×

zbMATH — the first resource for mathematics

Automation of primal and sensitivity analysis of transient coupled problems. (English) Zbl 1171.74043
Summary: The paper describes a hybrid symbolic-numeric approach to automation of primal and sensitivity analysis of computational models solved by finite element method. The necessary apparatus for the automation of steady-state, steady-state coupled, transient and transient coupled problems is introduced as combination of a symbolic system, an automatic differentiation (AD) technique and an automatic code generation. For this purpose the paper extends the classical formulation of AD by additional operators necessary for a highly abstract description of primal and sensitivity analysis of the typical computational models. At the end, we present an appropriate abstract description for the fully implicit primal and sensitivity analysis of hyperelastic and elasto-plastic problems and a symbolic input for the generation of necessary user subroutines for the two-dimensional, hyperelastic finite element.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
68W30 Symbolic computation and algebraic computation
PDF BibTeX Cite
Full Text: DOI
References:
[1] Amberg G, Tonhardt R, Winkler C (1999) Finite element simulations using symbolic computing. Math Comput Simul 49: 257–274
[2] Bartholomew-Biggs M, Brown S, Christianson B, Dixon L (2000) Automatic differentiation of algorithms. J Comput Appl Math 124(1–2): 171–190 · Zbl 0994.65020
[3] Beall MW, Shephard MS (1999) Object-oriented framework for reliable numerical simulations. Eng Comput 15(1): 61–72 · Zbl 05468055
[4] Bischof C, Hovland P, Norris B (2002) Implementation of automatic differentiation tools. In: Norris C, Fenwick JB Proceedings of the ACM SIGPLAN workshop on partial evaluation and semantics-based program manipulation. ACM Press, New York
[5] Bischof CH, Buecker HM, Lang B, Rasch A, Risch JW (2003) Extending the functionality of the general-purpose finite element package SEPRAN by automatic differentiation. Int J Numer Methods Eng 58: 2225–2238 · Zbl 1032.76589
[6] Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1, Linear systems. Springer, New York, p 446
[7] Eyheramendy D, Zimmermann Th (1999) Object-oriented symbolic derivation and automatic programming of finite elements in mechanics. Eng Comput 15(1): 12–36 · Zbl 05468053
[8] Fritzson P, Fritzson D (1992) The need for high-level programming support in scientific computing applied to mechanical analysis. Comput Struct 45: 387–395 · Zbl 0964.68584
[9] Gonnet G (1986) New results for random determination of equivalence of expression. In: Char BW (ed) Proceedings of 1986 ACM symposium on symbolic and algebraic computation, Waterloo, pp 127–131
[10] Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, Philadelphia · Zbl 0958.65028
[11] Keulen F, Haftka RT, Kim NH (2005) Review of options for structural design sensitivity analysis. Part 1: linear systems. Comput Methods Appl Mech Eng 194: 3213–3243 · Zbl 1091.74040
[12] Kleiber M, Antunez H, Hien TH, Kowalczyk P (1997) Parameter sensitivity in nonlinear mechanics. Wiley, New York
[13] Kirby RC, Knepley M, Logg A, Scott LR (2005) Optimizing the evaluation of finite element matrices. SIAM J Sci Comput 27: 741–758 · Zbl 1091.65114
[14] Korelc J (1997) Automatic generation of finite-element code by simultaneous optimization of expressions. Theor Comput Sci 187: 231–248 · Zbl 0893.68084
[15] Korelc J (2001) Hybrid system for multi-language and multi-environment generation of numerical codes. In: Proceedings of the ISSAC’2001 symposium on symbolic and algebraic computation. ACM Press, New York, pp 209–216 · Zbl 1356.65277
[16] Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18: 312–327 · Zbl 01993877
[17] Korelc J (2007) AceGen user manual, http://www.fgg.uni-lj.si/symech/
[18] Michaleris P, Tortorelli DA, Vidal CA (1994) Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int J Numer Methods Eng 37: 2471–2499 · Zbl 0808.73057
[19] Logg A (2007) Automating the finite element method. Arch Comput Methods Eng 14: 93–138 · Zbl 1158.74048
[20] Mathematica 6.0, Wolfram Research Inc., http://www.wolfram.com
[21] Pironneau O, Hecht F, Hyaric A (2008) FreeFem++, ftp://www.freefem.org/
[22] Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York · Zbl 0934.74003
[23] Wang PS (1986) Finger: a symbolic system for automatic generation of numerical programs in finite element analysis. J Symb Comput 2: 305–316 · Zbl 0604.65078
[24] Zienkiewicz OC, Taylor RL (1991) The finite element method, vols I, II. McGraw Hill, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.