具有多个不可穿透障碍物的多域有限元-边界元耦合
Multi-domain FEM-BEM coupling with several impenetrable obstacles
AI总结：
研究针对更一般几何和材料配置的多域有限元-边界元耦合问题，设计新变分公式，提出广义优化施瓦茨方法（GOSM），该方法可处理多域耦合及障碍物边界条件，是模拟声波传播的通用灵活框架。
AI中文摘要：
在最近的一篇论文中，我们分析了一种用于亥姆霍兹问题的有限元和边界元方法（FEM-BEM）耦合的新公式，该公式涉及多个异质有界子域和一个均匀无界子域。这种称为广义优化施瓦茨方法（GOSM）的公式是结构化的，其未知数与子域界面相关联。为了推导GOSM，第一步是证明亥姆霍兹问题的解满足特定的多域变分公式，该公式为每个子域涉及一个算子，每个算子相互独立。在本论文中，我们为更一般的几何和材料配置设计了一种这种类型的变分公式：允许有多个异质有界子域、不可穿透障碍物和均匀子域。我们假设只有一个子域是无界的，并且其边界是有界的。域划分可以有交叉点，即至少三个子域相邻的点。我们还证明了可以从多域变分公式的解中恢复初始亥姆霍兹问题的解。这表明GOSM是一个通用且灵活的框架来模拟声波传播，因为它既可以处理多域FEM-BEM耦合，也可以处理多个障碍物上的（弱施加）边界条件。
英文摘要：
In a recent paper we have analyzed a new formulation of the coupling of finite and boundary element methods (FEM-BEM) for Helmholtz problems, involving several heterogeneous bounded subdomains and one homogeneous unbounded subdomain. This formulation, called Generalized Optimized Schwarz Method (GOSM), is substructured, that is, its unknowns are associated with the subdomains interfaces. To derive the GOSM, the first step was to prove that a solution to the Helmholtz problem satisfies a specific multi-domain variational formulation, which involves one operator for each subdomain, each operator being independent of the others. In the present contribution, we design a variational formulation of that type for a more general geometrical and material configuration: several heterogeneous bounded subdomains, impenetrable obstacles and homogeneous subdomains are allowed. Note that, like in our recent paper, we assume that only one subdomain is unbounded, and that its boundary is bounded. The domain partition can have cross-points, that is, points where at least three subdomains are adjacent. We also prove that a solution to the initial Helmholtz problem can be recovered from a solution to the multi-domain variational formulation. This shows that the GOSM is a general and flexible framework to model acoustic wave propagation, as it can handle both multi-domain FEM-BEM coupling and (weakly imposed) boundary conditions on several obstacles.
