THB-Spline Approximations for Turbine Blade Design with Local B-Spline Approximations

Abstract

We consider two-stage scattered data fitting with truncated hierarchical B-splines (THB-splines) for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares variational spline approximations, exploiting a simple fairness functional to handle data distributions with a locally varying density of points. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the whole scattered data set and a suitable strategy to guide the adaptive refinement is introduced. A selection of examples on geometric models representing components of aircraft turbine blades highlights the performances of the scheme. The tests include a scattered data set with voids and the adaptive reconstruction of a cylinder-like surface.

Appendix

In this appendix we give the proof that, assuming d = (d ₁, d ₂) with d _i ≥ 1, i = 1, 2, the local vector spline s _J defined in Sect. 3.3 exists and is unique, provided that the sites in X _J are not collinear. First of all we observe that the objective function in (3.2) can be split in the sum of three analogous objective functions, one for each component $s_J^{(k)}, k=1,2,3,$ of s _J,

$$\displaystyle \begin{aligned} \sum_{i \in I_J} \Vert {\mathbf{s}}_J({\mathbf{x}}_i) - {\mathbf{f}}_i \Vert_2^2 + \mu \, E({\mathbf{s}}_J) = \sum_{k=1}^3 \big( \sum_{i \in I_j} \big( s_J^{(k)}({\mathbf{x}}_i) - ({\mathbf{f}}_i)_k \big)^2 + \mu \, \rho( s _J^{(k)} ) \big) \,,\end{aligned}$$

where

$$\displaystyle \begin{aligned}\rho \big( s _J^{(k)}\big) := \int_{\Omega_J} \left(\frac{\partial^2 s_J^{(k)}}{\partial x^2}\right)^2 + 2 \left(\frac{\partial^2 s_J^{(k)}}{\partial x \partial y}\right) ^2 +\left(\frac{\partial^2 s_J^{(k)}}{\partial y^2}\right)^2 \,\text{d}x\text{d}y\,.\end{aligned}$$

Thus the study can be developed in the scalar case and for brevity we remove the subscript or superscript k ranging between 1 and 3. The analysis is developed in the following theorem, where $s_J: \Omega _J \rightarrow \mathbb {R}$ denotes the local spline in $\mathcal {S}_J$ associated to the coefficient vector $\mathbf {c} \in \mathbb {R}^{\ell _J},$ with ℓ _J:= | Λ_J| ,

$$\displaystyle \begin{aligned}s_J(\mathbf{x}) = \sum_{I \in \Lambda_J} c_I B_I(\mathbf{x})\,.\end{aligned}$$

Theorem

Let the considered spline bi-degree d = (d ₁, d ₂) be such that d _i ≥ 1, i = 1, 2. When the points x _i ∈ Ω _J, i ∈ I _J are not collinear, there exists a unique local spline $s_J \in \mathcal {S}_J$ minimizing the following objective function,

$$\displaystyle \begin{aligned} \sum_{i \in I_J} \big(s_J({\mathbf{x}}_i) - f_i \big)^2 + \mu \, \rho(s_J)\,, \end{aligned} $$

(3.4)

where μ > 0. If such points in Ω _J are collinear, then such minimizer does not exist or is not unique.

Proof

Let us observe that ρ(s _J) = c ^T M c , where $M \in \mathbb {R}^{\ell _J \times \ell _J}$ is such that

$$\displaystyle \begin{aligned}M_{i,r} {:=} \int_{\Omega_J} \left(\frac{\partial^2 B_I }{\partial x^2}\right)\left(\frac{\partial^2 B_R }{\partial x^2}\right) + 2 \left(\frac{\partial^2 B_I}{\partial x \partial y}\right) \left(\frac{\partial^2 B_R}{\partial x \partial y}\right) +\left(\frac{\partial^2 B_I }{\partial y^2}\right)\left(\frac{\partial^2 B_R }{\partial y^2}\right)\text{d}x\text{d}y\,,\end{aligned}$$

