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On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

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Abstract

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.

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References

  • Azegami H, Fukumoto S, Aoyama T (2013) Shape optimization of continua using NURBS as basis functions. Struct Multidiscip Optim 47(2):247–258

    Article  MathSciNet  MATH  Google Scholar 

  • Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: a B-spline finite cell method combined with level set function. Comput Methods Appl Mech Eng 278:361–387

    Article  MathSciNet  Google Scholar 

  • Choi KK, Kim N-H (2006a) Structural sensitivity analysis and optimization 2: nonlinear systems, vol. 1 Springer

  • Choi KK, Kim N-H (2006b) Structural sensitivity analysis and optimization 1: linear systems, vol.1 Springer

  • Choi M-J, Cho S (2015) A mesh regularization scheme to upyear internal control points for isogeometric shape design optimization. Comput Methods Appl Mech Eng 285:694–713

    Article  Google Scholar 

  • Dems K, Rousselet B (1999a) Sensitivity analysis for transient heat conduction in a solid body—Part I: External boundary modification. Structural Optimization 17(1):36–45

  • Dems K, Rousselet B (1999b) Sensitivity analysis for transient heat conduction in a solid body—Part II: Interface modification. Structural Optimization 17(1):46–54

  • Dulikravich GS (1988) Inverse design and active control concepts in strong unsteady heat conduction. Appl Mech Rev 41(6):270–277

    Article  Google Scholar 

  • Gao Z, Grandhi RV (1999) Sensitivity analysis and shape optimization for preform design in thermo-mechanical coupled analysis. Int J Numer Methods Eng 45(10):1349–1373

    Article  MATH  Google Scholar 

  • Gillebaart E, De Breuker R (2016) Low-fidelity 2D isogeometric aeroelastic analysis and optimization method with application to a morphing airfoil. Comput Methods Appl Mech Eng 305:512–536

    Article  MathSciNet  Google Scholar 

  • Griewank A, Walther A (2000) Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans Math Softw (TOMS) 26(1):19–45

    Article  MATH  Google Scholar 

  • 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 

  • Ha YD (2015) Generalized isogeometric shape sensitivity analysis in curvilinear coordinate system and shape optimization of shell structures. Struct Multidiscip Optim 52(6):1069–1088

    Article  MathSciNet  Google Scholar 

  • Haftka RT (1981) Techniques for thermal sensitivity analysis. Int J Numer Methods Eng 17(1):71–80

    Article  MATH  Google Scholar 

  • Haftka RT, Grandhi RV (1986) Structural shape optimization—a survey. Comput Methods Appl Mech Eng 57(1):91–106

    Article  MathSciNet  MATH  Google Scholar 

  • Ho Lee D, Man Kwak B (1995) Shape sensitivity and optimization for transient heat diffusion problems using the BEM. Int J Numer Methods Heat Fluid Flow 5(4):313–326

    Article  MATH  Google Scholar 

  • Huang C-H, Chaing M-T (2008) A transient three-dimensional inverse geometry problem in estimating the space and time-dependent irregular boundary shapes. Int J Heat Mass Transf 51(21):5238–5246

    Article  MATH  Google Scholar 

  • Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  • Jarny Y, Ozisik M, Bardon J (1991) A general optimization method using adjoint equation for solving multidimensional inverse heat conduction. Int J Heat Mass Transf 34(11):2911–2919

    Article  MATH  Google Scholar 

  • Kane JH, Kumar B, Stabinsky M (1991) Transient thermoelasticity and other body force effects in boundary element shape sensitivity analysis. Int J Numer Methods Eng 31(6):1203–1230

    Article  MATH  Google Scholar 

  • Kiendl J, Schmidt R, WWüchner R, Bletzinger K-U (2014) Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Comput Methods Appl Mech Eng 274(0):148–167

    Article  MathSciNet  MATH  Google Scholar 

  • Kleiber M, SluŻalec A (1996) Material derivative and control volume approaches to shape sensitivity analysis of nonlinear transient thermal problems. Structural Optimization 11(1-2): 56–63

    Article  Google Scholar 

  • Korycki R (2006) Sensitivity analysis and shape optimization for transient heat conduction with radiation. Int J Heat Mass Transf 49(13):2033–2043

    Article  MATH  Google Scholar 

  • Korycki R (2007) Shape optimization and shape identification for transient diffusion problems in textile structures, Fibres & Textiles in Eastern Europe 1(60):43-49

  • Kostas K, Ginnis A, Politis C, Kaklis P (2017) Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver. Comput Aided Des 82:79–87

    Article  MathSciNet  Google Scholar 

  • Long C, Marsden A, Bazilevs Y (2014) Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk, Computational Mechanics 1–12

  • Manh ND, Evgrafov A, Gersborg AR, Gravesen J (2011) Isogeometric shape optimization of vibrating membranes. Comput Methods Appl Mech Eng 200(13):1343–1353

