Moving surfaces are ubiquitous in life and mathematics. Moving surfaces, like stationary surfaces, need their own language. We have shown that, in many ways, tensor calculus is an ideal language for describing stationary surfaces. For moving surfaces, the language of tensors must be extended to capture the particularities of moving surfaces. This extension comes in the form of the calculus of moving surfaces (CMS).The term moving surface likely invokes an image of dynamically deforming physical surfaces, such as waves in water, soap films, biological membranes, or a fluttering flag. However, in applications, moving surfaces arise in numerous other contexts. For example, in shape optimization-a branch of the calculus of variations where the unknown quantity is the shape of a domain-moving surfaces arise as a parametrized family of allowable variations. In shape perturbation theory-I think this term is self-descriptive enough-moving surfaces arise as evolutions from the unperturbed to the perturbed configurations. Finally, moving surfaces can be effectively introduced in problems where one may not think that moving surfaces can play any role at all. For example, in Chap. 17 we use the calculus of moving surfaces to prove a special case of the celebrated Gauss-Bonnet theorem which states that the integral of Gaussian curvature over a closed surface is independent of the shape of the surface and only depends on its genus (the number of topological holes).The fundamental ideas of the calculus of moving surfaces were introduced by the French mathematician Jacques Hadamard (Fig. 15.1
We propose dynamic nonlinear equations for free thin fluid films. The obtained numerical solutions display a number of features consistent with recent experiments with fluid films under large deformations. In particular, we observe dynamic thickening. Our analysis is based on a two-dimensional model. The film's thickness is represented by the two-dimensional density ρ. We show that a broad range of effects can be captured by a proper internal energy function e(ρ).
The difficulties are almost always at the boundary." That statement applies to the solution of partial differential equations (with a given boundary) and also to shape optimization (with an unknown boundary). These problems require two decisions, closely related but not identical: 1. How to discretize the boundary conditions 2. How to discretize the boundary itself. That second problem is the one we discuss here. The region Ω is frequently replaced by a polygon or polyhedron. The approximate boundary ∂Ω N may be only a linear interpolation of the true boundary ∂Ω. A perturbation theory that applies to smooth changes of domain is often less successful for a polygon. This paper concentrates on a model problem-the simplest we could find-and we look at eigenvalues of the Laplacian. The boundary ∂Ω will be the unit circle. The approximate boundary ∂Ω N is the regular inscribed polygon with N equal sides. It seems impossible that the eigenvalues of regular polygons have not been intensively studied, but we have not yet located an authoritative reference. The problem will be approached numerically from three directions, without attempting a general theory. Those directions are: 1. Finite element discretizations of the polygons ΩN 2. A Taylor series based on (non-smooth) perturbations of the circle 3. A series expansion of the eigenvalues in powers of 1/N. The second author particularly wishes that we could have consulted George Fix about this problem. His Harvard thesis demonstrated the tremendous improvement that "singular elements" can bring to the finite element method (particularly when Ω has a reentrant corner, or even a crack). His numerical experiments in [2] came at the beginning of a long and successful career in applied mathematics. We only wish it had been longer.
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