Surface Sensitivity

The surface sensitivity allows you to quantify the impact of changes to a part surface on an objective of interest. Cell faces that show large values of surface sensitivity have a large impact on the objective.

Conceptually, the surface sensitivity is similar to the Adjoint of Mesh Deformation. However, its formulation allows a much quicker calculation of sensitivity values at every face of the surface.

The surface sensitivity enables you to analyze large-scale trends in the design. You can use the surface sensitivity to guide a manual shape optimization. The surface sensitivity can also provide guidance for placing control points, which drive the displacement of mesh vertices when using the morpher.

Primal

The formulation for surface sensitivity is based on a spring-analogy mesh deformation strategy. However, Simcenter STAR-CCM+ does not use this strategy to actually deform the mesh. Instead, this method is used only to derive an approximate mesh adjoint that can provide surface sensitivities.

For this strategy, the coupling between neighboring cells is modeled using springs. Force equilibrium is used to solve for the displacement of the cell centers based on specified displacements on the boundary faces.

For cells in the interior of the domain, force equilibrium for cell $i$ requires that the forces that are associated with the adjacent cells sum up to zero:

Figure 1. EQUATION_DISPLAY

\sum_{j} k_{i j} (δ x_{j} - δ x_{i}) = 0

(5105)

where:

$j$ represents an adjacent cell.
$k_{i j}$ represents the spring constant of the spring connecting cell $i$ and $j$ . The spring constant $k_{i j}$ is inversely proportional to the distance between the centroids of cells $i$ and $j$ .
$δ x_{i}$ is the displacement at the centroid of cell $i$ .

For cells at the boundary of the domain, force equilibrium still requires that the forces balance. However, the displacement at the boundary face is a known quantity and can be moved to the right-hand side of the equation:

Figure 2. EQUATION_DISPLAY

\sum_{j, int e r i o r} k_{i j} (δ x_{j} - δ x_{i}) - k_{i b} {δ x}_{i} = - k_{i b} δ x_{b}

(5106)

where $δ x_{b}$ represents the displacement that is applied to the boundary face and $k_{ib}$ is the spring constant of the spring connecting cell $i$ and the boundary face. The summation is performed only over interior faces.

These equations can be solved simultaneously throughout the entire domain, yielding a linear system of the following form:

Figure 3. EQUATION_DISPLAY

K δ x = B

(5107)

where $B$ is defined as:

Figure 4. EQUATION_DISPLAY

B = {\begin{matrix} 0 \\ {- k}_{i b} δ x_{b} \end{matrix} \begin{array}{l} f o r int e r i o r c e l l s \\ fo r c e l l s o n b o u n d a r i e s \end{array}

(5108)

The displacement for each cell centroid is computed by inverting the matrix $K$ :

Figure 5. EQUATION_DISPLAY

δ x = K^{- 1} B

(5109)

Finally, the displacement for each vertex in the domain is calculated. For interior vertices, the vertex displacement is the average of the cell centroid displacements over all adjacent cells. For vertices on the boundaries, the displacement is an average of the specified displacements over all adjacent boundary faces.

Adjoint

To calculate the surface sensitivity, the above primal formulation is differentiated in reverse order. The input to this adjoint calculation is the cost function sensitivity with respect to the mesh $\frac{d L}{d X}$ . The output is the cost function sensitivity with respect to the surface displacements $\frac{d L}{d (δ x_{b})}$ . The output is associated with each face on the surface and represents the change of the objective value due to a displacement of the respective face centroid.

The first step is calculating the sensitivity with respect to the cell centroid and boundary face centroid displacements based on $\frac{d L}{d X}$ . The derivative with respect to the cell centroids is determined by performing a transpose average over the vertices of the cells:

Figure 6. EQUATION_DISPLAY

\frac{\partial L}{\partial (δ x_{i})} = \sum_{k} \frac{1}{n_{k}^{(c)}} {(\frac{d L}{d X})}_{k}

(5110)

where $n_{k}^{(c)}$ is the number of cells adjacent to vertex $k$ .

The derivative with respect to the boundary face centroids $\frac{\partial L}{\partial (δ x_{b})}$ is given by a transpose average of $\frac{d L}{d X}$ over the vertices of the boundary faces:

Figure 7. EQUATION_DISPLAY

\frac{\partial L}{\partial (δ x_{b, i})} = \sum_{k} \frac{1}{n_{k}^{(b)}} {(\frac{d L}{d X})}_{k}

(5111)

where $n_{k}^{(b)}$ is the number of boundary faces adjacent to vertex $k$ .

Because this mesh deformation process is linear, the mesh adjoint $Λ_{X}$ is given by inverting the transpose of the matrix $K$ onto the $\frac{\partial L}{\partial (δ x)}$ vector.

