Minimizing Compliance under Volume Constraint

This standard optimization task corresponding to maximizing the stiffness is defined as the minimization of the compliance that is the reciprocal value of the stiffness.

Before you begin:

The optimization problem can be solved with the controller-based approach (which needs about 15 iterations to solve the problem) and with the sensitivity approach (where the number of iterations is not previously known).

The controller-based approach works with equality constraints, so that the problem looks like:

min⁡(∑i=1,nU)

∑i=1,n   Vol⁢=vol_restrict

with U being the strain energy, Vol the element volume and vol_restrict the value of the volume constraint.

  1. Define two design responses in order to set up the optimization problem:
    • The design response for the sum of the strain energy over all elements (if not all elements are selected, then the problem will not represent the maximization of the total stiffness).
    • 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.
  2. Use the strain energy as an objective function term.

  3. The answer to the question whether to minimize or to maximize the sum of the strain energy depends on the loading types and boundary conditions, respectively:
    • If the loads for the model are applied as external forces or pressure, then the objective function must be minimized.
    • If only prescribed displacements are assigned and no external forces, then the objective function must be maximized.
    • If simultaneously prescribed displacements and external loadings are assigned, a new energy stiffness measure ENERGY_STIFF_MEASURE is available for stiffness optimization using sensitivity-based topology optimization.

    For more information, see Energy Stiffness Measure.

  4. Use the relative material volume in the equality constraint.
    This leads to an optimization resulting the stiffest model that has the specified material volume (and thus weight). Without the constraint, the stiffest structure will use as much material as possible.

Important: The same optimization task can be solved using the sensitivity-based topology optimization. Then, the relative material volume constraint should be set to "less or equal", that is, the LE_VALUE parameter should be used instead of EQ_VALUE parameter.

SIMULIA Tosca Structure Parameter File

The commands in the parameter file for this problem look like:


DRESP
 ID_NAME        = DRESP_SUM_ENERGY
 DEF_TYPE       = SYSTEM
 TYPE           = STRAIN_ENERGY
 UPDATE         = EVER
 EL_GROUP       = ALL_ELEMENTS
 GROUP_OPER     = SUM
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_SUM_ENERGY
 TARGET         = MIN
END_

CONSTRAINT
 ID_NAME        = volume_constraint
 DRESP          = DRESP_VOL_TOPO
 MAGNITUDE      = REL
 EQ_VALUE       = 0.45
END_

OPTIMIZE
 ID_NAME        = topology_optimization
 DV             = design_variables
 OBJ_FUNC       = maximize_stiffness
 CONSTRAINT     = volume_constraint
 STRATEGY       = TOPO_CONTROLLER
END_