About Constraints

Tosca Structure allows inequality constraints in all sensitivity-based algorithms. Equality constraints are allowed only in the controller-based approaches.

This means that the item EQ_VALUE defining the equality value might only be used for controller-based optimization. LE_VALUE and GE_VALUE are to be used for the upper and lower values of constraints in sensitivity-based optimization algorithms.

An equality constraint is given as:


$Ψ = φ$

where $φ$ is the value of the design response. Inequality constraints are given as:


$Ψ \leq φ$ $Ψ \geq φ$

The MAGNITUDE can be defined as ABS or REL, short for absolute or relative value of the design response in the constraint. When using the relative value the design response is normalized with respect to the initial value of the design response (design response value of optimization iteration 0).

Constraint Formulations

The following tables describe how a constraint for the respective design response is formulated. The design responses can be constrained using equality constraint, lower equal constraint, or greater equal constraint.

Important: The equality constraint is allowed only for the controller-based topology optimization. The lower equal or greater equal constraints are allowed for the sensitivity-based approach. The constraint values can be defined as absolute values or relative with respect to the corresponding values of the optimization start model.

Moreover, for the sensitivity-based approach new combinations using VAR_OPER or GROUP_OPER for the DRESP can also be applied in the constraints.


Compliance terms for constraints	Material volume terms for constraints	Eigenfrequency terms for constraints
$C_{k} \leq C_{k} $ $[C_{1} \geq C_{1} ]$	$V \leq V $ $[V \geq V ]$	$f_{k} \geq f_{k} $ $f_{1} \leq f_{1} $ $f_{k} - f_{1} \geq △ f $ $(f_{k} - f_{1} \leq △ f )$ $- \frac{1}{p} \ln (\sum_{k = 1}^{n} e^{- p f_{k}}) \geq f *$


Displacement terms for constraints	Reaction force terms for constraints	Internal force terms for constraints
$u_{i} \leq u_{i} $ $u_{i} \geq u_{i} $ $θ_{i} \leq θ_{i} $ $θ_{i} \geq θ_{i} $ $\sqrt{{u_{i}}^{2}} \leq u_{i} $ $[\sqrt{{u_{i}}^{2}} \geq u_{i} ]$ $\sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} \leq u $ $[\sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} \geq u ]$ $u_{i, 1} - u_{i, 2} \leq u_{i} $ $u_{i, 1} - u_{i, 2} \geq u_{i} $ $\sqrt{{(u_{i, 1} - u_{i, 2})}^{2}} \leq u_{i} $ $[\sqrt{{(u_{i, 1} - u_{i, 2})}^{2}} \geq u_{i} ]$ $α_{i, 1} u_{i, 1} + α_{i, 2} u_{i, 2} + ... \geq u_{i} $ $α_{i, 1} u_{i, 1} + α_{i, 2} u_{i, 2} + ... \leq u_{i} $	$R_{i} \leq R_{i} $ $R_{i} \geq R_{i} $ $M_{i} \leq M_{i} $ $M_{i} \geq M_{i} $ $\sqrt{{R_{i}}^{2}} \leq R_{i} $ $[\sqrt{{R_{i}}^{2}} \geq R_{i} ]$ $\sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}} \leq R $ $\sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}} \geq R $ $R_{i, 1} - R_{i, 2} \leq R_{i} $ $R_{i, 1} - R_{i, 2} \geq R_{i} $ $\sqrt{{(R_{i, 1} - R_{i, 2})}^{2}} \leq R_{i} $ $[\sqrt{{(R_{i, 1} - R_{i, 2})}^{2}} \geq R_{i} ]$	$F_{i} \leq F_{i} $ $F_{i} \geq F_{i} $ $M_{i} \leq M_{i} $ $M_{i} \geq M_{i} $ $\sqrt{{F_{i}}^{2}} \leq F_{i} $ $[\sqrt{{F_{i}^{}}^{2}} \geq F_{i} ]$ $\sqrt{F_{x}^{2} + F_{y}^{2} + F_{z}^{2}} \leq F $ $\sqrt{F_{x}^{2} + F_{y}^{2} + F_{z}^{2}} \geq F $ $F_{i, 1} - F_{i, 2} \leq F_{i} $ $F_{i, 1} - F_{i, 2} \geq F_{i} $ $\sqrt{{(F_{i, 1} - F_{i, 2})}^{2}} \leq F_{i} $ $[\sqrt{{(F_{i, 1} - F_{i, 2})}^{2}} \geq F_{i} ]$


Von Mises stress terms for constraints	Center of gravity terms for constraints	Moment of inertia terms for constraints
$M a x \| \frac{{(σ}_{v {M i s e s}^{) ²}}}{(f (ρ_{i}) σ_{r e f}) ²} \cdot σ_{r e f} \| \leq σ_{r e f}$	$i_{i} \leq i_{i} $ $i_{i} \geq i_{i} $	$I_{i j} \leq I_{i j} $ $I_{i j} \geq I_{i j} $


Plastic strain terms for constraints
$\sqrt{\frac{2}{3} ε^{pl} : ε^{pl}} \leq ε^{pl} $ $[\sqrt{\frac{2}{3} ε^{pl} : ε^{pl}} \geq ε^{pl} ]$

Important:

Constraints defined using relative values always refer to the design response of the start model for the optimization.
The element densities in the optimization start model might be modified compared to your original model (for example, when no volume constraint is present they are set to 50% of the original density). Take this into account when defining, for example, relative displacement or frequency constraints. This behavior can be controlled by the user with the parameter DENSITY_INITIAL in the OPT_PARAM command. For more information, see OPT_PARAM. In general, it is not recommended that you use relative constraints in a topology optimization, with an exception for volume and mass constraints.
The values marked with * are the constraint values defined by the user.
Design responses marked with ** are only allowed using Abaqus sensitivities.
Constraint expressions in [] are supported as constraint but do not apply to the standard cases.