IsogEometric Tearing and Interconnecting

Transcription

1 IsogEometric Tearing and Interconnecting Christoph Hofer and Ludwig Mitter Johannes Kepler University, Linz Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation Hofer, Mitter JKU Linz 1 / 24

2 IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 2 / 24

3 : Motivation In 2D, using vertex primal variables works quite well. In 3D, condition number grows with H/h(1 + log H/h) 2. 2D 3D #dofs H/h κ It. #dofs H/h κ It Using continuous edge/face averages gives (1 + log(h/h)) 2. Implementation gets a bit more tricky. Present method for arbitrary linear primal variables. Pechstein, C. (2012). Finite and boundary element tearing and interconnecting solvers for multiscale problems (Vol. 90). Springer Science & Business Media. Hofer, Mitter JKU Linz 3 / 24

4 : Motivation In 2D, using vertex primal variables works quite well. In 3D, condition number grows with H/h(1 + log H/h) 2. 2D 3D #dofs H/h κ It. #dofs H/h κ It Using continuous edge/face averages gives (1 + log(h/h)) 2. Implementation gets a bit more tricky. Present method for arbitrary linear primal variables. Pechstein, C. (2012). Finite and boundary element tearing and interconnecting solvers for multiscale problems (Vol. 90). Springer Science & Business Media. Hofer, Mitter JKU Linz 3 / 24

5 : Problem formulation - cg setting Find u h V D,h : a(u h, v h ) = F, v h v h V D,h, where V D,h is a conforming discrete subspaces of V D, e.g. a(u, v) = α u v dx, F, v = fv dx + g N v ds Ω Ω Γ N V D = {u H 1 : γ 0 u = 0 on Γ D }, V D,h = k span{n (k) i,p } H1 (Ω). The variational equation is equivalent to Ku = f. Hofer, Mitter JKU Linz 4 / 24

6 IETI-DP : Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 4 / 24

7 IETI-DP : IETI-DP Given K (k) and f (k), we can reformulate [ Ke B Ku = f T ] [ ] u = B 0 λ where K e = diag(k (k) ) and f e = [f (k) ]. [ ] fe, 0 Since K e is not invertible, we need additional primal variables incorporated in K e K, B, f: continuous vertex values continuous edge/face averages Hofer, Mitter JKU Linz 5 / 24

8 IETI-DP : IETI-DP Given K (k) and f (k), we can reformulate [ Ke B Ku = f T ] [ ] u = B 0 λ where K e = diag(k (k) ) and f e = [f (k) ]. [ ] fe, 0 Since K e is not invertible, we need additional primal variables incorporated in K e K, B, f: continuous vertex values continuous edge/face averages Hofer, Mitter JKU Linz 5 / 24

9 IETI-DP : IETI-DP Find (u, λ) [ K BT B 0 ] [u ] = λ [ ] fe. 0 K is SPD, hence, we can define: F := B K BT d := B K f The saddle point system is equivalent to solving: Using the preconditioner M sd find λ U : F λ = d., we obtain: κ(m sd F ker( B T ) ) C max 1 k N ( 1 + log ( Hk h k )) 2, Hofer, Mitter JKU Linz 6 / 24

10 IETI-DP : A bit more on primal variables W (k) := V (k) h, W := k W (k), Ŵ := V h. Intermediate space W : Ŵ W W, K := K W is SPD. Let Ψ Vh be a set of linearly independent primal variables, W := {w W : ψ Ψ : ψ(w i ) = ψ(w j )} n W := W (k) k=0 W = W Π W, W Π Ŵ (there are many choices for W Π) where W (k) := {w W (k) : ψ Ψ : ψ(w k ) = 0} If W ker(k) = {0}, then K is invertible. Hofer, Mitter JKU Linz 7 / 24

