Design sensitivity analysis for static stress analysis

Abaqus/Design supports design sensitivity analysis (DSA) for static stress problems.

The following topics are discussed:

Total displacement DSA formulation for nonlinear equilibrium problems
Incremental displacement DSA formulation for history-dependent equilibrium problems
Computational approach

Total displacement DSA formulation for nonlinear equilibrium problems

Let R and P be the numbers of design responses and design parameters, respectively. Let each response $ϕ_{r}$ , $r = 1, \dots, R$ , be a function of design parameters $h_{p}$ , $p = 1, \dots, P$ and depend on them both explicitly and via the displacement field represented here by the nodal displacement vector $u^{N}$ (see the definition of finite element interpolation in About procedures and basic equations),

ϕ_{r} = ϕ_{r} (u^{N} (h_{p}), h_{p}) .

The dependence $u^{N} (h_{p})$ is only implicit; i.e., it is implied only by the design dependence of coefficients in the equilibrium equation system whose solution is $u^{N}$ .

Assume that we have solved an equilibrium problem defined by Equation 2 at the end of an increment and that we have the converged solution $u^{N}$ as well as values of all responses. Sensitivity of a response $ϕ_{r}$ with respect to design parameter $h_{p}$ is defined as

\begin{matrix} (1) & \frac{d ϕ_{r}}{d h_{p}} = \frac{\partial ϕ_{r}}{\partial h_{p}} + \frac{\partial ϕ_{r}}{\partial u^{N}} \frac{d u^{N}}{d h_{p}} . \end{matrix}

All but one quantity in the above equation can be determined explicitly given the equilibrium solution. The only unknown is $d u^{N} / d h_{p}$ ; to compute it, an additional system of equations has to be solved.

Rewrite Equation 2 in the form

\begin{matrix} (2) & F^{N} (u^{M}) = 0, \end{matrix}

where

F^{N} = - \int_{V^{0}} β_{N} : τ^{c} d V^{0} + \int_{S} N_{N}^{T} \cdot t d S + \int_{V} N_{N}^{T} \cdot f d V .

All the quantities in the above equation are assumed to depend on design parameters $h_{p}$ explicitly or via displacement field $u^{N}$ . Differentiation of the above two equations with respect to design parameters leads to the following equation:

\begin{matrix} (3) & K_{M N} \frac{d u^{N}}{d h_{p}} = - \frac{\partial F^{M}}{\partial h_{p}}, \end{matrix}

in which

K_{M N} = \frac{\partial F^{M}}{\partial u^{N}}

is the tangent stiffness (Jacobian) matrix defined in Equation 4 and $\partial F^{M} / \partial h_{p}$ is an explicitly determinable quantity. Substituting Equation 3 into Equation 1, we obtain

\begin{matrix} (4) & \frac{d ϕ_{r}}{d h_{p}} = \frac{\partial ϕ_{r}}{\partial h_{p}} - \frac{\partial ϕ_{r}}{\partial u^{N}} K_{N M}^{- 1} \frac{\partial F^{M}}{\partial h_{p}}, \end{matrix}

which is the solution of the total displacement DSA problem.

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 $\partial ϕ_{r} / \partial h_{p}$ , $\partial ϕ_{r} / \partial u^{M}$ , and $\partial F^{M} / \partial h_{p}$ have to be computed in an element-by-element manner. $\partial F^{M} / \partial h_{p}$ 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 3 for each $p = 1, \dots, P$ with respect to the unknown vectors of nodal displacement sensitivity $d u^{N} / d h_{p}$ . The displacement sensitivities are then substituted into Equation 1 to compute $d ϕ_{r} / d h_{p}$ .

The coefficient matrix $K_{M N}$ 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.

Incremental displacement DSA formulation for history-dependent equilibrium problems

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 $F^{N}$ in Equation 2 or to define design responses do not lend themselves to be expressed as functions of total displacement $u^{N}$ . 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, $Δ u^{N}$ :

F^{N} = F^{N} (α^{t} (h_{p}), Δ u^{N} (h_{p}), h_{p}),

ϕ_{r} = ϕ_{r} (α^{t} (h_{p}), Δ u^{N} (h_{p}), h_{p});

see, for example, Kleiber et al. (1997). The notation $α^{t}$ stands for a set of state variables $α$ that may include tensors (stress, back stress, etc.) as well as scalar quantities (equivalent plastic strain, etc.) defined for a particular material point at time t. Some responses may also depend directly on the displacement $u^{N}$ , and the beginning-of-the-increment value of $u^{N}$ will, generally, also enter into the set $α^{t}$ .

In such a case Equation 4 takes the following form:

\begin{matrix} (5) & \frac{d ϕ_{r}}{d h_{p}} = \frac{D ϕ_{r}}{D h_{p}} - \frac{\partial ϕ_{r}}{\partial Δ u^{M}} K_{M N}^{- 1} \frac{D F^{N}}{D h_{p}}, \end{matrix}

where

\frac{D (\cdot)}{D h_{p}} \overset{def}{=} \frac{\partial (\cdot)}{\partial h_{p}} + \frac{\partial (\cdot)}{\partial α^{t}} \frac{d α^{t}}{{dh}_{p}}

denotes the explicit design derivative of a quantity $(\cdot)$ .

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 $α$ 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 $D ϕ_{r} / D h_{p}$ , $D F^{M} / D h_{p}$ (the pseudoload), and the derivatives with respect to displacements $\partial ϕ_{r} / \partial Δ u^{M}$ are assembled in the element loop. The set of design responses $ϕ_{r}$ , $r = 1, \dots, R$ , includes in this case all the scalars and tensor components of $α$ . In the direct differentiation method the following system of equations is solved for each design parameter $h_{p}$ :

K_{M N} \frac{d Δ u^{N}}{d h_{p}} = - \frac{D F^{M}}{D h_{p}},

and the solution vectors are substituted into Equation 5.

Computational approach

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 $K_{M N}$ . 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 $D F^{M} / D h_{p}$ and $d ϕ_{r} / d h_{p}$ numerically by finite differencing. For simplicity, assume that the finite difference technique is central difference such that for a given function $A (x)$ , the derivative of A with respect to x is

\frac{d A}{d x} = \frac{A (x + δ x) - A (x - δ x)}{2 δ x},

where $δ x$ is the perturbation of x.

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

F^{M} + δ F^{M} = F^{M} (α^{t} (h_{p} + δ h_{p}), Δ u (h_{p}), h_{p} + δ h_{p}) .

The change in the state corresponding to a perturbation in the design parameters is approximated by

α^{t} (h_{p} + δ h_{p}) = α^{t} (h_{p}) + \frac{d α^{t}}{d h_{p}} δ h_{p} .

The above process is repeated for a negative perturbation ( $- δ h_{p}$ ), after which the results are differenced to arrive at the explicit design derivative $D F^{M} / D h_{p}$ .

Once the (incremental) displacement sensitivities are found, the response sensitivities $d ϕ_{r} / d h_{p}$ can be obtained using

ϕ_{r} + δ ϕ_{r} = ϕ_{r} (α^{t} (h_{p} + δ h_{p}), Δ u^{N} (h_{p} + δ h_{p}), h_{p} + δ h_{p}),

where

Δ u^{N} (h_{p} + δ h_{p}) = Δ u^{N} (h_{p}) + \frac{d Δ u^{N}}{d h_{p}} δ h_{p} .

The process is repeated for a negative perturbation of $h_{p}$ , and the results are differenced.

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.