Abstract
Point projection is an important geometric need when boundaries described by parametric curves and surfaces are immersed in domains. In problems where an immersed parametric boundary evolves with time as in solidification or fracture analysis, the projection from a point in the domain to the boundary is necessary to determine the interaction of the moving boundary with the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge, we present in this paper an algebraic procedure to project a point on to Non-uniform rational B-spline (NURBS) curves and surfaces. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bézier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain are then used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G 0 continuity—problems where the Newton–Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively.
Highlights
Given a test point and a parametric entity, the generalized point projection problem is to find the closest point on the entity as well as the corresponding parameter value
Algebraic point projection in two-dimensional physical space is validated in Figure 9, where the second-order point projection result is compared against first-order point projection as well as Newton–Raphson iterations for various test distances
The curve examples show the performance of algebraic point projection in simple tests to achieve the accuracy of the Newton–Raphson method
Summary
Given a test point and a parametric entity (curve or surface), the generalized point projection problem is to find the closest point (footpoint) on the entity as well as the corresponding parameter value. In any immersed boundary problem solution, capturing the interaction of the field approximation defined on the immersed (explicitly defined) boundary with the approximations on the enriched domain requires one to determine the nearest point on the boundary from any given point in the underlying domain This projection from the spatial point to the boundary is necessary to compute the influence of the domain approximation on those approximations defined on the boundary (see Figure 2).
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