where we are assuming that, in the adopted ordering of the B-spline basis elements of $\mathcal {S}_J,$ B _R and B _I are the r–th and the i–th ones. On the other hand it is

$$\displaystyle \begin{aligned}\sum_{i \in I_J} \big(s_J({\mathbf{x}}_i) - f_i \big)^2 = \Vert V \mathbf{c} - \mathbf{F} \Vert_2^2\, = {\mathbf{c}}^T A^TA \mathbf{c} - 2{\mathbf{F}}^TA \mathbf{c} + {\mathbf{F}}^T\mathbf{F}\,,\end{aligned}$$

where $\mathbf {F} \in \mathbb {R}^{|I_J|}$ denotes the vector collecting all the f _i, i ∈ I _J and A is the |I _J|× ℓ _J collocation matrix of the tensor-product B-spline basis generating $\mathcal {S}_J.$ Thus the objective function in (3.4) can be written also as the following quadratic function,

$$\displaystyle \begin{aligned}{\mathbf{c}}^T (A^TA + \mu M) \mathbf{c} - 2{\mathbf{F}}^TA \mathbf{c} + {\mathbf{F}}^T\mathbf{F}\,.\end{aligned}$$

As well known a quadratic function admits a global unique minimizer if and only if the symmetric matrix defining its homogeneous quadratic terms is positive definite and in such case the minimizer is given by its unique stationary point. In our case such matrix is A ^T A + μM and the stationary points are the solutions of the following linear system of ℓ _J equations in as many unknowns,

$$\displaystyle \begin{aligned}(A^TA + \mu M) \mathbf{c} = A^T \mathbf{F}\,.\end{aligned}$$

Now, for all positive μ the matrix A ^T A + μM is symmetric and positive semidefinite since, for any vector $\boldsymbol {\zeta } \in \mathbb {R}^{\ell _J}, \boldsymbol {\zeta } \ne \mathbf {0}$ it is ζ ^T A ^T A ζ ≥ 0 and ζ ^T M ζ ≥ 0, the last inequality descending from the fact that ζ ^T M ζ = ρ(s _ζ), where $s_{\boldsymbol {\zeta }}(\mathbf {x}) = \sum _{I \in \Lambda _J} \zeta _I B_I(\mathbf {x})\,.$ Now if the points x _i, i ∈ I _j are distributed in Ω_J along a straight line ax + by + c = 0, the proposition proved in Sect. 3.3 implies that it is possible to find $\boldsymbol {\zeta } \in \mathbb {R}^{\ell _J}, \boldsymbol {\zeta } \ne \mathbf {0}$ such that s _ζ(x) ≡ ax + by + c, ∀x ∈ Ω_J. This implies that s _ζ(x _i) = 0 , ∀i ∈ I _J, that is the vector $A \boldsymbol {\zeta } \in \mathbb {R}^{|I_J|}$ vanishes. On the other hand, clearly it is also 0 = ρ(s _ζ) = ζ ^T M ζ , since s _ζ| Ω_J is a linear polynomial. This proves that the symmetric positive semidefinite matrix (A ^T A + μM) is not positive definite when all the x _i, i ∈ I _J are collinear. This is the only possible data distribution associated to a non positive definite matrix. Indeed if the points x _i, i ∈ I _J are not collinear, if $\boldsymbol {\zeta } \in \mathbb {R}^{\ell _J}, \boldsymbol {\zeta } \ne \mathbf {0}$ is associated to a non vanishing linear polynomial, it is ζ ^T M ζ = ρ(s _ζ) = 0 but A ζ≠0 and so ζ ^T A ^T A ζ > 0; on the other hand if $\boldsymbol {\zeta } \in \mathbb {R}^{\ell _J}, \boldsymbol {\zeta } \ne \mathbf {0}$ is not associated to a linear polynomial, then ζ ^T M ζ = ρ(s _ζ) > 0. □