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Center for Nonlinear Studies |
In this talk, I will discuss some of our work on the development of polygonal interpolants and their use within a Galerkin method for the solution of partial differential equations. Wachspress proposed polygonal basis functions in 1975, but only recently has renewed interest in such interpolants re-surfaced in areas such as geometric modeling and computer graphics. The use of polygonal elements is a natural choice in applications where the material microstructure consists of polygonal subdomains (e.g., cellular foam, bone, polycrystalline alloys and piezoelectrics, etc.). Natural neighbor shape functions are used to construct a new polygonal interpolant on irregular polygons. The meshfree shape functions are defined on a regular reference polygon and an affine map (isoparametric transformation) is used to define the shape functions on arbitrary convex polygons. The natural neighbor shape functions are non-negative, interpolate nodal data, and are linear on the boundary of the domain, which permits the imposition of Dirichlet boundary conditions. Convergence and accuracy of the method will be presented for two-dimensional elliptic boundary-value problems. As an application of polygonal shape functions, we construct conforming approximations on quadtree meshes. Quadtree is a spatial data structure that is very efficient for fast data storage and retrieval. Classical finite elements are non-conforming on such meshes due to the presence of hanging nodes, whereas the proposed method alleviates this shortcoming. Residual-based error estimators are developed to conduct adaptive computations on quadtree meshes. Numerical examples will be presented for linear and nonlinear problems involving sharp gradients, singularities, and crack discontinuities to demonstrate the performance of the h-adaptive finite element method.