Radial Basis Functions Theory Background

This entry is adapted from the peer-reviewed paper 10.3390/fluids7090310

Radial Basis Functions were born as interpolation methods for scattered data. They provide a tool to interpolate everywhere in a generic n-dimensional space, a scalar function defined at discrete points providing the exact values at the original points.

mesh morphing radial basis functions shape optimization

1. Introduction

A design problem consists of a series of activities aimed at modifying the information that characterizes the object to be designed ^[1]. Such information is related to entities that quantify the dimension and define the shape. The designer changes the state of the entity until the result obtained is considered satisfactory. The adoption of computers as design tools enables the possibility of creating more sophisticated relationships with such information. Models that adopt algorithms to define the entities that qualify and quantify the object to be designed are called parametric. The parameters are non-geometric features defined by dimensional, geometric, or algebraic constraints ^[2]. They are used to shape the objects according to rules that determine the relationship between design intent and design response ^[3]. The result is a parent–child interdependency between the features, allowing the rapid alteration of existing models by simply editing the values of some parameters ^[4].

Parameterization is the key aspect of all procedures in which a shape variation is involved. In automatic workflows, as in numerical optimization environments, it plays a crucial role, but also direct design processes significantly take advantage of the availability of tools able to generate new geometries with moderate user manual intervention rapidly. In the context of mechanical design, the implementation of feature-based parametric modeling paradigms within CAD (Computer-Aided Design) systems provides a significant impulse to the development of more efficient design approaches. When coupling parametric geometric models with CAE (Computer-Aided Engineering) analysis tools involving discretized domains, a procedure that updates the numerical configuration following the shape variation is required. Such an approach imposes a remeshing technique ^[5] and the development of a set of scripts and batch procedures that couple/guide the code execution in sequence in an automatic workflow ^[6]. It allows very large flexibility in implementing complex combinations of constraints and variables, exploiting the great potentialities modern CAD systems provide. Advanced implementation might also incorporate a topology control of the geometry by involving the generation of new CAD features to the model allowing the use of the newly added parameters ^[7]. Nevertheless, when dealing with aerodynamic optimization, or in general with numerical problems involving large computational domains, the remeshing action of new candidates might be very time-consuming. Remeshing also introduces numerical noise when comparing different solutions due to the inconsistency of the regenerated mesh with the old one. To mitigate these drawbacks, a strategy that can be implemented is to adopt structured meshes and/or overlapping grids ^[8], but such methods are usually limited to simple geometries. Very highly skilled users in both parametric solid modeling and in CAE analyses are always required ^[9].

From an industrial point of view, fast and easier shape optimization methods that require fewer efforts in setup activities are strongly desirable. An approach acting in this direction is to operate the parameterization directly on the numerical domain using mesh morphing techniques ^[10]. This strategy allows bypassing both the CAD model coupling and the mesh regeneration process with significant advantages in terms of solution consistency, workflow robustness, and time to setup. Several numerical techniques to solve the mesh morphing problem are possible. Some of the most popular in the past were mainly based on the Free-Form Deformation (FFD) ^[11] and the elastic models ^[12] proposed in both research and commercial codes. Today, Radial Basis Functions (RBF) have become a well-established tool to interpolate scattered data ^[13] and are considered one of the most efficient mathematical frameworks to face the problem of mesh morphing. RBFs, in fact, provide a better precision allowing exact control of nodes movement and exact preservation of surfaces ^[14]. The power of RBF mesh morphing is demonstrated in ^[15], with an aerospace application concerning the optimization of a wing.

The first commercial mesh morphing tool based on radial basis functions was RBF Morph (www.rbf-morph.com, accessed on 15 August 2022) ^[16]. A description of the software can be found in ^[17]. The code proved its efficiency in several fields of engineering. Examples of aerospace applications can be found in ^[18], where it is coupled to an adjoint solver for an external aerodynamic optimization problem, and in ^[19], where it is applied to model the ice accretion on an aerofoil. In ^[20], RBFs are used to generate the surface of a measured wind tunnel model. Several problems were successfully faced with RBF mesh morphing in other fields of engineering such as nautical ^[21], biomedical ^[22], automotive ^[23], train ^[24], and oil and gas ^[25]. Examples of applications to structural problems are reported in ^[26]^[27].

