2.3. Finite Element Discretization
By means of finite element approximation, Eq.6 can be stated symbolically as the generalized eigenvalue problem,
Eq. 8
In this problem, represents the eigenvector or mode, and denote the stiffness and geometric matrices, respectively. represents the vector of the design variables that are assumed to be constant for each finite element. Vector is the fundamental displacement, which is the solution of a finite element approximation of the linear elasto-static problem,
Eq. 9
where is the load vector. The generalized eigenvalue problem is solved by the subspace iteration method using the Sparse Equation method (ANSYS® Academic Research, ANSYS® 14.0 Help //
Theory Reference // 15. Analysis Tools // 15.16. Eigenvalue and Eigenvector Extraction).
3. OPTIMIZATION OF LINEAR BUCKLING PROBLEMS AND DESIGN SENSITIVITY
The optimization problem previously laid out is solved by the Method of Moving Asymptotes (MMA). The documentation of the algorithm can be found in several Svanberg publications (Svanberg, 1987, 2002 and 2007).
3.1. Design sensitivity analysis of simple eigenvalues
The gradients are calculated from the direct differentiation and the adjoint method. The derivation of sensitivities of simple eigenvalues is made for finite element equations. The sensitivity of a simple eigenvalue is the derivative of with respect to design variables . Pre-multiplying Eq.8 by , the derivative of is:
Eq. 13
Applying the quotient rule derivative, and assuming that the eigenvectors are normalized with respect to , i.e., , we have (see references (Neves, Rodrigues, & Guedes, Generalized topology design of structures with a buckling load criterion, 1995; Rodrigues, Guedes, & Bendsoe, 1995)) :
Eq. 14
The derivative of the stiffness matrix and the initial stress matrix (i.e. and ) are determined analytically at the finite element level. The derivative of the stiffness matrix with respect to thickness, is trivial, since depends on thickness explicitly.
The initial stress matrix, , is a function that depends on thickness explicitly and implicitly through the displacement field, i.e. . According to the nomenclature of the adjoint method, if is the adjoint force, , then the adjoint variable, , is defined as (Neves, Rodrigues, & Guedes, Generalized topology design of structures with a buckling load criterion, 1995):
Eq. 15
This definition makes use of the symmetry property of the stiffness matrix . The calculation of the adjoint variable requires the inversion of matrix, i.e. . However, this inversion is computationally too expensive. Multiplying both sides of the equation by , the adjoint problem similar to static analysis with finite element approximation is obtained:
Eq. 16
We chose to solve the problem in MatLab® using the command “mldivide \”. The calculation of initial stress matrix sensitivity using the adjoint variable is:
Eq. 17
Finally, introducing Eq.17 into Eq.14, the sensitivity of the simple eigenvalue is given by
Eq. 18
The objective function is the inverse of (see Eq.10). Therefore, we calculate the sensitivity of the objective function using the quotient derivative rule:
Eq. 19
Until now, the sensitivity was derived for single eigenvalues. When there are repeated eigenvalues, the objective function is not differentiated. In order to overcome this difficulty, an auxiliary routine based on the Generalized Gradient concept (Clark, 1983) is introduced. The mathematical formulation is presented in the next section.
3.2. Design sensitivity analysis of multiple eigenvalues
This Generalized Gradient technique was used for dimensional optimization of plates and beams with critical buckling load criterion by Folgado and Rodrigues (Folgado & Rodrigues, 1998) and Rodrigues et al. (Rodrigues, Guedes, & Bendsoe, 1995).
Let consider a general iteration “ ”. At this iteration it is necessary to define the decreasing direction of the objective function in the current design. To characterize that direction let be a small number defined by the user (e.g. ) and let , which we will call - multiplicity of , be equal to the number of eigenvalues satisfying the inequality:
Eq. 20
After defining this criterion, sub-gradients, , which satisfy the previous condition (i.e.
with ) are calculated. With the sub-gradients, a convex space is constructed (Neves, Sigmund, & Bendsoe, 2002):
Eq. 21
The descent direction of the objective function is given by vector , which belongs to , and whose norm is minimum. is calculated by formulating the following minimization problem (Dem’yanov & Malozemov, 1990; Kiwiel, 1985):
Eq. 22
If , then , and there is no descent direction, lying in the objective function in a stationary point. However, note that does not imply that design point is not a stationary point, since is an approximation of the gradient. The problem of minimizing the norm was solved using the fmincon function in MatLab®. The algorithm used to solve the problem is the interior point (see references (Waltz, Morales, Nocedal, & Orban, 2006; Byrd, Mary, & Nocedal, 1999)).
3.3. Optimization scheme
The optimization scheme uses the commercial finite element software Ansys® to obtain the linear elastic stability response, namely the displacement field , the critical load factors and the respective eigenvectors . The MMA approach was adopted to solve the optimization problem.
The optimization method implemented in this work is represented by the flowchart shown in Figure 1.
Figure 1: Optimization scheme.
The code built consists of the interface written in MatLab® linking the finite element program Ansys® and MMA in its version of MatLab®. The value of the objective function, the constraints and sensitivities are inputted to MMA algorithm, which calculates a new design point and the new structural response (critical loads, buckling modes, displacements) is again obtained with Ansys®. The solution is converged when the convergence criterion is verified, i.e
Eq. 23
where is defined by the user.