Isogeometric simulation of acoustic radiation

TL;DR

This paper demonstrates that Isogeometric Analysis (IgA) effectively solves the Helmholtz radiation problem in 2D, especially at high wavenumbers, outperforming traditional FEM by reducing pollution error and producing smoother solutions.

Contribution

The paper introduces an IgA-based method for the Helmholtz radiation problem, showing its advantages over FEM in high-frequency scenarios.

Findings

IgA reduces pollution error at high wavenumbers.

IgA produces smoother solutions with fewer degrees of freedom.

IgA outperforms FEM in numerical accuracy for the radiation problem.

Abstract

In this paper we discuss the numerical solution on a simple 2D domain of the Helmoltz equation with mixed boundary conditions. The so called radiation problem depends on the wavenumber constant parameter k and it is inspired here by medical applications, where a transducer emits a pulse at a given frequency. This problem has been successfully solved in the past with the classical Finite Element Method (FEM) for relative small values of k. But in modern applications the values of k can be of order of thousands and FEM faces up several numerical difficulties. To overcome these difficulties we solve the radiation problem using the Isogeometric Analysis (IgA), a kind of generalization of FEM. Starting with the variational formulation of the radiation problem, we show with details how to apply the isogeometric approach in order to compute the coefficients of the approximated solution of radiation problem in terms of the B-spline basis functions. Our implementation of IgA using GeoPDEs software shows that isogeometric approach is superior than FEM, since it is able to reduce substantially the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.