Overview of Compliance

Analysis Types: Static Linear or Nonlinear Analysis

$Ku=F$

where K might be linear or nonlinear.

For compliance, the following table shows the allowed combinations between the strategy and the items OBJ_FUNC and CONSTRAINT.

	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC	C*,S*	C**, S	C,S	S
CONSTRAINT	S*	S	S	S

Where C indicates that compliance design responses are usable for controller-based optimization and S indicates that it is usable for sensitivity-based optimization. C* and S* say that nonlinearities as well as temperature loading is allowed. For STRAIN_ENERGY_DENSITY, only the entries with ** are relevant.

Compliance has a large popularity within scientific publications, and compliance is of large importance in engineering applications even though the expression might not be widely known outside the optimization community. Compliance might be expressed as the overall flexibility or “softness” of a structure given by the sum of elastic or strain energy in a structure. Thus, compliance can be seen as a stiffness measure or more correctly the reciprocal of stiffness.

To maximize the global stiffness, we therefore minimize compliance. Compliance is defined in Tosca Structure by the sum of strain energy of all elements.

Remarks:

1. In certain cases, including prescribed displacements or thermal fields "minimizing compliance" results in a stiff structure. If a load case is driven by prescribed displacements or a thermal field, the elastic energy / compliance only decreases if the structure is made softer. If only prescribed displacements are present without external loading, the strain energy should be maximized to obtain optimal results: $\text{max}\left(\frac{R\cdot u^{*}}{2}\right)$ with R = reaction force and u* = nodal prescribed displacements. If both external loading and prescribed displacements are present, a new energy stiffness measure is introduced. The "standard" strain energy does not lead to optimal results: $\frac{P\cdot u}{2}+\frac{R\cdot u^{*}}{2}$ with P = external loading and u = corresponding nodal deflections of the loaded nodes.
2. Compliance is equal to the overall strain energy. Therefore, the strain energy for all elements is required. Any other element group is not allowed, because in such cases the optimization problems are not self-adjoint.
3. If no LC_SET is specified (no load case is selected from the existing load cases) Tosca Structure always reads the last substep for each load case in case of nonlinear loading.