Meshing has been a longstanding impediment to the engineering design-analysis cycle. As designs become increasingly more complex, meshing is considered to be the major bottleneck that might affect the advancement of downstream applications. There are some contending approaches developed to bypass the meshing, for instance isogeometric analysis, immersed and particle methods. They, however, present challenges to the simulation of complex solid industrial-scale geometries.

Our work is motivated by two general ideas. The first idea is based on the realisation that points and Voronoi diagrams are closely related. This dualism allows a fast and reliable discretisation of the computational domain using the distribution of points. The generation of the Voronoi cells becomes even more straightforward when the cells are not required to fit the boundary, as in immersed methods. The second idea comes from the generalisation of the convolutional definition of B-splines. It is observed that convolution or mollification regularises piecewise functions with improved differentiability based on the continuity of the mollifier. Therefore convolution is suitable to generating smooth basis functions that reproduce global polynomial and is easy to construct on polytopic cells.

We propose the mollified basis functions of arbitrary degree and smoothness on non-overlapping partitions consisting of convex polytopes. On each polytope, an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolution of the local approximant with a mollifier. The mollifier is chosen to be smooth, have a compact support and a unit volume. The approximation properties of the obtained basis functions are determined by the local polynomial approximation order and the mollifier smoothness. The convolution integrals are evaluated numerically after computing the intersection between the mollifier and the polytope and then applying the divergence theorem. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, may not be aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems using finite element and collocation.