The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized, several assertions added, minor correction in the proofs of Lemma 2.5, 3.3, Example 3.6, Corollary 5.

Finite element methods for surface PDEs

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples

Asymptotic expansions for high-contrast linear elasticity

We study linear elasticity problems with high contrast in the coefficients using asymptotic limits recently introduced. We derive an asymptotic expansion to solve heterogeneous elasticity problems in terms of the contrast in the coefficients. We study the convergence of the expansion in the H1 norm

Existence theorems in the geometrically non-linear 6-parametric theory of elastic plates

In this paper we show the existence of global minimizers for the geometrically exact, non-linear equations of elastic plates, in the framework of the general 6-parametric shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field and the rotation tensor field (representing in total 6 independent scalar kinematic variables). For isotropic plates, we prove the existence theorem by applying the direct methods of the calculus of variations. Then, we generalize our existence result to the case of anisotropic plates. We also present a detailed comparison with a previously established Cosserat plate model.Comment: 19 pages, 1 figur

A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation

Compressible Mooney-Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.Comment: Revision of earlier version of paper. Submitted for publication in Engineering with Computers on 9 September 2010. Accepted for publication on 20 May 2011. Published online 11 June 2011. The final publication is available at http://www.springerlink.co

Relativistic Elasticity

Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the non-relativistic limit . The field equations are cast into a first -- order symmetric hyperbolic system. As a consequence one obtains local--in--time existence and uniqueness theorems under various circumstances.Comment: 23 page

Characterising a universal cloning machine by maximum-likelihood estimation

We apply a general method for the estimation of completely positive maps to the 1-to-2 universal covariant cloning machine. The method is based on the maximum-likelihood principle, and makes use of random input states, along with random projective measurements on the output clones. The downhill simplex algorithm is applied for the maximisation of the likelihood functional.Comment: 5 pages, 2 figure

Review of group A rotavirus strains reported in swine and cattle

Group A rotavirus (RVA) infections cause severe economic losses in intensively reared livestock animals, particularly in herds of swine and cattle. RVA strains are antigenically heterogeneous, and are classified in multiple G and P types defined by the two outer capsid proteins, VP7 and VP4, respectively. This study summarizes published literature on the genetic and antigenic diversity of porcine and bovine RVA strains published over the last 3 decades. The single most prevalent genotype combination among porcine RVA strains was G5P[7], whereas the predominant genotype combination among bovine RVA strains was G6P[5], although spatiotemporal differences in RVA strain distribution were observed. These data provide important baseline data on epidemiologically important RVA strains in swine and cattle and may guide the development of more effective vaccines for veterinary use

Nonlinear weakly curved rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of Γ-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h 4. This is the same approach as the one used in Föppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature