Multidisciplinary Objectives (Minmax and Maxmin Formulations)

It is possible to minimize the maximum term or to maximize the minimum term referenced in the objective function. In such cases, the value of the objective function is generated with a maximum or a minimum function over a set of terms defined by Design Responses.

If the maximum objective term should be minimized, the definition is as follows:


OBJ_FUNC
 ID_NAME = ...
 DRESP   = ...
 DRESP   = ...
 ...
 TARGET  = MINMAX
END_

If the minimum objective term should be maximized, the definition is as follows:


OBJ_FUNC
 ID_NAME = ...
 DRESP   = ...
 DRESP   = ...
 ...
 TARGET  = MAXMIN
END_

In both cases, the DRESP definitions are referring to the desired design responses for the objective function using the ID name of the defined responses. The defined responses that should be minimized or maximized must be valid design responses.

Multidisciplinary Objective (Minmax /Maxmin Formulation)

The following tables describe how the terms of design responses are handled in the Min-Max / Max-Min formulation:


Compliance terms in Min-Max/Max-Min	Material volume terms in Min-Max/Max-Min	Eigenfrequency terms in Min-Max
$\| α (C_{k} - C_{k} *) \|$	$\| α (V - V *) \|$	$\| α \frac{1}{f_{k} - f_{k} *} \|$ Not for Max-Min formulation.


Displacement terms in Min-Max/Max-Min	Reaction force terms in Min-Max/Max-Min
$\| α (u_{i} - u_{i} ) \|$ $\| α (θ_{i} - θ_{i} ) \|$ $\| α (\sqrt{u_{i}^{2}} - u_{i} ) \|$ $\| α (\sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} - u ) \|$ $\| α ((u_{i, 1} - u_{i, 2}) - u_{i} ) \|$ $\| α (\sqrt{(u_{i, 1} - u_{i, 2}) ²} - u_{i} ) \|$	$\| α (R_{i} - R_{i} ) \|$ $\| α (M_{i} - M_{i} ) \|$ $\| α (\sqrt{R_{i}^{2}} - R_{i} ) \|$ $\| α (\sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}} - R ) \|$ $\| α ((R_{i, 1} - R_{i, 2}) - R_{i} ) \|$ $\| α (\sqrt{(R_{i, 1} - R_{i, 2}) ²} - R_{i} ) \|$


Von Mises stress terms in Min-Max/Max-Min	Center gravity terms in Min-Max/Max-Min	Moment of inertia terms in Min-Max/Max-Min
$+ α M a x \| \frac{{(σ}_{v {M i s e s}^{) ²}}}{(f (ρ_{i}) σ_{r e f}) ²} \cdot σ_{r e f} \|$ (Constant temperature loading is supported.)	$α \| i_{c} - i_{c} * \|$	$α \| I_{i j} - I_{i j} * \|$


Plastic strain terms in objective**	Thermal terms in objective**
$\| α (\sqrt{\frac{2}{3} ε^{pl} : ε^{pl}} - ε^{pl} *) \|$	$\| α (c_{T} - c_{T} ) \|$ $\| α (T - T ) \|$ $\| α (q - q ) \|$ $\| α (\sum_{e} \sum_{n} Q_{e,n} - q - q ) \|$

where the index i and j are indicating one of the x-y-z-directions. The compliance, eigenfrequencies, displacements, and reaction forces can be from different load cases.

The REFERENCE values marked with "*" are defined by the use in the DRESP command in OBJ_FUNC. Normally, the reference values are zero. Hence, by default all reference values are set to zero. The reference can be set individually for each term in the objective function.

Multiple Objective Termns (Weighting Factors)

You define the weighting factors $α$ in the respective DRESP command item in OBJ_FUNC. The weighting factors can be different for each individual term in the objective function. By default, the weighting factors are all set to one, so that the individual objective terms are taken into account with their actual value (e.g. stresses in the range of 100 MPa and frequencies in GHz).

The reference value and weight factor can be changed, for example, to 2.5 and 14.0 for response id_dresp in the following way:


OBJ_FUNC
 ...
 DRESP = id_dresp, 2.5, 14.0
 ...
END_

Important:

The main difference between the scheme in Minimization or Maximization of an Objective and the present scheme is the way the frequency terms are defined in the objective.
A minimization optimization task can be converted into a maximization optimization task (or vice versa) by changing the sign of the weighting factors in the definition of the objective function.
Design responses marked with ** are allowed only using Abaqus sensitivities.