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An approximation theorem for sequences of linear strains and its applications

Kewei Zhang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

An energy-preserving Discrete Element Method for elastodynamics∗

Laurent Monasse, Christian Mariotti (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical...

An energy-preserving Discrete Element Method for elastodynamics∗

Laurent Monasse, Christian Mariotti (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical...

An equilibrium finite element method in three-dimensional elasticity

Michal Křížek (1982)

Aplikace matematiky

The tetrahedral stress element is introduced and two different types of a finite piecewise linear approximation of the dual elasticity problem are investigated on a polyhedral domain. Fot both types a priori error estimates O ( h 2 ) in L 2 -norm and O ( h 1 / 2 ) in L -norm are established, provided the solution is smooth enough. These estimates are based on the fact that for any polyhedron there exists a strongly regular family of decomprositions into tetrahedra, which is proved in the paper, too.

An optimal control problem for a fourth-order variational inequality

A. Khludnev (1992)

Banach Center Publications

An optimal control problem is considered where the state of the system is described by a variational inequality for the operator w → εΔ²w - φ(‖∇w‖²)Δw. A set of nonnegative functions φ is used as a control region. The problem is shown to have a solution for every fixed ε > 0. Moreover, the solvability of the limit optimal control problem corresponding to ε = 0 is proved. A compactness property of the solutions of the optimal control problems for ε > 0 and their relation with the limit problem...

An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure

Andrew Lorent (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.

Analysis and numerical approximation of an elastic frictional contact problem with normal compliance

Weimin Han, Mircea Sofonea (1999)

Applicationes Mathematicae

We consider the problem of frictional contact between an elastic body and an obstacle. The elastic constitutive law is assumed to be nonlinear. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We present two alternative yet equivalent weak formulations of the problem, and establish existence and uniqueness results for both formulations using arguments of elliptic variational inequalities and fixed point theory. Moreover, we show the continuous...

Analysis of a frictionless contact problem for elastic bodies

S. Drabla, M. Sofonea, B. Teniou (1998)

Annales Polonici Mathematici

This paper deals with a nonlinear problem modelling the contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled by the well-known Signorini conditions. Two weak formulations of the model are presented and existence and uniqueness results are established using classical arguments of elliptic variational inequalities. Some equivalence results are presented and a strong convergence result involving a penalized problem...

Analysis of a new augmented mixed finite element method for linear elasticity allowing ℝ𝕋 0 - 1 - 0 approximations

Gabriel N. Gatica (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a new stabilized mixed finite element method for the linear elasticity problem in 2 . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet...

Analysis of optimality conditions for some topology optimization problems in elasticity

Luis Trabucho (2002)

International Journal of Applied Mathematics and Computer Science

The subject of topology optimization has undergone an enormous practical development since the appearance of the paper by Bendso e and Kikuchi (1988), where some ideas from homogenization theory were put into practice. Since then, several engineering applications as well as different approaches have been developed successfully. However, it is difficult to find in the literature some analytical examples that might be used as a test in order to assess the validity of the solutions obtained with different...

Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution

Jaroslav Haslinger, Tomáš Ligurský (2009)

Applications of Mathematics

The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction which depends on a solution. It is shown that a solution exists for a large class of and is unique provided that is Lipschitz continuous with a sufficiently small modulus of...

Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method

Jindřich Nečas, Tomáš Roubíček (1990)

Aplikace matematiky

The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in...

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