Additionally, the performance and parallel scalability of the implementation are evaluated.

Figures (39)

• Figure 1: Fiber composite weave with two types of fibers and a binding matrix material. The matrix material is shown in off-white and partially removed for better visibility.
• Figure 2: Cross-section of the fiber weave shown in \ref{['fig:fiber_patch_3D']} with a coarse background mesh. Details: (A) Cut background element occupied by three different materials. (B) Non-trivially intersected background element with highlighted material interface $\Gamma^I_{1,2}$. (C) Sub-element scale material features in neighboring elements. Highlighted is a single basis function's support $\mathrm{supp}(N)$ which spans multiple features. (D) Cut background element where one material subdomain occupies only a small volume fraction.
• Figure 3: Enrichment strategy for a single basis function for the material layout shown in \ref{['fig:fiber_patch_cross_section']}, detail (C). The basis function's support is shown in dashed blue lines. The support contains four disconnected material subdomains $R_{B}^{\varepsilon}$ for each of which an enriched basis function is defined, where $\widetilde{N}_{B}^{\varepsilon} = \psi^{\varepsilon}_{B} N_{B}$ with $\mathrm{supp}(\widetilde{N}_{B}^{\varepsilon}) = R_{B}^{\varepsilon}$
• Figure 4: Concept of a cluster: a background element with a single active subdomain $S_{E}^{u}$, a set of enriched basis functions supported inside that subdomain, and a quadrature rule associated with either that subdomain (A, B) or any part of the subdomain's boundary $\partial S_{E}^{u}$ (C). Green indicates the volume or surface integrated over; blue crosses represent exemplary quadrature points. Additionally, linked pairs of clusters are constructed for interfaces (D).
• Figure 5: Ghost faces for an example material layout. (A) Material layout of a two-material problem with materials $\mathcal{M} = \left\{ 1,2 \right\}$ occupying a domain smaller than the ambient domain $\mathcal{A}$ filled by the background mesh. The white domain $\Omega_{0}$ is assumed to be void. (B) The set of interior element faces $\mathcal{F}_{G}$ which the ghost penalty is applied to. (C) Exemplary ghost facet $F$ and the material layout in its adjacent background elements.