The finite element method in the 1990’s pp 296-310 | Cite as

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

## 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

Design Variable Basis Vector Composite Plate Laminate Composite Plate Response Vector## Nomenclature

## Nomenclature

- d
_{ι} material and lamination parameters of the plate

- E
^{L}, E^{T} elastic moduli of the individual layers in the direction of fibers and normal to it, respectively

- G
^{LT}, G^{TT} 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

- P
_{o} 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

- u
_{1}, u_{2}, w displacement components in the coordinate directions

- x
_{l}, x_{2}, x_{3} Cartesian coordinate system (x

_{3}normal to the middle plane of the plate)- |Z}
response vector which includes both the nodal displacements and the stress parameters

- |Γ|[Γ̃
_{ι}] matrices of basis vectors, see Eqs. (3) and (4)

- θ
fiber orientation angle

- V
_{LT} major Poisson’s ratio of the individual layers

- Φ
_{1}, Φ_{2} rotation components of the middle plane of the plate, see Fig. 1

- {Ψ}, {Ψ̃
_{ι}} 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

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