Overview of Energy Stiffness Measure

Analysis Types: Static Linear or Nonlinear Analysis

For energy stiffness 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.

For Bead and Shape, this design response is available only with Abaqus sensitivities.

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

ENERGY_STIFF_MEASURE describes a new stiffness measure without physical meaning for handling of prescribed displacement in stiffness optimization.

For stiffness optimization of structures with only external loading, the strain energy should be minimized (see Compliance (Stiffness Optimization)): $\text{min}\left(\frac{P\cdot u}{2}\right)$ where P is the external loading and u is the corresponding nodal deflections of the loaded nodes.

If a load case is driven by prescribed displacements, the elastic energy/compliance will only decrease 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: $\text{max}\left(\frac{R\cdot u^{*}}{2}\right)$ where u* are the nodal prescribed displacements being different from zero and R are the corresponding nodal reaction forces.

The physical strain energy with both external load and prescribed displacements is described as follows but might not lead to optimal stiffness results: $\frac{P\cdot u}{2}+\frac{R\cdot u^{*}}{2}$

The new stiffness measure (ENERGY_STIFF_MEASURE) combines the first two approaches in the following optimization formulation: $\text{min}\left(\frac{P\cdot u}{2}-\frac{R\cdot u^{*}}{2}\right)$

Important: Always minimize energy stiffness measure (TARGET = MIN) in the objective function independent on external loading, prescribed displacement, and thermal loading. This requires less user effort compared to total strain energy. Always apply energy stiffness measure to ALL_ELEMENTS. Energy stiffness measure can also be applied in a min-max optimization formulation (TARGET = MINMAX). The energy stiffness measure is also allowed in constraints. Thermal loading is allowed as long the thermal loading is independent on material distribution. If only external loading is present in a given load case, the energy stiffness measure (TYPE = ENERGY_STIFF_MEASURE) is equal to the total strain energy (TYPE = STRAIN_ENERGY) also called the compliance. If only prescribed displacements are present in a given load case then the energy stiffness measure (TYPE = ENERGY_STIFF_MEASURE) is equal to the minus total strain energy (TYPE = STRAIN_ENERGY) also called the compliance. If both external loading and prescribed displacements are present in a given load case, the energy stiffness measure (TYPE = ENERGY_STIFF_MEASURE) is not equal to the total strain energy (TYPE = STRAIN_ENERGY). The energy stiffness measure is not a physical energy measure. The energy stiffness measure is not available for the controller-based optimization strategy.