Block-Diagonal Decomposition of Matrices based on -Algebra for Structural Optimization with Symmetry Property

Transcription

1 Block-Diagonal Decomposition of Matrices based on -Algebra for Structural ptimization with Symmetry Property Yoshihiro Kanno Kazuo Murota Masakazu Kojima Sadayoshi Kojima University of okyo (Japan) okyo Institute of echnology (Japan) June 16, 2008

2 simultaneous block-diagonal decomposition given: A 1,A 2,...,A m symmetric n n find: P R n n orthogonal A = 1 symm. A 2 = symm. A 3 = symm. simultaneous block diagonalization P A 1P= P A 2P= P A 3P= CJK-SM 08 2/14

3 simultaneous block-diagonal decomposition: application K(x) : stiffness matrix x : design variables given: K 1,K 2,...,K m find: P R n n orthogonal K(x)=x 1 K 1 +x 2 K 2 +x 3 K 3 CJK-SM 08 3/14

4 simultaneous block-diagonal decomposition: application K(x) : stiffness matrix x : design variables given: K 1,K 2,...,K m find: P R n n orthogonal K(x)=x 1 K 1 +x 2 K 2 +x 3 K 3 simultaneous block diagonalization P K 1 P P K 2 P P K(x)P =x1 +x 2 +x 3 P K 3 P CJK-SM 08 3/14

5 application: SDP min s.t. m p=1 b py p C m p=1 A py p variables : y 1,...,y m coefficients : b 1,...,b m, A 1,...,A m,c S n n n symmetric matrices X X is positive semidefinite nonlinear but convex CJK-SM 08 4/14

6 application: SDP min s.t. m p=1 b py p C m p=1 A py p truss optimization (with eigenvalue constraints) min s.t. vol(x) Ω 1 (x) Ω can be solved by SDP [hsaki et al. 99] as min s.t. c x m p=1 (K p ΩM p )y p optimization with buckling load factor constraints robust structural optimization CJK-SM 08 4/14

7 application: SDP min s.t. m p=1 b py p C m p=1 A py p P A 1P= P A 2P=... P CP= C (1) C (l) m p=1. m p=1 A (1) p y p A (l) p y p CJK-SM 08 4/14

8 application: SDP min s.t. m p=1 b py p C m p=1 A py p P A 1P= P A 2P=... P CP= group theory [Gatermann & Parrilo 04], [Bai & de Klerk 07] [de Klerk & Sotirov 07] matrix -algebra [de Klerk, Pasechnik & Scrijver 07] aim: numerical algorithm which can be executed without any knowledge of symmetry property in advance CJK-SM 08 4/14

9 matrix -algebra R n n (set of n n matrices) I n If A, B, α R, then A + B AB αa A is a matrix -algebra CJK-SM 08 5/14

10 structure theorem 1. Q : orthogonal s.t. Q Q = S 1 S S l, S j j :simple 2. If j is simple, then ˆP : orthogonal s.t. B j ˆP B j j ˆP =..., B j B j ˆ j : irreducible CJK-SM 08 6/14

11 structure theorem: transformations to simple components Q : orthogonal, for any A, Q AQ= S 1 no common eigenvalues S 2 & irreducible components ˆP : orthogonal, for any A, CJK-SM 08 7/14

12 structure theorem: transformations to simple components Q : orthogonal, for any A, Q AQ= S 1 & irreducible components ˆP : orthogonal, for any A, B 1 B1 S 2 no common eigenvalues muptiplicity = 3 muptiplicity = 2 no multiplicity P AP= B 1 B 2 no multiplicity B 2 CJK-SM 08 7/14

13 algorithm (1) input: A 1,A 2,...,A m 1. Generate random r 1,...,r m. Put X m i=1 r ia i. 2. Eigenvalue decomposition of X: α 1 I m1 Q α 2 I m2 XQ =..., Q = [ Q 1,Q 2,...,Q k ]. α k I mk 3. Compute Q A 1 Q,...,Q A m Q. CJK-SM 08 8/14

