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 1 Kazuki Niino, Olaf Steinbach TU Graz

2 Background • Space-time method for a BEM • Flexible mesh • Stability • Large coefficient matrix Fast methods are indispensable • Fast methods for BEMs • With iteration methods (Fast multipole method) • With fast direct methods (H-matrix and ACA etc.) • Calderon’s Preconditioning • A preconditioning specific to a BEM • Widely used (Laplace, Helmholtz, Maxwell, etc...) Calderon’s preconditioning for a BEM with space- time method for heat equation in 2D

3 Formulations

4 Heat equation in 2D • Governing equation Γ ∂ u ∂ t ( x , t ) − ∆ u ( x , t ) = 0 in Q := Ω × (0 , T ) Ω • Initial condition in Ω u ( x , 0) = f ( x ) • Boundary condition u ( x , t ) = g ( x , t ) on Γ × (0 , T )

5 Representation formula Z t ⇢ � Z K ( x − y , t − τ ) ∂ u ( y , τ ) − ∂ K u ( x , t ) = ( x − y , t − τ ) g ( y , τ ) d y d τ ∂ n y ∂ n y 0 Γ Z x ∈ Ω × (0 , T ) K ( x − x 0 , t ) f ( x 0 )d x 0 + for Ω where ⇣ ⌘ − | x | 2 exp 4 t K ( x , t ) = H ( t ) 4 π t H ( t ) : Heaviside function

6 Boundary integral equation 1 Z K ( x − x 0 , t ) f ( x 0 ) d x 0 2 u ( x , t ) = S q − D u + Ω where Z t Z S ϕ = K ( x − y , t − τ ) ϕ ( y , τ )d y d τ 0 Γ Z t ∂ K Z D ϕ = ( x − y , t − τ ) ϕ ( y , τ )d y d τ ∂ n y 0 Γ q := ∂ u ∂ n y

7 Space-time method • Treat the time and space coordinate in the same way for discretisation Z Y X

8 Galerkin method • Discretised integral equation S q = b where Z T Z t ⇢ � Z Z ( S ) ij = d x d t φ i ( x , t ) K ( x − y , t − τ ) φ j ( y , τ )d y d τ 0 Γ 0 Γ Z T Z T ( ) 1 Z Z ∂ K K ( x − x 0 , t ) f ( x 0 )d x 0 ( b ) i = d x d t φ i ( x , t ) 2 g ( x , t ) + ( x − y , t − τ ) g ( y , τ )d y d τ − ∂ n y Γ Ω 0 0 • Integral operator • Matrix S , D , · · · S, D, · · ·

9 Galerkin method • Coefficient matrix Z T Z t ⇢ � Z Z ( S ) ij = d x d t φ i ( x , t ) K ( x − y , t − τ ) φ j ( y , τ )d y d τ 0 Γ 0 Γ φ i : piecewise linear element associated with the triangular mesh X q j φ j q = j

10 Calderon’s preconditioning

11 Preconditioning • Linear equation A x = b • Preconditioned equation AM − 1 y = b , x = M − 1 y (Right preconditioning) (Left preconditioning) M − 1 A x = M − 1 b : Preconditioner M • Choose so that M • M is “similar” to A in some sence • The inverse of M can be calculated easily

12 Integral equations 1 2 u = S q − D u + S 0 f e Γ = Γ × (0 , T ) 1 2 q = D ∗ q − N u + D 0 f Z Y X where Z ∂ K Z S ϕ = K ( x − y , t − τ ) ϕ ( y , τ )d S D ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n y e e Γ Γ ∂ K Z ∂ 2 K Z D ∗ ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S N ϕ = = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n x ∂ n x ∂ n y e e Γ Γ Z ∂ K Z K ( x − x 0 , t ) ϕ ( x 0 )d x 0 S 0 ϕ = ( x − x 0 , t ) ϕ ( x 0 ) d x 0 D 0 ϕ = ∂ n x Ω Ω

13 Representation formulae φ For any and ψ u = S φ + 1 2 ψ − D ψ + S 0 f e Γ = Γ × (0 , T ) q = 1 2 φ + D ∗ φ − N ψ + D 0 f Z Y X where Z ∂ K Z S ϕ = K ( x − y , t − τ ) ϕ ( y , τ )d S D ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n y e e Γ Γ ∂ K Z ∂ 2 K Z D ∗ ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S N ϕ = = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n x ∂ n x ∂ n y e e Γ Γ Z ∂ K Z K ( x − x 0 , t ) ϕ ( x 0 )d x 0 S 0 ϕ = ( x − x 0 , t ) ϕ ( x 0 ) d x 0 D 0 ϕ = ∂ n x Ω Ω

14 Calderon’s formulae Substitute representation formulae into integral equation ✓ ◆ 1 S φ + 1 2 ψ − D ψ + S 0 f 2 ✓ 1 ◆ ✓ ◆ S φ + 1 = S 2 φ + D ∗ φ − N ψ + D 0 f − D 2 ψ − D ψ + S 0 f + S 0 f ✓ ◆ DD − SN − 1 0 = ( SD ∗ − DS ) φ + 2 I ψ ✓ 1 ◆ 2 S 0 f + SD 0 f − DS 0 f + = 0

