A surface mesh deformation method near component intersections for high-fidelity design optimization

Abstract

Aerodynamic shape optimization based on computational fluid dynamics (CFD) requires three steps: updating the geometry based on the design variables, updating the CFD surface mesh for the new geometry, and updating the CFD volume mesh based on the new surface mesh. While there are many tools available for the first and third steps, the methods available for the second step are insufficient for geometries that have intersecting components. For these geometries, the CFD surface mesh needs to be updated near component intersections to conform to the component geometries and the updated intersection curves. To address this need, we introduce a method that can deform the CFD surface mesh nodes near component intersections. The method can handle arbitrary design changes for each intersecting component as long as the geometric topology is unchanged. Furthermore, the method is suitable for gradient-based optimization because it smoothly deforms every CFD surface node without introducing topological changes in the CFD surface mesh. In this paper, we detail each step of the proposed method and visualize the range of design changes that can be achieved with this approach. Finally, we use the proposed method in an aerodynamic shape optimization problem to optimize the wing-body intersection of the DLR-F6 configuration. These results demonstrate the effectiveness of the proposed method in a high-fidelity design optimization framework. The method applies to both structured and unstructured CFD meshes and makes it possible to use computer-aided design and conceptual design geometry tools within high-fidelity design optimization.

Appendix A: Computation of the curve-based deformation

The curve-based deformation step introduced in Sect. 2.5 relies on the computation of two integrals in the nominator and denominator in Eq. (6). The equation can be written again using a simplified notation for a general case as

$$\begin{aligned} \varDelta \mathbf{x} = \frac{\int \limits _{{\mathcal {C}}} w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t}}{\int \limits _{{\mathcal {C}}} w(\mathbf{x} , t) L {\text {d}}{t}}, \end{aligned}$$

(16)

where ${\mathcal {C}}$ represents the original location of the curve and $\varDelta {\mathcal {C}}$ is the displacement defined on this curve, such that ${\mathcal {C}} + \varDelta {\mathcal {C}}$ yields the updated coordinates of the curve. Using these displacements, the method computes the displacement of a given point $\mathbf{x}$, using a weighted averaging based on the inverse distance of the point to the curve. The parameter $t \in [0,1]$ is used to traverse the curve ${\mathcal {C}}$ with length L and the weight w can be written as

$$\begin{aligned} w(\mathbf{x} , t) = \frac{1}{\left| \left| \mathbf{x} - {\mathcal {C}}(t) \right| \right| _2^m}, \end{aligned}$$

(17)

where m is an exponent, which set to 3 in this work. In the next two sections, we inspect the behavior of this approach for the special case when the evaluation point $\mathbf{x}$ is located on the curve ${\mathcal {C}}$ and detail the computation of the terms for this method when the curve ${\mathcal {C}}$ is defined using line elements.

1.1 A.1: Special case for points located on curves

The weighting function has a finite value as long as the evaluation point $\mathbf{x}$ is not on the curve ${\mathcal {C}}$. For the special case where the evaluation point is on the curve and $m > 0$, the weighing function tends to infinity as the parameter $t \rightarrow t_*$, where the evaluation point $\mathbf{x}$ is co-located with the point ${\mathcal {C}}(t_*)$. This can be written as

$$\begin{aligned} \lim \limits _{ t \rightarrow t_*} w(\mathbf{x} , t) = \lim \limits _{ t \rightarrow t_*} \frac{1}{\left| \left| \mathbf{x} - {\mathcal {C}}(t) \right| \right| _2^m} \rightarrow \infty . \end{aligned}$$

(18)

As a result, the two integrands we define also tend to infinity as $t\rightarrow t_*$, which results in two improper integrals.

