Global Spline-Based Interpolation (GSB)

As alternatives to the existing FIE scheme, in the project MUNA, the spatial coup­ling schemes GSB and MLS were implemented in the ACM. They are closely related, as both cast the problem of projecting loads or deformations as an inter­polation problem: For a set of N points in space xn with dependent values f (xn) one seeks to find a functional approximation to f based on a suitable choice of interpol­ation functions. In this sense, the distribution of dependent values is the deformation u provided at the structural nodes. Its functional approximation is then evaluated at a second set of M points Xm, which are the points of the CFD surface mesh. The two projection methods differ in their choice of interpolation functions which dictates the solution process.

The GSB method was originally published by Beckert and Wendland [6]. The authors approximate the deformation on the whole domain with a global low-order polynomial with Q monomials. The monomial vectors are either

m = (1, x, y, z)T or m = (1, x, y, z, x2, y2, z2, xy, yz, zx)T. (6)

Superimposed are local contributions ф(х) that consist of radial basis functions (RBF). At a given coordinate, the interpolation function is

NS

s(x) = mT(x)в + X a(Xn) Ф(х, Xn). (7)

n=1

The coefficients a(Xn) of the local RBF contributions and the coefficient vectors в of the global polynomial are calculated simultaneously for all Ng support points with a weighted least-squares algorithm. The dependent values at the interpolation support points are reproduced exactly. The RBFs with compact support constructed by Wendland [30] serve as weighting functions. The C2-continuous Wendland-RBF with a support radius g is provided here as an example:

ф(х, x„) = (1 – x)+(4x+l) with x= ||x — x„ ||2 • (8)

The index + marks that the factor (1 – x)4 is set to zero for values of x > 1, whereby the compact support is realised. The functional approximation to the deformation distribution can be obtained from the linear system of equations

"[ф(х„ X,)] [mr (x0] ] ({a H = f {uX(xj ^ 1 1 < i ;< Ns (9)

L [m(Xj)] 0 J j в J = 0 }’ 1 — i ’ j – Ng ’ (9)

‘————– V————- ‘

=C

which has to be solved for each Cartesian displacement component X = x, y, z. (Here, scalar quantities that are combined to a vector are put in braces. Brackets denote that scalars or vectors that are assembled to form a matrix.) This process would have to be repeated in each coupling step; instead the inverse of the coeffi­cients matrix C is determined. The functional approximation can now be evaluated at the surface points, which yields the final projection matrix P:

In the GSB method the support radius g has to be the same all over the computa­tional domain, or else the interpolation scheme will not be consistent. In general, the number of support points Ng will differ from one CFD surface node to the next. The user has to define a minimum required number of support points and the projec­tion scheme searches the domain for the smallest radius g that contains this number. Because of the global contribution to the interpolation function the resulting pro­jection matrix is dense. Its definition here (and in the MLS method) differs from the definition in Eq. (2) in that here the deformations or forces are projected one spatial component X at a time. The GSB method is largely identical to the volume mesh deformation method presented by Barnewitz in this volume. The main differ­ence lies in the choice of weighting functions and the compact support of the local contributions to the interpolation function.