Efficient Absorbing Boundary Conditions for the Helmholtz Equation using Gauss-Legendre Quadrature Reduced Integrations

Core Concepts

The author introduces a new class of absorbing boundary conditions (ABCs) for the Helmholtz equation that generalize the perfectly matched discrete layers (PMDLs) and achieve high accuracy with efficient numerical implementation.

Abstract

The author introduces a new class of absorbing boundary conditions (ABCs) for the Helmholtz equation that generalizes the perfectly matched discrete layers (PMDLs). The proposed ABCs are obtained by using L discrete layers in the artificial domain and the QN Lagrange finite element in conjunction with the N-point Gauss-Legendre quadrature reduced integration rule. The key highlights are: The proposed ABCs are classified by a tuple (L, N) and achieve reflection error of order O(R^(2LN)) for some R < 1. This includes the PMDLs as a special case of type (L, 1). The new ABCs allow for monolithic discretizations over the computational domain even when the QN finite elements with N > 1 are used in the physical domain, unlike the PMDLs which require the Q1 element in the artificial domain. The author presents a detailed analysis of the proposed ABCs, motivated by techniques used to study dispersive and dissipative behavior of high-order discontinuous Galerkin methods. This provides more insight into the problem of ABCs. Numerical examples in 1D and 3D demonstrate the effectiveness of the new ABCs in efficiently solving the Helmholtz equation in unbounded domains.

Stats

The Helmholtz equation is given by: s^2 u - u,ii = f in R^d, where s is a complex number with Re(s) ≥ 0, and f is a spatial function. The mapped problem is: γ^2 u - u,xx = 0 in x > -δ, where γ = √(s^2 + k^2) and k = √(k_i k_i).

Quotes

"The proposed ABCs are classified by a tuple (L, N), and achieve reflection error of order O(R^(2LN)) for some R < 1." "The new ABCs generalise the perfectly matched discrete layers proposed by Guddati and Lim [Int. J. Numer. Meth. Engng 66 (6) (2006) 949-977], including them as type (L, 1)."

Key Insights Distilled From

Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations

by Koki Sagiyam... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2308.12255.pdf

Deeper Inquiries

How can the proposed ABCs be extended to handle more complex geometries and boundary conditions beyond the Helmholtz equation

The proposed ABCs can be extended to handle more complex geometries and boundary conditions by incorporating adaptive mesh refinement techniques. By dynamically adjusting the mesh resolution based on the local solution behavior, the ABCs can effectively handle irregular geometries and boundary conditions. Additionally, coupling the ABCs with domain decomposition methods can allow for the treatment of multi-domain problems with different boundary conditions. This approach enables the ABCs to be applied to a wider range of problems, including those with varying material properties, interface conditions, and geometric complexities.

What are the limitations of the Gauss-Legendre quadrature reduced integration approach, and how can it be further improved or combined with other techniques

The limitations of the Gauss-Legendre quadrature reduced integration approach include potential numerical instabilities when dealing with highly oscillatory solutions or steep gradients. To address this, a combination of adaptive quadrature techniques, such as adaptive Gauss-Kronrod quadrature, can be employed to accurately capture the solution behavior. Furthermore, incorporating higher-order quadrature rules or hybrid integration schemes, such as a combination of Gauss-Legendre and Gauss-Lobatto quadrature, can enhance the accuracy of the integration process. Additionally, utilizing advanced numerical stabilization techniques, such as artificial viscosity or filtering methods, can help mitigate oscillations and improve the stability of the integration scheme.

What are the potential applications of the new ABCs in fields beyond computational physics, such as in engineering or data science, where efficient handling of unbounded domains is crucial

The new ABCs have potential applications in various fields beyond computational physics, such as engineering and data science. In engineering, where simulations involving unbounded domains are common, the proposed ABCs can facilitate the accurate modeling of wave propagation phenomena, structural dynamics, and fluid flow problems. These ABCs can be particularly useful in acoustics, electromagnetics, and structural analysis, where the treatment of unbounded domains is essential for realistic simulations. In data science, the efficient handling of unbounded domains is crucial in applications such as signal processing, image analysis, and machine learning. By incorporating the new ABCs into numerical algorithms for data processing and analysis, researchers can improve the accuracy and reliability of their models when dealing with unbounded datasets or signals.