Mimetic Discretization of the High Frequency Helmholtz Equation using Hermite Interpolating Polynomials

Mimetic Discretization of the High Frequency Helmholtz Equation using Hermite Interpolating Polynomials

Lok, Peter (TU Delft Aerospace Engineering)

Gerritsma, Marc (mentor)

Delft University of Technology

Aerospace Engineering

2020-02-24

Numerical modelling of high frequency waves is a complex and challenging area. Although the underlying equation seems simple, −∆φ − k2φ = f, the numerical challenges are not. This time harmonic wave equation is known as the Helmholtz equation. The main challenge to be studied is the pollution error, which is the difference between the actual and numerical wavenumber. Due to this error, the solution of most numerical methods deteriorates rapidly when increasing the wavenumber. So far, either problem specific solutions have been found or solutions which require knowledge of the solution itself. In this thesis a numerical method is developed to approach this challenge. A mimetic discretization of the Helmholtz equation using C1-continuous Hermite interpolating polynomials is developed to model the Helmholtz equation in 2 dimensions. Mimetic theory is briefly introduced after which the Hermite polynomials are defined and analysed for their interpolating properties. A least-squares variational problem is defined and discretized using 49 basis function at its lowest order. The model is verified using a single sinusoidal wave after which
plane wave problems are solved and analysed for their wavenumber-dependence. A diffraction and interference problem is set-up and compared to analytical solutions. The proposed method shows no significant advantages over the use of C0-continuous Lagrange polynomials in terms of effectiveness. Both methods show deterioration at the same wave numbers and equal polynomial order. The Hermite polynomials require less unknowns to reach the same accuracy. In short, a p-accurate solution is found using Hermite polynomials
at the cost of a p − 1 model using Lagrange polynomials. This benefit in efficiency does not outweigh the additional effort and boundary information necessary to construct the problem. However, the variety of coefficients offer an opportunity for further research to find relations to the numerical wavenumber in the search of a wavenumber conserving discretization.

mimetic discretization
Helmholtz equation
Hermite polynomials

http://resolver.tudelft.nl/uuid:ec5c0fff-54d1-4cf3-a3e2-fef2b38c3f2c

Student theses

master thesis

© 2020 Peter Lok