Construction of minimal subdivision surface with a given boundary

Abstract

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.

Introduction

Minimal surfaces have several desirable properties because of which they are often used as models in architecture. First of all, minimal surfaces have the least surface area, which make them to be widely used in large scale and light roof constructions. Secondly, minimal surfaces have separable property, meaning: any sub-patch, no matter how small, cut from a minimal surface still has the least area of all surface sub-patches with the same boundary. Thirdly, minimal surfaces have balanced surface tension, which stabilizes the whole construction since the tension is in equilibrium at each point on a roof, as on a soap film. Finally, there are no umbilicus points on a minimal surface, hence no water could stay on a minimal surface roof. Architecture inspired from minimal surfaces embodies the unite of economy and beauty. The most spectacular buildings of that architectural style are the roofs of the Munich Olympic stadium, the former Kongreßhalle in Berlin, and many of the smaller tent roofs on pavilions in parks and other public places. In art world we can see plenty of ingenious sculpture works playing the ultimate of minimal surfaces. Scientists and engineers have anticipated the nanotechnology applications of minimal surfaces in areas of molecular engineering and materials science.

In the area of computer aided design (CAD), minimal surfaces are also studied. To construct a continuous surface with a given boundary, Monterde (see [1]) used a four-sided Bézier surface to approximate a minimal surface. The problem which is called Plateau–Bézier problem was solved by replacing the area functional with the Dirichlet functional. Cosin and Monterde [2] studied the properties that the control vertices must satisfy and showed that in the bicubical case all minimal surfaces are, up to an affine transformation, pieces of the Enneper surfaces. Triangular Bézier surface was constructed by Arnal et al. [3] using a variational approach. Excellent work has been done on the use of minimal surfaces in geometric modeling and shape design (see [4], [5], [6]). Discrete minimal surfaces were studied by Polthier in [7]. Minimal surfaces were also produced as the steady solution of the mean curvature flow (see [8]) for both continuous and discrete cases. For the continuous case, the constructed surfaces are usually Bézier surfaces, or $B$-spline surfaces.

Obviously, Bézier surfaces, $B$-spline or NURB surfaces have to be three or four sided. This is a serious limitation for designing minimal surfaces with any shaped boundaries. In this paper, our intention is to construct minimal subdivision surfaces with piecewise $B$-spline curve boundaries. There is no limitation on the number of spline pieces. $B$-spline has been widely accepted as a representation tool for curves and surfaces in industrial design. Using $B$-spline to represent a surface boundary is preferable and acceptable. To represent a surface patch with any topology, perhaps subdivision surfaces are the best candidates, since there is no limitation on the topology of the control mesh. However, subdivision surfaces, such as Loop’s subdivision surface and the Catmull–Clark subdivision surface are traditionally closed, which cannot be used directly for serving our purpose. For many surface modeling problems, such as the construction of bodies of cars, aircrafts, machine parts and roofs, surfaces are usually constructed in a piecewise manner with fixed boundaries. In such a case Loop’s subdivision scheme could not be applied near the boundaries of the control mesh. Therefore, extension of Loop’s subdivision scheme for the control mesh with boundaries is definitely required. On this aspect, an excellent work has been done by Biermann et al. [9] and that is just sufficient for our use.

Loop’s subdivision has been shown to be a very powerful tool in several areas. For instance, it is used successfully in smooth surface reconstruction from scattered data (see [10], [11]) and the thin-shell finite element analysis (see [12]) for describing the geometry and the displacement fields. Subdivision techniques can handle arbitrary topology surfaces. The variational subdivision method combines the two approaches for effective designing of freeform surfaces. Kobbelt et al. in [13], [14] proposed a mesh refinement method based on variational methods that try to overcome certain drawbacks of uniform subdivision. A new approach of subdivision based on the evolution of surfaces under curvature motion is presented in [15].

We intend to construct the minimal subdivision surfaces (see Fig. 1.1) using the mean curvature flow, a second order geometric partial differential equation. The equation is solved by a finite element method. The contribution of this paper includes the proposal of a method for constructing minimal subdivision surfaces with given boundaries. Several schemes, such as the extended Loop’s subdivision scheme, the fast evaluation of the basis functions and the finite element method for the initial-boundary problem of the mean curvature flow with the Dirichlet boundary condition, are combined together to form an efficient and mathematically sound approach.

The rest of this paper is organized as follows: In Section 2 we describe some related results of Loop’s subdivision scheme including interior surface patches and boundary surface patches. Section 3 shows the mean curvature flow used in this paper, the details of its discretization and the numerical computation for the solution of this flow. In Section 4 we give several examples of minimal subdivision surface construction and some comparison results for approximation error to illustrate the effects of our method. Section 5 is the conclusion.