Direct Design Sensitivity Analysis

The design sensitivity analysis (DSA) capability provides the derivatives of certain output variables with respect to specified design parameters. These derivatives are commonly referred to as sensitivities, because they provide a first-order measure of how sensitive the output variable is to a change in the design parameter. The output variables for which sensitivities are computed are called design responses or simply responses. Design parameters are chosen from a set of existing analysis parameters. As an example, you can choose to obtain the derivatives of stresses with respect to Young's modulus; stress is the response, and Young's modulus is the design parameter. The sensitivities are computed based on the direct differentiation method used in conjunction with the semi-analytical computational technique. In the semi-analytical technique some derivatives are computed using numerical (finite) differencing, thus requiring perturbations of the design parameters. For these derivatives by default Abaqus/Design will use a central differencing scheme and automatically determine appropriate perturbation sizes based on a heuristic algorithm. You can override these defaults by specifying the numerical differencing method and the perturbation sizes directly. A full discussion of DSA theory is given in About design sensitivity analysis.

You activate DSA on a step-by-step basis.

STEP, DSA=YES

You can define multiple parameters to be used in place of Abaqus input quantities for an analysis. You must indicate which of these parameters are to be considered as design parameters.

PARAMETER
par1=x
par2=y
$\mathrm{\dots }$

DESIGN PARAMETER
par1, par2, $\mathrm{\dots }$

Response requests are specified using a syntax analogous to that for specifying output requests to the output database. Except for eigenvalues and eigenfrequencies, there are no default responses—if no responses are requested, no response sensitivities will be output. If DSA is active in a frequency step, eigenvalue and eigenfrequency sensitivities will be output automatically. Specifying a response will cause output of both the response and the response sensitivities.

DESIGN RESPONSE, FREQUENCY=interval, MODE LIST
CONTACT RESPONSE, MAIN=main name, NSET=nset name, 
SECONDARY=secondary name
ELEMENT RESPONSE, ELSET=elset name
NODE RESPONSE, NSET=nset name

The DSA calculations require the gradients of the design-dependent input data with respect to the design parameters. For example, if Poisson's ratio, $\nu$, is made dependent on a design parameter, say h, the gradient $d\nu /dh$ is required. Design gradients with respect to shape design parameters are specified differently than those with respect to other design parameters.

Both total and incremental formulations are implemented for DSA. The choice of formulation depends on whether or not an analysis is history dependent. Below is a brief description of these formulation types. A more detailed discussion can be found in About design sensitivity analysis. By default, the incremental DSA formulation is used. You can specify the DSA formulation only for the entire model; this specification is ignored if given as part of a step definition.

The sensitivity of the perturbation response can be calculated in a linear perturbation step (see General and Perturbation Procedures). The perturbation response will include the effects of stress and load stiffening in the base state if geometric nonlinearity is considered. Since we need to calculate the sensitivity of an incremental (perturbation) response, the sensitivity of the stress and load stiffening effects must be known at the end of the base step. Thus, if geometric nonlinearity is considered in the base step, DSA must also be active in the base step, irrespective of the type of formulation (total or incremental).

The basis of the semi-analytic technique is the use of numerical differencing to obtain derivatives of certain element vectors and matrices (see About design sensitivity analysis). Abaqus/Design will automatically determine appropriate perturbation sizes to be used in the semi-analytic technique unless you specify them directly. Abaqus/Design determines the perturbation sizes using a heuristic perturbation sizing algorithm based on the behavior of a scalar s associated with an element. By default, the perturbation sizing algorithm is applied only for the first increment (static procedure) or first mode (frequency procedure) in each step for which DSA is active. The perturbation sizes are then reused for the remaining increments or modes in the step for which DSA calculations are done.

