On compliance and buckling objective functions in topology optimization of snap-through problems

Abstract

This paper deals with topology optimization of static geometrically nonlinear structures experiencing snap-through behaviour. Different compliance and buckling criterion functions are studied and applied for topology optimization of a point loaded curved beam problem with the aim of maximizing the snap-through buckling load. The response of the optimized structures obtained using the considered objective functions are evaluated and compared. Due to the intrinsic nonlinear nature of the problem, the load level at which the objective function is evaluated has a tremendous effect on the resulting optimized design. A well-known issue in buckling topology optimization is artificial buckling modes in low density regions. The typical remedy applied for linear buckling does not have a natural extension to nonlinear problems, and we propose an alternative approach. Some possible negative implications of using symmetry to reduce the model size are highlighted and it is demonstrated how an initial symmetric buckling response may change to an asymmetric buckling response during the optimization process. This problem may partly be avoided by not exploiting symmetry, however special requirements are needed of the analysis method and optimization formulation. We apply a nonlinear path tracing algorithm capable of detecting different types of stability points and an optimization formulation that handles possible mode switching. This is an extension into the topology optimization realm of a method developed, and used for, fiber angle optimization in laminated composite structures. We finally discuss and pinpoint some of the issues related to buckling topology optimization that remains unsolved and demands further research.

Appendix A: Design sensitivity analysis

1.1 A.1 Sensitivity of linear displacements

The displacement sensitivities $\frac{d\mathbf{D}}{d\rho_e}$ are computed by direct differentiation of the static equilibrium equation, (1), w.r.t. a design variable ρ _e , e = 1, ..., N _e .

$$ \mathbf{K_0} \frac{d\mathbf{D}}{d\rho_e}=-\frac{d\mathbf{K_0}}{d\rho_e}\mathbf{D} + \frac{d\mathbf{R}}{d\rho_e}, \quad e=1,\ldots,N_e $$

(20)

The displacement sensitivity $\frac{d\mathbf{D}}{d\rho_e}$ can be evaluated by backsubstitution of the factored global initial stiffness matrix in (20). 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 (20), called the pseudo load vector, need to be calculated. Note that the force vector derivative, $\frac{d\mathbf{R}}{d\rho_e}$, is zero for design independent loads. The global initial stiffness matrix derivative $\frac{d\mathbf{K_0}}{d\rho_e}$ only involve the derivative of the current local element stiffness matrix $\frac{d\mathbf{k}_{0}}{d\rho_e}$ which is determined by central difference approximations at the element level. The element stiffness derivative could easily be evaluated analytically, however for ease of programming a semi-analytical approach has been used.

1.2 A.2 Sensitivity of nonlinear displacements

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

$$ \mathbf{Q}^n(\mathbf{D}^n(\boldsymbol \rho),\boldsymbol \rho)=\mathbf{F}^n-\mathbf{R}^n=\boldsymbol 0 $$

(21)

where $\mathbf{Q}^n(\mathbf{D}^n(\boldsymbol \rho),\boldsymbol \rho)$ 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 ρ _e ,e = 1, ..., N _e , we obtain

$$ \frac{d\mathbf{Q}^n}{d\rho_e} = \frac{\partial\mathbf{Q}^n}{\partial\rho_e} +\frac{\partial\mathbf{Q}^n}{\partial\mathbf{D}^n}\frac{d\mathbf{D}^n}{d\rho_e}=\boldsymbol 0 $$

(22)

$$ \text{where \quad} \frac{\partial\mathbf{Q}^n}{\partial\mathbf{D}^n} = \frac{\partial\mathbf{F}^n}{\partial\mathbf{D}^n}-\frac{\partial\mathbf{R}^n}{\partial\mathbf{D}^n} $$

(23)

$$ \text{and \quad} \frac{\partial\mathbf{Q}^n}{\partial\rho_e} = \frac{\partial\mathbf{F}^n}{\partial\rho_e}-\frac{\partial\mathbf{R}^n}{\partial\rho_e} $$

(24)

We note that (23) reduces to the tangent stiffness matrix. Since it is assumed that the current load is independent of deformation, $\frac{\partial\mathbf{R}^n}{\partial\mathbf{D}^n}=\boldsymbol 0$, we obtain

$$ \frac{\partial\mathbf{F}^n}{\partial\mathbf{D}^n}= \mathbf{K}_{\mathbf{T}}^n $$

(25)

By inserting the tangent stiffness and (24) into (22), we obtain the displacement sensitivities $\frac{d\mathbf{D}^n}{d\rho_e}$ as

$$ \mathbf{K}_{\mathbf{T}}^n\frac{d\mathbf{D}^n}{d\rho_e}=\frac{\partial\mathbf{R}^n}{\partial\rho_e}-\frac{\partial\mathbf{F}^n}{\partial\rho_e} $$

