Stable and Efficient Decoupled Perfectly Matched Layer for 3D Acoustic Wave Propagation using Nodal Discontinuous Galerkin Method

Core Concepts

The study proposes and assesses a new stable and efficient decoupled perfectly matched layer (PML) formulation for the 3D acoustic wave equation using the nodal discontinuous Galerkin finite element method. The decoupled PML formulation involves fewer auxiliary variables and corresponding PDEs compared to other PML formulations, making it computationally efficient to solve.

Abstract

The paper presents a new perfectly matched layer (PML) formulation for the 3D acoustic wave equation using the nodal discontinuous Galerkin finite element method. The key highlights are:

The PML formulation is based on an efficient PML formulation that can be decoupled into three independent PML formulations along the Cartesian directions. This decoupled PML formulation involves fewer auxiliary variables and corresponding PDEs compared to other PML formulations, making it computationally efficient.

The decoupled PML formulation is demonstrated to be long-time stable. An optimization procedure of the damping functions is proposed to enhance the performance of the formulation.

The damping performance, accuracy, and stability of the PML formulation are assessed through various numerical experiments. The results show that the proposed PML formulation can effectively absorb outgoing waves, including those at grazing incidence, without polluting the computed solution in the inner domain.

The convergence of the PML formulation is studied by increasing the order of approximation in the discontinuous Galerkin method. The results indicate that higher-order approximations can significantly improve the accuracy of the solution, though at the cost of increased computational time.

The PML formulation is shown to be effective in absorbing waves at grazing incidence by considering an elongated computational domain. The decoupled PML formulation is able to maintain stability and accuracy even for such challenging wave propagation scenarios.

Overall, the proposed decoupled PML formulation provides a stable, efficient, and accurate approach for handling open boundaries when solving the 3D acoustic wave equation using the nodal discontinuous Galerkin finite element method.

Stats

The computational domain is a cube with an edge length of 5 m.
The PML widths δx, δy and δz are either equal to the peak wavelength of the source λpeak or half of λpeak.
Three peak frequencies fpeak are used: 343 Hz, 514.5 Hz and 646 Hz, with corresponding peak wavelengths λpeak of 1 m, 2/3 m and 1/2 m respectively.

Quotes

"The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method."
"The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy."
"The damping performance, accuracy and stability of the PML formulation is assessed."

Key Insights Distilled From

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

by Sophia Julia... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08464.pdf

Deeper Inquiries

How can the proposed decoupled PML formulation be extended to handle more complex geometries and boundary conditions beyond the simple cubic domain considered in this study

The proposed decoupled PML formulation can be extended to handle more complex geometries and boundary conditions by incorporating adaptive mesh refinement techniques and implementing multi-layered PML structures.

Adaptive Mesh Refinement: By dynamically adjusting the mesh resolution based on the local solution behavior, the PML formulation can adapt to complex geometries with varying acoustic properties. This adaptive approach ensures that the PML layers are appropriately positioned and sized to effectively dampen outgoing waves in regions of interest.

Multi-Layered PML: To address more intricate boundary conditions, multiple layers of PML with varying damping profiles and thicknesses can be implemented. Each layer can be optimized to absorb specific frequency ranges or handle different wave propagation characteristics, enhancing the overall absorption efficiency of the PML formulation.

Curved Boundaries: For domains with non-linear or curved boundaries, the PML formulation can be modified to account for the curvature and ensure accurate damping of waves approaching these complex surfaces. This may involve adjusting the damping functions and stretching transformations to align with the curved geometry of the domain.

By incorporating these strategies, the decoupled PML formulation can be tailored to effectively handle a wide range of complex geometries and boundary conditions encountered in real-world acoustic wave simulations.

What are the potential limitations or drawbacks of the decoupled PML approach compared to other PML formulations, and how can they be addressed

The decoupled PML approach offers several advantages in terms of stability, computational efficiency, and ease of implementation. However, there are potential limitations and drawbacks that should be considered:

Increased Computational Cost: The use of multiple decoupled PML formulations for different Cartesian directions may lead to a higher computational overhead compared to single-layer PML approaches. This can impact the overall efficiency of the simulation, especially for large-scale problems.

Complexity of Optimization: Optimizing the damping functions for each direction in the decoupled PML formulation may require additional computational resources and expertise. Balancing stability, accuracy, and efficiency across multiple damping profiles can be challenging.

Limited Absorption Efficiency: The decoupled PML approach may not fully eliminate reflections, especially for grazing incidence waves or complex geometries. Fine-tuning the damping functions and layer thicknesses is crucial to minimize reflection artifacts.

To address these limitations, advanced optimization algorithms, parallel computing techniques, and adaptive strategies can be employed to enhance the performance of the decoupled PML formulation and mitigate its drawbacks.

Given the focus on computational efficiency, how could the PML formulation be further optimized or integrated with other numerical techniques to enable large-scale acoustic wave simulations in real-world applications

To further optimize the PML formulation for large-scale acoustic wave simulations and real-world applications, the following strategies can be considered:

Parallelization and GPU Acceleration: Implementing parallel computing techniques and leveraging GPU acceleration can significantly improve the computational efficiency of the PML formulation. Distributing the workload across multiple processors and utilizing GPU resources can expedite the simulation process for complex geometries and high-resolution meshes.

Integration with Domain-Specific Solvers: Integrating the PML formulation with domain-specific solvers tailored for acoustic wave propagation can enhance the overall simulation performance. By combining the strengths of different numerical techniques, such as finite element methods or finite difference methods, the PML formulation can be optimized for specific application scenarios.

Dynamic Adaptation and Adaptive Mesh Refinement: Incorporating dynamic adaptation strategies and adaptive mesh refinement algorithms can optimize the resolution of the mesh based on the evolving solution characteristics. This adaptive approach ensures that computational resources are efficiently allocated to regions of interest, improving the accuracy and efficiency of the simulation.

By implementing these optimization strategies and integrating the decoupled PML formulation with advanced numerical techniques, large-scale acoustic wave simulations in real-world applications can be conducted with improved speed, accuracy, and scalability.