Development of a Method for Shape Optimization for a Gas Turbine Fuel Injector Design Using Metal-Additive Manufacturing

Gradient-based shape optimization using the adjoint method has been used to effectively compute design sensitivities in the case of complex objective functions with a large number of design variables [1–6]. Several applications including heat transfer [7–9], microfluidic flow [10–13], structural optimization [14–18], and aerodynamic optimization [19–29] have used the adjoint method to achieve optimal geometries through high-fidelity computational modeling.

Manufacturing and testing these novel, optimal geometries using traditional manufacturing methods can be challenging and expensive. The advent of additive manufacturing (AM) processes has empowered engineers with wider design flexibility. AM provides the opportunity to rapidly prototype and fabricate optimized geometries obtained from high-fidelity optimization routines employing sophisticated computational fluid dynamics (CFD) and adjoint solvers. Complex shapes such as lattice geometries, internal channels, and free-form geometries can be manufactured using AM. In 2020, 29% of all AM system sales revenue was contributed by aerospace, turbine, and helicopter industries [30]. The market potential of metal AM in the field of aerospace has been predicted to grow to $3.187 billion by 2025 at a compounded annual growth rate of 20.24% [31]. Therefore, there is growing interest in incorporating AM methods to fabricate novel, optimal designs. However, whether it is traditional or additive manufacturing, there are limitations to what can be produced [32–35]. Current and past adjoint optimization literature did very little to address the limitations of manufacturing. While there are papers that make use of geometric constraints imposed during adjoint optimization, the full ability of the computer aided design (CAD) parameterizations has not been used to incorporate process-specific manufacturability constraints [36–43]. CAD parameterizations are favorable for manufacturing because select parameterizations result in continuous surfaces that have been shown to accept geometric constraints that are reflective of manufacturability limitations [44].

While simultaneously optimizing a flow quantity using the adjoint method and imposing AM constraints, a key consideration is the difference in the parametrization schemes used by commercially available, high-fidelity adjoint CFD tools and the parameterization scheme that exists within CAD. Various kinds of shape parametrization schemes exist for different applications and their importance in shape optimization have been highlighted by Samareh [45]. Commercial CFD software typically use radial basis functions (RBFs) that discretize the geometry into elements and morph the mesh with respect to a given set of handles or control points [46–50], whereas CAD typically uses nonuniform rational basis splines (NURBS), which is a continuous representation for surfaces [51–54]. Currently, imposing geometric constraints during RBF-based mesh morphing remains a challenge within commercial software, which leads to creation of features such as sharp edges that might be optimal as per the numerical approximation of the computed solution but might be challenging, or even impractical, to reproduce during fabrication. NURBS surfaces handle such issues and ensure smoothness during deformation in addition to providing an exact representation of complex, freeform geometries that might result from such shape optimization routines. Several researchers [55–58] have proposed various methods of incorporating NURBS-based mesh morphing. However, implementation of the proposed methods has yet to be integrated with commercially available CFD software. Thus, in the current state of the process, the adjoint shape gradients are computed on the mesh defined by the RBF interpolation function where a point set controls the shape of the geometry under consideration. These shape gradients are then exported and used to guide shape changes in CAD. The use of different parameterization schemes used in CFD and CAD makes it challenging to exchange data to automate the process of simultaneously optimizing flow using CFD and imposing manufacturability constraints.

The objective of this study is to develop a method to enable simultaneous optimization of flow fields using the adjoint method and impose design for additive manufacturing (DfAM) constraints. This objective was achieved by augmenting a commercial CFD adjoint routine to externally modify CAD geometry and impose DfAM constraints that are incompatible with current commercial solvers. The key challenge of geometry definition was resolved by ensuring consistencies in the adjoint-guided shape deformation between NURBS-interpolated surfaces in CAD and RBF-interpolated surfaces within a commercially available CFD. Showing that this consistency in surface deformation lies within AM laser-powder bed fusion (L-PBF) process tolerances justifies the use of the adjoint shape gradients computed on an RBF interpolated surface within a commercially available CFD solver to externally modify a NURBS interpolated surface within CAD to ensure that the optimal design is also additively manufacturable. The graphical abstract highlighting the elements of the study is shown in Fig. 1.

