Thin-Plate Splines Generalized Interpolation Based on Duchon’s Semi-Norm Minimization Extended to CAD-Compliant Surface Mesh Deformation

Abstract

The Thin-Plate Splines (TPS) technique, using Kybic et al.’s generalized interpolation approach, is extended to differentiable manifolds. The initial application to surface mesh deformation resulting from parameterized Computer Aided Design (CAD) used in the framework of shape optimization is given. Coming to RSM (Response Surface Model), we give a comparison with Kriging and RBF (Radial Basis Functions), and an enrichment methodology is proposed. The new approach proposed is a breakthrough for UAV shape design of high-curvature areas.

1. Introduction

This work provides a missing link in the adjoint shape optimization loop: a surface mesh deformation compliant with CAD curvature information in order to provide the required smooth transition between various independent parameterized entities.

A key element of aerodynamic shape design and optimization framework [1] is the parametric shape modeler. In accordance with the needed differentiability of the full process (from CAD parameter to cost evaluation), mesh deformation is mandatory. A master–slave approach included in CAD modeler [2] generates for each deformed CAD the corresponding surface mesh. Subsequently a 3D-mesh deformation (PDE/elliptic operator or RBF approach) is applied using the new surface mesh to obtain the associated volume mesh. The next step is a RANS (Reynolds-Averaged Navier–Stokes) computation and a cost evaluation (e.g., drag integration).

In real life applications, only a part of the whole surface is suitable for the parametric CAD mesh, as the relevant local and global CAD characteristics used for parametrization in practice redefines part of the surface accordingly but generally needs to be blended in and merged with a more global CAD definition. So, the surface mesh deformation problem includes several local pure mesh deformation zones with Boundary Conditions (BCs) resulting from the CAD-driven mesh deformation. The classical approach uses only displacement given by CAD as the BC. Our new approach adds first- and second-order derivatives given by CAD as additional BCs. Especially in cases where large shape modifications in high-curvature area (a key point for UAV design) are suggested by the optimizer for simulation during the optimization process, the inclusion of first- and second-order derivative information becomes essential to generate relevant CAD-compliant meshes with the appropriate smoothness requirements.

2. TPS with Derivatives

Several extensions of RSM have been proposed to exploit derivative information, (see [3,4,5,6,7,8,9,10,11,12]). The present approach is in line with Kybic et al. [13,14]. More applications can be found in [15,16,17].

The objective is to reconstruct the evolution of a scalar function f, based only on a finite set of measures of f and of its derivatives (order 1 and order 2):

$$\begin{array}{ccc}{\displaystyle {\mathbb{R}}^n}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle \mathbb{R}}\hfill \\ {\displaystyle X=\left(\begin{array}{c}{x}_1,x_2,...,x_n\end{array}\right)}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle f\left(X\right)}\hfill \end{array}$$

(1)

The value of f (and its derivatives) is only known on D points ${\left[X_i\right]}_{1\le i\le D}$, as shown in the set $E_D$.

$${\displaystyle E_D=\left[{\left(X_i,f_i\right)}_{1\le i\le D},{\left(d{f}_i\right)}_{1\le i\le D_1},{\left(d^2{f}_i\right)}_{1\le i\le D_2}\right]}$$

Note that, for some points, not all partial derivatives are known.

The data avalaible are composed of the following:

• D values of the function.
• $D_1$ values of the first-order partial derivatives.
• $D_2$ values of the second-order partial derivatives.

We denote $F_{E_D}$ the set of functions F from ${\mathbb{R}}^n$ to $\mathbb{R}$, which respect the constraints of $E_D$.

Obviously, there is an infinite number of functions passing through the given points and respecting the imposed derivatives. We search for the most plausible function satisfying our interpolation conditions. The plausibility criterion J is the key concept of this approach.

As we will explain, this problem can return as a minimization problem.

Thin-Plate Splines (TPS) are a spline-based technique for data interpolation and smoothing based not only on the function but also on its derivatives. In order to take into account derivatives of order $\alpha -1$ in TPS process, the semi-norm of order $\alpha$ for f, namely ${\left|\right|f\left|\right|}_{D_{\alpha }}$, is considered.

$${\left|\right|f\left|\right|}_{D_{\alpha }}^2={\int }_{{\mathbb{R}}^n}\left|\right|{\overrightarrow{D}}^{\alpha }f{\left|\right|}_{L_2}^{2}d{x}_1\dots d{x}_n$$

(2)

where ${\overrightarrow{D}}^{\alpha }$ is the vector (size: $n^{\alpha }$) of all partial derivatives of f of order $\alpha$.

$${\overrightarrow{D}}^{\alpha }f={\left[{\displaystyle \frac{{\partial }^{\alpha }f}{\partial x_{i_1}\dots \partial x_{i_{\alpha }}}}\right]}_{1\le i_1\le n,\dots ,1\le i_{\alpha }\le n}$$

(3)

This semi-norm can be associated with a symetric bilinear form so that ${\left|\right|f\left|\right|}_{D_{\alpha }}^2=B(f,f)$.

$$B(f,g)={\int }_{{\mathbb{R}}^n}{\left({\overrightarrow{D}}^{\alpha }f\right)}^T\left({\overrightarrow{D}}^{\alpha }g\right)d{x}_1\dots d{x}_n$$

(4)

The criterion J could be restricted to a bilinear form B.

$$J\left(f\right)=B(f,f);\forall f\in F_{E_D}$$

and the problem can be written as a minimization problem as follows:

$${\displaystyle f^{*}=arg\underset{\begin{array}{c}f\in F_{E_D}\end{array}}{min}J\left(f\right)}$$

(5)

The kernel $K_{D_{\alpha }}$ associated with the semi-norm of order $\alpha$ contains only the polynomials of degree $\alpha -1$.

