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

TL;DR

This paper introduces an advanced interpolation-based immersed boundary method for multi-material and multi-physics problems, enabling high accuracy and efficiency in complex geometries without complex meshing.

Contribution

It extends immersed boundary methods to multi-material and multi-physics problems using a single foreground mesh and extraction techniques, with adaptive refinement for improved interface representation.

Findings

Achieved optimal convergence rates in 2D and 3D simulations.

Reduced degrees of freedom compared to classical boundary-fitted methods.

Demonstrated effectiveness in image-based composite analysis.

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.