11 IETI-DP : A bit more on primal variables W (k) := V (k) h, W := k W (k), Ŵ := V h. Intermediate space W : Ŵ W W, K := K W is SPD. Let Ψ Vh be a set of linearly independent primal variables, W := {w W : ψ Ψ : ψ(w i ) = ψ(w j )} n W := W (k) k=0 W = W Π W, W Π Ŵ (there are many choices for W Π) where W (k) := {w W (k) : ψ Ψ : ψ(w k ) = 0} If W ker(k) = {0}, then K is invertible. Hofer, Mitter JKU Linz 7 / 24

12 IETI-DP : Typical examples of Ψ and ψ Choices for ψ: Vertex evaluation: ψ V (v) = v(v) Edge averages: ψ E (v) = 1 E E v ds Face averages: ψ F (v) = 1 F Choices for Ψ: Algorithm A: Ψ = {ψ V } F v ds Algorithm B: Ψ = {ψ V } {ψ E } {ψ F } Algorithm C: Ψ = {ψ V } {ψ E } Hofer, Mitter JKU Linz 8 / 24

13 IETI-DP : Since W W, there is a natural embedding Ĩ : W W. We can define: B := BĨ : W U, B T = ĨT B T : U W, f := ĨT f W As before, we can write our saddle point problem as: Find (u, λ) W U : [ ] [u ] [ ] K BT f =, B 0 λ 0 Hofer, Mitter JKU Linz 9 / 24

14 IETI-DP : Since W W, there is a natural embedding Ĩ : W W. We can define: B := BĨ : W U, B T = ĨT B T : U W, f := ĨT f W As before, we can write our saddle point problem as: Find (u, λ) W U : [ ] [u ] [ ] K BT f =, B 0 λ 0 Hofer, Mitter JKU Linz 9 / 24

15 Implementation of primal variables : Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 9 / 24

16 Implementation of primal variables : CG Algorithm The equation F λ = d, is solved via the PCG algorithm: λ 0 given r 0 = d F λ 0, k = 0, β = 0 repeat s k = M sd r k β k = (r k,s k ) (r k,s k ) p k = s k + β k p k α k = (r k,s k ) (F p k,p k ) λ k+1 = r k + α k p k r k+1 = r k α k F p k k = k + 1 until stopping criterion fulfilled Hofer, Mitter JKU Linz 10 / 24

17 Implementation of primal variables : Required realizations In order to use the CG-algorithm, we need Application of F := B K BT Application of M sd := B DS e B T D Representation of W : K : W W, K : W W W = W Π W (k) representation of w W as {w Π, {w (k) } k} representation of f W as {f Π, {f (k) } k} Hofer, Mitter JKU Linz 11 / 24

18 Implementation of primal variables : Required realizations In order to use the CG-algorithm, we need Application of F := B K BT Application of M sd := B DS e B T D Representation of W : K : W W, K : W W W = W Π W (k) representation of w W as {w Π, {w (k) } k} representation of f W as {f Π, {f (k) } k} Hofer, Mitter JKU Linz 11 / 24

19 Implementation of primal variables : Required realizations W = W Π W (k) Construction of the primal space W Π and its basis. We choose the so called energy minimizing primal subspaces. The basis should be at least local and nodal. 2 possibilities to realize the dual space W : Transformation of basis: construction of basis, such that the primal variables vanishes. Realization with local constraints: constraints are added to the matrix to enforce vanishing of primal variables. Hofer, Mitter JKU Linz 12 / 24

20 Implementation of primal variables : Required realizations W = W Π W (k) Construction of the primal space W Π and its basis. We choose the so called energy minimizing primal subspaces. The basis should be at least local and nodal. 2 possibilities to realize the dual space W : Transformation of basis: construction of basis, such that the primal variables vanishes. Realization with local constraints: constraints are added to the matrix to enforce vanishing of primal variables. Hofer, Mitter JKU Linz 12 / 24

21 Implementation of primal variables : Required realizations In any case, a block LDL T factorization yields: [ ] KΠΠ K K = Π ( ) K Π K [ ] [ K I 0 S = K K Π I where S Π = K ΠΠ K Π K K Π. Π 0 0 K ] [ I KΠ K ], 0 I In order to apply K, one needs a realization of the individual subcomponents. If only continuous vertex values are use, one obtains ( ) just by reordering. (as in the previous talk) Hofer, Mitter JKU Linz 13 / 24