The minimization problem becomes

$$\begin{array}{ccc}{f}^{*}\hfill & =\hfill & arg\underset{\begin{array}{c}f\in F_{E_D}\end{array}}{min}B(f,f)\hfill \end{array}$$

(6)

Proposition: The solution $f^{*}$ exists and is unique in the space $K_{D_{\alpha }}^{\perp }$, orthogonal to the Duchon semi-norm kernel $K_{D_{\alpha }}$.

Proof: By definition, the semi-norm ${\left|\right|.\left|\right|}_{D_{\alpha }}$ becomes a norm in the space $K_{D_{\alpha }}^{\perp }$.

The bilinear form B verifies the two following conditions:

• Condition of continuity:

  $$B(f,g)\le {\left|\right|f\left|\right|}_{D_{\alpha }}{\left|\right|g\left|\right|}_{D_{\alpha }}\forall f,g\in K_{D_{\alpha }}^{\perp }$$

  (7)
• Condition of $K_{D_{\alpha }}^{\perp }$-ellipticity:

  $$B(f,f)={\left|\right|f\left|\right|}_{D_{\alpha }}^2\forall f\in K_{D_{\alpha }}^{\perp }$$

  (8)

Considering a linear form L defined in $K_{D_{\alpha }}^{\perp }$, Lax–Milgram lemma gives

$$\exists !f^{*}\in K_{D_{\alpha }}^{\perp };\forall g\in K_{D_{\alpha }}^{\perp };B(f^{*},g)=L\left(g\right)$$

(9)

With this condition, we can say that $f^{*}$ is also the unique solution in $K_{D_{\alpha }}^{\perp }$ of the minimization problem (2).

But in general, the solution $f^{*}$ is composed of the sum of two parts [14]:

• The fundamental part $f_{\mathrm{fundamental}}$, which corresponds to the unique solution in $K_{D_{\alpha }}^{\perp }$.
• The kernel part $f_{\mathrm{kernel}}\in K_{D_{\alpha }}$. This kernel part can be viewed as additional degrees of freedom in order to have a more precise approximation of the scalar function.

We can write

$$f^{*}\left(X\right)=f_{\mathrm{kernel}}\left(X\right)+f_{\mathrm{fundamental}}\left(X\right)$$

In the literature, such a TPS process only requires first-order derivatives ([13,18]). In the present work, second-order derivatives will be used.

3. Definition of TPS

The kernel part corresponds to a polynome $P^{\left(2\right)}$ of degree 2.

$$\begin{array}{ccc}{f}_{\mathrm{kernel}}\left(X\right)\hfill & =\hfill & P^{\left(2\right)}\left(X\right)\hfill \end{array}$$

(10)

To illustrate, an example is given below with $n=2$ (see [15] for general case). The coefficients $a_k$ are the unknowns.

$$\begin{array}{ccc}{f}_{\mathrm{kernel}}\left(X\right)\hfill & =\hfill & a_0+a_1{x}_1+a_2{x}_2+a_3{x}_1^2+a_4{x}_2^2+a_5{x}_1{x}_2\hfill \end{array}$$

(11)

The fundamental part is composed of a sum of three parts, corresponding respectively to the approximation of order 0, to the approximation of order 1, and to the approximation of order 2.

$$\begin{array}{ccc}{f}_{\mathrm{fundamental}}\left(X\right)\hfill & =\hfill & {\displaystyle \sum _{k=1}^D}{\lambda }_k^{\left(0\right)}\varphi \left(\right||X-X_k{\left|\right|}_2)+{\displaystyle \sum _{k=1}^D}{\displaystyle \sum _{i=1}^n}{\lambda }_{{\displaystyle k,i}}^{\left(1\right)}{\displaystyle \frac{\partial \varphi }{\partial x_i}}\left(\right||X-X_k{\left|\right|}_2)\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{k=1}^D}{\displaystyle \sum _{i_1=1}^n}{\displaystyle \sum _{i_2=i_1}^n}{\lambda }_{{\displaystyle k,i_1,i_2}}^{\left(2\right)}{\displaystyle \frac{{\partial }^2\varphi }{\partial x_{i_1}\partial x_{i_2}}}\left(\right||X-X_k{\left|\right|}_2)\hfill \end{array}$$

(12)

where

$$\left\{\begin{array}{ccc}\varphi \left(\right||X-X_k|{|}_2)\hfill & =\hfill & \left|\right|X-X_k{\left|\right|}_2^{\beta }log\left(\right||X-X_k{\left|\right|}_2)\mathrm{if}\beta \mathrm{is}\mathrm{even}(\beta =2\alpha -n=6-n).\hfill \\ \\ \varphi \left(\right||X-X_k|{|}_2)\hfill & =\hfill & \left|\right|X-X_k{\left|\right|}_2^{\beta }\mathrm{if}\beta \mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(13)

The coefficients ${\lambda }_k^{\left(0\right)}$, ${\lambda }_{{\displaystyle k,i}}^{\left(1\right)}$, and ${\lambda }_{{\displaystyle k,i_1,i_2}}^{\left(2\right)}$ are the unknowns of the fundamental part and polynomial coefficients $a_k$ of the kernel part.

A key point should be mentionned concerning the main difference between RBF and TPS interpolation.

