Overview of Optimization Terms

The two above statements are too unclear for defining an optimization. Therefore, we usually reduce the “what” to well-defined terms, say: Minimize the maximal nodal stresses of load case 1 and 2, or maximize the sum of the first 5 eigenvalues.

The goal or objective of an optimization is usually called the objective function; for example, when you want to minimize or maximize some well-defined terms. You might also want to enforce certain values; for example, a displacement of a given node must not exceed a certain value. This would be defined through a constraint.

In Tosca Structure the objective function depends on at least one term or more, whereas a constraint always depends on exactly one term. In Tosca Structure, these terms or responses are called design response(s), or short DRESP. DRESPs are the fundamental definitions of the optimization problem.

Most DRESP definitions depend on a node or element group, but there are exceptions such as eigenfrequencies (TYPE=DYN_FREQ). The node or element group might also consist of one single item, say one node; for example, displacement in X-direction of a node (TYPE=DISP_X). The optimization problem is summarized in the OPTIMIZE command and the dependencies can be visualized in following way:

Mathematical Formulation

An optimization problem can be stated as:


min $Φ (U (x), x)$ such that $Ψ_{i} (U (x), x) \leq 0$ such that $g_{i} (x) \leq 0$

where $Φ$ is the objective function that depends on the state variables, $U$ , as well as on the design variables $x$ . The problem might be constrained by the constraints $Ψ_{i}$ , and might have the design variable constraints $g_{i}$ . Note, that maximizing the objective is the same as minimizing $- Φ$ .

For minimization (MIN) or maximization (MAX), the objective function consists of a sum of design responses ( $φ_{i}$ ). Each design response can be given a weight ( $w_{i}$ ) and a reference value ( $φ_{i}^{ref}$ ). By minimizing or maximizing the objective, one gets the formulations:


$min (\sum_{i = 1}^{N}, w_{i}, (φ_{i} (U(x), x) - φ_{i}^{ref}))$ $max (\sum_{i = 1}^{N}, w_{i}, (φ_{i} (U(x), x) - φ_{i}^{ref}))$

Other important optimization formulations are to minimize the maximum design response and to maximize the minimum design response, the so-called MIN-MAX and MAX-MIN formulations. For controller-based optimization algorithms, the corresponding terms are:


${minmax}_{i = 1}^{N} \{w_{i} (φ_{i} (U(x), x) - φ_{i}^{ref})\}$ ${maxmin}_{i = 1}^{N} \{w_{i} (φ_{i} (U(x), x) - φ_{i}^{ref})\}$

For sensitivity-based algorithms, the absolute values of the corresponding terms are considered:


${minmax}_{i = 1}^{N} {w_{i} \| φ_{i} (U(x), x) - φ_{i}^{ref} \|}$ ${maxmin}_{i = 1}^{N} {w_{i} \| φ_{i} (U(x), x) - φ_{i}^{ref} \|}$

The MINMAX or MAXMIN formulations should always be used for controller-based optimization. For sensitivity-based optimization MIN or MAX is preferred because they tend to converge better and faster. Consider the remark in the next section regarding default reference value and shape optimization.