22 Implementation of primal variables : Required realizations In any case, a block LDL T factorization yields: [ ] KΠΠ K K = Π ( ) K Π K [ ] [ K I 0 S = K K Π I where S Π = K ΠΠ K Π K K Π. Π 0 0 K ] [ I KΠ K ], 0 I In order to apply K, one needs a realization of the individual subcomponents. If only continuous vertex values are use, one obtains ( ) just by reordering. (as in the previous talk) Hofer, Mitter JKU Linz 13 / 24

23 Implementation of primal variables : Choosing W Π and constructing the basis Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 13 / 24

24 Implementation of primal variables : Choosing W Π and constructing the basis A nice subspace W Π and its basis energy minimizing primal subspaces: W Π := W K W Π is K-orthogonal to W, i.e. Kw Π, w = 0 w Π W Π, w W. [ ] KΠΠ 0 K = = 0 K K = [ ] K ΠΠ 0 0 K Hofer, Mitter JKU Linz 14 / 24

25 Implementation of primal variables : Choosing W Π and constructing the basis A nice subspace W Π and its basis energy minimizing primal subspaces: W Π := W K W Π is K-orthogonal to W, i.e. Kw Π, w = 0 w Π W Π, w W. [ ] KΠΠ 0 K = = 0 K K = [ ] K ΠΠ 0 0 K Hofer, Mitter JKU Linz 14 / 24

26 Implementation of primal variables : Choosing W Π and constructing the basis Choosing W Π and constructing the basis Nodal basis φ : ψ i ( φ j ) = δ i,j. For each patch k we define: C (k) : W (k) R n Π,k The basis functions { φ (k) j } n Π,k j=1 [ (C (k) v) l = ψ i(k,l) (v) v W (k), l K (k) C (k)t C (k) 0 are the solution of: ] [ φ(k) ] µ (k) = [ ] 0 I For each patch the LU factorization is computed and stored. Hofer, Mitter JKU Linz 15 / 24

27 Implementation of primal variables : Choosing W Π and constructing the basis Choosing W Π and constructing the basis Nodal basis φ : ψ i ( φ j ) = δ i,j. For each patch k we define: C (k) : W (k) R n Π,k The basis functions { φ (k) j } n Π,k j=1 [ (C (k) v) l = ψ i(k,l) (v) v W (k), l K (k) C (k)t C (k) 0 are the solution of: ] [ φ(k) ] µ (k) = [ ] 0 I For each patch the LU factorization is computed and stored. Hofer, Mitter JKU Linz 15 / 24

28 Implementation of primal variables : Application of K Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 15 / 24

29 Implementation of primal variables : Application of K Application of K Given f := {f Π, {f (k) }} W, Find w := {w Π, {w (k) }} W : K = The application of K reduces to w = K f [ ] K ΠΠ 0 0 K w Π = K ΠΠ f Π w (k) = K(k) (k) f k = 0,..., n Hofer, Mitter JKU Linz 16 / 24

30 Implementation of primal variables : Application of K Application of K (k) The application of K (k) corresponds to K (k) w k = f (k) with the constraint C (k) w k = 0. This is equivalent to: [ ] [wk ] K (k) C (k)t C (k) = 0 [ ] f (k) 0 Hofer, Mitter JKU Linz 17 / 24

31 Implementation of primal variables : Application of K Application of K (k) The application of K (k) corresponds to K (k) w k = f (k) with the constraint C (k) w k = 0. This is equivalent to: [ ] [wk ] K (k) C (k)t C (k) = 0 [ ] f (k) 0 Hofer, Mitter JKU Linz 17 / 24

32 Implementation of primal variables : Application of K Application of K ΠΠ K ΠΠ can be assembled from the patch local matrices K (k) ΠΠ. Due to our special construction of φ (k), we have ( K (k) ΠΠ )i,j = K = = µ (k) i,j (k) φ(k) i, µ (k) i, C φ (k) j (k) φ(k) j = C (k)t µ (k) i, =, e (k) µ (k) i j (k) φ j Once K ΠΠ is assembled, one can calculate its LU factorization. Hofer, Mitter JKU Linz 18 / 24

