Measure for Intermediate Densities

The DRESP DENSITY_MEASURE is defined in the following way:


$\frac{4}{Volume} \int_{Volume} ρ (1 - ρ) d v \approx \frac{4}{N_{e l}} \sum_{e = 1}^{N_{e l}} ρ_{e} (1 - ρ_{e})$

where $ρ_{e}$ is the relative density in each element of the given user-defined element group and $N_{e l}$ is the number of elements in the defined element group, respectively.

Analysis-Independent Design Response

For the density measure the following table shows the allowed combinations between the strategy and the items OBJ_FUNC and CONSTRAINT with C for controller and S for sensitivity-based optimization.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC	S
CONSTRAINT	S

The DRESP named DENSITY_MEASURE can be applied for measuring the amount of intermediate densities. The DENSITY_MEASURE can be used in the OBJ_FUNC or CONSTRAINT for minimizing and surpassing the number of elements containing intermediate densities. This design response can be used for additional penalization of intermediate elements and obtain more clearly defined solid/void conceptual designs.

DENSITY_MEASURE is aimed at advanced dynamic applications like modal eigenfrequency analysis and frequency response analysis.


intermediate density of all elements	value of `DENSITY_MEASURE`
0.5	1.0
1.0	0.0
$ρ_{\min}$	$4 ρ_{\min} (1 - ρ_{\min}) \approx 4 ρ_{\min}$

DRESP DENSITY_MEASURE yields the value 1.0 for the intermediate density 0.5 of all elements.

DRESP DENSITY_MEASURE yields the value 0.0 for the intermediate density 1.0 of all elements.

DRESP DENSITY_MEASURE yields the value $4 ρ_{\min} (1 - ρ_{\min}) \approx 4 ρ_{\min}$ for the intermediate minimum density $ρ_{\min}$ of all elements.

The following figure shows the DRESP DENSITY_MEASURE depending on element density with all elements having the same intermediate density.

Command in Parameter File

The DRESP DENSITY_MEASURE is defined in the following way:


DRESP
 ID_NAME    = DRESP_DENSITY_MEASURE
 DEF_TYPE   = SYSTEM
 TYPE       = DENSITY_MEASURE
 EL_GROUP   = ………
 GROUP_OPER = SUM
END_

The DRESP DENSITY_MEASURE can then be applied in the objective function or in the constraint in the following way:


OBJ_FUNC>
 ID_NAME = DENSITY_MEASURE_ONJ
 DRESP   = ....
 DRESP   = DRESP_DENSITY_MEASURE, <weighting_factor>
 TARGET  = MIN
END_


CONSTRAINT
 ID_NAME   = DENSITY_MEASURE_constraint
 DRESP     = DRESP_DENSITY_MEASURE
 MAGNITUDE = ABS
 LE_VALUE  = 0.05
END_

The design response DENSITY_MEASURE drives the optimization towards a local minimum and therefore needs to be used with care.

Important:

The DRESP DENSITY_MEASURE is only for advanced applications. In particular, it is aimed at advanced dynamic applications like modal eigenfrequency analysis and frequency response analysis.
The weighting factor must be carefully chosen when applying the DRESP DENSITY_MEASURE in the objective function. Otherwise, only one of the objective terms would be optimized.
The constraint value must be carefully chosen when applying the DRESP DENSITY_MEASURE as constraint. Otherwise, there might be no feasible solution.
The constraint value for DRESP DENSITY_MEASURE cannot be lower than $4 \cdot \min d e n s i t y$ (DENSITY_LOWER) , which is default $4 \cdot 0.001 \cdot (1 - 0.001) \approx 0.004$ .
Usually, the DRESP DENSITY_MEASURE yields a significant higher number of optimization iterations.