Minimizing the Volume under Displacement Constraint

Formulation of the Optimization Problem

If the structure with the minimal volume (weight) subject to displacement constraints (corresponding to a restriction on the mechanical stiffness) is sought, the optimization task is formulated as follows:

where Vol is the relative material volume of an element in the design area, $u_1$ is the nodal displacement and $u_{restrict,j}$ is the restriction for the nodal displacement of the node.

This example shows a model for the minimization of the relative material volume under the displacement of the loaded node:

Necessary Definitions

Define two design responses:

• The first design response is the design response representing the relative material volume of the design area. This design response is the objective function that is to be minimized.
• The second design response is the displacement of the loaded node. The absolute displacement is used in this case. Alternatively, the displacement in the direction of the load (in the example it is z-direction of a local coordinate system) can be used. The restriction to one direction reduces the number of load case generated by the optimization system because using absolute displacement leads to the need of 3 additional load case in order to calculate the sensitivities while the displacement in z-direction only requires one additional pseudo load case for the calculation of the sensitivities. The constraint is then defined as an inequality constraint with an absolute value for the displacement value.

Result and Convergence

The objective function (relative material volume) is pictured in the following figure.

The displacement constraint value plot is presented in the next figure

The last figure presents the result of the optimization with displacement constraint.

SIMULIA Tosca Structure Parameter File

The commands in the parameter file look like follows:

DRESP
 ID_NAME       = DRESP_VOLUME
 DEF_TYPE      = SYSTEM
 EL_GROUP      = design_elem
 GROUP_OPER    = Sum
 TYPE          = VOLUME
END_

OBJ_FUNC
 ID_NAME        = MINIMIZE_VOLUME
 DRESP          = DRESP_VOLUME, ,  
 TARGET         = MIN
END_

DRESP
 ID_NAME        = MAX_DISPLACEMENT
 TYPE           = DISP_ABS
 DEF_TYPE       = SYSTEM
 GROUP_OPER     = Max
 NODE           = 300
END_

CONSTRAINT
 ID_NAME         = DISPLACEMENT_CONSTRAINT
 DRESP           = MAX_DISPLACEMENT
 MAGNITUDE       = ABS
 LE_VALUE       = 8.5
END_