Efficient Numerical Modeling of Acoustic Scattering in Heterogeneous Media with Multi-Domain FEM-BEM Coupling

Core Concepts

The authors propose and analyze new multi-domain FEM-BEM coupling formulations that combine boundary integral equations for homogeneous subdomains with volume variational formulations for heterogeneous subdomains, allowing for the presence of cross-points.

Abstract

The content presents a numerical approach for modeling time-harmonic acoustic scattering by an object composed of piecewise homogeneous parts and an arbitrarily heterogeneous part. The key highlights are:

The authors extend the classical Costabel FEM-BEM coupling to a multi-domain configuration, where the computational domain is subdivided into multiple subdomains, some of which are treated with boundary integral equations and others with finite elements.

The multi-domain formulations are designed using the multi-trace formalism, which allows for a clean treatment of cross-points (points where three or more subdomains are adjacent) from the perspective of function spaces.

The proposed formulations satisfy Gårding inequalities, ensuring stability and quasi-optimal convergence of conforming discretization methods, provided the associated operators are injective.

The authors identify conditions for injectivity and construct modified versions of the formulations that are immune to spurious resonances, a phenomenon that can affect the classical Costabel coupling.

The analysis covers both the case where only the exterior unbounded subdomain is treated with the BEM, as well as the more general case where all homogeneous parts of the scattering object are treated with the BEM.

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Key Insights Distilled From

Multi-domain FEM-BEM coupling for acoustic scattering

by Marcella Bon... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2305.09278.pdf

Deeper Inquiries

How can the proposed multi-domain FEM-BEM coupling formulations be extended to handle time-dependent acoustic scattering problems

The proposed multi-domain FEM-BEM coupling formulations can be extended to handle time-dependent acoustic scattering problems by incorporating the time variable into the formulation. In time-harmonic acoustic scattering, the wave equation is typically solved in the frequency domain, but for time-dependent problems, the wave equation needs to be solved in the time domain. This can be achieved by introducing a time-dependent term into the governing equations and adapting the boundary integral formulations accordingly. The time-dependent acoustic scattering problem can be solved using a similar approach as the one presented for the time-harmonic case, but with additional considerations for the temporal evolution of the acoustic waves. Time-stepping methods such as the finite difference method or the finite element method can be used to discretize the time domain, while the boundary element method can handle the spatial domain efficiently.

What are the computational advantages and limitations of the multi-domain approach compared to a single-domain FEM or BEM discretization for large-scale acoustic scattering problems

The multi-domain approach offers several computational advantages for large-scale acoustic scattering problems compared to a single-domain FEM or BEM discretization.
Advantages:

Efficient Handling of Heterogeneous Media: The multi-domain approach allows for the modeling of complex media with varying material properties in different subdomains. This flexibility is crucial for accurately representing real-world acoustic scattering scenarios.
Reduced Computational Cost: By dividing the computational domain into multiple subdomains, the problem can be solved more efficiently using domain-specific discretization methods. This can lead to a reduction in computational resources and time.
Improved Accuracy: The ability to tailor the discretization method to each subdomain can result in higher accuracy in capturing the acoustic scattering phenomena, especially in regions with significant material property variations.
Flexibility in Boundary Conditions: Different boundary conditions can be applied in each subdomain, allowing for a more realistic representation of the acoustic scattering problem.

Limitations:

Complexity of Implementation: Implementing and managing a multi-domain approach can be more complex than a single-domain approach, requiring careful coordination between different subdomains and boundary conditions.
Increased Memory Usage: Handling multiple subdomains simultaneously may require more memory compared to a single-domain approach, especially for large-scale problems with numerous subdomains.
Potential for Numerical Instabilities: The presence of interfaces between subdomains can introduce numerical challenges such as spurious resonances or stability issues if not properly addressed.

Can the multi-trace formalism and the techniques developed in this work be applied to other types of wave propagation problems, such as electromagnetic scattering or elastodynamics

The multi-trace formalism and techniques developed in this work can be applied to other types of wave propagation problems, such as electromagnetic scattering or elastodynamics, with appropriate modifications and adaptations.
For electromagnetic scattering problems, the multi-trace formalism can be utilized to handle the transmission conditions and interface interactions between different materials or domains. By extending the concepts of multi-trace spaces and boundary integral operators to electromagnetic wave equations, similar variational formulations can be developed for efficient and accurate simulations of electromagnetic scattering phenomena.
In elastodynamics, the multi-trace formalism can be employed to model the interaction of elastic waves in heterogeneous media. By considering the transmission conditions and boundary interactions between different elastic materials, the techniques developed for acoustic scattering can be adapted to solve elastodynamic wave propagation problems. This includes formulating appropriate boundary integral equations and variational formulations to capture the behavior of elastic waves across interfaces and subdomains.