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Semi-analytical sensitivity analysis for nonlinear transient problems

Abstract

Efficient analytical sensitivity computations are essential elements of gradient-based optimization schemes; unfortunately, they can be difficult to implement. This implementation issue is often resolved by adopting the semi-analytical method which exhibits the efficiency of the analytical methods and the ease of implementation of the finite difference method. However, care must be taken as semi-analytical sensitivities may exhibit errors due to truncation and round-off. Additional errors are introduced if the convergence tolerance of the primal analysis is not sufficiently small. This paper gives a general overview and some new developments of the analytical and semi-analytical sensitivity analyses for nonlinear steady-state, transient, and dynamic problems. We discuss the restrictive assumptions, accuracy, and consistency of these methods. Both adjoint and direct differentiation methods are studied. Numerical examples are provided.

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Notes

  1. 1.

    For α = 0, 1/2, or 1, we recover the forward Euler, Crank-Nicolson, and backward Euler strategies respectively.

References

  1. Adelman HM, Haftka RT (1986) Sensitivity analysis of discrete structural systems. AIAA J 24(5):823–832

    Article  Google Scholar 

  2. Barthelemy B, Haftka R (1990) Accuracy analysis of the semi-analytical method for shape sensitivity calculation. Mech Struct Mach 18(3):407–432

    Article  Google Scholar 

  3. Barthelemy B, Chon C, Haftka R (1988) Accuracy problems associted with semi-analytical derivatives of static response. Finite Elem Anal Des 4(3):249–265

    Article  Google Scholar 

  4. Bernard J, Kwon S, Wilson J (1993) Differentiation of mass and stiffness matrices for high order sensitivity calculations in finite element-based equilibrium problems. J Mech Des 115(4):829–832

    Article  Google Scholar 

  5. Bestle D, Seybold J (1992) Sensitivity analysis of constrained multibody systems. Arch Appl Mech 62 (3):181–190

    MATH  Google Scholar 

  6. de Boer H, van Keulen F (2000) Refined semi-analytical design sensitivities. Int J Solids Struct 37 (46-47):6961–6980

  7. de Boer H, van Keulen F, Vervenne K (2002) Refined second order semi-analytical design sensitivities. Int J Numer Methods Eng 55(9):1033–1051

  8. Botkin M (1982) Shape optimization of plate and shell structures. AIAA J 20(2):268–273

    Article  Google Scholar 

  9. Brüls O, Eberhard P (2008) Sensitivity analysis for dynamic mechanical systems with finite rotations. Int J Numer Methods Eng 74(13):1897–1927

    MathSciNet  Article  Google Scholar 

  10. Camarda C, Adelman H (1984) Static and dynamic structural-sensitivity derivative calculations in the finite-element-based engineering analysis language (eal) system. NASA TM-85743

  11. Chen B, Gu Y, Zhang H, Zhao G (2003) Structural design optimization on thermally induced vibration. Int J Numer Methods Eng 58(8):1187–1212

    Article  Google Scholar 

  12. Cheng G, Liu Y (1987) A new computation scheme for sensitivity analysis. Eng Optim 12(3):219–234

    Article  Google Scholar 

  13. Cheng G, Olhoff N (1993) Rigid body motion test against error in semi-analytical sensitivity analysis. Comput Struct 46(3):515–527

    Article  Google Scholar 

  14. Cheng G, Gu Y, Zhou Y (1989) Accuracy of semi-analytic sensitivity analysis. Finite Elem Anal Des 6 (2):113–128

    Article  Google Scholar 

  15. Deng Y, Liu Z, Zhang P, Liu Y, Wu Y (2011) Topology optimization of unsteady incompressible navier–stokes flows. J Comput Phys 230(17):6688–6708

    MathSciNet  Article  Google Scholar 

  16. Esping B (1984) Minimum weight design of membrane structures using eight node isoparametric elements and numerical derivatives. Comput Struct 19(4):591–604

    Article  Google Scholar 

  17. Fenyes P, Lust R (1991) Error analysis of semianalytic displacement derivatives for shape and sizing variables. Amer Inst Aeronaut Astronaut 29(2):271–279

    Article  Google Scholar 

  18. Gallagher R, Zienkiewicz O (1973) Optimum structural design: Theory and applications. Wiley, New York

    MATH  Google Scholar 

  19. Greene W, Haftka R (1991) Computational aspects of sensitivity calculations in linear transient structural analysis. Struct Optim 3(3):176–201

    Article  Google Scholar 

  20. Gu Y, Grandhi R (1998) Sensitivity analysis and optimization of heat transfer and thermal-structural designs. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, pp 4746

  21. Gu Y, Chen B, Zhang H, Grandhi R (2002) A sensitivity analysis method for linear and nonlinear transient heat conduction with precise time integration. Struct Multidiscip Optim 24(1):23–37

    Article  Google Scholar 

  22. Gunzburger MD (2003) Perspectives in flow control and optimization. Advances in Design and Control, Society for Industrial and Applied Mathematics

