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High-order Solution Transfer between Curved Meshes and Ill-conditioned Bézier Curve Intersection

  • Author(s): Hermes, Daniel Jerome
  • Advisor(s): Persson, Per-Olof
  • et al.

The problem of solution transfer between meshes arises frequently in computational physics, e.g. in Lagrangian methods where remeshing occurs. The interpolation process must be conservative, i.e. it must conserve physical properties, such as mass. We extend previous works --- which described the solution transfer process for straight sided unstructured meshes --- by considering high-order isoparametric meshes with curved elements. The implementation is highly reliant on accurate computational geometry routines for evaluating points on and intersecting Bézier curves and triangles.

Two ill-conditioned problems that occur evaluating points on and intersecting Bézier curves are then explored. This work presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. The resulting output is as accurate as the de Casteljau algorithm performed in K times the working precision. After compensated evaluation is considered, a compensated Newton's method is described, both for root-finding for polynomials in the Bernstein basis and for Bézier curve intersection.

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