Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
PinT 2015, Dresden, May 28, 2015
D. Lukas, M. Merta, J. Zapletal, and A. Veit
VSB–Technical University of Ostrava, Czech Rep.University of Chicago
email: [email protected]
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Parallel fast BEM and applications
Laplace equation with mixed boundary conditions
Ω ⊂ R2 lipschitz domain, Γ := ∂Ω = ΓD ∪ ΓN, ΓD ∩ ΓN = ∅
−u(x) = 0, x ∈ ΩγD u(x) := u(x) = g(x), x ∈ ΓD
γN u(x) :=dudn(x) = h(x), x ∈ ΓN
Fundamental solution
G(x,y) := −1
2πln ‖x− y‖ satisfies −yG(x,y) = δx in the distributional sense
Representation formula (”v(y) := G(x,y)”)
∀x ∈ Ω : u(x) =
∫
Γ
γNu(y)G(x,y) dl(y)−
∫
Γ
u(y) γN,yG(x,y) dl(y)
We are left to calculate u on ΓN and γNu on ΓD.
Parallel fast BEM and applications
Shape optimization of a DC electromagnet, FEM-BEM coupling
Ωi: ferromagnetic yoke
Ωi
Ωe: air
Ωe: air
Ωe
Jcoil
Ωm sample
Ωo: focusing optics
Γ: boundary
L., Postava, Zivotsky: J Magn Magn Mater ’10, Math Comput Simulat ’12
Parallel fast BEM and applications
Acoustics of a railway wheel
A joint work with J. Szweda, Department of mechanics, VSB–TU Ostrava
Parallel fast BEM and applications
Scattering of infrared light in a DLP projector
Ei
Es
Omegam Omegap
E
Parallel fast BEM and applications
Homogenization of periodic structures
L., Bouchala, Theuer, Zapletal: Numer Math (submitted)
Parallel fast BEM and applications
Parallel fast BEM using cyclic graph decompositions
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rotate
G0
G1
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L., Kovar, Kovarova, Merta: Numer Alg (online first)
Solution to the system of 2.7M DOFs on 273 cores in 16 minutes.
Parallel fast BEM and applications
3d wave equation, simult. space-time discretization
Veit, Merta, Zapletal, L.: IJNME (submitted)
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Boundary integral formulation of sound-hard scattering
Sound-hard scattering
Given the scatterer Ω and the causal incident wave uinc, we look for the scattered field u:
uinc
u
Γ Ω
u−u = 0 in Ω× [0, T ],
u(., 0) = u(., 0) = 0 in Ω,
∂u
∂n= −
∂uinc
∂non Γ× [0, T ],
+ rad. cond.
Boundary integral formulation of sound-hard scattering
Boundary integral ansatz
We search for u in the form of the retarded double-layer potential
u(x, t) = −1
4π
∫
Γ
n(y) · (x− y)
‖x− y‖
(φ(y, t− ‖x− y‖)
‖x− y‖2+φ(y, t− ‖x− y‖)
‖x− y‖
)dS(y),
which satisfies the wave equation and the initial conditions. It remains to fulfill theNeumann boundary condition
limΩ∋x→x∈Γ
n(x) · ∇xu(x, t)︸ ︷︷ ︸
=:(Wφ)(x,t)
= g(x, t) on Γ× [0, T ],
where g := −∂uinc
∂n.
Boundary integral formulation of sound-hard scattering
Weak boundary integral formulation [Bamberger, HaDuong ’86]
Find φ ∈ V such thata(ξ, φ) = b(ξ) ∀ξ ∈ V,
where
a(ξ, φ) :=
∫ T
0
∫
Γ
∫
Γ
n(x) · n(y)
4π‖x− y‖ξ(x, t) φ(y, t− ‖x− y‖)
+curlΓξ(x, t) · curlΓφ(y, t− ‖x− y‖)
4π‖x− y‖
dS(y) dS(x) dt,
b(ξ) :=
∫ T
0
∫
Γ
g(x, t) ξ(x, t) dS(x) dt.
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Time-domain boundary element method
Discrete ansatz
Replace V by a finite-dimensional subspace V h,∆t spanned by the tensor-product of Ntemporal and M spatial basis functions:
φh,∆t(x, t) :=N∑
l=1
M∑
j=1
αjl ϕj(x) bl(t).
We arrive at the (N M)× (N M) block linear systemA1,1 . . . A1,L... . . . ...
AL,1 . . . AL,L
α1...αL
=
b1...bL
,
where
(Ak,l)i,j := a (ϕi(x) bk(t), ϕj(y) bl(t)) , (bk)i := b (ϕi(x) bk(t)) , (αl)j := αjl .