• With RBF interpolation, each imposed point is associated with a kind of Gaussian (elementary RBF function), which is equal to 1 on the current point. The RBF approximation function is obtained by a sum of all these elementary RBF functions.
• With TPS approximation, it is the opposite. Each imposed point is associated with an elementary function (fundamental part), which is equal to zero on the current point. The information on the current point comes from all the other elementary functions.

The determination of the unknowns ${\lambda }_k^{\left(0\right)}$, ${\lambda }_{{\displaystyle k,i}}^{\left(1\right)}$, ${\lambda }_{{\displaystyle k,i_1,i_2}}^{\left(2\right)}$ and $a_k$ needs the resolution of a linear problem.

$$A_D{X}_D=V_D$$

(14)

To define the three components ($A_D$, $X_D$, and $V_D$) of this linear system, the following notations are adopted:

$$\left\{\begin{array}{ccc}{\displaystyle {\varphi }_{i,j}}\hfill & =\hfill & \varphi \left(\right||X_i-X_j|{|}_2)\hfill \\ {\displaystyle d_{i_1,\dots ,i_k}g}\hfill & =\hfill & {\displaystyle \frac{{\partial }^{k}g}{\partial x_{i_1},\dots ,\partial x_{i_k}}}\hfill \\ {\displaystyle P_k^{\left(\beta \right)}}\hfill & =\hfill & \mathrm{monomes}\mathrm{of}\mathrm{order}\mathrm{lower}\mathrm{or}\mathrm{equal}\mathrm{to}\beta ,\mathrm{with}1\le k\le n_{P^{\left(\beta \right)}}\hfill \\ {\displaystyle c_i\left(k\right)}\hfill & =\hfill & \mathrm{column}\mathrm{indice}\mathrm{going}\mathrm{from}1\mathrm{to}k\hfill \\ {\displaystyle l_i\left(k\right)}\hfill & =\hfill & \mathrm{line}\mathrm{indice}\mathrm{going}\mathrm{from}1\mathrm{to}k\hfill \end{array}\right.$$

(15)

Below, the use of the column and line indices ($c_i\left(k\right)$ and $l_i\left(k\right)$) is illustrated.

$$A_{l_1\left(3\right),c_1\left(2\right),c_2\left(3\right)}=\left(\begin{array}{cccccc}{A}_{1,1,1}& A_{1,2,1}& A_{1,1,2}& A_{1,2,2}& A_{1,1,3}& A_{1,2,3}\\ A_{2,1,1}& A_{2,2,1}& A_{2,1,2}& A_{2,2,2}& A_{2,1,3}& A_{2,2,3}\\ A_{3,1,1}& A_{3,2,1}& A_{3,1,2}& A_{3,2,2}& A_{3,1,3}& A_{3,2,3}\end{array}\right)$$

(16)

The terms $X_D$ and $V_D$ are given by the following expressions.

$$\left\{\begin{array}{ccc}{X}_D^T\hfill & =\hfill & {\left(\begin{array}{cccc}{a}_{c_1\left(n_{P^{\left(2\right)}}\right)}& {\lambda }_{c_1\left(D\right)}^{\left(0\right)}& {\lambda }_{c_1\left(D\right),c_2\left(n\right)}^{\left(1\right)}& {\lambda }_{c_1\left(D\right),c_2\left(n\right),c_3\left(n\right)}^{\left(2\right)}\end{array}\right)}^T\hfill \\ V_D^T\hfill & =\hfill & {\left(\begin{array}{cccc}{f}_{c_1\left(D\right)}& d_{c_2\left(n\right)}{f}_{c_1\left(D\right)}& d_{c_2\left(n\right),c_3\left(n\right)}{f}_{c_1\left(D\right)}& 0\end{array}\right)}^T\hfill \end{array}\right.$$

(17)

The matrix is given by the following expression.

$$A_D=\left(\begin{array}{cccc}{P}_1& \varphi & d_{c_2\left(n\right)}\varphi & d_{c_2\left(n\right),c_3\left(n\right)}\varphi \\ d_{l_2\left(n\right)}{P}_1& d_{l_2\left(n\right)}\varphi & d_{l_2\left(n\right),c_2\left(n\right)}\varphi & d_{l_2\left(n\right),c_2\left(n\right),c_3\left(n\right)}\varphi \\ d_{l_2\left(n\right),l_3\left(n\right)}{P}_1& d_{l_2\left(n\right),l_3\left(n\right)}\varphi & d_{l_2\left(n\right),l_3\left(n\right),c_2\left(n\right)}\varphi & d_{l_2\left(n\right),l_3\left(n\right),c_2\left(n\right),c_3\left(n\right)}\varphi \\ 0& P_2& d_{c_2\left(n\right)}{P}_2& d_{c_2\left(n\right),c_3\left(n\right)}{P}_2\end{array}\right)$$

(18)

where

$$\left\{\begin{array}{ccc}{P}_1\hfill & =\hfill & P_{c_1\left(n_{P^{\left(2\right)}}\right)}^{\left(2\right)}\left(X_{l_1\left(D\right)}\right)\hfill \\ P_2\hfill & =\hfill & P_{l_1\left(n_{P^{\left(2\right)}}\right)}^{\left(2\right)}\left(X_{c_1\left(D\right)}\right)\hfill \\ \varphi \hfill & =\hfill & {\varphi }_{c_1\left(D\right),l_1\left(D\right)}\hfill \end{array}\right.$$

(19)

The TPS2 (i.e., using first- and second-order derivatives) interpolation has been presented here. TPS0 (non-derivative) and TPS1 (using only first-order derivatives) interpolations could be deduced. The rank of the linear system $N^{rank}$ (representative of the numerical complexity) directly depends on the use of derivatives:

• $N^{rank}$ = $N_{kernel}+D$ for TPS0 (and also for RBF and Kriging).
• $N^{rank}$ = $N_{kernel}+D(1+n)$ for TPS1.
• $N^{rank}$ = $N_{kernel}+D(1+n+{\displaystyle \frac{n(n+1)}{2}})$ for TPS2.