The goal of the algorithm is to find perturbation sizes that are optimal for numerical differencing. Differencing formulas are based on Taylor series expansions, and the order of approximation of the derivative to be computed is reflected in the terms that are neglected in the series. The accuracy of the approximated derivatives often depends strongly on the perturbation size used in the differencing formula. Choosing a perturbation size that is too large will cause a truncation error, which occurs when the order of approximation is no longer valid (i.e., as a result of truncating higher-order terms in the Taylor series). A perturbation size that is too small will lead to inaccuracies in the differencing operations due to round-off, typically referred to as a cancellation error.

The algorithm attempts to find perturbation sizes giving the best compromise between cancellation and truncation errors by observing the behavior of s. For each design parameter s is computed for perturbation sizes spanning several orders of magnitude. The error in s between consecutive perturbation sizes is calculated as $e_i=\left|s_i-s_{i-1}\right|/s_i$. The perturbation size yielding an acceptable error, $e_a$, is chosen as the best perturbation size.

This scalar s is selected as follows:

• Static procedure. For static steps s is chosen as the norm of the element pseudoload (the partial derivative of the element residual with respect to the design parameters).

• Frequency procedure. For frequency steps s is computed from the element contribution to a matrix involving the derivatives of the mass and stiffness matrices (namely $\left(\frac{D{K}^{NM}}{Dh}-\lambda \frac{D{M}^{NM}}{Dh}\right)$, where $K^{NM}$ is the stiffness, $M^{NM}$ is the mass, h is the design parameter, and $\lambda$ is an eigenvalue). The scalar s is taken as the projection of this matrix onto an eigenvector ${\overline{\varphi }}^N$. If the perturbation sizing algorithm is applied to a mode with a distinct eigenvalue, ${\overline{\varphi }}^N$ is taken as the eigenvector associated with this mode. However, if a mode happens to be associated with a repeated eigenvalue, ${\overline{\varphi }}^N$ is taken as the sum of all the eigenvectors associated with the repeated eigenvalue. Thus, the entire set of modes associated with a repeated eigenvalue will be treated simultaneously by the perturbation sizing algorithm (the eigenvalue sensitivities of a repeated eigenvalue are obtained simultaneously from the same reduced eigenvalue system).

See About design sensitivity analysis for further details on the selection of s.

As can be seen in About design sensitivity analysis, the accuracy of the DSA solution is dictated by both the accuracy of the numerically computed derivatives and, for nonlinear static analysis, the accuracy of the tangent stiffness matrix. The accuracy of the numerically computed derivatives is governed by the semi-analytic DSA algorithm; you can control it by specifying DSA controls. In nonlinear static analysis DSA uses the tangent stiffness matrix formed during the last equilibrium iteration. It is possible that the accuracy of the tangent stiffness matrix needed to achieve an accurate equilibrium solution may be insufficient to achieve an accurate DSA solution. In such cases you can tighten the convergence tolerances during the equilibrium analysis so that a more accurate tangent stiffness matrix is obtained (see Commonly Used Control Parameters). Furthermore, an accurate equilibrium solution often can be obtained when unsymmetric terms in the tangent stiffness are ignored (i.e., the unsymmetric matrix storage and solution scheme is not used; see Defining an Analysis). However, even if mildly unsymmetric stiffness terms are neglected, the DSA solution may be inaccurate. Therefore, it is recommended that the unsymmetric solution scheme be used for DSA when the tangent stiffness matrix is known to be unsymmetric.

In some cases a response at a certain instant in time may be discontinuous with respect to a design parameter. For example, at this point of discontinuity a variation in the design parameter may cause a node to come into contact, frictional behavior to change from sticking to sliding, or a material point to transition from elastic to inelastic behavior. Since the DSA calculations make use of numerical differencing, it is possible that the perturbation of the design parameter used in the differencing scheme may result in values of the response to be differenced that lie on opposite sides of the discontinuity. If this occurs, the accuracy of the computed derivative cannot be guaranteed. Mathematically speaking, the derivative (sensitivity) of the response with respect to the design parameter does not exist at the point of discontinuity. Practically speaking, it is unlikely that the response at any given instant will lie precisely on the discontinuity. In cases where the response is near a discontinuity, if you choose to use the default perturbation sizing algorithm, the algorithm will attempt to choose design parameter perturbation sizes such that the values of the perturbed responses remain on the same side of the discontinuity. In addition, for contact elements DSA calculations are not performed in increments in which the associated contact node is open. Typically, the global results in any increment are not affected by a few discontinuous points in the model.