2. RBF Theory Background

Radial Basis Functions were born as interpolation methods for scattered data ^[28]. They provide a tool to interpolate everywhere in a generic n-dimensional space, a scalar function defined at discrete points providing the exact values at the original points ^[29]. RBFs are efficiently used to produce a mesh movement/morphing (for both surface shape changes and volume mesh smoothing) from a list of source points and their displacements. The interpolating function composed of RBFs, defining the motion of an arbitrary point inside or outside a domain (interpolation/extrapolation), is expressed as the linear combination of the radial contribution of each source point (if the point falls inside the domain of influence) by

s (x) = \sum_{i = 1}^{N} γ_{i} φ (‖ x - x_{S i} ‖) + h (x)

(1)

where

$φ$ is the selected interpolating radial function;
$N$ is the total number of contributing source points (also called centers);
$x_{S i} = \{x_{S i}, y_{S}_{i}, z_{S}_{i}\}$ is the vector of source points positions;
$γ_{i} = {\{γ_{1}, \dots, γ_{N}\}}^{T}$ is a vector of unknown coefficients;
$h$ is a correction polynomial.

The scalar function

s (x)

is defined for an arbitrary-sized variable

x

and represents a transformation defined in a multi-dimensional space

(ℝ^{N} \to ℝ)

. At a given point

x

, the value of the RBF is obtained by summing the interactions with all of the source points

x_{S i}

constituted by the radial distance between

x

and each

x_{S i}

multiplied by the radial interaction function

φ

(consisting of a transformation

ℝ \to ℝ

) and the weight

γ_{i}

. The latter term can be seen as the “intensity” of the source point. The radial contribution of each source point is specified without any special assumptions on their number or geometric position. This characteristic renders the formulation “meshless”.

The concept of radial interaction is explained in Figure 1. The source points are arranged in the plane. The coefficients of the RBF are computed so that the function is zero at all the source points along the square edges and is

\pm 1

at two internal points. The interaction of the point

x

with all source points

x_{S}

can be repeated many times, varying the position of the point

x

inside the square. The resulting scalar value

(ℝ^{2} \to ℝ)

can be represented as a height in the 3D plot.

Figure 1. RBF interactions between source points (a) and surrounding volume (b).

A linear system with an order equal to the number of source points introduced, needs to be solved for the calculation of coefficients

γ_{i}

. A radial basis fit exists if the coefficients and the weights of the polynomial

h

can be found such that the two following conditions are satisfied:

The value of $s (x_{S i})$ assumes the desired value at the point $x_{i}$ :

s (x_{S}_{i}) = g_{s}_{i}; 1 \leq i \leq N

(2)

The system is completed if the orthogonality condition of the polynomial terms is verified for all polynomials p with a degree less or equal to that of polynomial $h$ :

\sum_{i = 1}^{N} γ_{i} p (x_{s}_{i}) = 0

(3)

A unique interpolant exists if the basis function is a conditionally positive definite function. The minimal degree of the correction polynomial $h$ depends on the choice of the basis function (a table of polynomial augmentations is reported in ^[30]). If the basis functions are conditionally positive definite of order $m < 2$ , a linear polynomial can be used. The subsequent exposition will assume that the aforementioned hypothesis is valid. A consequence of using a linear polynomial is that rigid body translations are exactly recovered. In a 3D space, the linear polynomial has the form of

h (x) = β_{0} + β_{1} x + β_{2} y + β_{3} z

(4)

The orthogonality conditions of Equation (3) can be expressed as:

\sum_{i = 1}^{N} γ_{i} = \sum_{i = 1}^{N} γ_{i} x_{k}_{i} = \sum_{i = 1}^{N} γ_{i} y_{k}_{i} = \sum_{i = 1}^{N} γ_{i} z_{k}_{i} = 0

(5)

The values for the weights of RBF vector $γ$ and the coefficients vector $β$ of the linear polynomial can be obtained by solving the system obtained by imposing the conditions of Equations (2) and (5), which are compactly expressed as:

(\begin{matrix} M & P_{s} \\ P_{s}^{T} & 0 \end{matrix}) (\begin{matrix} γ \\ β \end{matrix}) = (\begin{matrix} g_{s} \\ 0 \end{matrix})

