In 1996 Hazard and Lenoir suggested a variational formulation of Maxwell’s equations using an overlapping integral equation and volume representation of the solution. They suggested a numerical scheme based on this approach, but no error analysis was provided. In this paper, we provide a convergence analysis of an edge finite element scheme for the method. The analysis uses the theory of collec...

A numerical code for modeling crack propagation using the Cell Method is proposed. The Leon failure surface is used to compute the direction of crack propagation, and the new crack geometry is realized by an intra-element propagation technique. Automatic remeshing is then activated. Applications in Mode I, Mode II and Mixed Mode are presented to illustrate the robustness of the implementation. ...

The ltering problem for discrete Volterra equations is a nontrivial task due to an increasing dimension of the equivalent single step process model A di erence equation of a moderate dimension is chosen as an approximate model for the original system Then the reduced Kalman lter can be used as an approximate but e cient estimator Using the duality theory of convex variational problems a level o...

Given a nonlinear electronic circuit, an associated linear time-varying small-signal circuit is formally derived by the tableau-method. It has the same topology as the original circuit while each original circuit element is replaced by an incremental one, evaluated along the signal-dependent nonlinear circuit solution. Since the variational circuit is linear in the first place, the designer is ...

Since the pioneering work by Treloar, many models based on polymer chain statistics have been proposed to describe rubber elasticity. Recently, Alicandro, Cicalese, and the first author rigorously derived a continuum theory of rubber elasticity from a discrete model by variational convergence. The aim of this paper is twofold. First we further physically motivate this model, and complete the an...

The purpose of this thesis is to develop variational integrators synchronous or asynchronous, which can be used as tools to study complex structures composed of plates and beams subjected to large deformations and stress. We consider the geometrically exact models of beam and plate, whose configuration spaces are Lie groups. These models are suitable for modeling objects subjected to large defo...

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplicticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these...

We construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangia...

This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties.The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting...

Complex molecules often have many structures (conformations) of the reactants and the transition states, and these structures may be connected by coupled-mode torsions and pseudorotations; some but not all structures may have hydrogen bonds in the transition state or reagents. A quantitative theory of the reaction rates of complex molecules must take account of these structures, their coupled-m...