Numerical integration on non-planar surface

meshes for finite element electromagnetics

Ryan Galagusz, Steve McFeeDepartment of Electrical & Computer Engineering, McGill University

Introduction

The objective of this research project is to test, extend and

improve upon a proposed numerical integration method for

arbitrary polygonal domains. Base-level numerical integration

plays a critical role in software packages used to simulate

electromagnetic field and wave problems. The motivation for this

goal is to produce robust, accurate and reliable adaptive

integration methods for generalized finite element applications.

Methodology

Using freely available software packages and a set of new

MATLAB scripts and functions, the generalized finite element

numerical integration method was extended to include adaption

and three-dimensional surface elements as follows:

INIT

• Given a non-planar, curvilinear triangle mesh defined on some arbitrary surface…

1• Construct a mesh of triangles in a planar, regular polygon that

preserves the connectivity of the original INIT phase mesh

2• Establish the transformation between the planar polygon in

Step 1 and the unit disk using Schwarz-Christoffel mapping

3• Locate Gauss-Legendre quadrature points adaptively over the

polar parameter rectangle

4• Trace back through the series of mappings in Steps 1-3 to locate

the integration points on the original surface

Results

Background

Generalized finite elements:

Incorporate information about the physical problem into the

basis functions of the element

Created to exploit parallel computational throughput

Can result in elements of arbitrary geometry

Schwarz-Christoffel conformal mapping:

Advantages:

Provides a one-to-one correspondence between an arbitrary

planar polygon and the unit disk

Integration points on the disk can be easily located

Disadvantages:

Introduces singularities in the Jacobian determinant at

prevertices

Restricts the type of generalized elements to two-dimensions

Previous integration method is non-adaptive

Conclusions

An adaptive numerical integration method on the unit disk is

more reliable and accurate than previous non-adaptive methods

for generalized finite element applications

The extension to numerical integration on surfaces in three-

dimensional space provides accurate integral estimates and

converges at roughly the same speed as classical techniques

Improvements to error control mechanisms that guide and tune

adaption are required to further reduce function evaluations

An illustrative example from electromagnetism:

The total radiated power for a half wave dipole is given by

Consider a symmetric octant of a spherical surface centered

on the dipole to visualize the proposed method:

A comparison of three integration methods yields:

Directed h-adaption outperforms uniform h-adaption by a

ratio of 4:1 function evaluations

There is little to separate classical methods from the

proposed directed h-adaption method on the disk

)(565.362

0 WIsdHEPS

INIT

1

3

2

4

MethodNumber of function

evaluationsRelative error tolerance (%)

Classical integration over

constituent triangles1183 0.2528

Proposed

integration over

the unit disk

Uniform

h-adaption 5456 0.5692

Directed

h-adaption1360 0.9875