This paper is organized as follows. Section 2 describes the proposed methodology of augmenting a commercial CFD adjoint shape optimization routine while emphasizing on shape parameterization consistencies, Sec. 3 demonstrates a test case on a simplified airfoil geometry, Sec. 4 presents the analysis on a complex airfoil geometry within the fuel injector of an industry relevant gas turbine engine, and Sec. 5 discusses the concluding statements and future work.

The methodology's core motivation was the integration of CAD-based DfAM constraints into physics-guided shape optimization. This section examines the current adjoint shape optimization routines within commercial CFD solvers, followed by delineating the steps to augment them with DfAM constraints. Additionally, this discussion addresses the assumptions and premises essential for ensuring consistency, forming the basis for justifying the exchange of information between software during the augmentation process.

The software star-ccm+ 2021.3 was used to demonstrate the augmentation process of the adjoint shape optimization routine. This software was chosen because of its widespread use to perform multifidelity physics simulations for rapid industry adoption. The process of adjoint shape optimization consists of the following steps:

1. Defining the design variables: The CAD file containing the fluid domain is imported. The inlets, outlets, and walls are assigned the initial and boundary conditions. The design variables are defined by importing a .csv table containing the three-dimensional coordinates of the control points for the surfaces to be modified. Most commercial software impose an RBF-based interpolation between the control points and the surface to be modified during the shape optimization routine [59].

2. Meshing: To accommodate the scale of the simulation domain while minimizing the computational time, structured prism cells were used for the near-wall regions while polyhedral meshing was used for the free-stream flow regions.

3. Setting up the primal solver: The primal solution establishes the relationship between the input design and the resulting flow variables by satisfying the principles of a physical continuum. The primal solver was set up for configuring a steady, non-reacting Reynolds-averaged Navier–Stokes (RANS) simulation to satisfy the mass and momentum conservation. The κ–ε turbulence model was used. The working fluid was defined as low-velocity incompressible air with constant density. Both the test cases used the same physics set up with appropriate initial and boundary conditions. The simulations were run for a user-specified number of iterations to achieve proper convergence in all simulations.

4. Setting up the adjoint solver: The adjoint solver computes the sensitivities of an objective function, which is a flow field variable, with respect to the geometry to either maximize or minimize its value. To represent a custom region of interest (ROI) over which the objective function was computed, a flag-based field function was created to compute flow quantities of interest and defined as the adjoint objective function. The adjoint solver was set to run for a specified number of iterations until the residuals converged.

5. Computing mesh sensitivity: The mesh sensitivity is computed using the primal and adjoint results. The mesh sensitivity provides the change in the objective function because of changes in the mesh geometry and is represented as $dJ/dX$ where J is the objective function and X represents the mesh vertices. When using design variables for mesh deformation, the mesh sensitivity is combined with the shape parameterization scheme being used to ensure a computationally efficient mesh movement operation. As discussed earlier, most commercial CFD solvers, uses RBF to parameterize the mesh movement. Therefore, the shape sensitivity is computed as

   $dJdα=dJdXdXdα$

   (1)

   where α is the control points that are used to parameterize the mesh movement and $dX/dα$ is the relationship between the control points and the mesh vertices (RBF in this case).
6. Deforming the mesh: The mesh is deformed by modifying the control points and obtaining new control points using

   $αnew =αold+(step size×dJdαold)$

   (2)

   Like any gradient-based optimization, a scalar step size is added to avoid over-shooting the optimal solution and specifically to avoid large mesh deformations in this case. The sign of the step-size also determines if the optimizer maximizes or minimizes the objective function. The objective function is maximized for a positive step size and minimized for a negative step size.

Steps 3–6 are repeated until the objective function is optimized, resulting in an optimized geometry.

During a standard adjoint shape optimization routine, there is limited provision in current commercial solvers to impose geometric constraints while deforming the mesh. Therefore, to simultaneously optimize fluid flow using the multiphysics adjoint while imposing DfAM constraints, augmentation of the standard adjoint shape optimization routine is proposed. After computing the shape sensitivities as described in step 6 of the adjoint shape optimization routine, the shape sensitivities along with the control point locations were exported from the CFD software. These shape sensitivities, combined with the defined step size, provide displacement vectors which were used to modify the CAD parametric data externally and obtain a modified CAD geometry. Since the modification is performed externally, the DfAM constraints can be incorporated as a CAD operation followed by re-inserting the modified CAD into the CFD adjoint shape optimization loop. The proposed method for augmentation of the adjoint shape optimization loop is shown in Fig. 2.

