A class of piecewise cubic interpolatory polynomials

A new class of C1 piecewise—cubic interpolatory polynomials is defined, by generalizing the definition of cubic X-splines given recently by Clenshaw and Negus (1978). It is shown that this new class contains a number of interpolatory functions which present practical advantages, when compared with the conventional cubic spline

The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking

Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a C1 continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bézier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field

Smooth Parametric Surfaces and n-Sided Patches

CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed. 1