A Delaunay approach to interactive cutting in triangulated surfaces

Our approach to deformation is based on the Finite Element Method (FEM). In this method, mesh size determines the computational requirements of a simulation. Larger meshes result in more degrees of freedom in the discretized problem, so solutions take more time to compute. This has motivated our work in Chapter 3, where we have tried a technique for simulating cuts that does not increase mesh size like subdivision techniques do. This technique creates flat elements in the mesh as artefacts, and we have found that they cause a considerable slowdown of the Linear CG algorithm. In Subsection 4.5.6 we have seen that element shape also affects the nonlinear CG and dynamic relaxation algorithms. Hence, mesh change operations, such as simulated cuts or cauterizations, should not only keep mesh size low; they should also keep elements of the mesh well-shaped. In this chapter we address this problem for cutting in triangulations, by presenting a method that keeps mesh size low and keeps mesh quality high. A FEM discretization is a form of interpolation: the continuous unknown in the problem is interpolated, so that the differential equation is transformed into a set of equations for a finite number of variables. The original problem is a differential equation, hence it is important that the derivative of the solution is approximated well by the interpolation. When we look at the influence of element shape on the derivative, we see that the large angles cause unbounded errors in the derivative. This is illustrated in 2D in Figure 5.1, and a similar argument also holds in 3D. Therefore, large angles should always be avoided. The convergence speed of iterative algorithms for linear problems