During the proposed augmentation process, two key considerations were addressed. The first consideration was to automate the CAD modification process to enable re-insertion of the modified CAD into the CFD adjoint shape optimization loop. The second consideration was to ensure that the control point displacement data exported from the CFD solver would result in a consistent shape change in CAD despite different parameterization schemes used in both operations.

The CAD modification was automated in Python using the NURBS-Python (geomdl) library [60]. To define a NURBS surface, the control point coordinates, degrees of polynomials, and knot vectors need to be specified [61]. These parameters are available through the initial graphics exchange specifications (IGES) file format under the “Type 128” data header [62]. The NURBS control point coordinates for the surfaces to be modified were extracted from the IGES file and defined as the design variables as defined in step 1 of the adjoint CFD shape optimization setup. Reading the control point coordinates stored in the IGES file, modifying each point using the displacement vectors from CFD software, and replacing the modified control points in the IGES file resulted in a modified CAD geometry. This process is illustrated in Fig. 3. Additionally, the IGES file format is a standard exchange file format that can be directly imported into CFD software for meshing. Since the input to the CFD software was the flow domain, the IGES file of the flow domain was used to extract the NURBS parameters and automate the CAD modification process. Having full control over the CAD parameterized geometry enabled the creation and enforcement of NURBS-based DfAM constraints during the Python modification step. The formulation of DfAM constraints like limits on thin walls and overhang angles has been discussed in detail in a separate publication along with ensuring geometric continuities during CAD shape modification [63].

As discussed, to augment the existing adjoint shape optimization process the control points from CAD must be defined within the CFD solver. Further, the adjoint shape sensitivities computed within the CFD solver must be used to externally modify the CAD geometry using Python while imposing DfAM constraints. The first step to ensure consistencies during shape change was defining the CAD NURBS control points as design variables in the CFD software. Since the adjoint shape sensitivities were computed on the NURBS control points, the displacement values for each control point were directly used to obtain the new set of NURBS control points. However, despite sharing the same control point coordinates, most commercial CFD solvers, use RBF to interpolate the underlying surface during mesh deformation. Therefore, the same displacement for colocated control points did not trivially imply the same resultant shape change due to different parameterization schemes.

In Eqs. (4) and (5), the first three terms on the right-hand side are built into the grid generation tools that are specific to the commercial software package. Hence, for a given geometry, it is the same in both cases. Therefore, if it can be shown that for α_RBF = α_NURBS, if $∂SRBF/∂αRBF≈ ∂SNURBS/∂αNURBS$⁠, then $(dJ/dα)RBF≈ (dJ/dα)NURBS$⁠. For the scope of this paper, the manufacturability tolerance using AM L-PBF was defined as the critical value within which the difference between $∂SRBF/∂αRBF$ and $∂SNURBS/∂αNURBS$ were considered insignificant. The AM L-PBF process tolerance was identified as 250 μm [64] for Inconel-based alloys, which are the choice of alloys for fabrication of gas turbine hot-section components. Thus, if $|∂SRBF/∂αRBF −∂SNURBS/∂αNURBS|<250 μm$⁠, then $(dJ/dα)RBF≈ (dJ/dα)NURBS$⁠.

Both RBF [65–68] and NURBS [69–73] have been studied extensively. Researchers have found that increasing the number of control points improves the accuracy of surface representation and provides better local control. This trend suggests that with fewer control points, each control point's displacement affects a larger area of the surface, leading to more significant variations between the two interpolation schemes (RBF and NURBS) for the same displacement on colocated control points. Conversely, with a higher number of control points, each surface point moves independently, resulting in minimal variation between the two schemes. By selecting an appropriate distribution of control points, the difference in errors between the interpolation schemes can be kept within the tolerance of the AM L-PBF process. Like a grid resolution study in CFD, a NURBS resolution study helps verify the assumptions being made during parameter selection for such shape optimization routines. With the difference between the two modified surfaces lying within the AM process tolerance, the reproducibility of both designs during the manufacturing process is indistinguishable. Showing this assumption justifies the use of the adjoint gradient computed on a high-resolution RBF parameterized geometry in CFD to modify the same high resolution NURBS parameterized geometry in CAD. One of the main advantages of using the adjoint method is that the computation time is independent of the number of design variables being used and rather depends on the complexity of the objective function being computed. This warrants the use of a sufficiently large number of design variables that balance the tradeoff between maintaining shape consistencies between CFD and CAD while being computationally efficient during CAD modification and DfAM constraint calculation.

