2D- and 3D-Shape Optimization with FEM and BEM

  • E. Schnack
  • G. Iancu
Part of the International Centre for Mechanical Sciences book series (CISM, volume 325)


The finite element formulation has been used by many researchers for shape optimization. Nonlinear programming with sensitivities obtained by implicitly differentiating the discretized equations has been used by Zienkiewicz and Campbell [47], Francavilla, Ramakrishnan and Zienkiewicz [07], Ramakrishnan and Francavilla [24] and Kristensen and Madsen [19] to solve this problem in two dimensions. The papers of Pedersen and Laursen [23], Zhang and Beckers [45] and Trompette and Marcelin [42] treat shape optimization of axisymmetric structures in a similar manner. Aspects associated with three-dimensional structures are discussed in this context by Botkin, Yang and Benett [03], Imam [15], and Kodiyalam and Vanderplaats [18]. A detailed description of the computation of structural response using the FE based discrete approach and numerical problems associated with this have been presented by Haftka [09], Wang, Sun and Gallagher [43] and Haftka and Barthelemy [10].


Boundary Element Boundary Element Method Distribute Parameter System Shape Optimization Problem Shape Sensitivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barone, M.R., R.J. Yang: A Boundary Element Approach for Recovery of Shape Sensitivities in Three-Dimensional Elastic Solids. Computer Methods in Applied Mechanics and Engng. 74 (1989), pp. 69–82.CrossRefMATHGoogle Scholar
  2. [2]
    Benett, J.A., M.E. Botkin: Shape optimization of Two-Dimensonal Structures with Geometric Problem Description and Adaptive Mesh Refinement, AIAA, 1983.Google Scholar
  3. [3]
    Botkin, M.E., R.J. Yang and J.A. Benett: Shape Optimization of Three-Dimensional Stamped and and Solid Automotive Components. Paper presented at the International Symposium on Optimum Shape, General Motors Research Labs, Warren, Michigan, 1985.Google Scholar
  4. [4]
    Choi, K.K., E.J. Haug: Shape Design Sensitivity Analysis of Elastic Structures. J. of Structural Mech. 11, No. 2 (1983), pp. 231–269.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Chun, Y.W., E.J. Haug: Two-Dimensional Shape Optimal Design. IJNME 13 (1978), pp. 311–336.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    Chun, Y.W., E.J. Haug: Shape Optimization of a Solid of Revolution. J. of Engng. Mech. 109, No. 1 (1983), pp. 30–46.Google Scholar
  7. [7]
    Francavilla, A., C.v. Ramakrishnan, 0.C. Zienkiewicz: Optimization of Shape to Minimize Stress Concentration. J. of Strain Analysis 10 /2 (1975), pp. 63–69.CrossRefGoogle Scholar
  8. [8]
    Ghosh, N., H. Rajiyah, S. Ghosh, S. Mukherjee: A New Boundary Element Method Formulation for Linear Elasticity. J. of Appl. Mech. 53 (1986), pp. 69–76.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Haftka, R.T.: Finite Elements in Optimal Structural Design. In: Computer Aided Optimal Design: Structural and Mechanical Systems. Ed.: C.A. Mota Soares. Springer-Verlag, Berlin, New York 1986, pp. 271–297.Google Scholar
  10. [10]
    Haftka, R.T., B. Barthelemy: On the Accuracy of Shape Sensitivity. In: Computer Aided Optimum Design of Structues: Recent Advances. Eds.: C.A. Brebbia, S. Hernandez. Springer-Verlag, Berlin, Heidelberg, New York 1989, pp. 327–336.Google Scholar
  11. [11]
    Haug, E.J., R.K. Choi, V. Komkov: Design Sensitivity Analysis of Structural Systems. Ed.: W.F. Ames. Mathematical Science and Engineering. Academic Press, Orlando, San Diego, New York, 1986.Google Scholar
  12. [12]
    Iancu, G.: Optimierung von Spannungskonzentrationen bei dreidimensionalen elastischen Strukturen. Doctoral Thesis, Karlsruhe University, 1991Google Scholar
  13. [13]
    Iancu, G., E. Schnack: Knowledge-Based Shape Optimization. First Conference on Computer Aided Optimum Design of Structures (CAOD) OPTI 89: Computer Aided Optimum Design of Structures: Recent Advances. Eds.: C.A. Brebbia and S. Hernandez. Springer-Verlag, Berlin, Heidelberg, New York 1989, pp. 71–83.Google Scholar