Figure 8. EQUATION_DISPLAY

Λ_{X} = K^{- T} \frac{\partial L}{\partial (δ x)}

(5112)

The total surface sensitivity is the sum of two components. The first component is the sensitivity of the boundary centroid displacement due to the vertex sensitivity given by Eqn. (5111). The second components is the sensitivity of the spring analogy equation with respect to the boundary displacement multiplied by the mesh adjoint. For boundary face $j$ connected to cell $i$ , the total surface sensitivity is given as:

Figure 9. EQUATION_DISPLAY

\frac{d L}{d (δ x_{b, j})} = \frac{\partial L}{\partial (δ x_{b, j})} - k_{b, i j} Λ_{X, i}

(5113)

Surface Sensitivity Filtering

The calculated surface sensitivity Eqn. (5113) directly depends on the mesh. If used directly for surface morph, this computed surface sensitivity can cause mesh distortion. To solve this issue, an explicit filtering technique is applied to the surface sensitivity.

In primal mode the explicit filter translates a scalar design field into a smooth displacement field. The adjoint of this process is used during the derivative calculation to essentially smooth the sensitivities. The displacement field is defined as:

Figure 12. EQUATION_DISPLAY

δ \vec{x} (\vec{x}) = \int_{Γ} \vec{A} (\vec{x}, {\vec{x}}^{'}) \cdot δ s (\vec{x}) ⅆ Γ

(5116)

where:

$δ s (\vec{x})$ is the control field.
$\vec{A} (\vec{x}, \vec{x} ´)$ is the filter.
$Γ$ is the boundary of domain.
$δ \vec{x} (\vec{x})$ is the displacement at a location.

The filter itself is defined as a Gaussian filter based on distance combined with the local normal vector:

Figure 13. EQUATION_DISPLAY

\vec{A} (\vec{x}, {\vec{x}}^{'}) = \frac{1}{\sqrt{2 π} \cdot σ} e^{- \frac{{‖ \vec{x} - {\vec{x}}^{'} ‖}^{2}}{2 σ^{2}}} \cdot \vec{n} ({\vec{x}}^{'})

(5117)

where:

$\vec{n} ({\vec{x}}^{'})$ is the normal vector.
$σ$ is the specified filter radius.

The filter has the effect of averaging the normal and the control field since it is evaluated using the second argument of the function. Decomposed into individual components, the filters become:

Figure 14. EQUATION_DISPLAY

\begin{matrix} {δ x (\vec{x}) = \int}_{Γ} \frac{1}{\sqrt{2 π} \cdot σ} e^{- \frac{{‖ \vec{x} - {\vec{x}}^{'} ‖}^{2}}{2 σ^{2}}} \cdot n_{x} ({\vec{x}}^{'}) \cdot δ s (\vec{x}) d Γ \\ \begin{matrix} {δ y (\vec{x}) = \int}_{Γ} \frac{1}{\sqrt{2 π} \cdot σ} e^{- \frac{{‖ \vec{x} - {\vec{x}}^{'} ‖}^{2}}{2 σ^{2}}} \cdot n_{y} ({\vec{x}}^{'}) \cdot δ s (\vec{x}) d Γ \\ {δ z (\vec{x}) = \int}_{Γ} \frac{1}{\sqrt{2 π} \cdot σ} e^{- \frac{{‖ \vec{x} - {\vec{x}}^{'} ‖}^{2}}{2 σ^{2}}} \cdot n_{z} ({\vec{x}}^{'}) \cdot δ s (\vec{x}) d Γ \end{matrix} \end{matrix}

(5118)

In this form, the filter condenses to standard filtering of three quantities: $n_{x} δ s, n_{y} δ s, n_{z} δ s$ . Given a set of boundary points, whose coordinates are ${\vec{x}}_{i}$ , the filter can be written in the following matrix form:

Figure 15. EQUATION_DISPLAY

\begin{matrix} δ X = [H] [n_{x}] δ S \\ \begin{matrix} δ Y = [H] [n_{y}] δ S \\ δ Z = [H] [n_{z}] δ S \end{matrix} \end{matrix}

(5119)

where $[n_{x}], [n_{y}], [n_{z}]$ are diagonal matrices with a component of the normal vector for each element. The elements of the filter matrix are:

Figure 16. EQUATION_DISPLAY

H_{i, j} = \frac{1}{\sqrt{2 π} \cdot σ} e^{- \frac{{‖ \vec{x_{i}} - {\vec{x_{j}}}^{'} ‖}^{2}}{2 σ^{2}}}

(5120)

Because the filter is based on the distance between points, it is symmetric. ${[H]}^{T} = [H]$ .

To compute the adjoint of this process, the filtering process for each component is transposed:

Figure 17. EQUATION_DISPLAY

{\frac{ⅆ L}{ⅆ S}}^{T} = {[n_{x}]}^{T} H^{T} {\frac{ⅆ L}{ⅆ X}}^{T} + {[n_{y}]}^{T} H^{T} {\frac{ⅆ L}{ⅆ Y}}^{T} + {[n_{z}]}^{T} H^{T} {\frac{ⅆ L}{ⅆ Z}}^{T}

(5121)

The final sensitivity includes the application of the filter in primal mode hence is a vector field.