Adjoint design sensitivity analysis for static procedures

Elements tested

C3D4H
C3D6H
C3D8
C3D8H
C3D8R
C3D10H
C3D10HS
C3D10MH
C3D15H
C3D20H

CPE4H
CPE4RH
CPE6H
CPE6M
CPE6MH
CPE8H
CPE8R
CPE8RH

CAX4
CAX4H
CAX6
CAX6H
CAX6M
CAX6MH
CAX8
CAX8H
CAX8R

CGAX3
CGAX3H
CGAX4H
CGAX4
CGAX6
CGAX6H
CGAX8
CGAX8H
CGAX8R

Features tested

This section includes a general set of simple tests to verify the adjoint design sensitivity analysis techniques for shell and solid continuum element types for a static procedure. The tests include general steps that are either geometrically linear or nonlinearly modeled, both with and without constitutive material nonlinearities. Linear perturbation steps with multiple load cases are also tested. A full range of design variables are used, including those related to topology, sizing, and shape/bead sensitivities. A number of different design responses, including user-defined element design responses based on either the stress or the plastic strain, are also tested.

Problem description

All tests are either one or four element models with elastic or elastic-plastic constitutive material properties. The models are loaded using either concentrated loads, distributed body loads, or prescribed temperature. The design parameters are one of the following: shell element thickness (for sizing sensitivities); nodal position (for shape/bead sensitivities); or material relative density that scales the constitutive material properties, physical density, and stress relaxation (for topology sensitivities).

Results and discussion

All adjoint sensitivity results are verified by comparing to results obtained using finite difference calculations.

Input files

S4_static_DV_Node.inp: Bead sensitivities for S4 elements with nodal positions as design variables.
S4_static_DV_Node_Oper.inp: Bead sensitivities for S4 elements with nodal positions as design variables. Uses operators in defining node and element design responses.
S4_static_DV_Thick.inp: Sizing sensitivities for S4 elements with shell thickness as design variables.
S4_static_DV_Thick_Oper.inp: Sizing sensitivities for S4 elements with shell thickness as design variables. Uses operators in defining node and element design responses.
S4_static_DV_Stiff.inp: Topology sensitivities for S4 elements with relative material density as design variables.
S4_static_DV_Stiff_Oper.inp: Topology sensitivities for S4 elements with relative material density as design variables. Uses operators in defining node and element design responses.
C3D10HS_static_LC_DV_Node.inp: Shape sensitivities for a multiple load case analysis with C3D10HS elements; nodal position as design variables.
C3D10HS_static_LC_DV_Stiff.inp: Topology sensitivities for a multiple load case analysis with C3D10HS elements; relative material density as design variables.
S4R_static_DV_Node_User_S.inp: Bead sensitivities for S4R elements with nodal position as design variables. Stress-based user-defined element response defined using user subroutine UELEMDRESP.
S4R_static_DV_Stiff_User_S.inp: Topology sensitivities for S4R elements with relative material density as design variables. Stress-based user-defined element response defined using user subroutine UELEMDRESP.
S4R_static_DV_Thick_User_S.inp: Sizing sensitivities for S4R elements with shell thickness as design variables. Stress-based user-defined element response defined using user subroutine UELEMDRESP.
Uelemdresp_S.f: User subroutine UELEMDRESP to define stress-based element design response.
C3D8R_static_DV_Node_User_PE.inp: Shape sensitivities for C3D8R elements with nodal position as design variables. Plastic strain–based user-defined element response defined using user subroutine UELEMDRESP.
C3D8R_static_DV_Stiff_User_PE.inp: Topology sensitivities for C3D8R elements with relative material density as design variables. Plastic strain–based user-defined element response defined using user subroutine UELEMDRESP.
Uelemdresp_PE.f: User subroutine UELEMDRESP to define plastic strain–based element design response.
C3D8_static_DV_Node_SINV.inp: Shape sensitivities of principal stresses and signed von Mises stress responses for C3D8 elements with nodal positions as design variables.
C3D8_static_DV_Stiff_SINV.inp: Topology sensitivities of principal stresses and signed von Mises stress responses for C3D8 elements with relative material density as design variables.
S4R_static_DV_Stiff_Neuber_Table.inp: Topology sensitivities of Neuber plasticity correction factors–based design responses for S4R elements with relative material density as design variables. The material data for the plasticity corrections are defined in tabular form.
S4R_static_DV_Node_Neuber_Table.inp: Bead sensitivities of Neuber plasticity correction factors–based design responses for S4R elements with nodal positions as design variables. The material data for the plasticity corrections are defined in tabular form.
S4R_static_DV_Thick_Neuber_Table.inp: Sizing sensitivities of Neuber plasticity correction factors–based design responses for S4R elements with shell thickness as design variables. The material data for the plasticity corrections are defined in tabular form.
S4R_static_DV_Stiff_NeuberGlinka_RO.inp: Topology sensitivities of Neuber and Glinka plasticity correction factors–based design responses for S4R elements with material relative density as design variables. The material data for the plasticity corrections are defined using the Ramberg-Osgood model.
S4R_static_DV_Node_NeuberGlinka_RO.inp: Bead sensitivities of Neuber and Glinka plasticity correction factors–based design responses for S4R elements with nodal positions as design variables. The material data for the plasticity corrections are defined using the Ramberg-Osgood model.
S4R_static_DV_Thick_NeuberGlinka_RO.inp: Sizing sensitivities of Neuber and Glinka plasticity correction factors–based design responses for S4R elements with shell thickness as design variables. The material data for the plasticity corrections are defined using the Ramberg-Osgood model.
C2D_static_DV_Stiff_H_M.inp: Topology sensitivities of various design responses for a number of 2D elements with regular, hybrid, modified, and modified-hybrid formulations, with material relative density as design variables.
CAX_static_DV_Stiff_H_M.inp: Topology sensitivities of various design responses for a number of axisymmetric elements with regular (that is, nonhybrid or nonmodified), hybrid, modified, and modified hybrid formulations, with material relative density as design variables.
CGAX_static_DV_Stiff_H.inp: Topology sensitivities of various design responses for a number of 2D axisymmetric elements with twist, with material relative density as design variables. Both regular and hybrid elements were tested.
C3D_static_DV_Stiff_H_M.inp: Topology sensitivities of various design responses for a number of 3D hybrid and modified elements, with material relative density as design variables.
C3D10MH_static_DV_Node_Hyperelas.inp: Shape sensitivities of various design responses for a C3D10MH element with hyperelastic material behavior, with nodal positions as design variables.
CPE6MH_static_DV_Node_Pert.inp: Shape sensitivities of various design responses for a CPE6MH element in a linear perturbation procedure, with nodal positions as design variables.
CAX6H_static_DV_Node_Linear.inp: Shape sensitivities of various design responses for a CAX6H element in a linear procedure, with nodal positions as design variables.