# Optimization Process

The following features are in this section.

## Minimize Objective Function

OptiStruct solves the following structural optimization problem:(1) $\mathrm{min}\left(f\left(x\right)=f\left({x}_{1},{x}_{2},...,{x}_{n}\right)\right)$

Subject to:

${g}_{j}\left(x\right)\le 0$ where $j=1,2,...,m$${x}_{i}^{L}\le {x}_{i}\le {x}_{i}^{U}$ where $i=1,2,...,n$

The objective function $f\left(x\right)$ and the functions $g\left(x\right)$ in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly ($g=0$); it is considered inactive if $g<0$; it is considered violated if $g>0$.

The selection of the vector of design variables x depends on the type of optimization being performed. In topology optimization, the design variables are element densities (Design Variables). In size optimization (including free-size), the design variables are properties of structural elements (Size (Parameter) Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (Topography Optimization and Shape Optimization).

The objective function is defined using a DESOBJ Subcase Information Entry. DESOBJ references a response defined by either the DRESP1, DRESP2, or DRESP3 Bulk Data Entry. Depending on the type of response, DESOBJ is located inside or outside of a SUBCASE. The constraints are defined using a DESSUB or DESGLB Subcase Information Entry, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD Bulk Data Entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.

## Minmax Objective Function

The minmax optimization problem is given as:(2) $\mathrm{min}\left[\mathrm{max}\left({f}_{1}\left(x\right)/{f}_{1},{f}_{2}\left(x\right)/{f}_{2},...,{f}_{K}\left(x\right)/{f}_{K}\right)\right]$
Subject to:(3) $\begin{array}{l}{g}_{j}\left(x\right)\text{ }\text{ }\text{ }j=1,\dots ,m\\ {x}_{i}^{L}\le {x}_{i}\le {x}_{i}^{U}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\dots ,n\end{array}$

Where, ${\overline{f}}_{K}$ are the reference values.

The reference values can take different values for positive or negative objective functions. These problems are solved using the Beta-method. In this method, the problem is transformed into a regular optimization problem through the introduction of an additional design variable $\beta$ such that:

min $\beta$

Subject to: (4) $\begin{array}{l}{f}_{i}\left(x\right)/{\overline{f}}_{i}\le \beta \text{ }\text{ }i=1,\dots ,k\\ {g}_{j}\left(x\right)\le 0\text{\hspace{0.17em}}\text{ }\text{ }\text{ }\text{\hspace{0.17em}}j=1,\dots ,m\end{array}$

The functions ${f}_{i}\left(x\right)$ and the functions $g\left(x\right)$ in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly ($g=0$); it is considered inactive if $g<0$; it is considered violated if $g>0$.

The selection of the vector of design variables x depends on the type of optimization being performed. In topology optimization, the design variables are element densities (Design Variables). In size optimization (including free-size), the design variables are properties of structural elements (Size (Parameter) Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (Topography Optimization and Shape Optimization).

The objective function of a minmax problem is defined using MINMAX or MAXMIN statements in the Subcase Information section. MINMAX or MAXMIN references a DOBJREF statement in the Bulk Data section, which again refers to a DRESP1, DRESP2, or DRESP3 response definition. The reference values are defined on the DOBJREF entry. The constraints are defined as stated above. The constraints are defined using a DESSUB or DESGLB Subcase Information Entry, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD Bulk Data Entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.

## System Identification

OptiStruct solves two structural optimization problems.

For system identification, following two structural optimization problems are solved:(5) $\mathrm{min}\sum _{i=1}^{n}{\left({W}_{i}\frac{{f}_{i}\left(x\right)-{T}_{i}}{{T}_{i}}\right)}^{2}$ (6)

The functions ${f}_{i}\left(x\right)$ and the functions ${g}_{j}\left(x\right)$ in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly ($g=0$); it is considered inactive if $g<0$; it is considered violated if $g>0$. The values TI are the target value for the particular response, Wi is a weighting factor.

The selection of the vector of design variables $g=0$ depends on the type of optimization being performed. In topology optimization, the design variables are element densities (Design Variables). In size optimization (including free-size), the design variables are properties of structural elements (Size (Parameter) Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (Topography Optimization and Shape Optimization).

The objective function is defined using a DESOBJ entry or a MINMAX, MAXMIN Subcase Information Entry. DESOBJ, MINMAX, or MAXMIN reference a DSYSID entry that defines target values for responses defined by either a DRESP1, DRESP2, or DRESP3 Bulk Data Entry. The constraints are defined using a DESSUB or DESGLB Subcase Information Entry, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD Bulk Data Entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.