  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.
