Finite cloud method: A true meshless technique based on fixed reproducing kernel approximation

We introduce fixed, moving and multiple fixed kernel techniques for the construction of interpolation functions over a scattered set of points. We show that for a particular choice of nodal volumes, the fixed, moving and multiple fixed kernel approaches are identical to the fixed, moving and multiple fixed least squares approaches. A finite cloud method, which combines collocation with a fixed kernel technique for the construction of interpolation functions, is presented as a true meshless technique for the numerical solution of partial differential equations. Numerical results are presented for several one-and two-dimensional problems, including examples from elasticity, heat conduction, thermoelasticity, Stokes flow and piezoelectricity.