(26)

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

$$ \frac{\partial\mathbf{R}^n}{\partial\rho_e}=\gamma^n \frac{\partial\mathbf{R}}{\partial\rho_e} + \frac{\partial\gamma^n}{\partial\rho_e} \mathbf{R} $$

(27)

For design independent loads $\frac{\partial\mathbf{R}}{\partial\rho_e}=\boldsymbol 0$ and for a fixed load level $\frac{\partial\gamma^n}{\partial\rho_e}=0$. The pseudo load vector, i.e. the right hand side of (26), is determined at the element level by central difference approximations and assembled to global vector derivatives. Again, the partial derivative only involves the element which is associated with the current design variable.

1.3 A.3 Linear compliance

The design sensitivity of linear compliance is obtained by the adjoint approach, see e.g. Bendsøe and Sigmund (2003; Lund and Stegmann (2005). The sensitivity with respect to any design variable ρ _e , e = 1, ..., N _e is

$$ \frac{dC_L}{d\rho_e} = - \mathbf{D}^\text{T} \, \frac{d\mathbf{K_0}}{d\rho_e} \, \mathbf{D} $$

(28)

The global initial stiffness matrix derivatives $\frac{d\mathbf{K_0}}{d\rho_e}$ are determined semi-analytically at the element level by central difference approximations and assembled to global matrix derivatives.

1.4 A.4 Nonlinear end compliance

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

$$ \frac{dC_{GNL}}{d\rho_e}= \boldsymbol{\lambda}^\text{T} \frac{\partial\mathbf{Q}^n}{\partial\rho_e} = \boldsymbol{\lambda}^\text{T} \left( \frac{\partial\mathbf{F}^n}{\partial\rho_e}-\frac{\partial\mathbf{R}^n}{\partial\rho_e} \right) $$

(29)

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

$$ \mathbf{K}_{\mathbf{T}}^n \, \boldsymbol{\lambda}=-\mathbf{R}^n $$

(30)

The partial derivatives in the right hand side of (29) are determined at the element level by central difference approximations and assembled to global vector derivatives.

1.5 A.5 Linear buckling

The linear buckling load factor sensitivities are determined by

$$ \frac{d\lambda_j}{d\rho_e}={\phi}_j^T \left( \frac{d\mathbf{K_0}}{d\rho_e} + \lambda_j \frac{d\mathbf{K}_{\boldsymbol \sigma}}{d\rho_e} \right) {\phi}_j $$

(31)

where the eigenvalue problem in (2) has been differentiated with respect to any design variable, ρ _e , e = 1, ..., N _e , assuming that λ _j is simple, see e.g. Courant and Hilbert (1953); Wittrick (1962). The global matrix derivatives of K ₀ and $\mathbf{K}_{\boldsymbol \sigma}$ are determined semi-analytically at the element level by central difference approximations and assembled to global matrix derivatives. The stress stiffness matrix is an implicit function of the displacement field, i.e. $\mathbf{K}_{\boldsymbol \sigma}(\mathbf{D}(\boldsymbol{\rho}),\boldsymbol{\rho})$, and thus depends on all elements within the model. 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 (20) such that $\Delta \mathbf{D} \approx \frac{d\mathbf{D}}{d\rho_e} \Delta \rho_e$.

1.6 A.6 Nonlinear buckling

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

$$ \frac{d\lambda_j}{d\rho_e}={\phi}_j^T \left( \frac{d\mathbf{K_0}}{d\rho_e} + \frac{d\mathbf{K}_{\mathbf{L}}^n}{d\rho_e} + \lambda_j \frac{d\mathbf{K}_{\boldsymbol \sigma}^n}{d\rho_e} \right) {\phi}_j $$

(32)

and

$$ \frac{d\gamma_j^c}{d\rho_e}=\frac{d\lambda_j}{d\rho_e} \, \gamma^n $$

(33)

where the eigenvalue problem in (11) has been differentiated with respect to any design variable, ρ _e , e = 1, ..., N _e , assuming that λ _j is simple, see Lindgaard and Lund (2010). 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 , $\mathbf{K}_{L}^n$, and $\mathbf{K}_{\boldsymbol \sigma}^n$ 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. The displacement field is perturbed via the calculated sensitivities of the nonlinear displacements in (26) such that $\Delta \mathbf{D}^n \approx \frac{d\mathbf{D}^n}{d\rho_e} \Delta \rho_e$.