Byoung-Wan Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab.

1. KAIST-Kyoto Univ. Joint Seminar on Earthquake Engineering Feb. 25, 2002. Matrix Power Lanczos Method and Its Application to the Eigensolution of Structures Byoung-Wan Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab. Department of Civil & Environmental Engineering, KAIST

2. Contents • Introduction • Matrix power Lanczos method • Numerical examples • Conclusions

3. Introduction • Background • Dynamic analysis of structures • - Direct integration method • - Mode superposition method  Eigenvalue analysis • Eigenvalue analysis • - Subspace iteration method • - Determinant search method • - Lanczos method • The Lanczos method is very efficient.

4. Literature review • The Lanczos method was first proposed in 1950. • Erricson and Ruhe (1980): • Lanczos algorithm with shifting • Smith et al. (1993): • Implicitly restarted Lanczos algorithm • Gambolati and Putti (1994): • Conjugate gradient scheme in Lanczos method • Kim and Lee (1999): • Lanczos-based algorithm for nonclassical damping system

5. In the fields of quantum physics, Grosso et al. (1993) • modified Lanczos recursion to improve convergence.

6. Objective • Application of Lanczos method using the power technique • to the eigensolution in structural dynamics •  Matrix power Lanczos method

7. Matrix power Lanczos method • Eigenproblem of structure

8. Modified Gram-Schmidt orthogonalization • Conventional Gram-Schmidt process: (i + 1) Lanczos vectors with i-iterated Krylov sequence • Gram-Schmidt process with power technique: (i + 1) Lanczos vectors with  i-iterated Krylov sequence

9. Modified Lanczos recursion

10. Reduced tridiagonal standard eigenproblem

11. Summary of algorithm and operation count

12. n = order of M and K m = halfband-width of M and K q = the number of calculated Lanczos vectors or order of T sj = the number of iterations of jth step in QR iteration

13. Numerical examples • Structures • Simple spring-mass system (Chen 1993) • Plan framed structure (Bathe and Wilson 1972) • Three-dimensional frame structure (Bathe and Wilson 1972) • Three-dimensional building frame (Kim and Lee 1999) • Physical error norm (Bathe 1996)

14. Simple spring-mass system (DOFs: 100) • System matrices

15. Number of operations  Failure in convergence due to numerical instability of high matrix power

16. Plane framed structure (DOFs: 330) • Geometry and properties A = 0.2787 m2 I = 8.63110-3m4 E = 2.068107Pa  = 5.154102kg/m3

17. Number of operations  Failure in convergence due to numerical instability of high matrix power

18. Three-dimensional frame structure (DOFs: 468) • Geometry and properties E = 2.068107Pa  = 5.154102kg/m3 Column in front building : A = 0.2787 m2, I = 8.63110-3m4 Column in rear building : A = 0.3716 m2, I = 10.78910-3m4 All beams into x-direction : A = 0.1858 m2, I = 6.47310-3m4 All beams into y-direction : A = 0.2787 m2, I = 8.63110-3m4

19. Number of operations

20. Three-dimensional building frame (DOFs: 1008) • Geometry and properties A = 0.01 m2 I = 8.310-6m4 E = 2.11011Pa  = 7850 kg/m3

21. Number of operations  Failure in convergence due to numerical instability of high matrix power

22. Conclusions • The convergence of matrix power Lanczos method is better than that of the conventional Lanczos method. • The optimal power of dynamic matrix that reduces the number of operations and gives numerically stable solution in matrix power Lanczos method is the second power.