To understand how the parameter selection affected the consistencies in shape changes between CFD and CAD, the two relevant parameters identified were the number of control points used as design variables and the step size of optimization. Geomagic Design X was used to obtain custom NURBS parameterization of designs to study the effect of number of control points. Step sizes of 1, 2, 5, 10, 25, 50, 75, and 100 μm were chosen to study the effect of step size.

where GV = 〈x_m − x_R, y_m − y_R, z_m − z_R〉.

In addition to the RMS value, the maximum difference was also noted to account for the extreme deviations between the two surfaces.

A simple airfoil geometry was chosen for performing an initial test to study the difference between NURBS- and RBF-based surface modifications with respect to number of control points and step sizes. Different NURBS configurations of the airfoil were created with 96, 128, and 640 control points each. The adjoint shape optimization was setup in the CFD software as described in Sec. 2.1 with the objective of maximizing the lift to drag ratio by changing the airfoil geometry using the control points. The primal solution was computed over 500 iterations and the adjoint solution was computed over 250 iterations. The step sizes were enforced during the mesh deformation stage after the first loop of primal and adjoint computations. The STLs of the RBF morphed mesh were exported from the CFD software along with the displacement vectors computed on the control points that were used to create the NURBS modified CAD geometry. The error metric for all configurations of number of control points and step sizes are shown in Fig. 4. As expected, it was observed that increasing the refinement of the control point set resulted in reduction of the error metric between the RBF and NURBS surfaces, thus displaying convergence. For all configurations, increasing step size resulted in increasing the error metric between the NURBS and RBF deformed shapes. However, for a larger number of control points, the error metric was below the AM L-PBF process tolerance for step sizes under 25 μm. Thus, having a greater number of control points provides the opportunity to use a larger step size while still ensuring consistencies between the two parametrization schemes remain under the AM L-PBF process tolerances. Using larger step sizes allows for fewer optimization loops, reducing the total computational time for the optimization. However, using a significantly large step size could lead to overshooting the optimal solution and mesh intersection errors. Thus, it is important to refine the step size selection process to choose the step size based on the tradeoff between computational time and solution accuracy.

To test the methodology on a gas-turbine relevant case study with sufficiently complex geometry parts suitable for AM L-PBF fabrication, we considered the issue of shape optimization to minimize the flame flashback propensity in an industrial gas turbine combustor. Due to rising environmental concerns such as climate change [75,76], there is an increasing trend within the energy sector to evaluate the use of alternative fuels for production of cleaner energy. One of the most promising alternative fuels has been the incorporation of hydrogen (H₂) in natural gas to reduce CO₂ emissions [77]. However, introduction of H₂ in the fuel mix leads to an increase in the turbulent flame speed. Flashback occurs when the flame propagates toward the upstream gases at velocities higher than the incoming flow velocity. This propagation propensity is the highest along the turbulent boundary layer region near the edge of the center body where the flame sits [78–85]. For this study, the chosen objective function was to maximize the volume-averaged velocity magnitude of the turbulent boundary layer region around the edge of the center body of the fuel injector. The maximization was to be achieved by modifying the shape of the external airfoil surfaces of the swirler vanes.

The baseline geometry of the swirler vanes was reverse engineered to fit 850 NURBS surface patches using Geomagic Design X. Each NURBS surface patch consisted of 64 control points (8 × 8 bidirectional net) with a degree-three polynomial in the u- and v-directions in order to maintain C2 continuity along the surface, resulting in 54,400 total control points. These points were also defined as the RBF control points in star-ccm+. Both parameterized geometries are shown in Fig. 5.