15 Calderon’s formulae 1 2 S 0 f + SD 0 f − DS 0 f ✓ ∂ ◆ = − 1 2 S 0 f + S ∂ n S 0 f − DS 0 f + S 0 f =0 f S 0 f (since is a solution of heat eq. with initial function ) • Integral equation ✓ ∂ u ◆ 0 = − 1 2 u + S − D u + S 0 f ∂ n

16 Calderon’s preconditioning • Calderon formulae DD − SN = 1 4 I SD ∗ − DS = 0 D ∗ D ∗ − NS = 1 4 I − D ∗ N + ND = 0 where Z ∂ K Z S ϕ = K ( x − y , t − τ ) ϕ ( y , τ )d S D ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n y e e Γ Γ ∂ K Z ∂ 2 K Z D ∗ ϕ = ( x − y , t − τ ) ϕ ( y , τ )d S N ϕ = = ( x − y , t − τ ) ϕ ( y , τ )d S ∂ n x ∂ n x ∂ n y e e Γ Γ

is expected to be a good “preconditioner” 17 Calderon’s preconditioning • Dirichlet problem S q = 1 Z K ( x − x 0 , t ) f ( x 0 ) d x 0 2 u ( x , t ) + D u − Ω given • Calderon’s formulae SN = − 1 4 I + DD compact N

18 Preconditioning in Galerkin’s Method φ , ψ ： arbitrary functions defined on e Γ ⇢ ψ = S φ φ ( x ) ≈ P N h ψ ( x ) ≈ P N h m =1 φ n t n ( x ) , n =1 ψ n t n ( x ) （ : a basis function) t n ( x ) N h Z X t m ( x )( S φ )( x )d S x S mn φ n ≈ e Γ n =1 Z = t m ( x ) ψ ( x )d S x e T ψ = S φ Γ N h Z X t m ( x ) ψ n t n ( x )d S x Z ≈ T := t m · t n d S e Γ n =1 e Γ N h Z X = t m ( x ) t n ( x )d S x ψ n e Γ n =1

14 Preconditioning in Galerkin’s Method • Operator • Matrix ψ = T − 1 S φ ψ = S φ T � 1 ST � 1 N = − 1 SN = − 1 4 I + K 0 4 I + K ST − 1 NT − 1 = − 1 4 I + K

14 Preconditioner • Discretised integral equation (Dirichlet problem) S q = b • Preconditioned equation ST � 1 NT � 1 q 0 = b , q = T � 1 NT � 1 q 0 where Z T Z ( T ) ij = φ i ( x , t ) φ j ( x , t )d x d t Γ 0 Z T Z t ∂ 2 K ⇢ � Z Z ( N ) ij = d x d t φ i ( x , t ) ( x − y , t − τ ) φ j ( y , τ )d y d τ ∂ n x n y 0 Γ 0 Γ

21 T can be easily inverted with iteration methods Inversion of T Z T Z ( T ) ij = φ i ( x , t ) φ j ( x , t )d x d t 0 Γ • The Gram matrix T is • Symmetry • Sparse • Diagonally dominant (if is piecewise linear) φ i

22 Numerical examples

23 Numerical examples • Iterative method • GMRES with error tolerance 10 − 5 • Preconditioning • Calderon’s preconditioning • 10 − 5 GMRES with error tolerance for inverting T • Point Jacobi (Scaling) • No preconditioning

0.0385 0.0117 0.0385 precond. No error L2 0.00360 0.0117 24 Jaboci error L2 0.00359 0.0385 0.00359 Calderon error L2 2362 204 114 Num. of element Num. of element 1237 118 35 DOF DOF 0.0117 Cylindrical domain • Initial condition u ( x , 0) = x 2 1 − x 2 2 • Boundary condition Z Y X u ( x , 0) = x 2 1 − x 2 2 Cylindrical domain

25 Cylindrical domain • Initial condition u ( x , 0) = x 2 1 − x 2 2 • Boundary condition Z Y X u ( x , 0) = x 2 1 − x 2 2 Cylindrical domain precon 3 / precon 1 / precon 0 / 45 40 Iteration Numbers 35 30 25 20 15 10 5 0 0 500 1000 1500 2000 2500 3000 Number of nodes

26 Cuboid domain • Initial condition u ( x , 0) = x 2 1 − x 2 2 • Boundary condition u ( x , 0) = x 2 1 − x 2 Z 2 Y X precon 3 / precon 1 / precon 0 / 60 Iteration Numbers 50 40 30 20 10 neumann Z Y X -2.47 -0.482 1.51 0 0 1000 2000 3000 4000 5000 6000 Number of nodes

27 Conclusion • 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 • Future works • Resolve the bad accuracy due to the continuous basis functions • Fast methods