Numerical integration on non-planar surface meshes for ... · PDF filemeshes for finite...
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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