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We investigate different techniques for fitting Bézier curves to surfaces in context of high-order curvilinear mesh generation. Starting from distance-based least-squares fitting we develop an incremental algorithm, which incorporates approximations of stretch and bending energy. In the process, the algorithm reduces the energy weight in favor of accuracy, leading to an optimized set of sampling points. This energy-minimizing fitting strategy is applied to analytically defined as well as triangulated surfaces. The results confirm that the proposed method straightens and shortens the curves efficiently. Moreover the method preserves the accuracy and convergence behavior of distance-based fitting. Preliminary application to surface mesh generation shows a remarkable improvement of patch quality in high curvature regions.

The present work is motivated by curvilinear mesh generation for high-order numerical methods such as spectral and hp element methods [

or of the same size as the radius of curvature, often in conjunction with a moderate polynomial degree. To the best of our knowledge, the generation of suitable meshes with higher order (say

As a first step of curvilinear mesh generation we consider the construction of polynomial curves from a given straight-sided surface mesh. This problem can be regarded as a special case of curve fitting, which, in turn, is a well established research area in computer aided geometric design [

In this paper we investigate techniques for fitting Bézier curves to surfaces for application with high-order mesh generation. Starting from squared distance minimization we develop an incremental algorithm leading to an accurate, energy-minimizing method that combines the ideas put forward in [

In spectral or hp element methods, each element face coinciding with the domain boundary constitutes a polynomial surface patch. Therefore, the construction of well behaved polynomial patches represents a natural building block in curvilinear mesh generation. Starting from a straight-sided initial mesh, the curvilinear mesh is often built in a hierarchical process consisting of the following steps: i) construction of boundary curves representing the edges of the boundary faces; ii) generation of patches defining the boundary faces; and iii) creation of curved volume elements. Here we focus on the first step, the construction of high-order polynomial boundary curves. Adopting the Bézier form, a curve of order

where

Curve fitting is used to construct a boundary edge

The procedure starts with the selection of

where, ideally, the operator

where

With coarse meshes purely distance-based fitting can lead to severe undulations in regions of high curvature. As a remedy one may look for curves that minimize a certain energy functional. Here we consider the

The norm of the first derivative (4) is related to the elastic stretch energy of a string [

where overbars indicate the normalization which has been introduced to compensate possible differences in the order of magnitude between the individual terms. In particular we define

and

where the superscript “0” refers to the curve obtained from distance-based fitting and “lin” to the straight edge. Note that

To obtain fair surface curves we employ an incremental approach, which is outlined in Algorithm 1. The basic idea is to start fitting with a high energy weight, which is successively reduced according to a generic shape function

Throughout the present study we used the shape function

which starts with 1 at

In the following we study the performance of the energy-minimizing fitting method in two different cases. As the first case we consider a coarse, but nearly uniform triangulation of an explicitly defined screw surface. In the second case the “exact” surface is defined as a patchwork of cubic triangles based on a fine mesh derived from CT scans of a rabbit aorta. For assessment we use the

and the curve energy norms

where again

As a first test case we consider the screw surface defined by the analytical expression

The projection

starting with

The energy-minimizing fitting procedure was examined over a wide range of polynomial degrees, ranging from

case, mere distance-based fitting yields a meandering curve, whereas energy-minimizing fitting straightens the path and removes undulations regardless of the chosen energy composition. As expected, the balanced energy mix,

Finally we look at the convergence behavior with respect to the fitting order

Scanning methods such as computed tomography (CT) or magnetic resonance imaging (MRI) provide scattered data that can be processed to give triangulated volume and surface representations of the investigated object. Here we consider a partition of a rabbit’s aortic arch, given as a fine mesh consisting of 24,644 triangles (

This representation was enhanced in two steps, yielding the “exact” surface: First, computing the vertex normals using the method of Max [

To evaluate the curve fitting methods we generated a curvature-dependent coarse mesh comprising 532 triangles (

Starting from the linear mesh we constructed curves of order

where

Remarkably, the energy-minimizing approach not only succeeds in reducing the energy norms, but also improves the approximation accuracy. Assuming a threshold of

In addition we constructed triangular Bézier patches in two steps [

We investigated different techniques for fitting Bézier curves to surfaces as a first step of curvilinear mesh generation for high-order discretization methods. As a starting point we examined a distance-based least-squares fitting method. This method achieves a high accuracy, but tends to produce distorted curves where the mesh spacing is large compared to the radius of curvature. As remedy, we included approximations of stretch and bending energy into an incremental algorithm, resulting in an energy-minimizing fitting method. Both approaches were evaluated using two examples: an analytically defined screw surface and a surface triangulation of a rabbit aorta. The results confirm that the energy-minimizing method straightens and shortens the curves efficiently. Moreover the method preserves the accuracy and convergence behavior of distance-based fitting. In accordance with previous work (see e.g. [

Method | |||||
---|---|---|---|---|---|

distance-based | - | 4.24 | 0.21 | ||

energy-minimizing | 0.00 | 4.03 | 0.47 | ||

energy-minimizing | 0.01 | 4.01 | 0.48 |

patch construction. This investigation shows a clear improvement in patch quality when using energy-minimized curves. Nonetheless distortion remains an issue in the patch interior. Therefore, future work should address the extension of energy-minimizing approach to surface patch fitting.

The authors gratefully acknowledge the funding of this project by the German Research Foundation (DFG, STI 157/4-1).

Karsten Bock,Jörg Stiller, (2014) Energy-Minimizing Curve Fitting for High-Order Surface Mesh Generation. Applied Mathematics,05,3318-3327. doi: 10.4236/am.2014.521309