Adjoint Design Sensitivity Analysis

The adjoint sensitivity approach for transient dynamic problems is based on the implicit dynamic procedure in Abaqus/Standard. Some classes of problems are better suited than others for such an approach. These problems include low frequency–mode bending-dominated deformations and low-velocity crash problems (such as side crashes and other bending-dominated crashes). Other problems in this category include battery frame crashes, pedestrian head impacts, dynamics of machinery, and compliant flexible dynamic mechanisms. Typically, such problems use 30–300 increments to capture the correct physics and applying Abaqus/Standard is appropriate. Therefore, the adjoint sensitivity method for transient dynamics is applicable for these situations.

However, for high frequency–mode crash problems (for example, axial deformation and crushing), very small time increments are required to capture the correct physics. The run time for many practical applications of this latter type are not feasible for Abaqus/Standard due to a very large number of time increments. Therefore, Abaqus/Explicit is better suited to solve this class of problems. Also, Abaqus/Standard might often fail to converge when the models have significant nonlinearities (many contacts and many buckling modes). Such problems are also solved more effectively using Abaqus/Explicit.

You activate an adjoint sensitivity analysis on a step-by-step basis.

Unlike direct design sensitivity analysis (which is commonly used for parametric design and optimization), adjoint design sensitivity analysis is commonly used for nonparametric design and optimization. The design parameters are more commonly referred to as design variables in this case. Unlike direct design sensitivity analysis (where the number of design parameters is relatively small in number), an adjoint design sensitivity analysis typically has one or more design variables associated with each element in the design domain.

The specific choices for design variables depend on the type of design sensitivities that you desire, as listed in the table below. For topology and sizing sensitivities, the design variables and the corresponding design sensitivities are element-based, while for shape and bead sensitivities these quantities are node-based.

Type of sensitivity	Design variables
Topology	Relative material density
Sizing with shells	Shell element thickness
Sizing with circular section beams	Beam element radius
Shape or bead	Nodal positions

An adjoint design sensitivity analysis supports three kinds of design responses: element responses, global responses, and node responses. The valid element responses are von Mises stress, signed von Mises stress (the sign of this quantity is the same as the sign of the trace of the stress tensor), maximum principal stress, minimum principal stress, plastic strain magnitude, Neuber and Glinka plasticity correction quantities (Plasticity Corrections), a user-defined response based on either the stress or the plastic strain, and internal forces. A stress- or plastic strain–based design response represents a scalar measure calculated using a specified operator (by default, the maximum) on the values in the specified element set. Abaqus/Standard uses the stresses or the plastic strains at the element integration points to compute stress- or plastic strain–based design responses and sensitivities. The internal forces response represents a scalar measure calculated using a specified operator (by default, the sum) on the internal nodal forces over a set of specified nodes from a specified set of elements.

For a user-defined design response based on stress or plastic strain, you must define the design response in user subroutine UELEMDRESP (see Defining the Design Response in User Subroutine UELEMDRESP).

Only two types of global responses are available: stiffness and eigenfrequency. The global stiffness response, available in a static procedure, provides a measure of the stiffness of the structure that accounts for both external loading and prescribed displacements. It is defined as

where $\mathbf{\text{P}}$ , $\mathbf{\text{U}}$ , ${\mathbf{\text{R}}}_F$ , and ${\mathbf{U}}_{pres}$ refer to vectors representing external loads, displacements, reaction forces, and prescribed displacements on the structure. For a structure that has only external loading and no prescribed displacements, the second term is identically zero, and only the first term produces a nonzero contribution. The global eigenfrequency response, available in an eigenfrequency procedure, refers to one or more eigenfrequencies of the structure.

The node responses refer to quantities such as displacements or reaction forces and moments at one or more nodes in the structure.

You request design responses using a syntax analogous to that for specifying output requests to the output database. There are no default responses. If no responses are requested, then no design responses and no sensitivities are output. Specifying a response outputs both the design response value and the response sensitivities.

For internal forces and internal moments in element responses, both an element and a node set must be specified to define the design response. Each node in the specified node set receives contributions from all the elements in the specified element set that are attached to it. For element responses and node responses, if the node set contains more than one node, the design response contains the sum of the contributions from all the nodes in the node set.

By default, the sensitivity for the stress and the plastic strain magnitude responses are approximated with aggregation functions over the specified element set, where the default is an aggregation function representing the maximum. Other aggregation functions approximating the minimum and the difference between the maximum and minimum values are also available. You apply these operator types for the design response. Additional operators for elemental design responses are the sum and the average over the elemental integration points of a given element set.

A transient dynamic adjoint design sensitivity analysis supports two kinds of design responses: node responses and element responses. The node responses refer to quantities such as displacements or reaction forces and moments at one or more nodes in the structure. Most of the available responses are node responses; the only valid element response is internal rate of work (power flow) for a given element set.

You request design responses using a syntax analogous to that for specifying output requests to the output database. There are no default responses. If no responses are requested, no design responses and no sensitivities are output. Specifying a response outputs both the design response value and its sensitivities. You must specify a node set or an element set as part of a node or an element design response definition. If the set contains more than one entity (node or element), the design response contains the sum of the contributions from all of the entities in the set.

Optionally, you can specify one or more time points at which a physical quantity of interest is to be treated as a design response. If no time point is specified, the time point corresponding to the end of the step is used for the design response.