Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

TL;DR

The paper introduces an interpolation-based immersed boundary method tailored for multi-material and multi-physics problems by combining level-set geometry, Heaviside enrichment, and truncated hierarchical B-splines (THB-splines). It interpolates enriched background bases onto a single boundary-fitted foreground mesh via Lagrange extraction, enabling coupling of fields with different discretization needs without remeshing. Numerical results in 2D and 3D for heat conduction, linear elasticity, and thermo-mechanical coupling demonstrate optimal convergence and reduced degrees of freedom compared to boundary-fitted FEM, including image-based composite analyses. The approach integrates easily with existing FEM codes and offers a practical path to accurate, mesh-efficient simulations of complex multi-material systems.

Abstract

Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed boundary method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchical B-splines is used to both improve interface geometry representations and resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom compared to classical boundary-fitted finite element methods.

Figures (15)

• Figure 1: Local refinement is applied through truncated hierarchically-refined B-splines (THB).
• Figure 2: A B-spline basis is defined on a structured background mesh and an example function $B_k$ is depicted in (a). Using the geometry description of the material subdomains in (b), Heaviside enrichment is applied to form the discontinuous functions $\psi^1_k B_k$ and $\psi^2_kB_k$, depicted in (c) and (d) respectively. A Lagrange foreground function space is defined on the boundary-fitted mesh in (e). The function space is used to interpolate the enriched background functions $\Hat{\psi}^1_k \Hat{B_k}$ and $\Hat{\psi}^2_k \Hat{B_k}$, depicted in (f) and (g), respectively.
• Figure 3: A series of uniformly refined meshes $\mathcal{K}^l$ are shown in the top row. A series of nested subdomains, $\Omega^{l+1}_T \subseteq \Omega^l_T$, shown in the second row, and $\Omega^{l+1}_u \subseteq \Omega^l_u$, shown in the third row, are defined for each state variable. For the decomposition mesh the series of subdomains $\Omega^l_{D}$, shown in the last row, is defined such that the domain on each level $l$ contains the union of the $l^{\text{th}}$ level domains for both each variable field. The hierarchically refined meshes $\mathcal{K}_T$, $\mathcal{K}_u$, and $\mathcal{K}_D$, shown in the rightmost column, are constructed with the mesh series $\mathcal{K}^l$ in the top row and their respective subdomain series.
• Figure 4: Foreground integration meshes are formed by triangulating cut elements of the decomposition mesh $\mathcal{K}_{D}$. (a) A cell is intersected by the isocontour of the discretized level set function $\phi^h = \phi_t$ defining a material interface. (b) The cell is subdivided into triangular cells and isocontour-edge intersections (indicated by black cicles) computed. (c) Using the intersection points as nodal points, the cell is further subdivided.
• Figure 5: The geometric configurations for the 2D (left) and 3D (right) domains. A three-material beam is embedded in a structured background grid and rotated such that material interfaces do not align with element edges (in 2D) or facets (in 3D) . Elements intersected by the level set functions defining the beam geometry are triangulated to form a boundary-fitted foreground mesh.