    Article  MathSciNet  MATH  Google Scholar 

  • Nagy AP, Abdalla MM, Gürdal Z (2010) Isogeometric sizing and shape optimisation of beam structures. Comput Methods Appl Mech Eng 199(17):1216–1230

    Article  MathSciNet  MATH  Google Scholar 

  • Nagy AP, Abdalla MM, Gürdal Z (2011) Isogeometric design of elastic arches for maximum fundamental frequency. Struct Multidiscip Optim 43(1):135–149

    Article  MathSciNet  MATH  Google Scholar 

  • Nagy AP, IJsselmuiden ST, Abdalla MM (2013) Isogeometric design of anisotropic shells: Optimal form and material distribution. Comput Methods Appl Mech Eng 264:145–162

    Article  MathSciNet  MATH  Google Scholar 

  • Nørtoft P, Gravesen J (2013) Isogeometric shape optimization in fluid mechanics. Struct Multidiscip Optim 48(5):909–925

    Article  MathSciNet  Google Scholar 

  • Park B-U, Seo Y-D, Sigmund O, Youn S-K (2013) Shape optimization of the stokes flow problem based on isogeometric analysis. Struct Multidiscip Optim 48(5):965–977

    Article  MathSciNet  Google Scholar 

  • Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Eng 199(29):2059–2071

    Article  MathSciNet  MATH  Google Scholar 

  • Qian X, Sigmund O (2011) Isogeometric shape optimization of photonic crystals via Coons patches. Comput Methods Appl Mech Eng 200(25):2237–2255

    Article  MathSciNet  MATH  Google Scholar 

  • SłuŻalec A, Kleiber M (1992) Shape optimization of thermo-diffusive systems. Int J Heat Mass Transfer 35(9):2299–2304

    Article  MATH  Google Scholar 

  • Song J, Shanghvi J, Michaleris P (2004) Sensitivity analysis and optimization of thermo-elasto-plastic processes with applications to welding side heater design. Comput Methods Appl Mech Eng 193(42):4541–4566

    Article  MATH  Google Scholar 

  • Sternberg J, Griewank A (2006) Reduction of storage requirement by checkpointing for time-dependent optimal control problems in ODE s. In: Automatic differentiation: applications, Theory, and Implementations. Springer, pp 99–110

  • Stumm P, Walther A (2010) New algorithms for optimal online checkpointing. SIAM J Sci Comput 32 (2):836–854

    Article  MathSciNet  MATH  Google Scholar 

  • Tortorelli DA, Haber RB (1989) First-order design sensitivities for transient conduction problems by an adjoint method. Int J Numer Methods Eng 28(4):733–752

    Article  MATH  Google Scholar 

  • Tortorelli DA, Haber RB, Lu SC (1989) Design sensitivity analysis for nonlinear thermal systems. Comput Methods Appl Mech Eng 77(1):61–77

    Article  MathSciNet  MATH  Google Scholar 

  • Van Keulen F, Haftka R, Kim N (2005) Review of options for structural design sensitivity analysis. Part 1: Linear systems. Comput Methods Appl Mech Eng 194(30):3213–3243

  • Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197(33):2976–2988

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Q, Moin P, Iaccarino G (2009) Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation. SIAM J Sci Comput 31(4):2549–2567

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Z. -P., Abdalla MM, Turteltaub S (2017a) NorMalization approaches for the descent search direction in isogeometric shape optimization. Comput Aided Des 82:68–78

  • Wang Z-P, Abdalla MM, Turteltaub S (2017b) Shape optimization and optimal control for transient heat conduction problems using an isogeometric approach, Comput Struct (accepted)

  • Wang Z-P, Turteltaub S (2015) Isogeometric shape optimization for quasi-static processes. Int J Numer Methods Eng 104(5):347–371. ISSN 1097-0207, nme.4940

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang W, Zhao L, Cai S (2015) Shape optimization of Dirichlet boundaries based on weighted B-spline finite cell method and level-set function. Comput Methods Appl Mech Eng 294:359–383

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Prof. M.M. Abdalla and Prof. S.Turteltaub at Delft University of Technology for the communications related to this work, and the reviewers for their constructive comments.

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Authors and Affiliations

  1. Department of Civil & Environmental Engineering, National University of Singapore, Engineering Drive 2, E1A 02-04, Singapore, 117576, Singapore

    Zhen-Pei Wang & Deepak Kumar

  2. Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629, HS, Delft, The Netherlands

    Zhen-Pei Wang

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  1. Zhen-Pei Wang
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  2. Deepak Kumar
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Correspondence to Zhen-Pei Wang.

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Wang, ZP., Kumar, D. On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach. Struct Multidisc Optim 56, 487–500 (2017). https://doi.org/10.1007/s00158-017-1669-5

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  • DOI: https://doi.org/10.1007/s00158-017-1669-5

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