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

• Published in: Computational mechanics Vol. 75; no. 1; pp. 301 - 325
• Main Authors: Fromm, Jennifer E., Wunsch, Nils, Maute, Kurt, Evans, John A., Chen, Jiun-Shyan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2025; Springer Nature B.V
• Subjects: B spline functions; Classical and Continuum Physics; Computational mechanics; Computational Science and Engineering; Conduction heating; Conductive heat transfer; Discretization; Engineering; Finite element method; Interpolation; Methods; Original Paper; Partial differential equations; Physics; Polynomials; Software; Theoretical and Applied Mechanics; Thermomechanical analysis; Immersed finite element method; XIGA; Lagrange extraction; Multi-material problems; Multi-physics problems
• ISSN: 0178-7675; 1432-0924; 1432-0924
• DOI: 10.1007/s00466-024-02506-z

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 structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured 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 isogeometric 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 hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to 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 partial differential equations representing 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 when compared to classical boundary-fitted finite element methods.