The injector design under consideration represented an annular configuration of axisymmetric swirler vanes with a proprietary nonaxisymmetric fuel delivery mechanism to accommodate industrial complexity. Furthermore, in order to investigate any effects of the upstream and downstream flow on the adjoint computation, the flow domain of the experimental test rig for an axisymmetric section of the turbine engine was modeled for the simulations. Surfaces representing the inlets for air and fuel were defined as mass flow inlets, the outlet for the exhaust was defined as a pressure outlet, and the remaining surfaces including the vanes were defined as adiabatic walls. A grid convergence study was performed using seven different mesh sizes ranging in a total cell size of 2–26 million cells. The final adaptive mesh representation contained ∼4.5 million cells and was achieved through a base size of 4 mm with varying custom meshing controls applied from the combustor section with a value of 10% (0.4 mm) of the base size to add refinement in the swirler regions to a value of 250% (10 mm) relative to the base size for the regions in the exhaust region. The turbulent boundary layer thickness for the given flow conditions was calculated as 3 mm using the Blasius wall friction profile as described in Ref. [78]. A 1:10 ratio for thickness to length ratio of the annulus region was used to represent the ROI to compute the adjoint objective function value. The mesh configuration with the objective function ROI is shown in Fig. 6.

The flow solver was setup using steady RANS with the while imposing the κ–ε turbulence model. A fully premixed flow of 0.0632 kg/s preheated at 523 K entered the inlet to maintain a bulk flow velocity of 40 m/s in the main duct. The boundary conditions were matched as per Ref. [86].

The optimization was performed for one loop using the adjoint CFD shape optimization method as well as the augmented method with external CAD modification. The primal solution was computed over 1500 iterations to achieve residual convergence for flow and the adjoint solution was computed over 500 iterations to achieve residual convergence for the objective function. The CFD adjoint shape optimization was performed through batch parallel processing using Penn State's ROAR supercomputing cluster using 96 cores. The primal solution took 1.5 h for 1500 iterations. The meshing, adjoint solver, computation of mesh sensitivity, and mesh deformation took 15, 45, 6, and 8 min, respectively. Modification of the IGES CAD file using Python was performed on a laptop notebook with four cores, 16 GB RAM and took 225 s from reading the file to creating the modified file.

Figure 7 shows the graph of error metric of the RBF and NURBS vanes surfaces for increasing step sizes. It was observed that for the chosen configuration of 54,400 control points the error metric between the two results of modified surfaces was below the AM L-PBF process tolerance of 250 μm for all step sizes. The maximum RMS difference observed was 0.0701 mm for the step size of 100 μm, while the minimum RMS difference was 0.0472 mm for the step size of 1 μm. Additionally, Fig. 8 shows the maximum difference between RBF and NURBS surfaces obtained from the CFD software and Python, respectively. It was found that the maximum difference between RBF and NURBS modified surfaces lied below the AM L-PBF process tolerance of 250 μm for step sizes less than 50 μm. The maximum values were affected by the large magnitudes of the adjoint gradients that were present in high sensitivity zones of the objective function with respect to the geometry. Overall, 25 μm was determined to be the largest step size using which the maximum difference between NURBS and RBF were consistent within AM L-PBF process tolerances.

In addition to showing geometric consistencies, the objective function changes due to the different parameterizations were also investigated. The CAD modified flow domain was re-imported into the CFD software's adjoint shape optimization loop. The objective function evolution for this modified flow domain and the baseline flow domain. The temperature profiles for a plane section of the flow region were also compared between the adjoint CFD shape optimization and the augmented method after re-inserting the CAD modified flow domain. This comparison was performed by exporting the coordinates of the mesh vertices for the plane section geometry of the NURBS and the RBF deformed flow domains and its associated physical quantities in a .csv file. Since the NURBS modified flow domain had to be re-imported and remeshed, the coordinates of the mesh vertices slightly varied from the baseline. Thus, for each mesh vertex, the closest mesh vertex in the baseline plane section was found and their associated physical quantities were compared.

Figure 9 shows the evolution of the objective function for the baseline (RBF deformation) and the NURBS modified flow domain restarted after the first loop shape changes. The objective function for both geometries were consistent within 1% variation with the value of the volume averaged velocity magnitude computed over the objective function region for the baseline optimization loop being 64.1 m/s and for the NURBS parameterized geometry of the equivalent shape change being 64.5 m/s. The subsequent loops showed consistent values for each shape change. Additionally, the NURBS modification resulted in a much smoother shape change for the same adjoint gradients and step size as compared with the RBF modification that had sharper edges.