  14. [14]
    Iancu, G., E. Schnack: Shape Optimization Scheme for Large Scale Structures. Proceedings of the Second World Congress on Computational Mechanics, 27–31 August 1990, Stuttgart/Germany. to appearGoogle Scholar
  15. [15]
    Imam, M.H.: Three-Dimensional Shape Optimization. IJNME 18 (1982), pp. 661–673.CrossRefMATHGoogle Scholar
  16. [16]
    Kane, J., S. Saigal: Design-Sensitivity Analysis of Solids Using BEM. J. of Engng. Mech. 114, No. 10 (1988), pp. 1703–1722.Google Scholar
  17. [17]
    Kikuchi, N., K.Y. Chung, T. Torigaki and J.E. Taylor: Adaptive Finite Element Methods for Shape Optimization of Linearly Elastic Structures. Comp. Meth. in Appl. Mech. and Engng. 57 (1986), pp. 67–89.CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Kodiyalam, S. and G.N. Vanderplaats: Shape Optimization of Three-Dimensional Continuum Structures via Force Approximation Techniques. AIAA Journal, 27 No. 9 (1989), pp. 1256–1263.CrossRefGoogle Scholar
  19. [19]
    Kristensen, E.S. N.F. Madsen: On the Optimum Shape of Fillets in Plates Subjected to Multiple In-Plane Loading Cases. IJNME 10 (1976), pp. 1007–1019.CrossRefMATHGoogle Scholar
  20. [20]
    Leal, R.P.: Boundary Elements in Bidimensional Elasticity. Master Sc. Thesis, Technical University of Lisbon, 1985.Google Scholar
  21. [21]
    Mota Soares, C.A, H.C. Rodrigues, L.M. Oliveira Faria, E.J. Haug: Optimization of the Geometry of Shafts Using Boundary Elements. ASME J. of Mechanisms, Transmissions and Automation in Design 106 (1984), pp. 199–203.CrossRefGoogle Scholar
  22. [22]
    Mota Soares, C.A., H.C. Rodriguez, L.M. Oliviera Faria, E.J. Haug: Boundary Elements in Shape Optimal design of Shafts. In: Optimization in Computer Aided Design. Ed. J.S. Gero. North-Holland 1985, pp. 155–175.Google Scholar
  23. [23]
    Pedersen, P. and L.L. Laursen: Design for Minimum Stress Concentration by Finite Element Elements and Linear Programming. J. of Struct. Mech. 10/4 (1982–83), pp. 375–391.Google Scholar
  24. [24]
    Ramakrishnan C.V., A. Francavilla: Structural Shape Optimization Using Penalty Funktions. J. of Struct. Mech. 3/4 (1974–1975), pp. 1974–1975.Google Scholar
  25. [25]
    Rodriguez, H.C., C.A. Mota Soares: Shape Optimization of Shafts. Third National Congress of Theoretical and Applied Mechanics, Lisbon 1983.Google Scholar
  26. [26]
    Rodriguez, H.C.: Shape Optimization of Shafts Using Boundary Elements. Master Sc. Thesis, Technical University of Lisbon, 1984.Google Scholar
  27. [27]
    Schnack, E.: Ein Iterationsverfahren zur Optimierung von Spannungskonzentrationen. Habilitationsschrift, Univ. Kaiserslautern, 1977.Google Scholar
  28. [28]
    Schnack, E.: Ein Iterationsverfahren zur Optimierung von Kerboberflächen. VDI-Forschungsheft, Nr. 589, Düsseldorf, VDI-Verlag 1978.Google Scholar
  29. [29]
    Schnack, E.: An Optimization Procedure for Stress Concentrations by the Finite Element Technique. IJNME 14, No. 1 (1979), pp. 115–124.CrossRefMATHGoogle Scholar
  30. [30]
    Schnack, E.: Optimierung von Spannungskonzentrationen bei Viellastbeanspruchung. ZAMM 60 (1980), T151 - T152.Google Scholar
  31. [31]
    Schnack, E.: Optimal Designing of Notched Structures without Gradient Computation. Proceedings of the 3rd IFAC-Symposium, Toulouse/France, 29 June–2nd July 1982. In: Control of Distributed Parameter Systems. Eds: J.P. Barbary and L. Le Letty. Pergamon Press, Oxford, New York, Toronto, Sydney, Paris, Frankfurt 1982, pp. 365–369.Google Scholar
  32. [32]
    Schnack, E.: Computer Simulation of an Experimental Method for NotchShape-Optimization. Proceedings of the Int. Symp. of IMACS, 9–11 May 1983 in Nantes. In: Simulation in Engineering Sciences, Tome 2. Eds: J. Burger and Y. Janny. Elsevier Science Publishers B.V. ( North Holland ), Amsterdam 1985, pp. 269–275.Google Scholar
  33. [33]
    Schnack, E.: Local Effects of Geometry Variation in the Analysis of Structures. Studies in Applied Mechanics 12: Local Effects in the Analysis of Structures. Ed.: P. Ladevèze. Elsevier Science Publisher, Amsterdam 1985, pp. 325–342.Google Scholar
