About Minimizing Displacement / Rotation under Volume Constraint | ||
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Formulation of the Optimization Problem
The optimization problem can be solved with the sensitivity-based approach.
The sensitivity-based approach works with an inequality constraint, and the optimization problem is
with being the displacement in a given coordinate or the total displacement, Vol the element volume and vol_restrict the value of the volume constraint.
Necessary Definitions
The user has to define two design responses in order to set up the optimization problem:
- The design response for the displacement of the given node. The displacement in a given direction (x, y or z) or the absolute value of the displacement is chosen according to the value of TYPE parameter. For more information, see Displacement and Rotation.
- The design response for the relative volume defined as the sum of volumes of elements multiplied with their relative densities and divided through the original volume.
- The objective function is the minimization of the displacement design response. If more than one node is used in the design response definition, an individual design response is created for each node. In this case, a large number of nodes leads to many objective function terms. The target of the objective function is to minimize the nodal displacement of a single node, or, if more than one node is specified in the displacement design response, the target should be set to the minimization of the largest displacement. Please check the TOSCA.OUT file for the list of generated design responses.
- The relative material volume is used in the inequality constraint, so that the optimization results the stiffest model that has the material volume (and thus weight) less than a certain value. Without the constraint, the stiffest structure will use as much material as possible.
SIMULIA Tosca Structure Parameter File
The commands in the parameter file for this problem look like:
DRESP ID_NAME = DRESP_DISP_X DEF_TYPE = SYSTEM TYPE = DISP_X UPDATE = EVER NODE = 557 GROUP_OPER = MAX LC_SET = STATIC,2,ALL LC_SEL = MAX END_
DRESP ID_NAME = DRESP_VOL_TOPO DEF_TYPE = SYSTEM TYPE = VOLUME UPDATE = EVER EL_GROUP = ALL_ELEMENTS GROUP_OPER = SUM END_ OBJ_FUNC ID_NAME = maximize_stiffness DRESP = DRESP_DISP_X TARGET = MIN END_ CONSTRAINT ID_NAME = volume_constraint DRESP = DRESP_VOL_TOPO MAGNITUDE = REL LE_VALUE = 0.45 END_ OPTIMIZE ID_NAME = topology_optimization DV = design_variables OBJ_FUNC = maximize_stiffness CONSTRAINT = volume_constraint STRATEGY = TOPO_SENSITIVITY END_
The following example deals with the minimization of displacements of more than one node:
DRESP ID_NAME = DRESP_DISP_X_1 DEF_TYPE = SYSTEM TYPE = DISP_X UPDATE = EVER NODE = 557 GROUP_OPER = MAX LC_SET = STATIC,2,ALL LC_SEL = MAX END_
DRESP ID_NAME = DRESP_DISP_X_2 DEF_TYPE = SYSTEM TYPE = DISP_X UPDATE = EVER NODE = 1997 GROUP_OPER = MAX LC_SET = STATIC,1,ALL LC_SEL = MAX END_ DRESP ID_NAME = DRESP_VOL_TOPO DEF_TYPE = SYSTEM TYPE = VOLUME UPDATE = EVER EL_GROUP = ALL_ELEMENTS GROUP_OPER = SUM END_ OBJ_FUNC ID_NAME = maximize_stiffness DRESP = DRESP_DISP_X_1 DRESP = DRESP_DISP_X_2 TARGET = MINMAX END_ CONSTRAINT ID_NAME = volume_constraint DRESP = DRESP_VOL_TOPO MAGNITUDE = REL LE_VALUE = 0.45 END_ OPTIMIZE ID_NAME = topology_optimization DV = design_variables OBJ_FUNC = maximize_stiffness CONSTRAINT = volume_constraint STRATEGY = TOPO_SENSITIVITY END_