Structural and Multidisciplinary Optimization

, Volume 50, Issue 6, pp 975–1000 | Cite as

Local continuum shape sensitivity with spatial gradient reconstruction

  • David M. CrossEmail author
  • Robert A. Canfield


Utilizing gradient-based optimization for large scale, multidisciplinary design problems requires accurate and efficient sensitivity or design derivative analysis. In general, numerical sensitivity methods, such as the finite difference method, are easy to implement but can be computationally expensive and inaccurate. In contrast, analytic sensitivity methods, such as the discrete and continuum methods, are highly accurate but can be very difficult, if not infeasible, to implement. A popular compromise is the semi-analytic method, but it too can be highly inaccurate when computing shape design derivatives. Presented here is an alternative method, which is easy to implement and can be as accurate as conventional analytic sensitivity methods. In this paper a general local continuum shape sensitivity method with spatial gradient reconstruction (SGR) is formulated. It is demonstrated that SGR, a numerical technique, can be used to solve the continuous sensitivity equations (CSEs) in a non-intrusive manner. The method is used to compute design derivatives for a variety of applications, including linear static beam bending, linear transient gust analysis of a 2-D beam structure, linear static bending of rectangular plates, and linear static bending of a beam-stiffened plate. Analysis is conducted with Nastran, and both displacement and stress design derivative solutions are presented. For each example the design derivatives are validated with either analytic or finite difference solutions.


Sensitivity analysis Structural optimization Fluid-structure interaction Shape optimization Aeroelasticity 



This material is based on research sponsored by Air Force Research Laboratory under agreement number FA8650-09-2-3938 and Air Force Office of Scientific Research under agreement number FA9550-09-1-0354. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The authors gratefully acknowledge the support of AFRL Senior Aerospace Engineers Dr. Raymond Kolonay, Dr. Ned Lindsley, and Dr. Jose Camberos and the AFOSR program manager Dr. Fariba Fahroo.


  1. Akbari J, Kim N, Ahmadi MT (2010) Shape sensitivity analysis with design-dependent loadingsequivalence between continuum and discrete derivatives. Struct Multidiscip OptimGoogle Scholar
  2. Arora J, Haug E (1978) Design sensitivity analysis of elastic mechanical systems. Comput Methods Appl Mech Eng 15: 35–62CrossRefzbMATHGoogle Scholar
  3. Arora J, Haug E (1979) Methods of design sensitivity analysis in structural optimization. AIAA J 17(9), Article No 79-4109Google Scholar
  4. Barthelemy B, Haftka RT (1988) Accuracy analysis of the semi-analytical method for shape sensitivity calculation. AIAA-88-2284Google Scholar
  5. Bhaskaran R, Berkooz G (1997) Optimization of fluid-structure interaction using the sensitivity equation approach. Fluid-Structure Interaction, Aeroelasticity, Flow-Induced Vibrations and Noise, vol 1, No. 53-1Google Scholar
  6. Borggaard J, Burns J (1994) A sensitivity equation approach to shape optimization in fluid flows. Technical Report, Langley Research CenterGoogle Scholar
  7. Borggaard J, Burns J (1997) A pde sensitivity equation method for optimal aerodynamic design. J Comput Phys 136Google Scholar
  8. Choi K, Kim NH (2005) Structural sensitivity analysis and optimization. Springer Science + Business MediaGoogle Scholar
  9. Cross D, Canfield RA (2012a) Continuum shape sensitivity with spatial gradient reconstruction of nonlinear aeroelastic gust response. 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, Indiana. AIAA 2012-5597Google Scholar
  10. Cross D, Canfield RA (2012b) Solving continuum shape sensitivity with existing tools for nonlinear aeroelastic gust analysis. 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii. AIAA 2012-1923Google Scholar
  11. Dems K, Haftka R (1989) Two approaches to sensitivity analysis for shape variation of structures. Mech Struct Mach 16(4)Google Scholar
  12. Dems K, Mroz Z (1985) Variational approach to first- and second-order sensitivity analysis of elastic structures. Int J Numer Methods Eng 21Google Scholar
  13. Duvigneau R, Pelletier D (2006) On accurate boundary conditions for a shape sensitivity equation method. Int J Numer Methods Fluids 50Google Scholar
  14. Etienne S, Pelletier D (2005) General approach to sensitivity analysis of fluid-structure interactions. J Fluids Struct 20(2)Google Scholar
  15. Haftka R, Gurdal Z (1992) Elements of structural optimization. Kluwer Academic PublishersGoogle Scholar
  16. Haftka RT, Adelman HM (1989) Recent developments in structural sensitivity analysis. Structural Optimization IGoogle Scholar
  17. Liu S, Canfield RA (2012) Continuum shape sensitivity method for fluid flow around an airfoil. 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii. AIAA 2012-1426Google Scholar
  18. Liu S, Canfield RA (2013a) Boundary velocity method for continuum shape sensitivity of nonlinear fluid-structure interaction problems. J Fluids StructGoogle Scholar
  19. Liu S, Canfield RA (2013b) Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations. To Appear in Structural and Multidisciplinary Optimization JournalGoogle Scholar
  20. Liu S, Canfield RA (2013c) Two forms of continuum shape sensitivity method for fluid-structure interaction problems. To Appear in AIAA JournalGoogle Scholar
  21. Liu S, Wickert PD, Canfield RA (2010) Fluid-structure transient gust response sensitivity for a nonlinear joined wing model. 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Orlando, Florida. AIAA 2010-3118Google Scholar
  22. MSC SoftwareCorporation Mdnastran. (2010) User defined services. MD Nastran DocumentationGoogle Scholar
  23. Stanley L, Stewart D (2002) Design sensitivity analysis: computational issues of sensitivity equation methods. Academic PressGoogle Scholar
  24. Timoshenko S (1940) Theory of plates and shells. McGraw-Hill Book Company Inc.Google Scholar
  25. Turgeon E, Pelletier D, Borggaard J (1999) A continuous sensitivity equation approach to optimal design in mixed convection. AIAA 99–3625Google Scholar
  26. Wickert DP (2009) Least-squares, continuous sensitivity analysis for nonlinear fluid-structure interaction. Dissertation, Air Force Institute of TechnologyGoogle Scholar
  27. Wickert DP, Canfield RA (2008) Least-squares continuous sensitivity analysis of an example fluid-structure interaction problem. 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumberg, Illinois, AIAA 2008–1896Google Scholar
  28. Wickert DP, Canfield RA, Reddy JN (2008) Continuous sensitivity analysis of fluid-structure interaction problems using least-squares finite elements. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, British Columbia, Canada. AIAA 2008–5931Google Scholar
  29. Wickert DP, Canfield RA, Reddy JN (2009) Fluid-structure transient gust sensitivity using least-squares continuous sensitivity analysis. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California. AIAA 2009–2535Google Scholar
  30. Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. part 1: The recovery technique. Int J Numer Methods Eng 33Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA
  2. 2.Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA

Personalised recommendations