The integrations over the curve ${\mathcal {C}}$, which is defined in $t=[0,1]$, can be split into two at $t = T$, which results in

$$\begin{aligned} \varDelta \mathbf{x} = \frac{\int \limits _{0}^{T} w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t} + \int \limits _{T}^1 w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t} }{\int \limits _{0}^{T} w(\mathbf{x} , t) L {\text {d}}{t} + \int \limits _{T}^1 w(\mathbf{x} , t) L {\text {d}}{t}}. \end{aligned}$$

(19)

Then, we substitute $t = -s$ for the two integrals that go from T to 1, and reverse the integration limits, which yields

$$\begin{aligned} \varDelta \mathbf{x} = \frac{\int \limits _{0}^{T} w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) \varDelta {\mathcal {C}}(-s) L {\text {d}}{s} }{\int \limits _{0}^{T} w(\mathbf{x} , t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) L {\text {d}}{s}}. \end{aligned}$$

(20)

For the special case where $\mathbf{x}$ is co-located with ${\mathcal {C}}(t_*)$ and $m\ge 1$, these integrals do not converge; however, $\varDelta \mathbf{x}$ can be computed using the L’Hôpital’s rule as

$$\begin{aligned} \begin{aligned} \varDelta \mathbf{x}&= \lim \limits _{T \rightarrow t_*^-} \frac{\int \limits _{0}^{T} w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) \varDelta {\mathcal {C}}(-s) L {\text {d}}{s} }{\int \limits _{0}^{T} w(\mathbf{x} , t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) L {\text {d}}{s}} \\&=\lim \limits _{T \rightarrow t_*^-} \frac{ \frac{ {\text {d}}{} }{ {\text {d}}{T}} \left( \int \limits _{0}^{T} w(\mathbf{x} , t) \varDelta {\mathcal {C}}(t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) \varDelta {\mathcal {C}}(-s) L {\text {d}}{s} \right) }{ \frac{{\text {d}}{} }{{\text {d}}{T}}\left( \int \limits _{0}^{T} w(\mathbf{x} , t) L {\text {d}}{t} + \int \limits _{-1}^{-T} w(\mathbf{x} , -s) L {\text {d}}{s} \right) } \\&=\lim \limits _{T \rightarrow t_*^-} \frac{ 2w(\mathbf{x} , T) \varDelta {\mathcal {C}}(T) L}{ 2w(\mathbf{x} , T) L} \\&= \varDelta {\mathcal {C}}(t_*). \end{aligned} \end{aligned}$$

(21)

This result means that when $\mathbf{x}$ is initially co-located with ${\mathcal {C}}(t_*)$, the displacement $\varDelta \mathbf{x}$ computed using Eq. (16) results in the displacement of the curve at $t=t_*$, which is the desired behavior demonstrated in Fig. 10. With this approach, the surface mesh nodes that are initially located on the intersection and feature curves directly follow the displacements of these curves.

1.2 A.2: Computation of the integrals over piecewise linear curves

In our application, the curve ${\mathcal {C}}$ is defined using a union of line elements. Using this definition, the two integrals introduced in Eq. (16) can be computed as a sum over every line element, where we compute the integral for each line element separately.

Without loss of generality, we consider the case where the curve ${\mathcal {C}}$ is defined using a single line element, with endpoints $\mathbf{x} _A$ and $\mathbf{x} _B$. In this case, we use the parameter $t \in [0,1]$ to traverse this line element, and the coordinate along this line element can be written as

$$\begin{aligned} {\mathcal {C}}(t) = \mathbf{x} _A (1-t) + \mathbf{x} _B t, \end{aligned}$$

(22)

such that ${\mathcal {C}}(0) = \mathbf{x} _A = (x_A, y_A, z_A)$ and ${\mathcal {C}}(1) = \mathbf{x} _B = (x_B, y_B, z_B)$ in Cartesian coordinates. Similarly, we can write the displacement of the curve as

$$\begin{aligned} \varDelta {\mathcal {C}}(t) = \varDelta \mathbf{x} _A (1-t) + \varDelta \mathbf{x} _B t, \end{aligned}$$

(23)

where $\varDelta \mathbf{x} _A$ and $\varDelta \mathbf{x} _B$ are the displacements of the points that define the line element.

Using these definitions, we can re-write $\varDelta \mathbf{x}$ as

$$\begin{aligned} \varDelta \mathbf{x} = \frac{\varDelta \mathbf{x} _A \int \limits _{0}^1 w(\mathbf{x} , t) L {\text {d}}{t} + (\varDelta \mathbf{x} _B - \varDelta \mathbf{x} _A) \int \limits _{0}^1 w(\mathbf{x} , t) L t {\text {d}}{t} }{\int \limits _{0}^1 w(\mathbf{x} , t) L {\text {d}}{t}}. \end{aligned}$$

(24)

