A system of radial functions is used to produce a solution for the mesh movement/morphing, from a list of source points and their displacements. This approach is valid for both surface shape changes and volume mesh smoothing.
The RBF problem definition does not depend on the mesh.
Radial Basis Function interpolation is used to derive the displacement in any location in the space, so it is also available in every grid node.
Radial basis were born as an interpolation tool for scattered data. They solve a very useful mathematical problem because they are able to interpolate everywhere in the space a function defined at discrete points giving the exact value at original points. The behaviour of the function between points depends on the kind of basis adopted.
The radial function can be defined with compact support, or valued everywhere, a polynomial corrector is added to guarantee compatibility for rigid modes.
Typical radial functions are reported in the following table.
As will be shown in detail a linear system (of order equal to the number of source point introduced) need to be solved for coefficients calculation. Once the unknown coefficients are calculated the motion of an arbitrary point inside or outside the domain (interpolation/extrapolation) is expressed as the summation of the radial contribution of each source point (if the point falls inside the influence domain).
Details of the theory need to be given using some equations. We look for an interpolation function composed by a radial basis and a polynomial as follows:
The degree of the polynomial has to be chosen depending on the kind of radial function adopted. A radial basis fit exists if the coefficients g and the weight of the polynomial can be found such that the desired function values are obtained at source points and the polynomial terms gives 0 contributions at source points:
The minimal degree of polynomial depends on the choice of the basis function . A unique interpolant exists if the basis function is a conditionally positive definite function. If the basis functions are conditionally positive definite of order less or equal than 2, a linear polynomial can be used. The subsequent exposition will assume valid the aforementioned hypothesis. A consequence of using a linear polynomial is that rigid body translations are exactly recovered. The values for the coefficients γ and the coefficients β of the linear polynomial can be obtained by solving the system:
Where g are the know values at the source points. M is the interpolation matrix defined calculating all the radial interactions between source points:
And P is a constraint matrix that arise balancing the polynomial contribution and contains a column of “1” and the x y z positions of source point in the others three columns:
Radial basis interpolation works for scalar fields. For the smoothing problem each component of the displacement field prescribed at the source points is interpolated as follows:
Radial basis method has several advantages that make it very attractive in the area of mesh smoothing. The key point is that being a meshless method only grid points are moved regardless of the mesh-element connected and so the method is naturally suitable for parallel implementation. In fact once the solution is known and shared in the memory of each calculation node, each partition has the ability to smooth its nodes without taking care of what happens outside; this is possible because the smoother is a global point function and the continuity at interfaces is implicitly guaranteed.