Minimization or Maximization of an Objective Function

The optimization formulation consists of an objective function and a set of constraints as shown in the first equation of Mathematical Formulation.

The objective can be minimized or maximized using the MIN and MAX in the TARGET parameter of the OBJECTIVE command, respectively. In these cases, the values for the objective function defined by the DRESPs are summed up.

For example, if the objective should be minimized (or maximized),


OBJ_FUNC
 ID_NAME = ...
 DRESP   = ...
 DRESP   = ...
 ...
 TARGET  = MIN (or MAX)
END_

where the DRESP definitions are referring to the desired 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.

Minimization or Maximization of an Objective

The following tables describe how design responses are handled in an objective function that is to be minimized or maximized:


Compliance terms for objective	Material volume terms for objective	Eigenfrequency terms for objective
$+ α c_{k}$	$+ α (V - V *)$	$+ α f_{k}$ $- \frac{α}{k} \ln (\sum_{j = 1}^{n} e^{- k f_{j}})$


Displacement terms for objective	Reaction force terms for objective	Internal force terms for objective
$+ α u_{i}$ $+ α θ_{i}$ $+ α \sqrt{u_{i}^{2}}$ $+ α \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}}$ $+ α (u_{i, 1} - u_{i, 2})$ $+ α \sqrt{{(u_{i, 1} - u_{i, 2})}^{2}}$	$+ α R_{i}$ $+ α M_{i}$ $+ α \sqrt{R_{i}^{2}}$ $+ α \sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}}$ $+ α (R_{i, 1} - R_{i, 2})$ $+ α \sqrt{{(R_{i, 1} - R_{i, 2})}^{2}}$	$+ α F_{i}$ $+ α M_{i}$ $+ α \sqrt{F_{i}^{2}}$ $+ α \sqrt{F_{x}^{2} + F_{y}^{2} + F_{z}^{2}}$ $+ α (F_{i, 1} - F_{i, 2})$ $+ α \sqrt{{(F_{i, 1} - F_{i, 2})}^{2}}$


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


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

In this table the indexes i and j indicate a direction of an axis (x, y, or z) of the global or a local coordinate system. The compliance, eigenfrequencies, displacements, and forces can be based on results from different load cases.

The reference values marked with "*" are defined in the DRESP parameter of the OBJ_FUNC command. By default, the reference values are zero. The reference values can be set individually for each term in the objective function.

You define the weight factors $α$ in the DRESP parameter of the OBJ_FUNC command. The weight factors are set individually for each term of the objective function. By default, the weight factors are set to 1.

In the example below, the weight factor is set to 1.5 and the reference value is set to 4.0:


OBJ_FUNC
...
DRESP = id_dresp, 1.5, 4.0
...
END_

Important:

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.