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Optimization formulations for the maximum nonlinear buckling load of composite structures

Abstract

This paper focuses on criterion functions for gradient based optimization of the buckling load of laminated composite structures considering different types of buckling behaviour. A local criterion is developed, and is, together with a range of local and global criterion functions from literature, benchmarked on a number of numerical examples of laminated composite structures for the maximization of the buckling load considering fiber angle design variables. The optimization formulations are based on either linear or geometrically nonlinear analysis and formulated as mathematical programming problems solved using gradient based techniques. The developed local criterion is formulated such it captures nonlinear effects upon loading and proves useful for both analysis purposes and as a criterion for use in nonlinear buckling optimization.

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Acknowledgement

The authors gratefully acknowledge the support from the Danish Center for Scientific Computing (DCSC) for the hybrid Linux Cluster “Fyrkat” at Aalborg University, Denmark.

Author information

Correspondence to Esben Lindgaard.

Appendix A Design sensitivity analysis

Appendix A Design sensitivity analysis

A.1 Linear displacement sensitivity

The displacement sensitivities \(\frac{d \mathbf{D}}{d a_i}\) are computed by direct differentiation of the static equilibrium equation, see (1), w.r.t. a design variable a i , i = 1,..., I.

$$\label{eqn:displacement_sensitivity_pseudoload_elo} \mathbf{K_0} \frac{d {\bf D}}{d a_i}=-\frac{d \mathbf{K_0}}{d a_i}{\bf D} + \frac{d {\bf R}}{d a_i}, \quad i=1,\ldots,I $$
(15)

The displacement sensitivity \(\frac{d {\bf D}}{d a_i}\) can be evaluated by backsubstitution of the factored global initial stiffness matrix in (15). The initial stiffness matrix has already been factored when solving the static problem in (1) and can here be reused, whereby only the new terms on the right hand side of (15), called the pseudo load vector, need to be calculated. Note that the force vector derivative, \(\frac{{d \bf R}}{d a_i}\), is zero for design independent loads as in the case for CFAO. The global initial stiffness matrix derivatives \(\frac{d \mathbf{K_0}}{d a_i}\) are determined semi-analytically at the element level by central difference approximations and assembled to global matrix derivatives.

$$\frac{d {\bf k}_{\bf 0}}{d a_i} \approx \frac{\mathbf{k_{0}}(a_i+\Delta a_i) - \mathbf{k_{0}}(a_i-\Delta a_i)}{2 \Delta a_i} \label{eqn:k0_sensitivity_elementlevel_elo}\\ $$
(16)
$$\frac{d \mathbf{K_0}}{d a_i}=\sum_{n=1}^{N_e^{as}}\frac{d \mathbf{k_0}}{d a_i}, \quad i=1,\ldots,I \label{eqn:assembly_global_stiffness_matrix_elo} $$
(17)

k 0 is the element initial stiffness matrix, Δa i is the design perturbation, and \(N_e^{as}\) is the number of elements in the finite element model associated to the design variable a i .

A.2 Nonlinear displacement sensitivity

The nonlinear displacement sensitivities are computed by considering the residual or force unbalance equation at a converged load step n,

$${\bf Q}^n({\bf D}^n({\bf a}),{\bf a})={\bf F}^n-{\bf R}^n=\boldsymbol 0 $$
(18)

where Q n(D n(a),a) is the so-called residual or force unbalance, F n is the global internal force vector, and R n is the global applied load vector. Taking the total derivative of this equilibrium equation with respect to any of the design variables a i ,i = 1,...,I, we obtain

$$\frac{d {\bf Q}^n}{d a_i}=\frac{\partial {\bf Q}^n}{\partial a_i} +\frac{\partial {\bf Q}^n}{\partial {\bf D}^n}\frac{d {\bf D}^n}{d a_i}=\boldsymbol 0 \label{eqn:total_deriv_unbalance_desvar} \\ $$
(19)
$$\text{where \quad} \frac{\partial {\bf Q}^n}{\partial {\bf D}^n}=\frac{\partial {\bf F}^n}{\partial {\bf D}^n}-\frac{\partial {\bf R}^n}{\partial {\bf D}^n} \label{eqn:partial_unbalance_D_elo}\\ $$
(20)
$$\text{and \quad} \frac{\partial {\bf Q}^n}{\partial a_i}=\frac{\partial {\bf F}^n}{\partial a_i}-\frac{\partial {\bf R}^n}{\partial a_i} \label{eqn:partial_forceunbalance_desvar} $$
(21)