The terms $\varDelta \mathbf{x} _A$ and $\varDelta \mathbf{x} _B$ correspond to the displacements defined at the two endpoints of the line and can be taken out of the integrals because they do not depend on the parameter t. This expression can be evaluated by evaluating the two integrals that appear, which can be defined as

$$\begin{aligned} I_1 = \int \limits _{0}^1 w(\mathbf{x} , t)L {\text {d}}{t}, \quad I_2 = \int \limits _{0}^1 w(\mathbf{x} , t)L t {\text {d}}{t}. \end{aligned}$$

(25)

In this work, $m=3$ in the definition of w (17); therefore, we specialize the exact integration of these terms for this value. In this case, we can write the two integrals as

$$\begin{aligned} \begin{aligned} I_1&= \int \limits _{0}^1 \frac{L}{\left| \left| \mathbf{x} - {\mathcal {C}}(t) \right| \right| _2^3} {\text {d}}{t} = \int \limits _{0}^1 \frac{L}{\left| \left| \mathbf{x} - \left( \mathbf{x} _A (1-t) + \mathbf{x} _B t, \right) \right| \right| _2^3} {\text {d}}{t} , \\ I_2&= \int \limits _{0}^1 \frac{Lt}{\left| \left| \mathbf{x} - {\mathcal {C}}(t) \right| \right| _2^3} {\text {d}}{t} = \int \limits _{0}^1 \frac{Lt}{\left| \left| \mathbf{x} - \left( \mathbf{x} _A (1-t) + \mathbf{x} _B t, \right) \right| \right| _2^3} {\text {d}}{t} . \end{aligned} \end{aligned}$$

(26)

By expressing the evaluation point $\mathbf{x}$ as (x, y, z) in Cartesian coordinates, the 2-norm that appears in the denominator of both integrals can be written as

$$\begin{aligned} \left| \left| \mathbf{x} - \left( \mathbf{x} _A (1-t) + \mathbf{x} _B t, \right) \right| \right| _2 = (a t^2 + b t + c)^{1/2}, \end{aligned}$$

(27)

where

$$\begin{aligned} \begin{aligned} a&= L^2 = ( x_B - x_A )^2 + ( y_B - y_A )^2 + ( z_B - z_A )^2, \\ b&= 2 (( x_B - x_A )( x_A - x ) + ( y_B - y_A )( y_A - y ) + ( z_B - z_A )( z_A - z ) ), \\ c&= ( x_A - x )^2 + ( y_A - y )^2 + ( z_A - z )^2 . \end{aligned} \end{aligned}$$

(28)

With these definitions, the two integrals can be analytically computed as

$$\begin{aligned} \begin{aligned} I_1&= \left. -\frac{ 2 (2at+b) \sqrt{a} }{ (b^2 - 4ac) \sqrt{t(at+b) + c} } \right| _0^1 \\&= -\frac{ 2 (2a+b) \sqrt{a} }{ (b^2 - 4ac) \sqrt{a+b+c} } + \frac{ 2 b \sqrt{a} }{ (b^2 - 4ac) \sqrt{c} } , \\ I_2&= \left. \frac{ 2 (2bt+c) \sqrt{a} }{ (b^2 - 4ac) \sqrt{t(at+b) + c} } \right| _0^1 \\&= \frac{ 2 (2b+c) \sqrt{a} }{ (b^2 - 4ac) \sqrt{a+b+c} } - \frac{ 2c \sqrt{a} }{ (b^2 - 4ac) \sqrt{c} }. \end{aligned} \end{aligned}$$

(29)

In this formulation, the terms $a+b+c$ and c can have small negative values because of machine precision errors. Therefore, when computing the square roots of these terms, we set them to zero if they are less than zero to prevent results with nonzero imaginary parts. Finally, the terms $(b^2 - 4ac) \sqrt{a+b+c}$ and $(b^2 - 4ac) \sqrt{c}$ can have a value of zero. This is problematic because these terms appear in the denominators. Therefore, we subtract a small number $\epsilon = 10^{-50}$ from the denominators. We subtract $\epsilon$ instead of adding it, because the result of $\sqrt{a+b+c}$ and $\sqrt{c}$ are guaranteed to be non-negative, while the term $b^2 - 4ac \le 0$. With these considerations, we can robustly compute the terms required for the curve-based displacement approach introduced in Eq. (6).