Here, $N_{kernel}$ is the rank of the kernel.

Coming to CPU time cost, we observe a ratio dealing with $N^{rank}$ cubed, due to Gauss factorization complexity. Further optimization could be applied for a linear system. For example, H-matrix reduction, parallelized computation, and GPU acceleration could be used and even combined. In the framework of our shape optimization loop, thousands of processors are involved (useful for Reynolds-Averaged Navier–Stokes computation). Using domain decomposition and iterative linear solver, we can efficiently apply TPS approach with Message-Passing Interface methodology and obtain a dramatic CPU decrease.

Standard interpolation techniques (kriging, RBF) have similar complexity (number and size of linear systems to solve) compared to TPS0, and so CPU cost becomes very close.

Concerning numerical stability, the eventually ill-conditioning of the system matrix can be improved using adequate preconditioners [14]. Generally, quality of the operator conditioning worsens for high-order TPS.

4. CAD-Compliant Surface Mesh

The aim of our work is to exploit the power of the CAD modeler, giving access to precise derivatives. Considering only a surface mesh based on planar elements (quads or triangles), computing derivatives on a given vertex gives only poor accuracy. In contrast, when using CAD, derivatives are computed in the CAD framework (at least $C^2$) and provided at each vertex of the surface mesh.

We consider an oriented manifold $\mathsf{\Sigma }$, supposed $C^2$. Such a surface could be defined by the following parametrization:

$$\begin{array}{ccc}{\displaystyle {\mathbb{R}}^2}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle {\mathbb{R}}^3}\hfill \\ {\displaystyle (u,v)}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle \overrightarrow{\sigma }(u,v)}\hfill \end{array}$$

(20)

Thanks to the $C^2$ class of the surface, the $u-v$ curvatures could be estimated (see Figure 1 for an illustration on a generic wing).

This $u-v$ parametrization gives access to the definition of the two principal directions and the two principal curvatures, using, for example, the Weingarten method [19].

We consider a point $P\in \mathsf{\Sigma }$, such that $P=\overrightarrow{\sigma }(u,v)$. The following two matrices $N_1\left(P\right)$ and $N_2\left(P\right)$ are defined as follows:

$$N_1\left(P\right)=\left(\begin{array}{cc}E& F\\ F& G\end{array}\right)N_2\left(P\right)=\left(\begin{array}{cc}L& M\\ M& N\end{array}\right)$$

(21)

where

$$\left\{\begin{array}{c}\overrightarrow{{\tau }_1}={\displaystyle \frac{\partial \overrightarrow{\sigma }(u,v)}{\partial u}}\overrightarrow{{\tau }_2}={\displaystyle \frac{\partial \overrightarrow{\sigma }(u,v)}{\partial v}}\hfill \\ E=\left|\right|\overrightarrow{{\tau }_1}\left|\right|G=\left|\right|\overrightarrow{{\tau }_2}\left|\right|F=<\overrightarrow{{\tau }_1},\overrightarrow{{\tau }_2}>\hfill \\ L=<\overrightarrow{n},{\displaystyle \frac{{\partial }^2\overrightarrow{\sigma }(u,v)}{\partial u^2}}>N=<\overrightarrow{n},{\displaystyle \frac{{\partial }^2\overrightarrow{\sigma }(u,v)}{\partial v^2}}>M=<\overrightarrow{n},{\displaystyle \frac{{\partial }^2\overrightarrow{\sigma }(u,v)}{\partial u\partial v}}>\hfill \end{array}\right.$$

(22)

The vector $\overrightarrow{n}$ designs the normal vector to the surface at the point P.

The Weingarten matrix $W\left(P\right)=N_1^{-1}\left(P\right)N_2\left(P\right)$ gives access to the following:

• the two principal directions ${\overrightarrow{W}}_1$ and ${\overrightarrow{W}}_2$, corresponding to the two eigen-vectors of $W\left(P\right)$.
• the two principal curvatures ${\kappa }_1$ and ${\kappa }_2$, corresponding to the two eigen-values of $W\left(P\right)$.

5. Extension of TPS Method to CAD-Compliant Surface Mesh Deformation

Given that $\mathsf{\Sigma }$ is a surface, let $\mathsf{\Gamma }=\partial \mathsf{\Sigma }$ and $\mathsf{\Omega }=\mathsf{\Sigma }\setminus \mathsf{\Gamma }$.

The $1D$ frontier $\mathsf{\Gamma }$ is discretized with $N_{\mathsf{\Gamma }}$ points ${\left(M_i^{\mathsf{\Gamma }}\right)}_{1\le i\le N_{\mathsf{\Gamma }}}$. A displacement $\Delta M_i^{\mathsf{\Gamma }}=\left(\begin{array}{ccc}\Delta x_i^{\mathsf{\Gamma }}& \Delta y_i^{\mathsf{\Gamma }}& \Delta z_i^{\mathsf{\Gamma }}\end{array}\right)$ is imposed for each of these limit nodes.