  34. [34]
    Schnack, E.: Free Boundary Value Problems in Elastostatics. Proceedings of the 4th Int. Symp. on Numerical Methods in Engineering, Atlanta, Georgia/ USA, 24–28 March 1986. In: Innovative Numerical Methods in Engineering. Eds.: R.P. Shaw, J. Periaux, A. Chaudouet, J. Wu, C. Marino, C.A. Brebbia. Computational Mechanics Publications Southampton, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1986, pp. 435–440.Google Scholar
  35. [35]
    Schnack, E.: A Method of Feasible Direction with FEM for Shape Optimization. Invited lecture: Proceedings of the IUTAM-Symp. on Structural Optimization, Melbourne, 9–13 February 1988. In: Structural Optimization. Eds.: G.I.N. Rozvany, B.L: Karihaloo. Kluwer Academic Publishers, Dordrecht, Boston, London 1988, pp. 299–306.Google Scholar
  36. [36]
    Schnack, E., G. Iancu: Control of the von Mises Stress with Dynamic Programming. GAMM-Seminar on Discretization Methods and Structral Optimization–Procedures and Applications, 5–7 October 1988, University of Siegen. In: Proceedings of the GAMM-Seminar, Vol. 43. Eds.: H.A. Eschenauer and G. Thierauf. Springer-Verlag, Berlin, Heidelberg 1989, pp. 154–161.Google Scholar
  37. [37]
    Schnack, E., G. Iancu: Shape Design of Elastostatics Structures Based on Local Perturbation Analysis. Structural Optimization 1 (1989), pp. 117–125.CrossRefGoogle Scholar
  38. [38]
    Schnack, E., G. Iancu: Non-Linear Programming Applicable for the Control of Elastic Structures. In: Preprints of the 5th IFAC Symposium on Control of Distributed Parameter Systems, 26–29 June 1989 in Perpignan/France. Eds.: A. El Jai and M. Amouroux. Institut de Science et de Génie des Matériaux et Procédés (CNRS), Groupe d’Automatique, Université de Perpignan 1989, pp. 163–168.Google Scholar
  39. [39]
    Schnack, E., U. Spörl: A Mechanical Dynamic Programming Algorithm for Structure Optimization. IJNME 23, No. 11 (1986), pp. 1985–2004.CrossRefMATHGoogle Scholar
  40. [40]
    Schnack, E., U. Spörl, G. Iancu: Gradientless Shape Optimization with FEM. VDI Forschungsheft 647 /88 (1988), pp. 1–44.Google Scholar
  41. [41]
    Spörl, U.: Spannungsoptimale Auslegung elastischer Strukturen. Doctoral Thesis, Karlsruhe University, 1985.Google Scholar
  42. [42]
    Trompette, Ph. and J.L. Marcelin: On the Choice of the Objectives in Shape Optimization. In: Computer Aided Optimal Design: Structural and Mechanical Systems. Ed. C.A. Mota Soares, Springer-Verlag, Berlin, New-York 1986, pp. 247–261.Google Scholar
  43. [43]
    Wang, S.-Y., Y. Sun, R.H. Gallagher: Sensitivity Analysis in Shape Optimization of Continuum Structures. Computer Structures 20, No. 5 (1985), pp. 855–867.CrossRefMATHGoogle Scholar
  44. [44]
    Zhang, Q., S. Mukherjee: Design Sensitivity Coefficients for Linear Elasticity Problems by Boundary Element Methods. Proceedings of the IUTAM/IACM Symposium on Discretized Methods in Structural Mechanics, 5–9 June 1989, Vienna/Austria. Eds.: G. Kuhn and H. Mang. Springer-Verlag, Berlin, Heidelberg 1990, pp. 283–289.Google Scholar
  45. [45]
    Zhang, W.H., P. Beckers: Comparison of Different Sensitivity Analysis Approaches for Structural Shape Optimization. In: Computer Aided Optimum Design of Structures: Recent Advances. Eds. C.A. Brebbia and S. Hernandez. Springer-Verlag, Berlin, Heidelberg, New-York 1989, pp.346356.Google Scholar
  46. [46]
    Zolesio, J.-P.: The Material Derivative (or speed) Method for Shape Optimization of Distributed Parameter Structures. Eds.: Haug E.J., J. Cea. Sijthoff and Noordhoff, Alphen aan den Rhijn (1981), pp. 10891151.Google Scholar
  47. [47]
    Zienkiewicz, O.C. and J.S. Campbell: Shape Optimization and Sequential Linear Programming. In: Optimum Structural Design. Eds. R.H. Gallagher, and O.C. Zienkiewicz. John Wiley Sons, London, New-York, Sydney, Toronto, 1973.Google Scholar

Copyright information

© Springer-Verlag Wien 1992

Authors and Affiliations

  • E. Schnack
    • 1
  • G. Iancu
    • 1
  1. 1.Karlsruhe UniversityKarlsruheGermany

Personalised recommendations