Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities or other non-trivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromecanics model.

翻译：多物理场模拟通常需要在具有非匹配空间离散的子问题之间传递解场，通常采用插值技术。标准方法通常基于欧氏距离衡量点之间的接近程度，但欧氏距离无法考虑曲率、切割、空洞或其他非平凡的几何或拓扑特征。这可能导致插值函数在这些特征附近产生虚假振荡。为解决此问题，我们提出了一种对重新缩放局部径向基函数（RL-RBF）插值的改进方法，通过适应几何和拓扑特征的一致性及保真度来考虑插值域的几何特性。所提出的方法被称为RL-RBF-G，其依赖于测量数据点之间的测地距离。RL-RBF-G消除了RL-RBF插值中出现的虚假振荡，从而在具有复杂几何的域中提高了精度。我们在理想化设置下通过收敛性研究展示了RL-RBF-G插值的有效性。此外，我们讨论了在分布式内存并行框架中RL-RBF-G插值的算法方面和实现，并给出了接近理想结果的强可扩展性测试结果。最后，通过考虑应用于全心心脏电机械模型的多物理场模拟，展示了RL-RBF-G插值的有效性。