Objective Function for Sensitivity-Based Shape Optimization

Overview

The OBJECTIVE FUNCTION is the function, which value can be maximized or minimized during the optimization. This function depends on the results of the FE analysis. Therefore, the values of interest must be derived from the FE results to define an objective function and functional constraints. The results of the FE analysis (total displacement, energy, etc.) for determining the objective function and functional constraints are called ’design responses’. A single value from the design response must be specified for the definition of the objective function using the command DRESP(Design Response).

Objective Function Terms

Tosca Structure.shape controller allows optimization on different stress hypotheses, strain formulations, and damage results. The most used equivalent stress is von Mises (SIG_MISES).

The supported types of design responses usable in an objective function for sensitivity-based shape optimization are listed in the table below.


Static analysis	Description
CENTER_GRAVITY_X CENTER_GRAVITY_Y CENTER_GRAVITY_Z	Center of gravity design responses
DISP_ABS DISP_X DISP_Y DISP_Z DISP_X_ABS DISP_Y_ABS DISP_Z_ABS	Displacement design responses
INERTIA_XX INERTIA_XY INERTIA_XZ INERTIA_YY INERTIA_YZ INERTIA_ZZ	Moment of inertia design responses
INTERNAL_FORCE_ABS INTERNAL_FORCE_X INTERNAL_FORCE_Y INTERNAL_FORCE_Z INTERNAL_FORCE_X_ABS INTERNAL_FORCE_Y_ABS INTERNAL_FORCE_Z_ABS	Internal force design responses
INTERNAL_MOMENT_X INTERNAL_MOMENT_Y INTERNAL_MOMENT_Z INTERNAL_MOMENT_X_ABS INTERNAL_MOMENT_Y_ABS INTERNAL_MOMENT_Z_ABS	Internal moment design responses
PEMAG	Plastic strain magnitude design response**
REACTION_FORCE_ABS REACTION_FORCE_X REACTION_FORCE_Y REACTION_FORCE_Z REACTION_FORCE_X_ABS REACTION_FORCE_Y_ABS REACTION_FORCE_Z_ABS	Reaction force design responses
REACTION_MOMENT_X REACTION_MOMENT_Y REACTION_MOMENT_Z REACTION_MOMENT_X_ABS REACTION_MOMENT_Y_ABS REACTION_MOMENT_Z_ABS	Reaction moment design responses
ROT_ABS ROT_X ROT_Y ROT_Z ROT_X_ABS ROT_Y_ABS ROT_Z_ABS	Rotation design responses
STRAIN_ENERGY
SIG_1	Maximum principal stress.
SIG_3	Minimum principal stress.
SIG_MISES SIG_SENS_MISES	Von Mises Stress design responses For `SIG_MISES`, `SIG_SENS_MISES` is used for sensitivity calculation.
SIG_SIGNED_MISES	Signed von Mises stress failure criteria.
SIG_GLINKA_EEQ SIG_GLINKA_PEEQ SIG_GLINKA_SEQ SIG_NEUBER_EEQ SIG_NEUBER_PEEQ SIG_NEUBER_SEQ	Glinka and Neuber formulations for equivalent strain (_EEQ), stress (_SEQ) and plastic strain (_PEEQ) using the plastic correction factor, respectively**.
STRAIN_ENERGY	Strain energy design responses
WEIGHT	Weight design response
Modal analysis	Description
DYN_FREQ	Dynamic frequency design response
Analysis independent	Description
VOLUME	Volume design response

Additional design response types


Modal analysis	Description
DYN_FREQ_KREISSEL	Dynamic frequency design response

Note:

Stress responses (*SIG*) are only supported by Tetrahedron 4 and 10 as well as Hexahedron 8. This does not hold if solver sensitivities are used. Refer the related topics.
GBL_SIG_MISES_SENS responses are based on element groups.
SIG_* generates a pseudo load for EACH element.
Design responses marked with ** are only allowed using Abaqus sensitivities.

Differences in Objective Target Formulation

The objective formulation for the optimization varies depending on the objective target such as minimization/ maximization or a Min-Max/ Max-Min definition. For further details and formulas, see Minimization or Maximization of an Objective Function and Multidisciplinary Objectives (Minmax and Maxmin Formulations).