The open surface $\mathsf{\Omega }$ is also discretised, with $N_{\mathsf{\Omega }}$ points ${\left(M_i^{\mathsf{\Omega }}\right)}_{1\le i\le N_{\mathsf{\Omega }}}$.

The objective is to propagate the imposed displacement from the frontier points $({\left(M_i^{\mathsf{\Gamma }}\right)}_{1\le i\le N_{\mathsf{\Gamma }}}$ to the interior points ${\left(M_i^{\mathsf{\Omega }}\right)}_{1\le i\le N_{\mathsf{\Omega }}}$.

The displacement can be viewed as the combination of three scalar functions (${\Delta }_x$, ${\Delta }_y$ and ${\Delta }_z$), defined as follows:

$$\begin{array}{ccc}{\displaystyle \mathsf{\Sigma }}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle {\mathbb{R}}^3}\hfill \\ {\displaystyle M^{\mathsf{\Sigma }}=\left(x^{\mathsf{\Sigma }},,,y^{\mathsf{\Sigma }},,,z^{\mathsf{\Sigma }}\right)}\hfill & {\displaystyle \mapsto }\hfill & {\displaystyle \left({\Delta }_x\left(M^{\mathsf{\Sigma }}\right),{\Delta }_y\left(M^{\mathsf{\Sigma }}\right),{\Delta }_z\left(M^{\mathsf{\Sigma }}\right)\right)}\hfill \end{array}$$

(23)

With TPS interpolation, for each limit nodes $M^{\mathsf{\Gamma }}$, not only the displacement $\Delta M^{\mathsf{\Gamma }}$ but also the first-order derivatives of the displacement ($D_{w_1}(\Delta M^{\mathsf{\Gamma }}))$ and $D_{w_2}(\Delta M^{\mathsf{\Gamma }}))$) and the second-order derivatives of the displacement ($D_{w_1}^2(\Delta M^{\mathsf{\Gamma }}))$ and $D_{w_2}^2(\Delta M^{\mathsf{\Gamma }}))$) are imposed. These constraints can be written as follows (illustration for $x^{\mathsf{\Gamma }}$):

$$\left\{\begin{array}{c}{\Delta }_x\left(M^{\mathsf{\Gamma }}\right)=\Delta x^{\mathsf{\Gamma }}\hfill \\ \\ {\partial }_{{\overrightarrow{W}}_{1,M^{\mathsf{\Gamma }}}}{\Delta }_x\left(M^{\mathsf{\Gamma }}\right)=D_{w_1}(\Delta x^{\mathsf{\Gamma }}){\partial }_{{\overrightarrow{W}}_{2,M^{\mathsf{\Gamma }}}}{\Delta }_x\left(M^{\mathsf{\Gamma }}\right)=D_{w_2}(\Delta x^{\mathsf{\Gamma }})\hfill \\ \\ {\partial }_{{\overrightarrow{W}}_{1,M^{\mathsf{\Gamma }}}}^2{\Delta }_x\left(M^{\mathsf{\Gamma }}\right)=D_{w_1}^2(\Delta x^{\mathsf{\Gamma }}){\partial }_{{\overrightarrow{W}}_{2,M^{\mathsf{\Gamma }}}}^2{\Delta }_x\left(M^{\mathsf{\Gamma }}\right)=D_{w_2}^2(\Delta x^{\mathsf{\Gamma }})\hfill \end{array}\right.$$

(24)

where

$$\left\{\begin{array}{ccc}{\overrightarrow{W}}_{1,M}\hfill & =\hfill & \left(\begin{array}{c}{W}_{1,M,x}\\ W_{1,M,y}\\ W_{1,M,z}\end{array}\right)\mathrm{is}\mathrm{the}\mathrm{first}\mathrm{principal}\mathrm{direction}\mathrm{for}\mathrm{the}\mathrm{point}M\hfill \\ {\overrightarrow{W}}_{2,M}\hfill & =\hfill & \left(\begin{array}{c}{W}_{2,M,x}\\ W_{2,M,y}\\ W_{2,M,z}\end{array}\right)\mathrm{is}\mathrm{the}\mathrm{second}\mathrm{principal}\mathrm{direction}\mathrm{for}\mathrm{the}\mathrm{point}M\hfill \\ {\partial }_{{\overrightarrow{W}}_{1,M}}f\hfill & =\hfill & W_{1,M,x}{\displaystyle \frac{\partial f}{\partial x}}+W_{1,M,y}{\displaystyle \frac{\partial f}{\partial y}}+W_{1,M,z}{\displaystyle \frac{\partial f}{\partial z}}\hfill \\ {\partial }_{{\overrightarrow{W}}_{1,M}}^{2}f\hfill & =\hfill & W_{1,M,x}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{1,M}}f\right)}{\partial x}}+W_{1,M,y}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{1,M}}f\right)}{\partial y}}+W_{1,M,z}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{1,M}}f\right)}{\partial z}}\hfill \\ {\partial }_{{\overrightarrow{W}}_{2,M}}f\hfill & =\hfill & W_{2,M,x}{\displaystyle \frac{\partial f}{\partial x}}+W_{2,M,y}{\displaystyle \frac{\partial f}{\partial y}}+W_{2,M,z}{\displaystyle \frac{\partial f}{\partial z}}\hfill \\ {\partial }_{{\overrightarrow{W}}_{2,M}}^{2}f\hfill & =\hfill & W_{2,M,x}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{2,M}}f\right)}{\partial x}}+W_{2,M,y}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{2,M}}f\right)}{\partial y}}+W_{2,M,z}{\displaystyle \frac{\partial \left({\partial }_{{\overrightarrow{W}}_{2,M}}f\right)}{\partial z}}\hfill \end{array}\right.$$

(25)

The TPS method has to be applied three times, according to the three scalar functions ${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$. The geometry should be as smooth as possible. To avoid oscillations, the TPS kernel part is reduced to a constant $a_0$.

