Reduced Basis Technique for Calculating Sensitivity Derivatives of the Nonlinear Structural Response

  • A. K. Noor
  • J. M. Peters

Summary

An efficient reduced basis technique is presented for calculating the sensitivity of nonlinear structural response to variations in the design variables. The structure is discretized by using two-field mixed finite element models. The vector of structural response and its sensitivity derivatives, with respect to design variables, are each expressed as a linear combination of a small number of global approximation (or basis) vectors. The Bubnov-Galerkin technique is then used to approximate each of the finite element equations governing the response, and the sensitivity derivatives, by a small number of algebraic equations in the amplitudes of these vectors. The path derivatives (derivatives of the response vector with respect to path parameters — e.g., load parameter) are used as basis vectors for approximating the response. A combination of the path derivatives and their derivatives with respect to the design variables is used for approximating the sensitivity derivatives. The potential of the proposed technique is discussed and its effectiveness is demonstrated by means of a numerical examples of laminated composite plates subjected to uniform transverse loading.

Keywords

Lamination 

Nomenclature

Nomenclature

dι

material and lamination parameters of the plate

EL, ET

elastic moduli of the individual layers in the direction of fibers and normal to it, respectively

GLT, GTT

shear moduli in the plane of fibers and normal to it, respectively

|K|

global linear structure matrix which includes the flexibility and the linear strain-displacement matrices, see Eqs. (1) and A.2 (Appendix A)

{G(Z)}

vector of nonlinear terms, see Eqs. (1)

{G̃{Ψ},{Ḡ(Ψ)}

vectors of nonlinear terms of the reduced systems, see Eqs. (5) and (6)

h

thickness of the plate

|K̃|,|K̄|

linear matrices of the reduced systems, see Eqs. (5) and (6)

L

side length of the plate

Po

intensity of uniform transverse loading

q

load parameter

{Q}

normalized load vector, see Eqs. (1)

{Q̃},{Q̄}

load vectors of the reduced systems, see Eqs. (5) and (6)

r

number of basis vectors used in evaluating the response

U

total strain energy of the plate

u1, u2, w

displacement components in the coordinate directions

xl, x2, x3

Cartesian coordinate system (x3 normal to the middle plane of the plate)

|Z}

response vector which includes both the nodal displacements and the stress parameters

|Γ|[Γ̃ι]

matrices of basis vectors, see Eqs. (3) and (4)

θ

fiber orientation angle

VLT

major Poisson’s ratio of the individual layers

Φ1, Φ2

rotation components of the middle plane of the plate, see Fig. 1

{Ψ}, {Ψ̃ι}

vectors of undetermined coefficients of the reduced equations, see Eqs. (5) and (6)

Superscripts

t

denotes matrix transposition

Range of Indices

I,J,L

1 to the total number of degrees of freedom (nodal displacements and stress parameters)

i,j

1 to the total number of reduced degrees of freedom used in evaluating the sensitivity derivatives

