An Efficient Solver to Approximate CAD Curves with Super-Convergent Rates

We present a specific-purpose solver to approximate curves with super-convergent rates. To obtain super-convergence, we minimize a disparity measure in terms of a piece-wise polynomial approximation and a curve re-parametrization. We have numerical evidence that the disparity converges with 2p order for planar curves and [3/2(p-1)]+2 for 3D curves, p being the mesh polynomial degree. To meet these rates, we exploit the quadratic convergence of a globalized Newton’s method with the help of three main ingredients. First, we employ a nonmonotone line search reducing the number of nonlinear iterations. The second ingredient is to introduce a log barrier function preventing element inversion in the curve re-parameterization. Third, we propose a constrained optimization of the disparity functional where the element interfaces are fixed, improving the computational efficiency whilst preserving super-convergence. We approximate analytic curves as well as CAD models with meshes of several polynomial degrees. We conclude that the solver is well-suited to obtain super-convergent approximations to curves at reasonable computational times.