Abstract
An exact reanalysis method named Indirect Factorization Updating (IFU) is proposed for the structure with local modifications, including boundary modifications. The IFU method is developed from the Independent Coefficients (IC) and Sherman-Morrison-Woodbury (SMW) formula. Due to the local modifications, the modified equations are divided into two parts: balanced equations and unbalanced equations. Using the theory of the IC, extra constraints are enforced on the unbalanced Degree of Freedoms (DOFs), so that the fundamental solution system of the balanced equations can be obtained by using the SMW formula. Then, a unique solution is derived from the general solution of the balanced equations to satisfy the unbalanced equations. In order to use the SMW formula directly, the change of the stiffness matrix is converted to a low-rank form by using the Cholesky factorization of the initial stiffness matrix. The Cholesky factorization is indirectly updated according to the stiffness matrix of the balanced equations but not directly according to the modified equations. Three examples are presented to verify the performance of the suggested IFU method. The results show that the IFU can efficiently obtain the exact solution of the structures with boundary modifications.
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References
Akgün MA, Garcelon JH, Haftka RT (2001) Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int J Numer Methods Eng 50:1587–1606
Cheikh M, Loredo A (2002) Static reanalysis of discrete elastic structures with reflexive inverse. Appl Math Model 26:877–891
Chen S, Yang Z (2004) A universal method for structural static reanalysis of topological modification. Int J Numer Methods Eng 61:673–686
Chen S, Ma L, Meng G (2008) Dynamic response reanalysis for modified structures under arbitrary excitation using epsilon-algorithm. Comput Struct 86:2095–2101
Davis TA, Hager WW (1999) Modifying a sparse Cholesky factorization. SIAM J Matrix Anal Appl 20:606–627
Davis TA, Hager WW (2001) Multiple-rank modifications of a sparse Cholesky factorization. SIAM J Matrix Anal Appl 22:997–1013
Gao G, Wang H, Li G (2013) An adaptive time-based global method for dynamic reanalysis. Struct Multidiscip Optim 48:355–365
Huang G, Wang H, Li G (2014) A reanalysis method for local modification and the application in large-scale problems. Struct Multidiscip Optim 49:915–930
Kirsch U (2000) Combined approximations – a general reanalysis approach for structural optimization. Struct Multidiscip Optim 20:97–106
Kirsch U (2003) A unified reanalysis approach for structural analysis, design, and optimization. Struct Multidiscip Optim 25:67–85
Kirsch U, Moses F (1998) An improved reanalysis method for grillage-type structures. Comput Struct 68:79–88
Kirsch U, Papalambros PY (2001a) Structural reanalysis for topological modifications – a unified approach. Struct Multidiscip Optim 21:333–344
Kirsch U, Papalambros PY (2001b) Exact and accurate reanalysis of structures for geometrical changes. Eng Comput 17:363–372
Kirsch U, Kocvara M, Zowe J (2002) Accurate reanalysis of structures by a preconditioned conjugate gradient method. Int J Numer Methods Eng 55:233–251
Kirsch U, Bogomolni M, Sheinman I (2006) Nonlinear dynamic reanalysis of structures by combined approximations. Comput Meth Appl Eng 195:4420–4432
Liu HF, Wu BS, Li ZG (2013) Method of updating the Cholesky factorization for structural reanalysis with added degrees of freedom. J Eng Mech 140(2):384–392
Liu HF, Wu BS, Li ZG, Zheng SP (2014) Structural static reanalysis for modification of supports. Struct Multidiscip Optim 50:425–435
Rong F, Chen S, Chen Y (2003) Structural modal reanalysis for topological modifications with extended Kirsch method. Comput Meth Appl Mech Eng 192:697–707
Sherman J, Morrison WJ (1949) Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix. Ann Math Stat 20(4):621
Song Q, Chen P, Sun S (2014) An exact reanalysis algorithm for local non-topological high-rank structural modifications in finite element analysis. Comput Struct 143:60–72
Tuckeman LS (1989) Divergence-free velocity fields in nonperiodic geometries. J Comput Phys 80:403–441
Wang H, Zeng Y, Li E et al (2016) “Seen Is Solution” a CAD/CAE integrated parallel reanalysis design system. Comput Methods Appl Mech Eng 299:187–214
Woodbury M (1950) Inverting modified matrices. Memorandum Report, 42. Statistical Research Group. Princeton University, Princeton
Wu B, Li Z (2005) Reanalysis of structural modifications due to removal of degrees of freedom. Acta Mech 180:61–71
Wu B, Li Z (2006) Static reanalysis of Structures with added degrees of freedom. Commun Numer Methods Eng 22:269–281
Wu B, Li Z, Li S (2003) The implementation of a vector-value rational approximate method in structural reanalysis problems. Comput Meth Appl Mech Eng 192:1773–1784
Wu B, Lim C, Li Z (2004) A finite element algorithm for reanalysis of structures with added degrees of freedom. Finite Elem Anal Des 40:1791–1801
Yang X, Chen S, Wu B (2001) Eigenvalue reanalysis of structures using perturbations and Padé approximation. Mech Syst Signal Process 15(2):257–263
Yang X, Lian H, Chen S (2002) An adaptive iteration algorithm for structural modal reanalysis of topological modifications. Commun Numer Methods Eng 18:373–382
Yang Z, Chen X, Kelly R (2009) An adaptive static reanalysis method for structural modifications using epsilon algorithm. Comput Sci Optim: Theory Simul Exp 2:897–899
Zhang G, Nikolaidis E, Mourelatos ZP (2009) An efficient re-analysis methodology for probabilistic vibration of large-scale structures. J Mech Des 131:1–13
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This work has been supported by Project of the Key Program of National Natural Science Foundation of China under the Grant Numbers 11572120, 11302266 and 61232014; Hunan Provincial Innovation Foundation for Postgraduate under grant number 521293021.
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Huang, G., Wang, H. & Li, G. An exact reanalysis method for structures with local modifications. Struct Multidisc Optim 54, 499–509 (2016). https://doi.org/10.1007/s00158-016-1417-2
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DOI: https://doi.org/10.1007/s00158-016-1417-2