ι

1 to the total number of lamination and material parameters

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References

  1. 1.
    RYU, Y. S., HARIRIAN, M., WU, C. C. and ARORA, J. S., Structural Design Sensitivity Analysis of Nonlinear Response, Computers and Struct, 21(1/2), 245–55(1985).MATHCrossRefGoogle Scholar
  2. 2.
    MROZ, Z., KAMAT, M. P. and PLAUT, R. H., Sensitivity Analysis and Optimal Design of Nonlinear Beams and Plates, Journal of Struct. Mech., 13(3/4), 245–66 (1985).MathSciNetCrossRefGoogle Scholar
  3. 3.
    HAFTKA, R. T. and MROZ, Z., First- and Second-Order Sensitivity Analysis of Linear and Nonlinear Structures, AIAA J., 24(7), 1187–92 (1986).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    WU, C. C. and ARORA, J. S., Design Sensitivity Analysis and Optimization of Nonlinear Structural Response Using Incremental Procedure, AIAA J., 25(8), 1118–25(1987).MATHCrossRefGoogle Scholar
  5. 5.
    CARDOSO, J. B. and ARORA, J. S., Variational Method for Design Sensitivity Analysis in Nonlinear Structural Mechanics, AIAA J., 26(5), 595–603 (1988).MathSciNetCrossRefGoogle Scholar
  6. 6.
    TORTORELLI, D. A., HABER, R. B. and LU, S. C.Y., Design Sensitivity Analysis for Nonlinear Thermal Systems, Comp. Meth. in Appl. Mech. and Eng., 77,61–77(1989).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    ARORA, J. S. and CARDOSO, J. E. B., A Design Sensitivity Analysis Principle and Its Implementation into ADINA, Computers and Struct., 32(3/4), 691–705 (1989).MATHCrossRefGoogle Scholar
  8. 8.
    TORTORELLI, D. A., HABER, R. B. and LU, S. C.-Y., Adjoint Sensitivity Analysis for Nonlinear Dynamic Thermoelastic Systems, AIAA J., 29(2), 253–263 (1991).CrossRefGoogle Scholar
  9. 9.
    PHELAN, D. G., VIDAL, C. and HABER, R. B., An Adjoint Variable Method for Sensitivity Analysis of Nonlinear Elastic Systems, Int. Journal for Num. Meth. in Eng., 31, 1649–1667 (1991).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    NOOR, A. K., TANNER, J. A. and PETERS, J. M., Sensitivity of Tire Response to Variations in Material and Geometric Parameters, Finite Elements in Design and Analysis (to appear).Google Scholar
  11. 11.
    ADELMAN, H. M. and HAFTKA, R. T., Sensitivity Analysis of Discrete Structural Systems, AIAA J., 24(5), 823–832 (1986).CrossRefGoogle Scholar
  12. 12.
    HAFTKA, R. T. and ADELMAN, H.M., ‘Recent Developments in Structural Sensitivity Analysis,’ Structural Optimization I, Springer-Verlag, 1989, pp. 137–151.Google Scholar
  13. 13.
    HABER, R. B., TORTORELLI, D. A., VIDAL, C. A. and PHELAN, D. G. -’Design Sensitivity Analysis of Nonlinear Structures — I: Large-Deformation Hyperelasticity and History-Dependent Material Response,’ Structural Optimization: Status and Promise, Ed. Kamat, M. P., AIAA Series: Progress in Astronautics and Aeronautics (to appear).Google Scholar
  14. 14.
    ARORA, J. S., LEE, T. H. and KUMAR, V. — ‘Design Sensitivity Analysis of Nonlinear Structures — III: Shape Variation of Viscoplastic Structures,’ Structural Optimization: Status and Promise, AIAA Series: Progress in Astronautics and Aeronautics (to appear).Google Scholar
  15. 15.
    NOOR, A. K., Recent Advances in Reduction Methods for Nonlinear Problems, Computers and Struct, 13(1/2), 31–44 (1981).MATHCrossRefGoogle Scholar
  16. 16.
    NOOR, A. K., On Making Large Nonlinear Problems Small, Comp. Meth. in Appl. Mech. and Eng., 34, 955–985 (1982).MATHCrossRefGoogle Scholar
  17. 17.
    NOOR, A. K. and PETERS, J. M., Recent Advances in Reduction Methods for Instability Analysis of Structures, Computers and Struct., 16(1–4), 67–80 (1983).MATHCrossRefGoogle Scholar
  18. 18.
    NOOR, A. K. and PETERS, J. M., Multiple-Parameter Reduced Basis Technique for Bifurcation and Postbuckling Analyses of Composite Plates, Int. Journal for Num. Meth. in Eng.,19, 1783–1803 (1983).MATHCrossRefGoogle Scholar
  19. 19.
    NOOR, A. K. and CAMIN, R. A., Symmetry Considerations for Anisotropic Shells, Comp. Meth. in Appl. Mech. and Eng., 9, 317–335 (1976).MATHCrossRefGoogle Scholar
  20. 20.
    NOOR, A. K. and ANDERSEN, C. M., Mixed Models and Reduced/Selective Integration Displacement Models for Nonlinear Shell Analysis, Int. Journal for Num. Meth. in Eng., 18, 1429–1454 (1982)MATHCrossRefGoogle Scholar
  21. 21.
    CHOI, K. K. and TWU, S. L., On Eigenvalue of Continuum and Discrete Methods of Shape Design Sensitivity Analysis, AIAA J., 27(10), 1418–1424 (1989).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • A. K. Noor
    • 1
  • J. M. Peters
    • 1
  1. 1.Center for Computational Structures Technology, NASA Langley Research CenterUniversity of VirginiaHamtonUSA

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