The equations are developed for ${\Delta }_x$.

The TPS approximation is the following:

$$\begin{array}{ccc}{\Delta }_x\left(M^{\mathsf{\Sigma }}\right)\hfill & =\hfill & a_0+{\displaystyle \sum _{M^{\mathsf{\Gamma }}}}{\lambda }_{1,M^{\mathsf{\Gamma }}}{\varphi }_{M^{\mathsf{\Gamma }},M^{\mathsf{\Sigma }}}+\hfill \\ \hfill & \hfill & {\displaystyle \sum _{M^{\mathsf{\Gamma }}}}{\lambda }_{2,M^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M}}{\varphi }_{M^{\mathsf{\Gamma }},M^{\mathsf{\Sigma }}}+{\displaystyle \sum _{M^{\mathsf{\Gamma }}}}{\lambda }_{3,M^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M}}{\varphi }_{M^{\mathsf{\Gamma }},M^{\mathsf{\Sigma }}}+\hfill \\ \hfill & \hfill & {\displaystyle \sum _{M^{\mathsf{\Gamma }}}}{\lambda }_{4,M^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M}}^2{\varphi }_{M^{\mathsf{\Gamma }},M^{\mathsf{\Sigma }}}+{\displaystyle \sum _{M^{\mathsf{\Gamma }}}}{\lambda }_{5,M^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M}}^2{\varphi }_{M^{\mathsf{\Gamma }},M^{\mathsf{\Sigma }}}\hfill \end{array}$$

(26)

where

$${\varphi }_{M_1,M_2}=\varphi \left(\right||M_1-M_2{\left|\right|}_2)$$

(27)

The determination of the unknowns $a_0$ and ${\lambda }_{i,M^{\mathsf{\Gamma }}}$ requires solving the linear problem (14).

The constitutive elements of the linear problem (the matrix $A_D$, the unknown vector $X_D$, and the right-hand-side vector $V_D$) are explicated for only two imposed points $M_1^{\mathsf{\Gamma }}$ and $M_2^{\mathsf{\Gamma }}$.

Concerning the matrix $A_D$ (the kernel part is reduced to polynome of order 0 for simplicity, so the matrix rank is 11), the two first lines $L_1$ and $L_2$ are the following:

$$\left\{\begin{array}{ccc}{L}_1\hfill & =\hfill & (1{\varphi }_{M_1^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}{\varphi }_{M_1^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}{\varphi }_{M_1^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}\dots \hfill \\ \hfill & \hfill & \dots {\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}^2{\varphi }_{M_1^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}^2{\varphi }_{M_1^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}^2{\varphi }_{M_2^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}^2{\varphi }_{M_2^{\mathsf{\Gamma }},M_1^{\mathsf{\Gamma }}})\hfill \\ L_2\hfill & =\hfill & (1{\varphi }_{M_1^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}{\varphi }_{M_1^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}{\varphi }_{M_1^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}{\varphi }_{M_2^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}\dots \hfill \\ \hfill & \hfill & \dots {\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}^2{\varphi }_{M_1^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}^2{\varphi }_{M_1^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}^2{\varphi }_{M_2^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}}{\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}^2{\varphi }_{M_2^{\mathsf{\Gamma }},M_2^{\mathsf{\Gamma }}})\hfill \end{array}\right.$$

(28)

The matrix line numbers 3, 4, 5, and 6 are devoted to the first-order derivatives. They are obtained by a first-order differentiation of the two first lines, as explicated below:

$$L_3={\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}{L}_1{L}_4={\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}{L}_1{L}_5={\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}{L}_2{L}_6={\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}{L}_2$$

(29)

The matrix line numbers 7, 8, 9, and 10 are devoted to the second-order derivatives. They are obtained by a second-order differentiation of the two first lines, as explicated below:

$$L_7={\partial }_{{\overrightarrow{W}}_{1,M_1^{\mathsf{\Gamma }}}}^2{L}_1{L}_8={\partial }_{{\overrightarrow{W}}_{2,M_1^{\mathsf{\Gamma }}}}^2{L}_1{L}_9={\partial }_{{\overrightarrow{W}}_{1,M_2^{\mathsf{\Gamma }}}}^2{L}_2{L}_{10}={\partial }_{{\overrightarrow{W}}_{2,M_2^{\mathsf{\Gamma }}}}^2{L}_2$$

(30)

Finally, the last line of the matrix $L_{11}$ corresponds to the reverse of the first colomn of the same matrix.

