Timo Betcke Matthew Scroggs Wojciech Ś migaj Supported by EPSRC Grants EPI0300421 EPK03829X1 Boundary integral equations in a nutshell Neumann data Dirichlet data Two equations for Neumann and ID: 532737
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Slide1
Fast boundary element solution of Maxwell problems with BEM++
Timo Betcke, Matthew Scroggs,Wojciech ŚmigajSupported by EPSRC Grants EP/I030042/1,EP/K03829X/1Slide2
Boundary integral equations in a nutshell
Neumann data
Dirichlet
data
Two equations for Neumann and
Dirichlet
dataSlide3
Electromagnetic Scattering
Theoretical formulations
Fast assembly
Software design
Example computationsSlide4
Potential Operators
Representation FormulaJump of tangential trace
Jump of
normal trace
Electric Field Potential Operator
Magnetic Field Potential OperatorSlide5
Interlude: Spaces for Maxwell boundary operators.
Define anti-symmetric dual pairingSpace of div-conforming tangential traces
Right space for representing boundary operatorsAnti-symmetric pairing causes computational issues (deal with them later)
A.
Buffa
, R.
Hiptmair
:
Galerkin
boundary element methods for electromagnetic scattering (2003)Slide6
Maxwell boundary operators
Electric field integral operatorAverage of interior and exterior trace
Magnetic
Field Integral OperatorSlide7
Div and Curl conforming spaces
Div
-Conforming
Curl-Conforming
Raviart
-Thomas (RWG) basis
Nédélec
basisSlide8
Discretizing the electric field operator
Curl-conforming
Div
-conforming
O(N
2
) complexity for matrix-vector product
Fast BEM achieves
O(N log N) complexity
O(N
2
)
O(N log N)
Evaluate weakly singular integrals
Naive assembly requires O(N^2) operationsSlide9
Compressed assembly
Fast Multipole MethodsH/H2/HSS - Matrix Techniques
Assembly/matvec
in O(N log N) complexity
Cluster Tree T
Balanced Clustering of
dofs
BEM++ implements
H-Matrix compression
W.
Hackbusch
: Hierarchical Matrices
:
Algorithms and Analysis, Springer, 2015Slide10
…Compressed Assembly
Form block cluster tree T x T
Stop with a leaf when eligibility criterion between row and column cluster node is satisfied.
Standard admissibility condition:Slide11
…Compressed Assembly
Low-Rank compression of admissible blocks via adaptive cross approximationOnly a fraction of the elements in each blocks is actually computedO(N log N) complexity for compression and evaluation of non-oscillatory integral operators.
Randomized recompression techniquesFully task-parallel implementationSlide12
Assembly at high-frequencies
Memory Growth
1
Low-rank
approximability
of the Green’s function
2
1. T. Betcke, E. van ‘t
Wout
, P.
Gélat
: Computationally efficient boundary element methods for high-frequency Helmholtz problems in unbounded domains
2. B. Engquist
, J.Zhao: Approximate Separability of Green’s Function for high-frequency Helmholtz equationsSlide13
The BEM++ library
C++ Lines of Code: 48768Python Lines of Code: 6972Github Commits: 3008Growing collection of tutorials
Binary versions available for Ubuntu
Install on Mac via Homebrew
Virtual machine image for all operating systems
W.
Śmigaj
, et. al.
: Solving boundary integral problems with BEM++, ACM Trans. Math. Software, 41 (2015)Slide14
BEM++ Structure
FIBER Collection of low-level templated integrators on elementsHigh-Level operator assembly
H-Matrix Interfaces
Grid Management vie DUNE
User-visible data structures
GridFunction
,
BoundaryOperator
,
DiscreteBoundaryOperator
High-Level functions
Import/Export,
Iterative Solvers,
Preconditioners
Python
C++Slide15
Solving the EFIE in BEM++
Electric field integral equationElectric
Field Operator
Tangential trace
of incident wave
Unknown density
rt_space
=
bempp.api.function_space
(grid, “RT”, 0)
nc_space
=
bempp.api.function_space
(grid, “NC”, 0)
e
fie
=
bempp.api.operators.maxwell.efie
(
rt_space
,
rt_space
,
nc_space
)
Defining the operator in BEM++
domain
range
d
ual_to_rangeSlide16
Calderon Preconditioning for the EFIE
Standard EFIE extremely ill-conditionedIterative solvers converge slowly or not at allAvoid by using dense discretization + LU. However, O(N^3) computational cost.