33 Implementation of primal variables : Application of K Summary for application of F = BK BT Given λ U: 1. Application of B T : {f (k) } n k=0 = BT λ 2. Application of ĨT : {f Π, {f (k) }n k=0 } = ( ĨT {f (k) } n ) k=0 3. Application of K : w Π = K w (k) ΠΠ f Π (k) f = K(k) k = 0,..., n ( ) 4. Application of Ĩ : {w(k) } n k=0 = Ĩ {w Π, {w (k) }n k=0 } 5. Application of B : ν = B ( {w (k) } n ) k=0 It remains to investigate Ĩ and ĨT. Hofer, Mitter JKU Linz 19 / 24

34 Implementation of primal variables : Application of K Application of Ĩ and ĨT embedding operator: Ĩ : W W {w Π, {w (k) } k} ΦA T w Π + w partial assembling operator: ĨT : W W f {AΦ T f, {(I C T Φ T )f} k } Φ... basis of W Π (block version) C... matrix representation of primal variables W Π (block version) A... assembling operator A T... distribution operator Hofer, Mitter JKU Linz 20 / 24

35 Implementation of primal variables : Application of the preconditioner Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 20 / 24

36 Implementation of primal variables : Application of the preconditioner Application of M sd The application of the preconditioner M sd = B DSBD T the application of S: is basically S = diag(s (k) ) S (k) = K (k) BB K(k) BI (K(k) II ) K (k) IB The calculation of v (k) = S (k) w (k) consists of 2 steps: 1. Solve: K (k) II x(k) = K (k) IB w(k) (Dirichlet problem) 2. v (k) = K (k) BB w(k) + K (k) BI x(k) Again, a LU factorization of K (k) II can be computed and stored. Hofer, Mitter JKU Linz 21 / 24

37 Implementation of primal variables : Application of the preconditioner Application of M sd The application of the preconditioner M sd = B DSBD T the application of S: is basically S = diag(s (k) ) S (k) = K (k) BB K(k) BI (K(k) II ) K (k) IB The calculation of v (k) = S (k) w (k) consists of 2 steps: 1. Solve: K (k) II x(k) = K (k) IB w(k) (Dirichlet problem) 2. v (k) = K (k) BB w(k) + K (k) BI x(k) Again, a LU factorization of K (k) II can be computed and stored. Hofer, Mitter JKU Linz 21 / 24

38 Numerical examples : Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 21 / 24

39 Numerical examples : Example with α 1, p = 4 2D 3D V V #dofs H/h κ It. #dofs H/h κ It V + E V + E + F Hofer, Mitter JKU Linz 22 / 24

40 Numerical examples : p-dependence: 2D + 3D & different multiplicity keeping multiplicity & increasing smoothness ( ) increasing multiplicity & keeping smoothness ( ) increase mult. keep mult. cg - dependence on p (2D) increase mult. keep mult. cg - dependence on p (3D) condition number - κ condition number - κ degree - p degree - p Hofer, Mitter JKU Linz 23 / 24

41 Conclusion : Overview IETI-DP Implementation of primal variables 1. Choosing W Π and constructing the basis 2. Application of K 3. Application of the preconditioner Numerical examples Conclusion Hofer, Mitter JKU Linz 23 / 24

42 Conclusion : Conclusion and Extensions Also other primal variables can be realized in an efficient way. Provides application of IETI-DP to 3D problems. With suitable scaling robustness wrt. jumping coefficients. Method can be combined with dg-formulation. Parallelization wrt. patches (distributed memory setting). Instead of LU-factorization, one can use Multigrid (inexact IETI). Extension to nonlinear problems Apply IETI to linearized equation Apply IETI to non-linear equation and use Newton on each subdomain. Hofer, Mitter JKU Linz 24 / 24