The boundary element method discretised with the space-time method - - PowerPoint PPT Presentation

The boundary element method discretised with the space-time method for the heat equation in 2D

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Kazuki Niino, Olaf Steinbach

TU Graz

Background

• Calderon’s Preconditioning
• Flexible mesh
• Stability
• Large coefficient matrix Fast methods are indispensable
• Space-time method for a BEM

2

Calderon’s preconditioning for a BEM with space- time method for heat equation in 2D

• A preconditioning specific to a BEM
• Widely used (Laplace, Helmholtz, Maxwell, etc...)
• Fast methods for BEMs
• With iteration methods (Fast multipole method)
• With fast direct methods (H-matrix and ACA etc.)

Formulations

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Heat equation in 2D

• Governing equation

4

∂u ∂t (x, t) − ∆u(x, t) = 0

• Initial condition

u(x, 0) = f(x) u(x, t) = g(x, t)

• Boundary condition

Ω Γ in

• n

in Ω Γ × (0, T)

Q := Ω × (0, T)

Representation formula

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+ Z

Ω

K(x − x0, t)f(x0)dx0 where H(t) : Heaviside function

u(x, t) = Z t Z

Γ

⇢ K(x − y, t − τ) ∂u ∂ny (y, τ) − ∂K ∂ny (x − y, t − τ)g(y, τ)

• dydτ

K(x, t) = exp ⇣ − |x|2

4t

⌘ 4πt H(t) x ∈ Ω × (0, T) for

Boundary integral equation

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where q := ∂u ∂ny 1 2u(x, t) = Sq − Du + Z

Ω

K(x − x0, t)f(x0) dx0 Sϕ = Z t Z

Γ

K(x − y, t − τ)ϕ(y, τ)dydτ Dϕ = Z t Z

Γ

∂K ∂ny (x − y, t − τ)ϕ(y, τ)dydτ

Space-time method

• Treat the time and space coordinate in

the same way for discretisation

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X Y Z

Galerkin method

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• Discretised integral equation

Sq = b

where

(S)ij = Z T Z

Γ

dxdt ⇢ φi(x, t) Z t Z

Γ

K(x − y, t − τ)φj(y, τ)dydτ

• (b)i =

Z T dxdtφi(x, t) ( 1 2g(x, t) + Z T Z

Γ

∂K ∂ny (x − y, t − τ)g(y, τ)dydτ − Z

Ω

K(x − x0, t)f(x0)dx0 )

• Integral operator
• Matrix

S, D, · · · S, D, · · ·

Galerkin method

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φi : piecewise linear element associated with the triangular mesh

• Coefficient matrix

q = X

j

qjφj

(S)ij = Z T Z

Γ

dxdt ⇢ φi(x, t) Z t Z

Γ

K(x − y, t − τ)φj(y, τ)dydτ

Calderon’s preconditioning

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Preconditioning

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• Linear equation

Ax = b

• Preconditioned equation

AM −1y = b, x = M −1y M −1Ax = M −1b (Right preconditioning) (Left preconditioning) : Preconditioner

M

• Choose so that
• M is “similar” to A in some sence
• The inverse of M can be calculated easily

M

Integral equations

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where

1 2u = Sq − Du + S0f

1 2q = D∗q − Nu + D0f

Dϕ = Z

e Γ

∂K ∂ny (x − y, t − τ)ϕ(y, τ)dS

D∗ϕ = Z

e Γ

∂K ∂nx (x − y, t − τ)ϕ(y, τ)dS

Nϕ = = Z

e Γ

∂2K ∂nx∂ny (x − y, t − τ)ϕ(y, τ)dS

S0ϕ = Z

Ω

K(x − x0, t)ϕ(x0)dx0

D0ϕ = Z

Ω

∂K ∂nx (x − x0, t)ϕ(x0) dx0

e Γ = Γ × (0, T)

Sϕ = Z

e Γ

K(x − y, t − τ)ϕ(y, τ)dS

Representation formulae

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For any and

φ

ψ where

Dϕ = Z

e Γ

∂K ∂ny (x − y, t − τ)ϕ(y, τ)dS

D∗ϕ = Z

e Γ

∂K ∂nx (x − y, t − τ)ϕ(y, τ)dS

Nϕ = = Z

e Γ

∂2K ∂nx∂ny (x − y, t − τ)ϕ(y, τ)dS

S0ϕ = Z

Ω

K(x − x0, t)ϕ(x0)dx0

D0ϕ = Z

Ω

∂K ∂nx (x − x0, t)ϕ(x0) dx0 Sϕ = Z

e Γ

K(x − y, t − τ)ϕ(y, τ)dS

u = Sφ + 1 2ψ − Dψ + S0f

q = 1 2φ + D∗φ − Nψ + D0f

e Γ = Γ × (0, T)

Calderon’s formulae

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Substitute representation formulae into integral equation