Time-domain boundary element method
Matrix: a deeper look
(Ak,l)i,j =
∫
suppϕi
∫
suppϕj
n(x) · n(y)
4π‖x− y‖ϕi(x)ϕj(y)
=:Ψk,l(‖x−y‖)︷ ︸︸ ︷∫ T
0
bk(t) bl(t− ‖x− y‖) dt dS(y) dS(x)
+
∫
suppϕi
∫
suppϕj
curlΓϕi(x) · curlΓϕj(y)
4π‖x− y‖
∫ T
0
bk(t) bl(t− ‖x− y‖) dt︸ ︷︷ ︸
=:Ψk,l(‖x−y‖)
dS(y) dS(x),
Piecewise smooth time-ansatz expensive quadrature
due to nontrivial intersection of the light cone
suppΨk,l, supp Ψk,l
withsuppϕi × suppϕj.
Time-domain boundary element method
C∞-smooth (partition of unity) temporal basis [Sauter, Veit ’14]
0.5 1.5 2.5
0.2
0.4
0.6
0.8
t [s]
b0 b1 b2 b3
allows for Sauter-Schwab quadrature
over suppϕi × suppϕj.
(Ak,l)i,j =
∫
suppϕi
∫
suppϕj
n(x) · n(y)
4π‖x− y‖ϕi(x)ϕj(y)
=:Ψk,l(‖x−y‖)∈C∞(R)︷ ︸︸ ︷∫ T
0
bk(t) bl(t− ‖x− y‖) dt dS(y) dS(x)
+
∫
suppϕi
∫
suppϕj
curlΓϕi(x) · curlΓϕj(y)
4π‖x− y‖
∫ T
0
bk(t) bl(t− ‖x− y‖) dt︸ ︷︷ ︸
=:Ψk,l(‖x−y‖)∈C∞(R)
dS(y) dS(x),
To accelerate the assembly, Ψ and Ψ are replaced by piecewise Chebyshev interpolants.
Time-domain boundary element method
Convergence of ‖φh,∆t(x, .)− φanalytical(x, .)‖L2(0,T ) on the sphere
Ω the unit sphere, φanalytical a spherical harmonic function, x ∈ Γ
1st-order time-basis functions 2nd-order time-basis functions
101
102
10−3
10−2
10−1
1/N
Number of timesteps
Err
or
5 10 20 40
10−5
10−4
10−3
1/N 2
Number of timesteps
Err
or
Time-domain boundary element method
Matrix structure
The matrix is sparse and it has a block-Hessenberg structure.
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Parallelization, preconditioning, numerical experiments
Parallel implementation
• For equidistant time stepping only the blueparts have to be assembled.
• We employ a hybrid MPI-OpenMP model:
– pairs of triangles corresponding tononzero entries are evenly distributed toMPI nodes,
– on each node the assembly is performedusing OpenMP.
• We use up to 64 Intel Xeon E5 nodes (1024cores) of the cluster Anselm, VSB-TU Os-trava.
Parallelization, preconditioning, numerical experiments
Weak parallel scalability of the assembly
Submarine surface decomposed into 5604 triangles, 40 time steps
Parallelization, preconditioning, numerical experiments
Preconditioning
We approximate the upper Hessenberg matrixby an inexact lower triangular preconditioner:
A :=
(AI,I AI,II
AII,I AII,II
)≈
(AI,I 0
AII,I AII,II
)=: A
in the recursive fashion so that AI,I and AII,II
are again the inexact lower triangular precondi-tioners to the upper Hessenberg matrices AI,I
and AII,II , respectively.
Parallelization, preconditioning, numerical experiments
Numerical experiments
We compare convergence of preconditioned restarted GMRES, deflated GMRES, andflexible GMRES.
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Outline
• Parallel fast BEM and applications
• Boundary integral formulation of sound-hard scattering
• Time-domain boundary element method
• Parallelization, preconditioning, numerical experiments
• Conclusion, outlook, references
Parallel Time-Domain Boundary Element Methodfor 3-Dimensional Wave Equation
Conclusion, outlook
X Time-domain BEM for 3d wave equation, adaptivity in time,
X parallely scalable assembly and postprocessing,
→ mapping properties of the operator, preconditioning,
→ extension to the elastic wave equation.
References
• Analysis: Bamberger, Ha Duong, Math. Meth. Appl. Sci. ’86
• Numerics: Sauter, Veit, Numer. Math. ’14
• Parallelization: Veit, Merta, Zapletal, L., Int. J. Numer. Meth. Eng., submitted
• Parallel fast BEM: L., Kovar, Kovarova, Merta, Numer. Alg. ’15 (online first)
http://homel.vsb.cz/∼luk76