Let R and P be the numbers of design responses and design parameters, respectively. Let each response , , be a function of design parameters , and depend on them both explicitly and via the displacement field represented here by the nodal displacement vector (see the definition of finite element interpolation in “Procedures: overview and basic equations,” Section 2.1.1),

Assume that we have solved an equilibrium problem defined by Equation 2.1.1–2 at the end of an increment and that we have the converged solution as well as values of all responses. Sensitivity of a response with respect to design parameter is defined as

Rewrite Equation 2.1.1–2 in the form

The DSA algorithm used in ABAQUS is known as the direct differentiation method (DDM) and consists of the following operations. After the converged equilibrium solution is obtained, the three arrays , , and have to be computed in an element-by-element manner. is often called the pseudoload since it becomes the right-hand side of the DSA problem. The final DSA solution is obtained by solving the system of Equation 2.18.1–3 for each with respect to the unknown vectors of nodal displacement sensitivity . The displacement sensitivities are then substituted into Equation 2.18.1–1 to compute .

The coefficient matrix used in the DSA computations is simply the last tangent stiffness matrix used in the equilibrium iterative algorithm. At the stage of the DSA computations this matrix is still available in the decomposed form and can be retrieved easily to perform the back substitutions for the DSA right-hand-side vectors. This makes the DSA module a very efficient add-on to the equilibrium analysis enabling sensitivity computations at a relatively low cost.

The formulation of DSA presented above provides a brief introduction to the way DSA is implemented in ABAQUS; however, due to some simplifications, the discussion is not relevant to a large number of nonlinear mechanical problems, especially those involving history-dependent behavior of the structure modeled. The main difficulty in such problems is that many quantities necessary to compute the residual in Equation 2.18.1–2 or to define design responses do not lend themselves to be expressed as functions of total displacement . Rather, at each time increment they are functions of certain state variables at the beginning of the increment (referred to as the time instant t) and of the incremental displacements, :

In such a case Equation 2.18.1–4 takes the following form:

The fundamental difference, from the point of view of the DSA solution algorithm, between the total and incremental approach is that in the latter case all state variables effectively become additional, or internal, design responses, whose sensitivities must be computed and updated at the end of each time increment to proceed with the DSA in the next increment. The number of such internal responses may be significant with obvious effects both on the computational time and memory requirement.

The DSA solution procedure is similar to that in the total displacement approach. After the equilibrium computations are complete, the arrays of explicit design derivatives , (the pseudoload), and the derivatives with respect to displacements are assembled in the element loop. The set of design responses , , includes in this case all the scalars and tensor components of . In the direct differentiation method the following system of equations is solved for each design parameter :

The derivatives required for DSA can be computed analytically or numerically. In the analytical approach the finite element equations are differentiated exactly, following the theory described in the previous sections. This approach is difficult to implement, but it is efficient and yields exact sensitivities. In the numerical approach some or all of the required derivatives are computed using the finite difference technique. The numerical approach can be further subdivided into the overall or global finite difference approach and the semi-analytic approach. In the global finite difference approach the response sensitivities with respect to a particular design parameter are obtained by perturbing that design parameter a number of times (depending on the finite difference technique) and performing an entire equilibrium analysis for each perturbation. The responses are retained for each analysis and then differenced to obtain the response sensitivities. This approach is computationally expensive since an entire equilibrium problem must be solved for each perturbation, but it is easily implemented. The semi-analytic approach is used in ABAQUS and can be viewed as a compromise between the analytic and global finite difference approaches. In the semi-analytic approach the DSA element vectors are obtained by differencing; but, like the analytic approach, the DSA solution is obtained by back-substitution against . The advantage of the semi-analytic approach is that it is much easier to implement than the analytic approach and much more efficient than the global finite difference approach. The details of this method are described in the following paragraphs.

The objective of the semi-analytic approach is to compute the DSA vectors and numerically by finite differencing. For simplicity, assume that the finite difference technique is central difference such that for a given function , the derivative of A with respect to x is

For generality, consider the history-dependent case. To approximate the explicit design derivatives of , the incremental displacement is held constant while a positive perturbation is applied to each design parameter . In this way perturbed values of are obtained as

Once the (incremental) displacement sensitivities are found, the response sensitivities can be obtained using

The finite difference interval must be chosen carefully. If the interval is too small, round-off or cancellation errors occur due to loss of precision during the differencing operations. On the other hand, if the interval is too large, truncation errors may occur. Truncation errors arise from the fact that differencing formulas are based on truncated Taylor series expansions. ABAQUS will automatically choose a perturbation size that provides the best compromise between cancellation and truncation errors.

The theory presented below assumes that all the eigenvalues are distinct (i.e., no repeated eigenvalues). If this is not the case, further manipulations are required to obtain the eigenvalue and eigenvector sensitivities corresponding to the repeated eigenvalues, and the following equations for the sensitivities will be incorrect. The repeated eigenvalue case is considered in the next section.

Performing a frequency analysis means solving the following eigenvalue problem (see “Eigenvalue extraction,” Section 2.5.1):

Pre-multiplying by , making use of Equation 2.18.1–6, and manipulating the result gives the eigenvalue sensitivities:

This section outlines the formulation used to obtain eigenvalue sensitivities for repeated eigenvalues. Further information can be found in the papers by Mills-Curran (1988) and Shaw (1991). When an eigenvalue repeats R times, the eigenvectors associated with are linearly independent but not unique—any linear combination of these eigenvectors is also an eigenvector. Because of this non-uniqueness, the eigenvectors may not be continuous or differentiable in the design parameter. However, a set of R eigenvectors that are continuous and differentiable can be obtained by an orthogonal transformation:

A semi-analytic approach is used to compute the eigenvalue sensitivities. The basic idea of this approach, as outlined in the section on static DSA, is to compute some of the required intermediate derivatives using finite differencing. In the context of DSA for frequency procedures this means that the derivatives of the mass and stiffness matrices are computed using finite differencing.