Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

iSci

the informatics of scientific computing

Introduction

The quasiperiodic minimal surface investigated in Chapter 2 was defined as the limit of a process intended to stretch a 5D minimal surface in perp space. The discrete approximation of the Voronoi construction essentially sidestepped the approach to this limit by assuming that the topology of the surface had stabilized before the limit of infinite stretch had been realized and then sacrificing the minimality of the surface for the simplicity of a discrete structure. Whether one uses the discrete construction or the best analytic approximation available for Ψ ( x ) , say by Fourier transforming the discrete data, further steps must still be taken to arrive at a minimal surface since the approximation will at best be an almost-minimal surface. That is, it will in general have small mean curvature, H , which is simply a way of saying that the principal curvatures are roughly equal and opposite and that their sum is much less than the magnitude of either principal curvature.

So in a sense, the evolution of an almost-minimal surface to a minimal surface can be viewed as an economical way of ultimately achieving the limit t → 0 . All that is required is a starting surface with the same topology as the quasiperiodic minimal surface. Of course, complete modeling of the surface in physical space is impossible, but the cell complex formed by the Voronoi cells of the extrema of $\tilde{{\Psi}}$ ( x ) (Section 2.6.3) provides an efficient way to produce reasonable discrete approximations to the minimal surface (should it indeed exist) that capture the topology of the minimal surface. These approximations can then be relaxed and refined to more closely capture the geometry of the limit surface in physical space.

The relaxation of an initial surface to a nearby minimal surface is known as motion by mean curvature[ 48 ], and this motion can be followed by minimizing the bending energy of an interface, given in terms of the principal curvatures k ₁ and k ₂ , as

E _bend = $\displaystyle \int$ ( k ₁ ² + k ₂ ² ) d A ,

( 44 )

or equivalently, by minimizing the Willmore energy[ 49 ]

E _W = $\displaystyle \int$ H ² d A = ¼ $\displaystyle \int$ ( k ₁ ² + k ₂ ² +2 k ₁ k ₂ ) d A .

( 45 )

These two expressions differ only by a scale factor and the integral of Gaussian curvature K ≡ k ₁ k ₂ , which is a topological invariant for any surface and contributes only a constant term to the energy[ 49 ]. Clearly, a surface with E _W = 0 is minimal.

To adopt a similar approach for discrete data, an approximate Willmore energy, E _h must be associated with a polyhedral surface, and it is assumed that successively finer triangulations will result in increasingly more accurate approximations to a continuous surface. A triangulated surface's evolution to a minimal surface can then be followed by minimizing E _h over the triangulated surface, but to successfully implement this idea, suitable approximations for the mean curvature of a triangulated surface are needed.

Subsections

Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu

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