  23. Haftka R (1993) Semi-analytical static nonlinear structural sensitivity analysis. AIAA J 31(7):1307–1312

    MathSciNet  Article  Google Scholar 

  24. Haftka R, Adelman H (1989) Recent developments in structural sensitivity analysis. Struct Optim 1(3):137–151

    Article  Google Scholar 

  25. Haftka RT, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business Media, Berlin

  26. Haug EJ (1987) Design sensitivity analysis of dynamic systems. In: Computer aided optimal design: Structural and Mechanical Systems. Springer, Berlin, pp 705–755

    Google Scholar 

  27. Hooijkamp EC, van Keulen F (2018) Topology optimization for linear thermo-mechanical transient problems: Modal reduction and adjoint sensitivities. Int J Numer Methods Eng 113(8):1230–1257

    MathSciNet  Article  Google Scholar 

  28. Jensen J, Nakshatrala P, Tortorelli D (2014) On the consistency of adjoint sensitivity analysis for structural optimization of linear dynamic problems. Struct Multidiscip Optim 49(5):831–837

    MathSciNet  Article  Google Scholar 

  29. van Keulen F, Haftka R, Kim R (2005) Review of options for structural design sensitivity analysis. part 1: Linear systems. Comput Methods Appl Mech Eng 194(30-33):3213–3243. Structural and Design Optimization

  30. Kiendl J, Schmidt R, Wüchner R, Bletzinger KU (2014) Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Comput Methods Appl Mech Eng 274:148–167

    MathSciNet  Article  Google Scholar 

  31. Kramer JL, Stockman NO (1963) Effect of variable thermal properties on one-dimensional heat transfer in radiating fins. Technical report, NASA TN D-1878

  32. Kreissl S, Pingen G, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87(13):1229–1253

    MathSciNet  MATH  Google Scholar 

  33. Meric R (1988) Shape design sensitivity analysis of dynamic structures. AIAA J 26(2):206–212

    MathSciNet  Article  Google Scholar 

  34. 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(14):2471–2499

    Article  Google Scholar 

  35. Mlejnek H (1992) Accuracy of semi-analytical sensitivities and its improvement by the natural method. Struct Optim 4(2):128–131

    Article  Google Scholar 

  36. Mróz Z, Haftka R (1994) Design sensitivity analysis of non-linear structures in regular and critical states. Int J Solids Struct 31(15):2071–2098

    MathSciNet  Article  Google Scholar 

  37. Olhoff N, Rasmussen J (1991) Study of inaccuracy in semi-analytical sensitivity analysis a model problem. Struct Optim 3(4):203–213

    Article  Google Scholar 

  38. Olhoff N, Rasmussen J, Lund E (1993) A method of exact numerical differentiation for error elimination in finite-element-based semi-analytical shape sensitivity analyses. Mech Struct Mach 21(1):1–66

    MathSciNet  Article  Google Scholar 

  39. Oral S (1996) An improved semianalytical method for sensitivity analysis. Struct Optim 11(1-2):67–69

    Article  Google Scholar 

  40. Pedersen P, Cheng G, Rasmussen J (1989) On accuracy problems for semi-analytical sensitivity analyses. Mech Struct Mach 17(3):373–384

    Article  Google Scholar 

  41. Ray D, Pister KS, Polak E (1978) Sensitivity analysis for hysteretic dynamic systems: theory and applications. Comput Methods Appl Mech Eng 14(2):179–208

    MathSciNet  Article  Google Scholar 

  42. Tortorelli D, Michaleris P (1994) Design sensitivity analysis: Overview and review. Inverse Probl Eng 1(1):71–105

    Article  Google Scholar 

  43. Tortorelli DA, Haber RB, Lu SCY (1991) Adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems. AIAA J 29(2):253–263

    Article  Google Scholar 

  44. Tromme E, Tortorelli D, Brüls O, Duysinx P (2015) Structural optimization of multibody system components described using level set techniques. Struct Multidiscip Optim 52(5):959–971

    MathSciNet  Article  Google Scholar 

  45. Van Keulen F, De Boer H (1998) Rigorous improvement of semi-analytical design sensitivities by exact differentiation of rigid body motions. Int J Numer Methods Eng 42(1):71–91

    Article  Google Scholar 

  46. Wang W, Clausen PM, Bletzinger KU (2015) Improved semi-analytical sensitivity analysis using a secant stiffness matrix for geometric nonlinear shape optimization. Comput Struct 146:143–151

    Article  Google Scholar 

  47. Zhong WX, Williams FW (1994) A precise time step integration method. Proc Inst Mech Eng Part C: J Mech Eng Sci 208(6):427–430

    Article  Google Scholar 

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Correspondence to Felipe Fernandez.

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This work was partially performed under the auspices of the US Department of Energy by Lawrence Livermore Laboratory under contract DE-AC52-07NA27344, cf. ref number LLNLCONF-717640.

Responsible Editor: Hai Huang

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Fernandez, F., Tortorelli, D.A. Semi-analytical sensitivity analysis for nonlinear transient problems. Struct Multidisc Optim 58, 2387–2410 (2018). https://doi.org/10.1007/s00158-018-2096-y

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Keywords

  • Non-linear
  • Semi-analytical
  • Sensitivity analysis
  • Dynamic
  • Adjoint
  • Direct