= 0

1 2 ✓ Sφ + 1 2ψ − Dψ + S0f ◆ =S ✓1 2φ + D∗φ − Nψ + D0f ◆ − D ✓ Sφ + 1 2ψ − Dψ + S0f ◆ + S0f

0 = (SD∗ − DS)φ + ✓ DD − SN − 1 2I ◆ ψ + ✓1 2S0f + SD0f − DS0f ◆

Calderon’s formulae

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• Integral equation

0 = −1 2u + S ✓ ∂u ∂n ◆ − Du + S0f

1 2S0f + SD0f − DS0f = − 1 2S0f + S ✓ ∂ ∂nS0f ◆ − DS0f + S0f =0

(since is a solution of heat eq. with initial function )

S0f

f

Calderon’s preconditioning

• Calderon formulae

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where

Dϕ = Z

e Γ

∂K ∂ny (x − y, t − τ)ϕ(y, τ)dS

D∗ϕ = Z

e Γ

∂K ∂nx (x − y, t − τ)ϕ(y, τ)dS

Nϕ = = Z

e Γ

∂2K ∂nx∂ny (x − y, t − τ)ϕ(y, τ)dS

Sϕ = Z

e Γ

K(x − y, t − τ)ϕ(y, τ)dS

DD − SN = 1 4I SD∗ − DS = 0 −D∗N + ND = 0 D∗D∗ − NS = 1 4I

Calderon’s preconditioning

• Calderon’s formulae

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• Dirichlet problem

given compact

is expected to be a good “preconditioner”

Sq = 1 2u(x, t) + Du − Z

Ω

K(x − x0, t)f(x0) dx0

SN = −1 4I + DD

N

：arbitrary functions defined on e Γ

⇢ ψ = Sφ φ(x) ≈ PNh

m=1 φntn(x),

ψ(x) ≈ PNh

n=1 ψntn(x)

Tψ = Sφ

T := Z

e Γ

tm · tndS

φ, ψ

Nh

X

n=1

Smnφn ≈ Z

e Γ

tm(x)(Sφ)(x)dSx = Z

e Γ

tm(x)ψ(x)dSx ≈ Z

e Γ

tm(x)

Nh

X

n=1

ψntn(x)dSx =

Nh

X

n=1

Z

e Γ

tm(x)tn(x)dSxψn

（ : a basis function) tn(x)

Preconditioning in Galerkin’s Method

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ψ = Sφ

ψ = T −1Sφ

SN = −1 4I + K T 1ST 1N = −1 4I + K0 ST −1NT −1 = −1 4I + K

Preconditioning in Galerkin’s Method

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• Operator
• Matrix

Sq = b

ST 1NT 1q0 = b, q = T 1NT 1q0

(N)ij = Z T Z

Γ

dxdt ⇢ φi(x, t) Z t Z

Γ

∂2K ∂nxny (x − y, t − τ)φj(y, τ)dydτ

• (T)ij =

Z T Z

Γ

φi(x, t)φj(x, t)dxdt

Preconditioner

• Discretised integral equation (Dirichlet problem)
• Preconditioned equation

14

where

Inversion of T

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• The Gram matrix T is

T can be easily inverted with iteration methods

• Symmetry
• Sparse
• Diagonally dominant (if is piecewise linear)

φi

(T)ij = Z T Z

Γ

φi(x, t)φj(x, t)dxdt

Numerical examples

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Numerical examples

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• Iterative method
• Preconditioning
• Calderon’s preconditioning
• Point Jacobi (Scaling)
• No preconditioning
• GMRES with error tolerance

10−5

10−5

• GMRES with error tolerance for inverting T

Cylindrical domain

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DOF DOF 35 118 1237

• Num. of element
• Num. of element

114 204 2362 L2 error Calderon 0.0385 0.0117 0.00359 L2 error Jaboci 0.0385 0.0117 0.00360 L2 error No precond. 0.0385 0.0117 0.00359

X Y Z

• Initial condition
• Boundary condition

u(x, 0) = x2

1 − x2 2

u(x, 0) = x2

1 − x2 2

Cylindrical domain

Cylindrical domain

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X Y Z

• Initial condition
• Boundary condition

5 10 15 20 25 30 35 40 45 500 1000 1500 2000 2500 3000

Iteration Numbers Number of nodes

precon3/ precon1/ precon0/

u(x, 0) = x2

1 − x2 2

u(x, 0) = x2

1 − x2 2

Cylindrical domain

Cuboid domain

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X Z Y Y 1.51

• 2.47

neumann

• 0.482

X Z

• Initial condition
• Boundary condition

u(x, 0) = x2

1 − x2 2

u(x, 0) = x2

1 − x2 2

10 20 30 40 50 60 1000 2000 3000 4000 5000 6000

Iteration Numbers Number of nodes

precon3/ precon1/ precon0/

Conclusion

• The BEM discretised with the space-time method for

the heat equation in 2D is discussed

• Calderon’s preconditioning successfully reduces the

number of iteration

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• Conclusion
• Future works
• Resolve the bad accuracy due to the continuous basis

functions

• Fast methods