Objective Function

Overview

The OBJECTIVE FUNCTION is the function, which value can be maximized or minimized during the optimization. This function depends on the results of the FE analysis. Therefore, the values of interest must be derived from the FE results in order to define an objective function and functional constraints. The results of the FE analysis (total displacement, energy, etc.) for determining the objective function and functional constraints are called ’design responses’. A single value from the design response must be specified for the definition of the objective function using the command DRESP (Design Response).

Minimization or Maximization of an Objective

The following tables describe which design response types are valid as the terms of 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.)

$+ α i_{i}$

$+ α I_{i j}$

In this table, the indices 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 upon results from different load cases.

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

The weight factors $α$ are defined by the user in DRESP parameter of 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_

Note:

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.

Multidisciplinary Objective (Minmax /Maxmin Formulation)

The following tables describe which terms and response types are valid 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.)

$α | i_{c} - i_{c} * |$

$α | I_{i j} - I_{i j} * |$

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.

The WEIGHT factors $α$ are defined by the user in the DRESP command in OBJ_FUNC. The weight factors can be different for each individual term in the objective function. By default the weighting factors are all set to one.

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

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

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.