We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation written on the whole domain, strongly...

The numerical solution of granular dynamics problems with Coulomb friction leads to the problem of minimizing a convex quadratic function with semidefinite Hessian subject to a separable conical constraints. In this paper, we are interested in the numerical solution of this problem. We suggest a modification of an active-set optimal quadratic programming algorithm. The number of projection steps is decreased by using a projected Barzilai-Borwein method. In the numerical experiment, we compare our...

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of $S^1$; the minimizer $u$ is $C^1$ and is such that $det\nabla u$ vanishes at one point.

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S1; the minimizer u is C1 and is such that $det\nabla u$ vanishes at one point.


A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of...

This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution...

This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution...

A mixed formulation is given for elastic problems. Existence and uniqueness of the discretized problem are given for conformal continuous interpolations for the stress tensor components and for the components of the displacement vector. A counterpart of the problem is discussed in the case of an even-dimensional Euclidean space with an associated Hamiltonian vector field and the Poisson structure. For conformal interpolations of the same order the question remains open.

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element...

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element...

In this paper we prove that every weak and strong local minimizer $u\in W^{1,2}(\Omega ,{ℝ}^3)$ of the functional $I\left(u\right)={\int }_{\Omega }{\left|Du\right|}^2+f\left(\mathrm{Adj}Du\right)+g\left(\det Du\right),$ where $u:\Omega \subset {ℝ}^3\to {ℝ}^3$, f grows like ${\left|\mathrm{Adj}Du\right|}^p$, g grows like ${\left|\det Du\right|}^q$ and 1<q<p<2, is $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of Ω such that $\mathrm{𝑚𝑒𝑎𝑠}(\Omega \setminus {\Omega }_0)=0$. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers. ...