Sensitivity with Respect to Design Parameters

The optimization of objectives on the basis of design modifications requires the sensitivities of the objectives with respect to the design parameters, that is $\frac{d L}{d D}$ .

The derivative

\frac{d L}{d D}

is given by the chain of the Jacobians of each operation in the sequence

(X (D); Q (X); L (Q, X))

Figure 1. EQUATION_DISPLAY

\frac{d L}{d D} = [\frac{\partial L}{\partial X} + \frac{\partial L}{\partial Q} \frac{\partial Q}{\partial X}] \frac{d X}{d D}

(5064)

Consider

n

design variables

D

and a mesh

X

with

m

points. The Jacobian of the operation

X (D)

is the matrix of all first-order partial derivatives of this operation:

Figure 2. EQUATION_DISPLAY

\frac{d X}{d D} = [\begin{matrix} \frac{\partial X_{1}}{\partial D_{1}} & \dots & \frac{\partial X_{1}}{\partial D_{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial X_{m}}{\partial D_{1}} & \dots & \frac{\partial X_{m}}{\partial D_{n}} \end{matrix}] = J_{X}

(5065)

where each column of this matrix is a tangent and each row is a gradient of the Jacobian.

For a large number of design variables $D$ , the computation of Eqn. (5064) becomes expensive with respect to memory. To alleviate this problem the derivative of the system can be composed from tangents of $J_{X}$ . That is, Simcenter STAR-CCM+ computes:

Figure 3. EQUATION_DISPLAY

\frac{d L}{d D_{i}} = [\frac{\partial L}{\partial X} + \frac{\partial L}{\partial Q} \frac{\partial Q}{\partial X}] \frac{d X}{d D_{i}}

(5066)

for all columns of $J_{X}$ .

In alleviating one problem, this approach introduces another, in that computational time grows linearly with the number of design variables $D$ . Whilst for many engineering problems the number of design variables $D$ is large, which would rule out the tangent approach, there are often only a few outputs $L$ which are required. If the tangents approach were to be inverted, then this would yield an efficient way of computing the derivatives of the whole system. This goal can be achieved by taking the transpose of the derivative of the system for a particular gradient of $J_{L}$ :

Figure 4. EQUATION_DISPLAY

{\frac{d L_{j}}{d D}}^{T} = {\frac{d X}{d D}}^{T} [{\frac{\partial L_{j}}{\partial X}}^{T} {+ \frac{\partial Q}{\partial X}}^{T} {\frac{\partial L_{j}}{\partial Q}}^{T}]

(5067)

Eqn. (5067) represents the basic form of how the actual final sensitivity $\frac{d L}{d D}$ is built up by the adjoint of the sequence of steps listed in the introduction.

The steps in solving Eqn. (5067) are:

Computing the sensitivity of the report objectives with respect to the solution ${\frac{\partial L_{j}}{\partial Q}}^{T}$ . This solution analysis also computes any direct sensitivities with respect to the mesh, that is ${\frac{\partial L_{j}}{\partial X}}^{T}$ . See Adjoint of Solution Analysis.
Evaluating the sensitivity of the report objectives with respect to the mesh, that is the rightmost term in Eqn. (5067):
${\frac{d L_{j}}{d X}}^{T} = {\frac{\partial L_{j}}{\partial X}}^{T} + {\frac{\partial Q}{\partial X}}^{T} {\frac{\partial L_{j}}{\partial Q}}^{T}$

See Adjoint of Physics Solution.
Performing the adjoint-differentiated mesh deformation algorithm that evaluates ${\frac{d L_{j}}{d D}}^{T}$ , that is the product of ${\frac{d X}{d D}}^{T}$ and the result of the previous step ${\frac{d L_{j}}{d X}}^{T}$ . The evaluated term describes the sensitivity of the report objectives with respect to the mesh design parameters. Adjoint of Mesh Deformation provides the derivation of the adjoint of the Radial Basis Function morpher.