Large-scale topology optimization using preconditioned Krylov subspace methods with recycling

International Journal for Numerical Methods in Engineering

Volumn 69, Issue 12, 2007, Pages 2441-2468

  Wang, Shun ^a   de Sturler, Eric ^b   Paulino, Glaucio H ^a  

^a University of Illinois at Urbana Champaign   (United States)

^b VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Iterative methods; Krylov subspace recycling; Large scale computation; Preconditioning; Three dimensional analysis; Topology optimization

Indexed keywords

COMPUTER SIMULATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; LINEAR SYSTEMS; SHAPE OPTIMIZATION;

KRYLOV SUBSPACE RECYCLING; LARGE-SCALE COMPUTATION; PRECONDITIONING; THREE-DIMENSIONAL ANALYSIS;

OPERATIONS RESEARCH;

COMPUTER SIMULATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; LINEAR SYSTEMS; OPERATIONS RESEARCH; SHAPE OPTIMIZATION;

EID: 33947216833     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1798     Document Type: Article

References (34)

1. Borrvall T, Petersson J. Large-scale topology optimization in 3D using parallel computing. Computer Methods in Applied Mechanics and Engineering 2001; 190:6201-6229.
2. Vemaganti K, Lawrence WE. Parallel methods for optimality criteria-based topology optimization. Computer Methods in Applied Mechanics and Engineering 2005; 194:3637-3667.
3. Kim TS, Kim JE, Kim YY. Parallelized structural topology optimization for eigenvalue problems. International Journal of Solids and Structures 2004; 41:2623-2641.
4. Parks ML, de Sturler E, Mackey G, Johnson DD, Maiti S. Recycling Krylov subspaces for sequences of linear systems. Technical Report UIUC DCS-R-2004-2421, available from http://www-faculty.cs.uiuc.edu/~sturler, 2004, accepted for publication in SISC.
5. Kilmer ME, de Sturler E. Recycling subspace information for diffuse optical tomography. SIAM Journal on Scientific Computing 2006; 27(6):2140-2166.
6. Gersborg-Hansen A, Sigmund O, Haber RB. Topology optimization of channel flow problems. Journal for Structural and Multidisciplinary Optimization 2005; 30:181-192.
7. Sigmund O, Jensen JS. Systematic design of phononic band gap materials and structures. Philosophical Transactions of the Royal Society London, Series A (Mathematical, Physical and Engineering Sciences) 2003; 361:1001-1019.
8. Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis 1975; 12:617-629.
9. Guest JK, Prévost JH. Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. International Journal for Numerical Methods in Engineering 2006; 66:461-484.
10. Bendsøe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer: Berlin, Germany, 2003.
11. Sigmund O. Topology optimization: a tool for the tailoring of structures and materials. Transactions of the Royal Society: Science into the next Millennium (Issue III, Mathematics, Physics and Engineering) 2000; 358(1765):211-228 (Special issue).
12. Rozvany GIN. Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Journal for Structural and Multidisciplinary Optimization 2001; 21:90-108.
13. Mackerle J. Topology and shape optimization of structures using FEM and BEM: a bibliography (1999-2001). Finite Elements in Analysis and Design 2003; 39:243-253.
14. Paulino GH, Page III RC, Silva ECN. A Java-based topology optimization program with web access: nodal design variable approach. In Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization (WCSMO6), International Society for Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May 30-June 3, 2005.
15. Stump FV, Paulino GH, Silva ECN. Material distribution design of functionally graded rotating disks with stress constraint. In Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization (WCSMO6), International Society for Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May 30-June 3 2005.
16. Bendsøe MP. Optimal shape design as a material distribution problem. Structural Optimization 1989; 1:193-202.
17. Bendsøe MP, Sigmund O. Material interpolation schemes in topology optimization. Archives of Applied Mechanics 1999; 69(9-10):635-654.
18. Matsui K, Terada K. Continuous approximation of material distribution for topology optimization. International Journal for Numerical Methods in Engineering 2004; 59:1925-1944.
19. Paulino GH, Silva ECN. Design of functionally graded structures using topology optimization. Materials Science Forum 2005; 492-493:435-440.
20. Jog CS, Haber RB. Stability of finite element models for distributed-parameter optimization and topology design. Computer Methods in Applied Mechanics and Engineering 1996; 130:203-226.
21. Rahmatalla S, Swan CC. A Q4/Q4 continuum structural optimization implementation. Journal for Structural and Multidisciplinary Optimization 2004; 27:130-135.
22. Sigmund O. On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machines 1997; 25:495-526.
23. Rozvany GIN. Structural Design via Optimality Criteria. Kluwer Academic Publishers: Dordrecht, 1989.
24. Svanberg K. The method of moving asymptotes: a new method for structural optimization. International Journal for Numerical Methods in Engineering 1987; 24(2):359-373.
25. Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1986; 7(3):856-869.
26. Arnoldi WE. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics 1951; 9:17-29.
27. Saad Y. Iterative Methods for Sparse Linear Systems (2nd edn). SIAM: Philadelphia, PA, U.S.A., 2003.
28. de Sturler E. Nested Krylov methods based on GCR. Journal of Computational and Applied Mathematics 1996; 67:15-41.
29. de Sturler E. Truncation strategies for optimal Krylov subspace methods. SIAM Journal on Numerical Analysis 1999; 36:864-889. Electronically available from http://epubs.siam.org
30. van de Vorst HA. Iterative Krylov Methods for Large Linear Systems. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press: Cambridge, MA, 2003.
31. Greenbaum A. Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, PA, 1997.
32. Meijerink J, van der Vorst HA. An iterative solution method for linear equations systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation 1977; 31:148-162.
33. Axelsson O, Barker VA. Finite Element Solution of Boundary Value Problems. Classics in Applied Mathematics, vol. 35. SIAM: Philadelphia, PA, U.S.A., 2001.
34. Toledo S. TAUCS: A Library of Sparse Linear Solvers. School of Computer Science, Tel-Aviv University, September 2003.