Solution of the 3D-Helmholtz equation in exterior domains of arbitrary shape using hp -finite-infinite elements

TLDR: In this article, a convergence and performance study of finite-infinite element discretizations for the Helmholtz equation in exterior domains of arbitrary shape was performed, and the proposed approximation applies to arbitrary geometries, combining an hp-FE discretization between the object and a surrounding sphere and an hp infinite element (IE) discretisation outside the sphere with a spectral-like representation.

About: This article is published in Finite Elements in Analysis and Design.The article was published on 1998-05-15 and is currently open access. It has received 19 citations till now. The article focuses on the topics: Inversion in a sphere & Fuzzy sphere.

Figures

Figure Initial Finite Element Mesh for the rigid scattering on a sphere within the unit sphere with p q layers of nite elements Three quarters of the mesh are displayed

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Solution of the 3d-helmholtz equation in exterior domains of arbitrary shape using hp-finite infinite elements" ?

This work is devoted to a convergence and performance study of nite in nite element discretizations for the Helmholtz equation in exterior domains of arbitrary shape The sphere problem admits an exact solution and serves as a basis for the convergence study Solutions to the other two problems are compared with those obtained using the Boundary Element Method Introduction This is also the approach which the authors investigated in and continue to use in this work In between the scatterer and the truncating sphere the authors use D hp FE approx imations described in Outside the sphere the compatible hp discretization on the sphere is combined with a spectral like approximation in the radial direction Numerical results are con ned to these geometries a sphere a nite cylinder and a nite cylinder with spherical incaps The numerical solutions to the cylinder problems are compared with solutions obtained by the BEM discussed in The plan of the presentation is as follows The authors begin by formulating the Helmholtz equation and describing the IE in section Numerical experiments are presented in section Section discusses some preliminary results on a posteriori error estimation and hp adaptivity and they nish the presentation with concluding remarks in section Coupled FE IE Discretizations for the Helmholtz Equation in Arbitrary Exterior Domains Notation I R is a domain occupied by the rigid scatterer and contained in the unit sphere e I R is the domain exterior to the scatterer s fx I R jxj g is the surface of the unit sphere es fx I R jxj g is the domain exterior to the unit sphere is the surface of the rigid scatterer s fx I R jxj g is the domain between the unit sphere and the rigid scatterer 

Q2. What is the corresponding order of approximation of j functions?

The corresponding shape functions are tensor products of the D triangle shape functions i discussed in and D incremental shape functionsjk i j k k i jFor j functions j are the regular linear shape functions Given a particular order of approximation q in the vertical direction functions j j q coincide with the regular D Lagrange shape functions of order q vanishing at the endpointsConsequently the mid side and the middle node have two corresponding orders of approximations a horizontal p and a vertical order q 

Q3. What is the proposed variational formulation for the sphere problem?

As for the sphere problem the proposed variational formulation corresponds to an extension of the operator setting of Leis where the domain of the oper ator is restricted to a subspace of H w e consisting of all functions for which the Helmholtz operator value is in the weighted L w e space 

Q4. What is the top base of each prism?

The bottom base of each of the prisms coincides with a triangle or a rectangle on the scatterer and the top base lies on the unit sphere 

Q5. what is the simplest way to generate a mesh?

For all the details concerning the geometric modeling the authors refer toThe initial mesh is generated by using the idea of an algebraic mesh generator and hp interpolation Given for the reference prism numbers n and l of divisions in thehorizontal and vertical directions respectively compatible for neighboring elementsthe initial mesh is always regular the reference blocks are covered with uniform regular grids consisting of elements K 

Q6. how does the hp mesh re nement work?

Scattering of a plane wave on a rigid cylinder with spherical incaps k Imaginary part of the scattered pressure obtained with a mesh of D nite elements l n of order p q and in nite elements with terms in the radial directionh-p BEM pressure k = 1.0imag component 

Q7. how much pressure is the hp mesh re nement?

Scattering of a plane wave on a rigid cylinder with spherical incaps k Real part of the scattered pressure obtained with a mesh of D nite elements l n of order p q and in nite elements with terms in the radial directionh-p BEM pressure k = 1.0real component 

Q8. What is the simplest way to calculate the stiness matrices?

The computation of the D nite element sti ness matrices is also standard and the element contributions are then assembled by a frontal solver adapted to handle complex matrices 

Q9. what is the real component of the hp mesh re nement?

Scattering of a plane wave on a rigid cylinder with spherical incaps k Real part of the scattered pressure obtained with a mesh of D nite elements of order ph-p IEM/FEM pressurek = 1.0imag componentFigure FEM!IEM 

Q10. what is the error indicator for a nite element?

The error indicator given in the previous sec tion is computed for every D FE after solving the scattering problem using the FE IE methodology Each element l for which the relative error indicator rl is greater than a treshhold value will be re ned i e ifrl l twhere t denotes the maximum element error indicator over the whole mesh 

Q11. how is the re nement of a plane wave obtained?

Scattering of a plane wave on a rigid cylinder k Imaginary part of the scattered pressure obtained with a mesh of D nite elements l n of order p q and in nite elements with terms in the radial directionh-p BEM pressure k = 1.0imag component 

Q12. what is the re ned i e ifrl l?

Scattering of a plane wave on a rigid cylinder k Absolute value of the scattered pressure obtained with a mesh of D nite elements l n of order p q and in nite elements with terms in the radial directionadjacent to the re ned nite elements