# Stress Responses for Topology and Free-Size Optimization

The following approaches are available to handle stress constraints for Topology and Free-size Optimization.

Note: Augmented Lagrange Method is only supported for Topology Optimization.

## Norm-based Approach (Topology and Free-size Optimization)

The Norm-based approach is the default method for handling stress responses for Topology and Free-Size Optimization. This method is used when corresponding stress response RTYPE’s on the DRESP1 Bulk Data Entry are input.

The Response-NORM aggregation is internally used to calculate the stress responses for groups of elements in the model. Solid, shell, bar stresses and solid corner stresses are supported with the Response-Norm aggregation approach (Free-size Optimization is only supported for shell stresses). Refer to NORM Method for more information.

## Augmented Lagrange Method (ALM) (Topology Optimization only)

The Augmented Lagrange Method (ALM) is an alternative method for handling stress responses for Topology Optimization. It can be activated using DOPTPRM,ALMTOSTR,1 when DRESP1 Bulk Data Entry is used to specify local stress responses.

ALM is also an alternative method to efficiently solve topology optimization problems with local stress constraints, which is stated in Equation 1.

$\mathrm{min}f\left(X\right)$ (1)
$s.t.\text{\hspace{0.17em}}{g}_{j}\left(X\right)\le 0,j=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{1}$

${g}_{j}\left(X\right)\le 0,j={m}_{1}+1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{1}$

Where,
$X$
Vector of topology design variables
$f\left(X\right)$
Objective function
${g}_{j}\left(X\right)$
jth constraint
${m}_{1}$
Number of local stress constraints
$m$
Total number of constraints
Typically, ${m}_{1}$ is equivalent to the number of elements and is a large number. ALM converts the optimization Equation 1 to the following format:(2)
(3)EQ 2
$s.t.\text{\hspace{0.17em}}{g}_{j}\left(X\right)\le 0,j={m}_{1}+1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{1}$
Where,
${h}_{j}\left(X\right)=max\left[{g}_{j}\left(X\right),-\frac{{\lambda }_{j}}{\mu }\right]$
${g}_{j}\left(X\right)={E}_{j}\left(\frac{{\sigma }_{j}}{{\sigma }_{lim}}-1\right)\left[1+{\left(\frac{{\sigma }_{j}}{{\sigma }_{lim}}-1\right)}^{2}\right]$
${E}_{j}=\epsilon +\left(1-\epsilon \right){\left[\frac{tanh\left(\beta \eta \right)+tanh\left(\beta \left({\rho }_{j}-\eta \right)\right)}{tanh\left(\beta \eta \right)+tanh\left(\beta \left(1-\eta \right)\right)}\right]}^{p}$
${\rho }_{j}$
Element density
${\sigma }_{j}$
Element von Mises stress
${\sigma }_{\mathrm{lim}}$
stress upper bound
${\lambda }_{j}$
Lagrange multiplier estimator
$\mu >0$
The penalty factor is typically updated as:(4)
${\mu }^{\left(k+1\right)}=min\left(\alpha {\mu }^{\left(k\right)},{\mu }_{max}\right)$
Where,
$\alpha >1$
Update parameter
${\mu }_{\mathrm{max}}$
Upper limit to prevent numerical instabilities

The Lagrange multiplier estimators are updated as ${\lambda }_{j}^{\left(k+1\right)}={\lambda }_{j}^{\left(k\right)}+{\mu }^{\left(k\right)}{h}_{j}\left(X\right)$ .

In general, the number of local stress constraints ${m}_{1}$ is very large. Directly solving Equation 1 is computationally time consuming. By penalizing the stress constraints onto the objective function, the total number of constraints can be significantly reduced. As a result, the optimization problem can be efficiently solved.

The parameters are set as ${\mu }^{\left(0\right)}=10$ , ${\mu }_{\mathrm{max}}={10}^{6}$ , $\alpha =1.3$ , $\eta =0.5$ , $\epsilon ={10}^{-6}$ , $p=3$ . $\beta$ is initially set to 1.0 and multiplied by 1.3 in every five iterations, with an upper limit of 30.0. Based on this process, Topology Optimization is carried out within one phase.
Note: Except for this ALM, OptiStruct generally uses a multi-phase strategy to solve topology optimization problems.

The default Stress-norm (P-norm) method continues to efficiently solve Topology Optimization problems with local stress constraints. The ALM 1 is a good alternative for such models.

## Global von Mises Stress Response (Topology and Free-size Optimization)

The von Mises stress constraints may be defined for topology and free-size optimization through the STRESS optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint:
• The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, that is, the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space.
• Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards.
• The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization.
• Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh.
• Stress constraints do not apply to 1D elements.
• Stress constraints may not be used when enforced displacements are present in the model.

The buckling factor can be constrained for shell topology optimization problems with a base thickness not equal to zero. Constraints on the buckling factor are not allowed in any other cases of topology optimization.

The following responses are currently available as the objective or as constraint functions for elements that do not form part of the design space:
 Composite Stress Composite Strain Composite Failure Criterion Frequency Response Stress Frequency Response Strain Frequency Response Force