Review the following for the theory behind the solution methodology for density-based optimization analyses:

Sequential Convex Programming

The Sequential Convex Programming method (SCP), see Zillober^[1][2][3], is an extension of the method of moving asymptotes (MMA), see Svanberg^[4]. The Sequential Convex Programming method requires the derivatives of all functions present in the Topology Optimization problem. MMA is a nonlinear programming algorithm that approximates a solution for a Topology Optimization problem by solving a sequence of convex and separable subproblems. These subproblems can be solved efficiently due to their special structure.

The Sequential Convex Programming method extends MMA to ensure convergence by rejecting steps that do not lead to an optimal solution of the underlying problem. The test for acceptance is done by a merit function and a corresponding line search procedure, see Zillober^[5]. The goal of the merit function is to measure the progress and enable the objective function and the constraints to be combined in a suitable way.

Optimality Criteria

The Optimality Criteria method can be used to solve density-based optimization problems with a simple compliance objective that uses a volume or mass constraint. The Optimality Criteria method is an iterative solver, see Bensoe and Sigmund^[6]. The Optimality Criteria method should not be used for a Modal Analysis.

Solution Methodology for Natural Frequencies

When performing density-based optimization analysis with supported natural frequencies, you can specify the frequency as either an objective or as a constraint. A single natural frequency or a weighted combination of several natural frequencies can be defined using the Objective object. The aim of the optimization is to maximize these frequencies according to their weights (as defined in the Worksheet).

In addition, you can add a single natural frequency as a constraint and define a lower and an upper bound on the frequency. The solver will guarantee, if possible, that this frequency lies within the specified range.

If the design objective is to optimize a frequency, then all of the repeating frequencies are optimized simultaneously.

It is important to note that the mode shapes will change during the iterative solution procedure and that there is no tracking with respect to the initial mode shape. Only the actual value of the specified natural frequency is considered. This means at the final iteration the mode shape may change dramatically in comparison to the initial shape of the optimized mode.

Because the underlying solver is sensitivity based, problems with natural frequencies have to be handled with care. The problem is not differentiable in the common sense, such as a case of multiple eigenvalues. Instead, derivatives for multiple eigenvalues have to be calculated in a special way. Since the mode shapes are not unique for multiple eigenvalues, additional effort is necessary to get sensitivities that are independent of the mode shapes. In order to obtain unique sensitivities for these eigenvalues, an additional eigenvalue problem has to be solved for each optimized element, see Seyranian^[7].

Solution Methodology for Stress Constraints

When working with topological optimization for global stress constraints, and local stress constraints applied to more than one element, you can specify an upper bound on the stress that has to be satisfied by all elements. Theoretically, this requires the solution of an optimization problem with n stress constraints, where n denotes the number of optimized elements taken into account. Because the computational effort would be too great to achieve this, a relaxed reformulation has to be applied. In order to keep the complexity of the optimization problem low, a set of elements is represented by one constraint instead of individual ones. This technique divides the original design space into clusters. The maximum stress value with respect to all elements in the cluster/set S has to satisfy the following:

Where is the elemental mean value of the equivalent (von-Mises) stress of element e in set S. Since the maximum leads to a non-differentiable problem formulation, the p-norm is used to approximate the actual maximum instead. Applying the differentiable p-norm leads to:

Where denotes the vector of all stress values of the elements in set S. Note that the p-norm overestimates the actual maximum. To stabilize the solver different regularization techniques are used in the literature. In Holmberg^[8], a fixed scaling parameter is introduced. With factor: that leads to:

where n_S is the number of elements in the considered set. In previous releases this approach was used. Since at the final iteration, the maximum stress of some optimized elements might be greater than the user-defined upper bound of the global/local stress constraint, the validation might fail.

To improve the accuracy of the approximation, a different regularization techniques is available. In Le^[9], the nnormalized maximum approximation is used to measure the stress value of a cluster/set. Here the p-Norm is also applied but instead of using a fixed factor an adaptive factor is introduced. In each iteration the factor is modified. This technique leads to:

Where denotes the iteration. This approach improves accuracy as well as the estimate of the stress value.

Solution Convergence Criteria

The density-based optimization solver approaches a stationary point where all constraints are satisfied within a tolerance of 0.1 percent of the defined bound. This tolerance is defined by the Convergence Accuracy property (see Specify Analysis Settings).

To simplify the notation, assume that only one constraint exists. The optimality conditions of the Topology Optimization problem can be stated with the following equation:

Where denotes the Lagrange function. The Lagrange function is defined by:

Where is the Lagrange multiplier corresponding to the constraint , and is the objective function to be either maximized or minimized. The solver will stop as soon as the desired tolerance is achieved, where: , as defined here:

Because approaching this stationary point can require a large number of iterations, a relaxed convergence criterion is used. The optimization stops as soon as the following equation has three successive iterations. In this equation, denotes the vector of pseudo densities of the iteration.

Note that three successive iterations are considered as the underlying solver is stabilized by a line search procedure. This line search procedure might lead to small changes with respect to the pseudo densities as well as small changes to the objective function. It is possible that the convergence tolerance is satisfied for one iteration but the next iteration leads to a significant improvement of the objective function. Due to the relaxed stopping criterion, the optimization might terminate too early. In this case, the optimization should be rerun with a smaller tolerance.

Topology Optimization with Thermal Condition

The optimization is influence by the thermal condition according to the following equation^[10]:

Linear static equilibrium in finite element system including both mechanical and thermal loading is given by:

,

Where:

= stiffness matrix

= displacement vector

= externally applied mechanical loading

= thermal load vector.

The nodal load vector due to temperature effects for the element may be written as:

Here is the element strain-displacement matrix, is the element elasticity matrix, and is the thermal strain vector for the element given by:

With is the thermal expansion coefficient of the material, is the temperature change on the element, and is [1,1,1,0,0,0] for three-dimensions and [1,1,0] for two-dimension.

References