paralleltime-domainboundary elementmethod for 3 ...naumann/pint2015/lukas.pdf ·...

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]


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.


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


Acoustics of a railway wheel

A joint work with J. Szweda, Department of mechanics, VSB–TU Ostrava


Acoustics of a railway wheel


Fast BEM: geometry clustering, H-matrices, ACA/FMM


Scattering of infrared light in a DLP projector

Ei

Es

Omegam Omegap

E


Homogenization of periodic structures

L., Bouchala, Theuer, Zapletal: Numer Math (submitted)


Parallel fast BEM using cyclic graph decompositions

01

2

3

4

5

67

8

9

10

11

12

rotate

G0

G1

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

L., Kovar, Kovarova, Merta: Numer Alg (online first)

Solution to the system of 2.7M DOFs on 273 cores in 16 minutes.


3d wave equation, simult. space-time discretization

Veit, Merta, Zapletal, L.: IJNME (submitted)


Outline






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 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.


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.


Outline






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 .


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.


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.


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


Matrix structure

The matrix is sparse and it has a block-Hessenberg structure.


Outline






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.


Weak parallel scalability of the assembly

Submarine surface decomposed into 5604 triangles, 40 time steps


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.


Numerical experiments

We compare convergence of preconditioned restarted GMRES, deflated GMRES, andflexible GMRES.


Outline







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

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