We note that (20) reduces to the tangent stiffness matrix. Since it is assumed that the current load is independent of deformation, \(\frac{{\partial\bf R}^n}{{\partial\bf D}^n}=\boldsymbol 0\), we obtain

$$ \frac{\partial {\bf F}^n}{\partial {\bf D}^n}= {\bf K}_{{\bf T}}^n $$
(22)

By inserting the tangent stiffness and (21) into (19), we obtain the displacement sensitivities \(\frac{{d \bf D}^n}{d a_i}\) as

$$\label{eqn:nonlinear_displacement_sensitivity} {\bf K}_{{\bf T}}^n\frac{d {\bf D}^n}{d a_i}=\frac{\partial {\bf R}^n}{\partial a_i}-\frac{\partial {\bf F}^n}{\partial a_i} $$
(23)

The partial derivative of the load vector, \(\frac{\partial {\bf R}^n}{\partial a_i}\), can explicitly be expressed by two terms by taking the partial derivative to (6)

$$ \frac{\partial {\bf R}^n}{\partial a_i}=\gamma^n \frac{\partial {\bf R}}{\partial a_i} + \frac{\partial \gamma^n}{\partial a_i} \mathbf{R} $$
(24)

For design independent loads \(\frac{\partial {\bf R}}{\partial a_i}=\boldsymbol 0\) and for a fixed load level \(\frac{\partial \gamma^n}{\partial a_i}=0\). The pseudo load vector, i.e. the right hand side to (23), is determined at the element level by central difference approximations and assembled to global vector derivatives.

A.3 Linear compliance

The design sensitivity of linear compliance is obtained by applying the adjoint approach, see e.g. Bendsøe and Sigmund (2003) and Lund and Stegmann (2005), and obtaining the sensitivity with respect to any design variable a i , i = 1,...,I as

$$\label{eqn:DSA_linear_compliance} \frac{d C_L}{d a_i} = - \mathbf{D}^\text{T} \, \frac{d \mathbf{K_0}}{d a_i} \, \mathbf{D} $$
(25)

The global initial stiffness matrix derivatives \(\frac{d \mathbf{K_0}}{d a_i}\) are determined semi-analytically at the element level by central difference approximations and assembled to global matrix derivatives as in (16) and (17).

A.4 Nonlinear end compliance

The design sensitivity of nonlinear end compliance at a converged load step n with respect to any design variable, a i , i = 1,...,I, is obtained by the adjoint approach, see e.g. Bendsøe and Sigmund (2003)

$$\label{eqn:DSA_nonlinear_compliance} \frac{d C_{\rm GNL}}{d a_i}= \boldsymbol{\lambda}^\text{T} \frac{\partial {\bf Q}^n}{\partial a_i} = \boldsymbol{\lambda}^\text{T} \left( \frac{\partial {\bf F}^n}{\partial a_i}-\frac{\partial {\bf R}^n}{\partial a_i} \right) $$
(26)

Assuming the end load fixed and independent of design changes we have that \(\frac{\partial {\bf R}^n}{\partial a_i}=\boldsymbol 0\). The adjoint vector \(\boldsymbol{\lambda}\), which is not to be confused with the eigenvector, is obtained as the solution to the adjoint equation

$$\label{eqn:adjointvector_nonlinear} {\bf K}_{{\bf T}}^n \, \boldsymbol{\lambda}=-{\bf R}^n $$
(27)

The partial derivatives in the right hand side of (26) are determined at the element level by central difference approximations and assembled to global vector derivatives.

A.5 Nonlinear first principal element strain

The design sensitivities of the first principal element strain, \(\frac{d \varepsilon_1}{d a_i}\), are determined semi-analytically by forward differences at the element level.

$$\label{eqn:DSA_Principal_Strain} \frac{d \varepsilon_1}{d a_i} \approx \frac{\varepsilon_1(\mathbf{D}^n + \Delta \mathbf{D}^n) - \varepsilon_1(\mathbf{D}^n)}{\Delta a_i} $$
(28)

The displacement field is perturbed via the calculated displacement sensitivities in (23) such that \(\Delta \mathbf{D}^n \approx \frac{d \mathbf{D}^n}{d a_i} \Delta a_i\).