(6)

where $g_{s} = {\{g_{S}_{1}, \dots, g_{S}_{N}\}}^{T}$ are the known values of the interpolating function at the source points and $β = \{β_{0}, β_{1}, β_{2}, β_{3}\}$ are the coefficients of the polynomial $h$ . $M$ is the interpolation matrix defined by calculating all of the radial interactions between the source points

M_{i j} = φ (‖ x_{s}_{i} - x_{s}_{j} ‖); 1 \leq i \leq N, 1 \leq j \leq N

(7)

$P_{s}$ is a constraint matrix that arises when balancing the polynomial contribution

P_{s} = (\begin{matrix} 1 & x_{s 1} & y_{s}_{1} & z_{s}_{1} \\ 1 & x_{s}_{2} & y_{s}_{2} & z_{s}_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & x_{s}_{N} & y_{s}_{N} & z_{s}_{N} \end{matrix})

(8)

Radial basis interpolation works for scalar fields, but the fit can be repeated many times using the same interpolation and constraint matrixes. In this case, the $g_{s}$ vector is replaced by a rectangular matrix and solved in a column-wise fashion, computing the coefficients $γ$ and $β$ related to each column. If a deformation vector field has to be fitted, each component of the displacement prescribed at the source points is interpolated as follows:

\{\begin{matrix} s_{x} (x) = \sum_{i = 1}^{N} γ_{i}^{x} φ (‖ x - x_{s}_{i} ‖) + β_{0}^{x} + β_{1}^{x} x + β_{2}^{x} y + β_{3}^{x} z \\ s_{y} (x) = \sum_{i = 1}^{N} γ_{i}^{y} φ (‖ x - x_{s}_{i} ‖) + β_{0}^{y} + β_{1}^{y} x + β_{2}^{y} y + β_{3}^{y} z \\ s_{z} (x) = \sum_{i = 1}^{N} γ_{i}^{z} φ (‖ x - x_{s}_{i} ‖) + β_{0}^{z} + β_{1}^{z} x + β_{2}^{z} y + β_{3}^{z} z \end{matrix}

(9)

The RBF fit guarantees the passage of the interpolated function through all the points of the original dataset with the prescribed value. The behavior (and smoothness) of the function between points (interpolation) or outside the dataset (extrapolation) depends on the radial function used. Several formulations exist in the literature.

When global support is used, the probe point interacts with all of the points of the cloud, and very high accuracy is achieved. The numerical problem becomes very challenging because a dense, fully populated interpolation matrix has to be solved in the fit stage. Each RBF evaluation requires interaction with all of the sources. Fast methods can be used to accelerate both the fit process and the evaluation of global supported RBF but only for specific cases. The Fast Multipole Method (FMM) is very well established for poly-harmonic splines ^[31]. The method is based on the concepts of the far and near field. It consists of approximating the evaluation of the interpolating function

s (x)

, considering just a few points close to the probe (near field). The effect of the remaining points (far field) is included by adding a function

s_{f a r} (x)

defined by substituting the cluster of points with an analytical expression that is independent of the number of points (Multipole Expansion). Further details about the adoption of FMM to accelerate the computation of an RBF problem are reported in ^[32].

Compact supported RBFs are used when long-distance interaction is not desired. The interaction radius can be defined. The Wendland functions with different continuity classes are collected in ^[33]. The function is active only inside the sphere, identified by the interaction radius

R_{s u p}

, while it is zero outside. It is worth noticing that the compact support affects not only the interpolation behavior but it can also simplify the fitting process as the interpolation matrix of the system becomes sparse and can be managed with numerical algorithms specifically conceived for this class of problems.

Solving large RBF problems requires a high computational cost. Nevertheless, it is suitable for a very efficient parallel implementation. Once the solution is known and shared in the memory of each cluster node, in fact, each partition can smooth its nodes without taking care of what happens outside (high scalability), implicitly guaranteeing the continuity at the interfaces. The acceleration achieved can be really very high (close to 100% efficiency) because the main cost is due to the evaluation of the inter-distance of the points (square root computation). Such a cost can be avoided by implementing specific algorithms in which space partitioning is used to rearrange the problem as the combination of sub-problems adopting Partition of Unity Methods (POU) or using the aforementioned expansion of the far field by Fast Multipole Methods (FMM). The reader can find further details regarding parallel implementations of RBF solutions in ^[32], including a benchmark on the acceleration achieved with parallel-solving technology.

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1. Introduction

2. RBF Theory Background

References

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