The unknwon vector $X_D$ and the right-hand-side vector $V_D$ are given below:

$$\left\{\begin{array}{ccc}{X}_D^T\hfill & =\hfill & \left[a_0{\lambda }_{1,M_1^{\mathsf{\Gamma }}}{\lambda }_{1,M_2^{\mathsf{\Gamma }}}{\lambda }_{2,M_1^{\mathsf{\Gamma }}}{\lambda }_{3,M_1^{\mathsf{\Gamma }}}{\lambda }_{2,M_2^{\mathsf{\Gamma }}}{\lambda }_{3,M_2^{\mathsf{\Gamma }}}{\lambda }_{4,M_1^{\mathsf{\Gamma }}}{\lambda }_{5,M_1^{\mathsf{\Gamma }}}{\lambda }_{4,M_2^{\mathsf{\Gamma }}}{\lambda }_{5,M_2^{\mathsf{\Gamma }}}\right]\hfill \\ V_D^T\hfill & =\hfill & [\Delta x_1^{\mathsf{\Gamma }}\Delta x_2^{\mathsf{\Gamma }}{D}_{w_1}(\Delta x_1^{\mathsf{\Gamma }})D_{w_2}(\Delta x_1^{\mathsf{\Gamma }})D_{w_1}(\Delta x_2^{\mathsf{\Gamma }})D_{w_2}(\Delta x_2^{\mathsf{\Gamma }})\dots \hfill \\ \hfill & \hfill & \dots D_{w_1}^2(\Delta x_1^{\mathsf{\Gamma }})D_{w_2}^2(\Delta x_1^{\mathsf{\Gamma }})D_{w_1}^2(\Delta x_2^{\mathsf{\Gamma }})D_{w_2}^2(\Delta x_2^{\mathsf{\Gamma }})]\hfill \end{array}\right.$$

(31)

Note that the matrix remains the same for ${\Delta }_y$ and ${\Delta }_z$. The only difference impacts the right-hand-side vector.

Finally, the displacement $(\Delta x^{\mathsf{\Omega }},\Delta y^{\mathsf{\Omega }},\Delta z^{\mathsf{\Omega }})$ for the interior point $M^{\mathsf{\Omega }}$ is given by the following relation;

$$\Delta x^{\mathsf{\Omega }}={\Delta }_x\left(M^{\mathsf{\Omega }}\right)\Delta y^{\mathsf{\Omega }}={\Delta }_y\left(M^{\mathsf{\Omega }}\right)\Delta z^{\mathsf{\Omega }}={\Delta }_z\left(M^{\mathsf{\Omega }}\right)$$

(32)

6. Applications

6.1. TPS Interpolation

The potential of the TPS interpolation is illustrated by the analytical function $g_2$, and defined below:

$$\begin{array}{c}{g}_2(x_1,x_2)=sin\left(x_1\right)cos\left(x_2\right)+cos\left(x_2\right);(x_1,x_2)\in {[1,10]}^2\hfill \end{array}$$

(33)

The value of the function (and eventually the values of its derivatives) is only known at nine points (equally spaced within the interval ${[1,10]}^2$).

This test case was an opportunity to check the operator conditioning. The lowest eigen-value is, respectively, $0.5$, $0.02$, and $0.001$ for TPS0, TPS1, and TPS2. As expected, increasing TPS order makes the linear system more difficult to solve.

Figure 2 and Figure 3 (showing the 3D surface and the 2D iso-lines, respectively) highlight the importance of derivatives (and especially of second-order derivatives) to obtain a good approximation of the function. As we can see from the bottom-left of Figure 2, TPS1 approximation captures the peak close to $x_1=1$ and $x_2=6$, but the magnitude for the highest peak ($x_1=7$ and $x_2=6$) is lower than the reference. On the contrary, using TPS2 fits the two peaks with correct magnitudes. Considering TPS0, we can see that when using only function information on the boundary $x_1=1$, we obtain relatively constant behavior between $x_2=1$ and $x_2=5$ because the two points used for interpolation ($x_2=1$ and $x_2=5$ on the boundary $x_1=1$) are at approximately the same level. Conversely, the additional first-order derivatives indicate the correct slop to follow.

The TPS is compared with two other interpolation methods: kriging [5] (similar to RBF [3]) and sparse grid [4].

With addition of the 2D-function $g_2$, a 1D-function $g_1$ (with $g_1\left(x\right)=sin\left(x\right)cos\left(x\right)+cos\left(x\right)$) and a 4D-function $g_4$ (with $g_4(x_1,x_2,x_3,x_4)=sin\left(x_1\right)cos\left(x_2\right)+cos\left(x_3\right)cos\left(x_4\right)+sin\left(x_3\right)$) are considered. The value of the function (and eventually the values of its derivatives) is only known at $5^n$ points (equally spaced within the interval ${[1,10]}^2$), with n equal to the space dimension.

Table 1. ${\displaystyle l_1}$-errors for the interpolation methods using $N_p$ points.

	${\mathit{N}}_{\mathit{p}}$	${\mathit{D}}_0$	${\mathit{D}}_1$	${\mathit{D}}_2$	Krig. Exp.	Krig. Gauss.	Sparse Grid
${\displaystyle g_1}$	5	$0.51$	$0.16$	$0.025$	$0.52$	$0.40$	$0.51$
${\displaystyle g_2}$	25	$0.23$	$0.025$	$0.0056$	$0.36$	$0.13$	$0.75$
${\displaystyle g_4}$	81	$0.92$	$0.49$	$0.28$	$0.85$	$0.91$	$0.32$

(resp.

Table 2. $l_{\infty }$-errors for the interpolation methods using $N_p$ points.

	${\mathit{N}}_{\mathit{p}}$	${\mathit{D}}_0$	${\mathit{D}}_1$	${\mathit{D}}_2$	Krig. Exp.	Krig. Gauss.	Sparse Grid
${\displaystyle g_1}$	5	$1.19$	$0.37$	$0.062$	$1.14$	$1.05$	$1.19$
${\displaystyle g_2}$	25	$1.11$	$0.21$	$0.027$	$1.33$	$0.69$	$2.77$
${\displaystyle g_4}$	81	$2.90$	$2.54$	$1.17$	$2.67$	$2.80$	$1.06$

) gives the ${\displaystyle l_1}$-errors, i.e., the mean of the absolute difference between reference and interpolation for each node (resp. the $l_{\infty }$-errors, i.e., the maximum of the absolute difference between reference and interpolation for each node) for all the interpolation methods mentioned (for the kriging approach, two kinds of correlation function are compared: exponential and gaussian ones). Like the $D_0$ approach, kriging and sparse grid methods do not use any derivatives (extensions of the kriging and RBF using derivatives could be considered).

