Jonas Koko

A Matlab mesh generator for the two-dimensional finite element method

A new Matlab code for the generation of unstructured (3-node or 6-node) triangular meshes in two dimensions is proposed. The method is based on the Matlab mesh generator distmesh of Persson and Strang (2004). As input, the code takes a signed distance function for the domain geometry. A mesh size function, for the spatial node distribution, is constructed using an approximate medial axis. As outputs, the code generates a 3-node or a 6-node triangular mesh with boundary data (edges and nodes). The approach presented consists of three steps: (1) an initial nodes placement is obtained using a probabilistic node distribution, (2) an iterative smoothing is performed assuming the presence of an attractive/repulsive internode force, and (3) a fast refinement procedure is performed for 6-node triangular meshes or large scale meshes.

An Augmented Lagrangian Method for TVg+L1-norm Minimization

In this paper, the minimization of a weighted total variation regularization term (denoted TVg) with L1 norm as the data fidelity term is addressed using the Uzawa block relaxation method. The unconstrained minimization problem is transformed into a saddle-point problem by introducing a suitable auxiliary unknown. Applying a Uzawa block relaxation method to the corresponding augmented Lagrangian functional, we obtain a new numerical algorithm in which the main unknown is computed using Chambolle projection algorithm. The auxiliary unknown is computed explicitly. Numerical experiments show the availability of our algorithm for salt and pepper noise removal or shape retrieval and also its robustness against the choice of the penalty parameter. This last property is useful to attain the convergence in a reduced number of iterations leading to efficient numerical schemes. The specific role of the function g in TVg is also investigated and we highlight the fact that an appropriate choice leads to a significant improvement of the denoising results. Using this property, we propose a whole algorithm for salt and pepper noise removal (denoted UBR-EDGE) that is able to handle high noise levels at a low computational cost. Shape retrieval and geometric filtering are also investigated by taking into account the geometric properties of the model.

Uzawa Conjugate Gradient Domain Decomposition Methods for Coupled Stokes Flows

This paper deals with nonoverlapping domain decomposition methods for two coupled Stokes flows, based on the duality theory. By introducing a fictitious variable in the transmission condition and using saddle-point equations, the problem is restated as a linearly constrained maximization problem. According to whether constraints are uncoupled Stokes problems or uncoupled Poisson problems, two Uzawa-type domain decomposition algorithms are proposed. The results of some numerical experiments on a model problem are given.

Vectorized Matlab codes for linear two-dimensional elasticity

A vectorized Matlab implementation for the linear finite element is provided for the two-dimensional linear elasticity with mixed boundary conditions. Vectorization means that there is no loop over triangles. Numerical experiments show that our implementation is more efficient than the standard implementation with a loop over all triangles.

AN OPTIMIZATION-BASED DOMAIN DECOMPOSITION METHOD FOR NONLINEAR WALL LAWS IN COUPLED SYSTEMS

We study an optimization-based domain decomposition method for a nonlinear wall law in a coupled system. The problem is restated as a saddle-point problem by introducing as a new variable the displacement jump across the interface. Then the minimization step of the saddle-point problem corresponds to the equilibrium equations stated in each subdomain with Lagrange multiplier as interface force. The maximization step corresponds to maximizing a (nonlinear) strictly concave functional. This could have a lot of applications in geophysical flows such as coupling ocean and atmosphere, free surface and groundwater flows.

Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem

Abstract We propose a Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem. We introduce auxiliary variables to separate subdomains representing linear elastic bodies. Applying a Uzawa block relaxation algorithm to the corresponding augmented Lagrangian functional yields a domain decomposition algorithm in which we have to solve two uncoupled linear elasticity subproblems in each iteration while the auxiliary variables are computed explicitly using Kuhn–Tucker optimality conditions.

An Optimization-Based Domain Decomposition Method for a Two-Body Contact Problem

Abstract We study a domain decomposition method for the numerical simulation of a two-body contact problem in two-dimensional linear elasticity. The problem is restated as a decomposition-coordination problem by introducing a smooth fictitious rigid surface between the two elastic bodies. In the decomposition step, two independent problems are solved: a Signorini-like problem and a prescribed displacement problem. In the coordination step, the fictitious rigid surface is adjusted to minimize a auxiliary functional.

A Lagrange multiplier decomposition method for a nonlinear sedimentary basin problem

We study a Lagrange multiplier based non-overlapping domain decomposition method for the nonlinear over-pressure equation. Using an additional unknown, the problem is restated as a linearly constrained minimization problem. The resulting Uzawa-type algorithm requires at each iteration the solution of one uncoupled Poisson problem.

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Rieszs representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.

Domain Decomposition Method for Stokes Problem with Tresca Friction

Development of numerical methods for the solution of Stokes system with slip boundary conditions (Tresca friction conditions) is a challenging task whose difficulty lies in the nonlinear conditions. Such boundary conditions have to be taken into account in many situations arising in practice, in flow of polymers (see [10] and references therein).