面向多材料与多物理场问题的基于插值的浸入几何分析方法
Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems
AI总结：
本文将基于插值的浸入边界方法扩展到多材料与多物理场问题，通过Heaviside增强、截断分层B样条自适应细化以及单一前景网格提取技术，实现耦合场的高效离散，并在2D和3D数值算例中验证了最优收敛率及减少自由度的优势。
摘要：
浸入边界方法是一类高阶精确的计算工具，用于对计算力学中几何复杂的问题进行建模。传统有限元方法要求构造高质量的贴体网格，而浸入边界方法则将计算域嵌入背景网格中。基于插值的浸入边界方法通过提取（extraction）对现有有限元软件进行非侵入式扩展，以实现浸入边界能力。提取将背景基函数插值为定义在前景网格上的拉格朗日多项式的线性组合，从而生成一个可被现有方法轻松积分的插值基函数。本工作将基于插值的浸入边界方法扩展到多材料与多物理场问题。从域几何的水平集描述出发，实现了Heaviside增强以处理状态变量场在材料界面处的不连续性。采用截断分层B样条的自适应细化既用于改善界面几何表示，也用于解析界面附近的大解梯度。多物理场问题通常涉及耦合场，其中每个场具有独特的离散化要求。本工作通过应用提取，使用单一前景网格对所有场进行离散，提出了一种新颖的耦合问题离散化方法。数值算例表明，该方法在二维和三维热传导、线弹性以及耦合热-力学问题中均达到最优收敛率。通过对复合材料样本进行基于图像的分析，证明了该方法的实用性：除了规避典型的网格划分困难外，与经典贴体有限元方法相比，该方法还减少了所需的自由度数量。
英文摘要：
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.
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