Self-Regularising property of the EFIE on closed smooth domains
efie
=
bempp.api.operators.maxwell.efie
(
rt_space
,
rt_space
,
nc_space
)
efie_sq
= efie * efie
Not
inf
-sup stable!!
Could we use curl-conforming basis functions instead? No, but…Slide17
Almost Curl-Conforming Buffa-Christiansen basis functions
Div
-Conforming on
barycentric
mesh
Quasi-curl conforming on original mesh
Well-conditioned mass matrix
rwg_space
=
bempp.api.function_space
(grid, “B-RWG”, 0)
s
nc_space
=
bempp.api.function_space
(grid, “B-SNC”, 0)
bc_space
=
bempp.api.function_space
(grid, “BC”, 0)
rbc_space
=
bempp.api.function_space
(grid, “RBC”, 0)
efie1
=
bempp.api.operators.maxwell.efie
(
rwg_space
,
rwg_space
,
snc_space
)
e
fie2 =
bempp.api.operators.maxwell.efie
(
bc_space
,
rwg_space
,
rbc_space
)
efie_sq
=
efie2
*
efie1
F. P.
Andriulli
, et. al: A multiplicative Calderon
preconditioner
for the electric field integral equation, IEEE Antennas and Propagation, 56 (2008)
Efficient implementation with only one EFIE discretization possible.Slide18
Scattering from the NASA Almond shape
NASA benchmark shapeSimulation at 5GhzAssembly of 65k element meshAssembly time: 79s
Solver time: 72s (56 iterations)Slide19
Scattering from the
Sierpinski triangle
Important for wide-band antenna design
Investigating the efficiency of Calderon preconditioningSlide20
Magnetic Field Integral Equation
mfie = bempp.api.operators.boundary.maxwell.magnetic_field(
rwg_space
,
rwg_space
,
rbc_space
, k
)
ident
= bempp.api.operators.boundary.sparse.identity
( rwg_space
, rwg_space,
rbc_space)MFIE Convergence on a cube
Standard MFIE not stable on cornered domains
MFIE using stable anti-symmetric dual pairing requires quasi-curl conforming basis functions.
K. Cools, et. al.: Accurate and conforming mixed discretization of the MFIE, IEEE Antennas and Propagation, 10 (2011)Slide21
Optimal assembly spaces
mfie = bempp.api.operators.boundary.maxwell.magnetic_field( rt_space
, rt_space
,
rbc_space
, k
)
Spaces defined on
barycentric
refinement
Operator on
barycentric
space
Projection on RBC Space
Projection on RWG space
Optimal assembly and projection operators performed automatically!!Slide22
Light-Scattering from Ice Crystals
"CirrusField-color" by PiccoloNamek - Own work of user PiccoloNamek at the English language Wikipedia. Licensed under CC BY-SA 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/
File:CirrusField-color.jpg#mediaviewer/File:CirrusField-color.jpg
Cirrus clouds cover ~30% of the earth at any given time.
Understanding radiation balance of cirrus clouds makes necessary understanding the
radiative
properties of atmospheric ice crystals.
New NERC funded project!Slide23
Numerical results
Good agreement with T-Matrix methods for convex
scatterers
Allows arbitrary non-convex shapes
Simulations already useful but need better Maxwell high-frequency solvers.
S. P.
Groth
, et al. The boundary element method for scattering by ice crystals and its implementation in BEM++,
Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 167, December 2015, Pages 40-52Slide24
A real world application: BEM formulations for
HIFUNon-invasive cancer treatment modalityRib bones absorb and reflect ultrasoundDanger of overheating bone and surrounding tissueAberration of focal region due to diffraction
Develop efficient boundary element models for scattering from rib bones
Joint work with P.
Gelat
(UCL)Slide25
Application to scattering from a rib cage
E. Van’t
Wout et. al., A fast boundary element method for the simulation of High-Intensity Focused Ultrasound
,
J.
Acoust
. Soc. Am.
138
, 2726 (2015)
Slide26
Some recent results…
Assembly: 5 hours, 194 GBCompression
rate: 2.08% (single-layer)
3.31% (
hypersingular
)
GMRES Solves: 19
iterations,
6:59
minutesNumber of
dofs: 479124Slide27
Final remarks
Upcoming developments:Maxwell FEM/BEM coupling with FEniCS and BEM++Maxwell multitrace formulationsH^2 matrices, MLMDA accelerated H-Matrix solves
High-Quality Software
Exciting Mathematics