The net effect of the shape changes on the physical quantities after solving the primal physics on both NURBS and RBF modified geometries is shown in Fig. 10. The velocity magnitude and axial velocity plots were compared for both the shape changes. For this comparison, the values compared at corresponding vertices between the two shapes were mostly zero. The small non-zero difference values of the quantities were attributed to the minor shape changes between the two parameterization schemes. The larger non-zero difference values of the quantities were an artifact of the variability between different iterations within the residual convergence region of the solution approximation. The arguments that supported this claim were that the large variations were isolated mesh vertices and most of the instances were observed upstream of the swirler where no geometric changes were made or in regions where the mesh size was relatively coarse and hence any change in values were an artifact of small variations within the residual convergence region and remeshing differences. Additionally, the difference in the pressure drop computed across the injector between the RBF and NURBS modified geometry was 0.188%. Thus, the physical quantities obtained using NURBS deformed geometries were consistent with RBF deformed geometries while providing a smoother shape and a standardized CAD exchange file format.

This study proposed a method to enable simultaneous optimization of flow field and imposing DfAM constraints by justifying the use of adjoint gradients computed on RBF parameterized surfaces in CFD to guide the movement of NURBS parameterized surfaces in CAD as a motivation to perform constrained isogeometric optimization. Using a test case, it was verified that like a CFD grid resolution study, a NURBS control point density resolution study needs to be performed to obtain a sufficient middle ground between surface modification consistency and computational efficiency. The results of an adjoint shape optimization of the baseline swirler vanes geometry in CFD using RBF compared to NURBS modification in CAD showed that the RMS difference between the surfaces were consistent within the AM L-PBF process tolerances for all step sizes. It also showed that the maximum difference between the two surfaces were consistent within the AM L-PBF process tolerance below step sizes of 25 μm. To consolidate consistency in CFD behavior, the objective function value computed for both geometries were within 1% of each other as well as the physical quantities computed using the CFD primal solver were within residual convergence tolerances. The process for external modification of the CAD surfaces added minimal computation overhead, however, the resubmission of the modified geometry in the optimization loop within the CFD software added additional computational time due to remeshing of the newly imported geometry. This additional step added a 8.92% increase in computation time per optimization loop, which can be significant for problems requiring more number of loops to converge to a solution and for complex/large geometry mesh setups. Therefore, this warrants the current and future integration of NURBS into CFD flow software to not only enable NURBS-based design constraints but also improved computational efficiency.

Currently, the adjoint solver can accept limited flow field variables and user defined operations as the objective function and largely uses simplified steady nonreacting RANS physics models to compute those. Moreover, the shape change caused by imposing DfAM constraints can be favorable or against the displacements due to the adjoint physics depending on the geometry, chosen objective function, build orientation, and other dominating factors. The future work scope includes defining and imposing such DfAM constraints using the NURBS representation in the design loop to obtain novel optimal geometries and study its effects on the adjoint objective function. With developments in commercial solvers, more complex models will be compatible as adjoint cost functions in the future to address critical issues such as emission control, fuel flexibility, and management of thermoacoustic instabilities to enhance gas turbine performance through combustor design modifications. The method proposed can also be used to compare different parameterization schemes and show similar justifications across different commercial finite element solvers for a wide range of applications. As AM process tolerances improve, the error metric will become smaller to justify the use of different parameterization schemes. However, the adoption of NURBS-based mesh movement being integrated directly into commercial solvers would facilitate the justification of data exchange between finite element solvers and CAD programs. The proposed method augments a commercial finite element adjoint shape optimization routine and shows justification to exchange data to ensure that optimal designs are additively manufacturable for a wide range of applications.

The authors are grateful for the financial support provided by the U.S. Department of Energy University Turbine Systems Research (DoE UTSR) Program Grant No. DE-FE0031806 under contract monitor Mark Freeman. The authors would also like to thank Jinming Wu, Pratikshya Mohanty, and Drue Seksinsky for their help in setting up the simulations.

• U.S. Department of Energy University Turbine Systems Research (DoE UTSR) Program (Grant No. DE-FE0031806; Funder ID: 10.13039/100000015).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

J =

objective function

n_points =

number of points

P_m =

points on the measured surface

P_R =

points on the reference surface

S =

geometric surface

S_NURBS =

surface define using NURBS

S_RBF =

surface defined using radial basis function

V =

volumetric grid

X =

surface grid

x_m =

x-coordinate of the measured surface

x_R =

x-coordinate of the reference surface

y_m =

y-coordinate of the measured surface

y_R =

y-coordinate of the reference surface

z_m =

z-coordinate of the measured surface

z_R =

z-coordinate of the reference surface

α =

design variables

α_NURBS =

control point coordinates for NURBS surface

α_RBF =

control point coordinates for RBF surface