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.

翻译：沉浸边界法是计算力学中用于模拟几何复杂问题的高阶精确计算方法。传统有限元方法需要构建高质量边界适配网格，而沉浸边界法则将计算域嵌入背景网格中。基于插值的沉浸边界法通过提取技术，将背景基函数表示为前景网格上定义的拉格朗日多项式的线性组合，从而生成可被现有方法轻松集成的插值基函数，实现了对现有有限元软件的非侵入式沉浸边界功能扩展。本研究将基于插值的沉浸边界法拓展至多材料及多物理场问题。从域几何的水平集描述出发，引入海维赛德富集以处理状态变量场跨材料界面的不连续性。采用截断层次B样条的自适应细化，既能改善界面几何表示，又可解析界面附近的大梯度解。多物理场问题通常涉及耦合场，每个场具有独特的离散化需求。本文提出一种针对耦合问题的新型离散化方法，通过应用提取技术，对所有场统一使用单一前景网格。数值算例展示了该方法在二维和三维热传导、线弹性力学及热-力耦合问题中的最优收敛速率。通过复合材料试样的图像分析验证了该方法的应用价值：除规避传统网格生成难题外，相较于经典边界适配有限元法，该方法还能有效降低所需自由度。