Adaptive PUBEM for Helmholtz problems

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Adaptive PU-BEM for Helmholtz problems

• Jon Trevelyan Graham Coates
• School of Engineering, Durham University

2
The motivation
Short wave scattering
Limitation of polynomial basis elements (FEM/BEM)
10 DOF per wavelength
Partition of Unity Method (PUM) is just one
approach that aims to overcome the limitation.
The PUM is a general class of methods involving
enrichment of the approximation space with sets
of analytical functions known to populate the
solution space.
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Partition of Unity Method
Melenk Babuka, Comp. Meth. Appl. Mech. Engg.,
1996
Application to waves PU-FEM
Laghrouche Bettess, J. Comput. Acoust., 2000.
Ortiz Sanchez, Int. J. Num. Meth. Eng., 2000.
Farhat et al., various, discontinuous enrichment
method
Gamallo Astley, Int. J. Num. Meth. Eng., 2006.
Strouboulis et al, Comp Meth Appl Mech Eng., 2006.
Ultraweak variational formulation
Cessenat Despres, SIAM J. Numer. Anal., 1998.
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Partition of Unity Method in integral equations
Enrichment using plane wave(s) in integral
equations
de la Bourdonnaye, CRAS, 1994. microlocal
discretisation
Abboud et al., SIAM Wave propagation conf., 1995.
Perrey-Debain et al., J. Sound Vib., 2003.
Bruno et al., Phil Trans Royal Society A, 2003.
Langdon Chandler-Wilde, SIAM J. Numer. Anal.,
2006.
etc.
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The scattering problem
In the frequency domain we solve the Helmholtz
equation
? is the potential we seek and the wave number,
k, is given by
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The scattering problem
In the frequency domain we solve the Helmholtz
equation
? is the potential we seek and the wave number is
given by
For simplicity consider Neumann problem
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Collocation boundary integral equation
Boundary integral equation collocating at x0 on
smooth boundary ?
where the double layer potential operator is
and the Greens function (for 2D problems) is
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PU-BEM enriched basis
The PUM multiple plane wave expansion for
potential on element
Polynomial shape functions
Plane waves
Amplitudes
Reformulates the problem so that the unknowns
become the amplitudes.
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PU-BEM
Substituting the basis
into the boundary integral equation
and collocating at a sufficient number of points
x0 gives a system of equations that may be solved
for the amplitudes.
To overcome the BIE non-uniqueness problem we use
the CHIEF method of Schenck.
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Collocation method
Each element will now have many degrees of
freedom.
degrees of freedom gt number of
nodes So we generate an auxiliary set of
equations by collocating at a set of points
uniformly distributed over the element.
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PU-BEM efficiency
Define measure of efficiency ? DOF per
wavelength
Conventional polynomial basis requires ? ? 10
PU-BEM plane wave basis requires ? ? 2.5
Computational burden shifts from solver to
integration and assembly.
Important to optimise plane wave basis to
minimise number of evaluations of highly
oscillatory integrals.
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Adaptive scheme
We develop an adaptive scheme, of p-adaptive
character, by allowing the number of plane waves
at each node to vary.
With the PUM basis our BIE becomes
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Adaptive scheme
BIE
Define a residual error indicator
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Adaptive scheme
Implementation of residual error indicator
Evaluate at any point
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Adaptive scheme
R has useful local and global properties. Global
error indicator gives us a stopping criterion
Define a 1-norm in terms of the value of R over a
scatterer of perimeter P
This is expensive to compute, so we approximate
using
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Adaptive scheme
R has useful local and global properties. Global
error indicator gives us a stopping criterion
Numerical tests suggest a stopping criterion of
gives a converged solution of engineering
accuracy (1 relative error norm in L2)
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Adaptive scheme
Local properties of R are used to decide where to
add waves.
To verify this, plot norms of both R and L2
error, evaluated over each element
1.2
Norm of R and L2 error
1
normalised Rnorm
0.8
normalised L2
0.6
0.4
0.2
0
0
4
8
12
16
20
24
Element
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Adaptive scheme
We evaluate R only at points close to end nodes
0.015
R
0.01
0.005
0
x
-1
-0.5
0
0.5
1
Numerical tests suggest addition of a plane wave
at nodes adjacent to points where R gt 0.015
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Plane wave addition
At each adaptive step, we can introduce an extra
plane wave at selected nodes.
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Plane wave addition
Check accuracy of PU-BEM for non-uniformly
distributed wave directions.
12 elements
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Plane wave addition
Check accuracy of PU-BEM for non-uniformly
distributed wave directions.
0
uniform
-1
uniform 1
uniform 2
-2
-3
-4
-5
-6
0
0.5
1
1.5
2
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Adaptive scheme
The extra DOF require us to add extra rows and
columns
A
x

b
New terms in blue areas to be found from boundary
integrals
Big savings in numerically intensive oscillatory
integrals
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Illustration
Scattering of a plane wave by sound-hard cylinder
Initial model 24 boundary elements 48 nodes 6
waves/node 288 DOF ? 2.29
a 10

l 0.5
ka 125
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Illustration
?
first run 288 DOF
Error indicator suggests a wave to be added at 21
nodes
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Illustration
?
R
0.06
0.05
0.04
0.03
0.02
0.01
0
?
0
2
4
6
second run 309 DOF
first run 288 DOF
Error indicator suggests a wave to be added at 21
nodes
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?
Illustration
Converged solution
Numerical
Analytical
?
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Illustration
Initial run
Matrix 336 ? 288 ? 96768 matrix terms
The other alternative without adaptivity run
new analysis with 7 waves/node requiring 129024
terms
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Conclusions
Adaptive scheme for PU-BEM in wave scattering
Residual error indicator R
Add waves near nodes having high R
Norm of R gives stopping criterion
Illustration shows large savings in required
number of oscillatory integrals