Responses depend on design parameters explicitly and implicitly. Implicit design dependence is the dependence on the design parameter through the solution variables; therefore, this type of dependence can be quantified only after the DSA solution is obtained (recall that the DSA solution is the total displacement sensitivity for the total formulation and the incremental displacement sensitivity for the incremental formulation). All other design dependencies are explicit, meaning that they can be resolved without knowing the DSA solution. The types of dependencies can be identified by looking at the form of the sensitivity of a response, say r, with respect to a design parameter, say h. This sensitivity is expressed as

for the total formulation and

for the incremental formulation, where $u^N$ is a displacement degree of freedom and ${\mathit{\alpha }}^t$ represents state variables at the beginning of the increment (see About design sensitivity analysis for further details). The state variables include the displacements at the beginning of the increment. In both cases the last term on the right-hand side represents the implicit design dependence through the solution variables.

It is observed from the incremental equation above that the explicit design dependence consists of two terms. The first of these, $\frac{\partial r}{\partial h}$, represents a direct design dependence, because this term arises from the direct dependence of the response on the design parameter. The second explicit term, $\frac{\partial r}{\partial {\mathit{\alpha }}^t}\frac{d{\mathit{\alpha }}^t}{dh}$, represents the dependence on the design parameter through the state variables at the beginning of the increment. For the total formulation, it is seen that the explicit term involves only direct design dependence.

Any feature for which direct design dependence calculations are implemented in Abaqus will be referred to as supported for DSA. Supported and unsupported features can be mixed in an analysis, unless the supported features cause unsupported features to become directly design dependent (an example of this would be making the Young's modulus for a frame element design dependent, since frames are not supported for DSA).

To make a clearer distinction between the types of design dependencies, consider the more concrete example of a linear elastic truss element, fixed at one end and pulled with a concentrated load at the other end. Let $u=u^t+\mathrm{\Delta }u$ represent the displacement at the free end, E represent Young's modulus, and L represent the length of the truss. Consider the axial stress $S_{11}$ as the response. Although it is clear in this simple example that the stress can be computed easily as the load divided by the cross-sectional area, the finite element analysis computes the stress equivalently as $S_{11}=Eu/L$. Choosing Young's modulus, E, as the design parameter, the stress sensitivity is given by

for the total formulation and

for the incremental formulation. This example is a valid analysis since elastic materials and truss elements are supported for DSA. Suppose now that a frame element is added, extending the length of the structure. If the frame element shares the same Young's modulus, the analysis becomes invalid since the dependency on the design parameter E causes the frame element, which is unsupported, to become directly design dependent (i.e., the term $\frac{\partial S_{11}}{\partial E}$ would need to be computed). On the other hand, if the frame uses a different modulus, say $E_1$ that is not a design parameter, the analysis again becomes valid, since the frame no longer depends directly on the design parameter E.

Surface-based contact between deformable and rigid surfaces with small- or finite-sliding relative surface motion including friction is supported in a design sensitivity analysis. In all the friction models only the friction coefficients (no test data input) can be made design dependent. Shape design parameters are not valid for rigid surfaces. Contact between deformable surfaces is not supported.

A design sensitivity analysis can be restarted (see Restarting an Analysis). However, DSA must have been active in the base analysis, and no design parameter or gradient data can be modified in the restart run. The restarted analysis will follow all the DSA propagation rules that are applicable to a regular analysis. For total formulation DSA, you may choose to activate or deactivate DSA in any new step that is added to the restart run. However, for the incremental formulation DSA must have been active in the step at which restart is attempted for you to continue doing DSA in the restarted analysis.