A.6 Element nonlinearity factor

The design sensitivities of the element nonlinearity factor, \(\varepsilon_{\text{GNL}}\), are determined semi-analytically by forward differences at the element level.

$$\label{eqn:DSA_GNL_Factor} \frac{d \varepsilon_{\text{GNL}}}{d a_i} \approx \frac{\varepsilon_{\text{GNL}}(\mathbf{D}^1 + \Delta \mathbf{D}^1,\mathbf{D}^n + \Delta \mathbf{D}^n) - \varepsilon_{\text{GNL}}(\mathbf{D}^1,\mathbf{D}^n)}{\Delta a_i} $$
(29)

It is assumed that the initial load level and the final load level are fixed whereby the perturbed element nonlinearity factor is determined by

$$\begin{array}{lll} &\varepsilon_{\text{GNL}}(\mathbf{D}^1 + \Delta \mathbf{D}^1,\mathbf{D}^n + \Delta \mathbf{D}^n) \\ &{\kern4pt} = 1 + \left| \frac{\varepsilon_1^n(\mathbf{D}^n + \Delta \mathbf{D}^n)/\gamma^n-\varepsilon_1^1(\mathbf{D}^1 + \Delta \mathbf{D}^1)/\gamma^1}{\varepsilon_1^1(\mathbf{D}^1 + \Delta \mathbf{D}^1)/\gamma^1} \right| \end{array} $$
(30)

Since the element nonlinearity factor is determined by information at two equilibrium points, i.e. the initial load step and the final step n, the displacement sensitivities have to be calculated at both load steps by (23). The perturbation of the displacement fields at both equilibrium points may then be evaluated by \(\Delta \mathbf{D}^n \approx \frac{d \mathbf{D}^n}{d a_i} \Delta a_i\) and \(\Delta \mathbf{D}^1 \approx \frac{d \mathbf{D}^1}{d a_i} \Delta a_i\), respectively.

A.7 Linear buckling

The linear buckling load factor sensitivities may be determined by

$$\label{eqn:linear_buckling_DSA_elo} \frac{d \lambda_j}{d a_i}=\boldsymbol{\phi}_j^T \left( \frac{d \mathbf{K_0}}{d a_i} + \lambda_j \frac{d {\bf K}_{\boldsymbol \sigma}}{d a_i} \right) \boldsymbol{\phi}_j $$
(31)

where the eigenvalue problem in (2) has been differentiated with respect to any design variable, a i , i = 1,...,I, assuming that λ j is simple, see e.g. Courant and Hilbert (1953) and Wittrick (1962). The global matrix derivatives of K 0 and \({\bf K}_{\boldsymbol \sigma}\) are determined semi-analytically at the element level by central difference approximations and assembled to global matrix derivatives, see (16) and (17). The stress stiffness matrix is an implicit function of the displacement field, i.e. \({\bf K}_{\boldsymbol \sigma}(\mathbf{D}(\mathbf{a}),\mathbf{a})\), thus both displacement field and design variables need to be perturbed in the element central difference approximation. The displacement field is perturbed via the calculated displacement sensitivities in (23) such that \(\Delta \mathbf{D}^n \approx \frac{d \mathbf{D}^n}{d a_i} \Delta a_i\).

A.8 Nonlinear buckling

The nonlinear buckling load factor sensitivities at load step n are determined by

$$ \label{eqn:DSA_nonlin_eigprb_elo} \frac{d \lambda_j}{d a_i}=\boldsymbol{\phi}_j^T \left( \frac{d \mathbf{K_0}}{d a_i} + \frac{d {\bf K}_{{\bf L}}^n}{d a_i} + \lambda_j \frac{d {\bf K}_{\boldsymbol \sigma}^n}{d a_i} \right) \boldsymbol{\phi}_j $$
(32)

and

$$ \frac{d \gamma_j^c}{d a_i}=\frac{d \lambda_j}{d a_i} \, \gamma^n $$
(33)

where the eigenvalue problem in (11) has been differentiated with respect to any design variable, a i , i = 1,...,I, assuming that λ j is simple, see Lindgaard and Lund (2010a). It is assumed that the final load level is fixed and that the nonlinear buckling load has been determined at load step n by evaluation of (10) and (11). The global matrix derivatives of K 0 , \({\bf K}_{L}^n\), and \({\bf K}_{\boldsymbol \sigma}^n\) are determined in the same manner as for the linear buckling load sensitivities, i.e. semi-analytical central difference approximations at the element level and assembly to global matrix derivatives.

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Lindgaard, E., Lund, E. Optimization formulations for the maximum nonlinear buckling load of composite structures. Struct Multidisc Optim 43, 631–646 (2011). https://doi.org/10.1007/s00158-010-0593-8

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Keywords

  • Composite laminate optimization
  • Buckling
  • Structural stability
  • Design sensitivity analysis
  • Geometrically nonlinear
  • Composite structures