Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

TL;DR

This paper introduces an immersed isogeometric finite element framework using hierarchical B-splines and XFEM for accurate simulation of complex multi-material and multi-physics problems with local refinement and complex geometries.

Contribution

It develops a novel hierarchical B-spline discretization approach combined with XFEM for multi-material, multi-physics problems, enabling local refinement and complex geometry handling without conformal meshing.

Findings

Validated accuracy on 2D and 3D elastic and thermo-elastic problems

Demonstrated applicability to large, complex geometries

Enriched B-spline bases effectively handle small features and multi-material interfaces

Abstract

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an eXtended Finite Element Method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.