A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

We design and analyze a $$C^0$$ C 0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions.

Computational Technique of the Sliding Interface in Numerical Simulations of 3D Penetration

Based on the Lagrange finite element method, a simplified computational technique for sliding interface in oblique penetration is proposed in the paper. By applying the 3D finite element program developed on this technique to numerical analog computation of ogive-noded steel rod penetrating in the aluminum target, a result which is consistent with that of A.J.Piekutowskis experiment can be reached. Hence it proves the rationality and validity of the program and method discussed in this paper and that the method is useful in the numerical study of penetration and perforation.

Quasi Static Fracture – Global Minimizer of the Regularized Energy

Fracture mechanics has been revisited aimed at modeling brittle fracture based on Griffith viewpoint. The purpose of this work is to present a numerical computational method for solving the quasi static crack propagation based on the variational theory. It requires no prior knowledge of the crack path or of its topology. Moreover, it is capable of modeling crack initiation. At the numerical level, we use a standard linear (P1) Lagrange finite element method for space discretization. We perform numerical simulations of a piece of brittle material without initial crack. We show also the necessity of adding the backtracking algorithm to alternate minimizations algorithm to ensure the convergence of the alternate minimizations algorithm to a global minimizer.

Quasi Static Analysis of Anti-Plane Shear Crack

Fracture mechanics has been revisited by proposing different models of quasi static brittle fracture. In this work, the problem of the quasi static crack propagation is based on variational approach. It requires no prior knowledge of the crack path or of its topology. Moreover, it is capable of modeling crack initiation. In the numerical experiments, we use a standard linear (P1) Lagrange finite element method for discretization. We perform numerical simulations of a piece of brittle material without initial crack. An alternate minimizations algorithm is used. Based on these numerical results, we determine the influence of numerical parameters on the evolution of energies and crack propagation. We show also the necessity of considering the kinetic term and the crack propagation becomes dynamic.

High order optimal anisotropic mesh adaptation using hierarchical elements

Anisotropic mesh adaptation has made spectacular progress in the past few years. The introduction of the notion of a metric, directly linked to the interpolation error, has allowed to control the elongation of elements as well as the discretisation error. This approach is however essentially restricted to linear (P(1)) finite element solutions, though there exists some generalisations. A completely general approach leading to optimal meshes and this, for finite element solution of any degree, is still missing. This is precisely the goal of this work where we show how to estimate the error on a finite element solution of degree k using hierarchical basis for Lagrange finite element polynomials. We then show how to use this information to produce optimal anisotropic meshes in a sense that will be precised.