All the 0-order methods give a similar interpolation error (even if the Gaussian kriging is the best 0-order method for these analytical functions). As expected, the use of first-order and, above all, second-order derivatives significantly reduces the interpolation error.

6.2. TPS Enrichement

In order to illustrate our enrichment methodology, we first start with TPS interpolation for a 2D case. For $x_1$ and $x_2$ between 1 and 10, we consider $g_2$ (Equation (33)).

The target is to interpolate $g_2$ over 100 points (10-10 regular grid). We start with a 9-point DoE (Design of Experiments: 3-3 regular grid). Processus is the following:

• On each point of the target grid, we compute the absolute value of difference between TPS of order 2 and TPS of order 1 approximations.
• We add to the DoE the point where the error is maximum (obviously, at this stage, the best approximation is TPS2).

On the following graph (Figure 4), you can see convergence for the following:

• $l_{\infty }$-error using TPS of order 1 (red color).
• $l_{\infty }$-error using TPS of order 2 (green color).
• Criterion (blue color).

Obviously, criterion and TPS of order 1 error are very close, and TPS of order 2 error is globally less than TPS of order 1 error.

The behavior of TPS1 error in Figure 4 seems to give a linear decrease with respect to number of calculation points.

Remark: For a DoE equal to the target grid, error is equal to zero.

From a practical point of view, starting from Figure 4, we can see that, in order to reach a $l_{\infty }$-error less than ${10}^{-2}$, $30\%$ of the target grid should be calculated.

Comparison with the classical leave-one-out enrichment strategy shows that our strategy outperforms. Our new approach is cheaper (no extra-linear systems to solve) but requires second-order derivative calculation.

6.3. CAD-Compliant Surface Mesh Deformation

In real-life applications, surface mesh of an aircraft is based on 2 types of surfaces:

• The first one is the control surface type. A CAD parametric modeler (like our GANIMEDE code [2]) uses a master–slave approach to obtain the deformed surface mesh companion that corresponds to the deformed CAD. Using this strategy leads to a differentiable shape optimization process [1]. Surface intersections (like wing–fuselage) and parameters like twist angle, Leading Edge or Trailing Edge camber, span, scale, ...are routinely involved in gradient-based shape optimizations.
• The second one is a blending surface type. Typically, this kind of surface is needed to obtain a watertight surface for the aircraft. So, we are in a position to mesh the whole surface of the aircraft.

In order to illustrate our extension of TPS to 3D-surface mesh deformation, we will consider a manufactured solution for a mesh displacement of a wing. The right wing surface mesh is built of 3138 triangular elements and 1588 vertices. Only the part of the wing span position between 8 and 10 m is not driven by the CAD modeler, and it is redefined by interpolation. The manufactured solution allows us to compute the error with respect to the TPS interpolation for order 0, 1, and 2. The displacement field consists of bending and torsional deformation (Figure 5) from the root to tip of the wing.

• ${\Delta }_x={\displaystyle \frac{y-0.986}{1.330-0.986}}\ast sin(0.5\ast \pi \ast {\displaystyle \frac{y-0.986}{1.330-0.986}})$
• ${\Delta }_y=0$
• ${\Delta }_z=10\ast {\left({\displaystyle \frac{y-0.986}{1.330-0.986}}\right)}^4+{\displaystyle \frac{y-0.986}{1.330-0.986}}\ast cos(0.5\ast \pi \ast {\displaystyle \frac{y-0.986}{1.330-0.986}})$

With ${\Delta }_x$, ${\Delta }_y$ and ${\Delta }_z$ unity equal to m.

The use of first- and second-order derivatives allows a significant reduction in the interpolation error (Figure 6 and

Table 3. Absolute ($e_a$) and relative ($e_r$) error between analytical deformation and TPS0, TPS1, and TPS2.

	$\mathit{TPS}0$	$\mathit{TPS}1$	$\mathit{TPS}2$
${\displaystyle e_a}$	6.5 × 10−2	5.5 × 10−4	1.3 × 10−5
${\displaystyle e_r}$	2.0 × 10−3	1.7 × 10−5	3.8 × 10−7

).

In the interpolation area, we have 206 vertices. For each spatial component (x, y, and z), a linear system of rank 1144 (TPS0), 4973 (TPS1), and 11431 (TPS2) is solved. Using LINPACK library (full storage double precision matrix, Gauss factorization), we obtain CPU on one chore around 2 s for TPS0, 50 s for TPS1, and 720 s for TPS2. The CPU ratio between TPS2 and TPS1 is consistent with the prediction given at the end of Section 3.

7. Conclusions

The TPS approach (based on Duchon’s semi-norm minimization) is an efficient method to construct a surface deformation based on a discrete set of imposed values of the function, augmented with imposed values of the function first- and second-order derivatives.

The aim of this paper was twofold: Firstly, to assess and compare this TPS approach to classical RSM methods. Secondly, to propose and validate an extension of differentiable manifold blending. Compared to Kriging and RBF, the TPS approach gives the best results. An enrichment technique has been proposed for the TPS approach. Validation of differentiable manifold blending mesh extension on a wing mesh deformation led us to remarkable results.