14 algorithm (1) input: A 1,A 2,...,A m 1. Generate random r 1,...,r m. Put X m i=1 r ia i. 2. Eigenvalue decomposition of X: 3. Compute Q A 1 Q,...,Q A m Q. 4. Rearrange eigenvectors Q as ˆQ = [ Q[K 1 ],Q[K 2 ],...,Q[K l ] ] so that Q ^ A Q= ^ Q ^ 1 A 2Q= ^... which is the decomposition to simple components. CJK-SM 08 8/14

15 input: algorithm (2) Â 1, Â2,...,Âm simple 1. We know that the simple component is decomposed as P AP= ^ B B B muptiplicity = 3 2. e.g. B R 2 2, P ÂP can be rearranged as P Â P b 11 I b 12 I b 13 I = b 21 I b 22 I b 23 I, I = b 31 I b 32 I b 33 I [ ] 1 0. ( ) 0 1 We first find the transformation ( ). CJK-SM 08 9/14

16 input: algorithm (2) Â 1, Â2,...,Âm simple 1. In order to obtain P Â P = b 11 I b 12 I b 13 I b 21 I b 22 I b 23 I b 31 I b 32 I b 33 I, I = [ ] 1 0, ( ) 0 1 we use the form of b ij P i P j = Âij. 2. Put P 1 = I. hen, b 31 P 3 P 1 = Â31 = P 3 = Â31/b 31 b 23 P 2 P 3 = Â23 = P 2 = Â23P 3 /b 23 CJK-SM 08 9/14

17 algorithm (1) & (2) (1) simple components (2) irreducible components Q ^ A Q= ^ Q ^ 1 A 2Q= ^... P A 1P= P A 2P=... CJK-SM 08 10/14

18 algorithm (1) & (2) (1) simple components (2) irreducible components advantages: algorithm uses Q ^ A Q= ^ Q ^ 1 A 2Q= ^... P A 1P= P A 2P=... only linear algebraic computations (e.g. eigenvalue decomposition) no knowledge of algebraic structure (symmetry property) execution is free from group representation theory or matrix -algebra CJK-SM 08 10/14

19 ex.) spherical dome spherical lattice dome D 6 -symmetry CJK-SM 08 11/14

20 ex.) spherical dome spherical lattice dome D 6 -symmetry variable linking 3 kinds of members K(x) = x 1 K 1 + x 2 K 2 + x 3 K 3 21 DF Ki = symm CJK-SM 08 11/14

21 ex.) spherical dome spherical lattice dome D 6 -symmetry variable linking 3 kinds of members K(x) = x 1 K 1 + x 2 K 2 + x 3 K 3 21 DF P KiP = Ki = symm CJK-SM 08 11/14

22 ex.) cubic truss (tetrahedral symm.) d -symmetry (tetrahedral) CJK-SM 08 12/14

23 ex.) cubic truss (tetrahedral symm.) d -symmetry (tetrahedral) variable linking 4 kinds of members K(x) =x 1 K 1 + +x 4 K 4 24 DF Ki = symm CJK-SM 08 12/14

24 ex.) cubic truss (tetrahedral symm.) d -symmetry (tetrahedral) variable linking 4 kinds of members K(x) =x 1 K 1 + +x 4 K 4 24 DF P KiP = Ki = symm CJK-SM 08 12/14

25 ex.) cubic truss (sparsity) remove 4 members variable linking 3 kinds of members K(x) =x 1 K x 3 K 3 24 DF cf. original: CJK-SM 08 13/14

26 ex.) cubic truss (sparsity) remove 4 members variable linking 3 kinds of members K(x) =x 1 K x 3 K 3 24 DF cf. original: block-diagonal decomposition = geometric symmetry + sparsity of matrices CJK-SM 08 13/14

27 ex.) cubic truss (sparsity) remove 4 members variable linking 3 kinds of members K(x) =x 1 K x 3 K 3 24 DF cf. original: P KiP = sparsity gives finer decomposition 2 CJK-SM 08 13/14

28 conclusions simultaneous block-diagonalization algorithm input: A 1,A 2,...,A m : symmetric matrices output: block-diagonal decomposition (common block-structure) validity: matrix -algebra application: member stiffness matrices of structure with symmetric configuration uses only linear algebraic computation does not use any knowledge of symmetry property is free from group representation theory / matrix -algebra CJK-SM 08 14/14