Adaptive Strategies for High Order FEM in Elastoplasticity (original) (raw)
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A convergent adaptive finite element method for the primal problem of elastoplasticity
International Journal for Numerical Methods in Engineering, 2006
The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R-linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic-kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd.
Fast Solvers and A Posteriori Error Estimates in Elastoplasticity
Texts & Monographs in Symbolic Computation, 2011
The paper reports some results on computational plasticity obtained within the . Adaptivity and fast solvers are the ingredients of efficient numerical methods. The paper presents fast and robust solvers for both 2D and 3D plastic flow theory problems as well as different approaches to the derivations of a posteriori error estimates. In the last part of the paper higher-order finite elements are used within a new plastic-zone concentrated setup according to the regularity of the solution. The theoretical results obtained are well supported by the results of our numerical experiments.
Newton-Like Solver for Elastoplastic Problems with Hardening and its Local Super-Linear Convergence
Numerical Mathematics and Advanced Applications, 2000
We discuss a new solution algorithm for quasi-static elastoplastic problems with hardening. Such problems are described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as each one minimization problem with a convex energy functional, which depends smoothly on the displacement and non-smoothly on the plastic strain. There exists an explicit formula how to minimize the energy functional with respect to the plastic strain for a given displacement. Thus, by its substitution, an energy functional depending only on the displacement can be obtained. Our technique based on the well known theorem of Moreau from convex analysis shows that the energy functional is differentiable with an explicitely computable first derivative. The second derivative of the energy functional exists everywhere in the domain apart from the elastoplastic interface, which separates the deformed continuum in elastic and plastic parts. A Newton-like method exploiting slanting functions of the energy functional's first derivative is proposed and implemented numerically. The local super-linear convergence of the Newton-like method in the discrete case is shown and sufficient regularity assumptions are formulated to guarantee local super-linear convergence also in the continuous case.
Solution of One-Time-Step Problems in Elastoplasticity by a Slant Newton Method
SIAM Journal on Scientific Computing, 2009
We discuss a solution algorithm for quasi-static elastoplastic problems with hardening. Such problems can be described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as a minimization problem with a convex energy functional which depends smoothly on the displacement and nonsmoothly on the plastic strain. There exists an explicit formula for minimizing the energy functional with respect to the plastic strain for a given displacement. By substitution, the energy functional can be written as a functional depending only on the displacement. The theorem of Moreau from convex analysis states that the energy functional is differentiable with an explicitly computable first derivative. The second derivative of the energy functional does not exist, due to the lack of smoothness of the plastic strain across the elastoplastic interface, which separates the continuum in elastically and plastically deformed parts. A Newton-like method exploiting slanting functions of the energy functional's first derivative instead of the nonexistent second derivative is applied. Such a method is called a slant Newton method for short. The local superlinear convergence of the algorithm in the discrete case is shown, and sufficient regularity assumptions are formulated, which would guarantee the local superlinear convergence also in the continuous case.
An rp-adaptive finite element method for the deformation theory of plasticity
Computational Mechanics, 2007
In this paper, we present an rp-discretization strategy for physically non-linear problems based on a high order finite element formulation. In order to achieve convergence, the p-version leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally, whereas the r-method moves nodes and edges of an existing FE-mesh. The basic idea of our rp-version approach is to adjust the finite element mesh to the shape of the elastic-plastic interface in order to take into account the loss of regularity which arises along the curve of the plastic front. Numerical examples will demonstrate that this approach leads to an exponential rate of convergence and highly accurate results.
New Numerical Solver for Elastoplastic Problems based on the Moreau-Yosida Theorem
2006
We discuss a new solution algorithm for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth functional depending on unknown displacement smoothly and on the plastic strain nonsmoothly. It is shown that the functional structure allows the application of the Moreau-Yosida Theorem known in convex analysis. It guarantees that the substitution of the non-smooth plastic-strain as a function of the linear strain which depends on the displacement only yields an already smooth functional in the displacement only. Moreover, the second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. This allows the efficient implementation of a Newton-like method. For easy implementation most essential Matlab c © functions are provided. Numerical experiments in two dimensions state quadratic convergence of the Newton...
Implementation of an elastoplastic solver based on the Moreau–Yosida Theorem
Mathematics and Computers in Simulation, 2007
We discuss a technique for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth objective. We actually show that its objective structure satisfies conditions of the Moreau-Yosida Theorem known from convex analysis. Therefore, the substitution of the nonsmooth plastic-strain p as a function of the total strain ε(u) yields an already smooth functional in the displacement u only. The second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. The numerical experiment states super-linear convergence of a Newton method or even quadratic convergence as long as the interface is detected sufficiently.
Numerical Mathematics and Advanced Applications, 2006
The main subject of this paper is the detailed description of an algorithm solving elastoplastic deformations. Our concern is a one time-step problem, for which the minimization of a convex but non-smooth functional is required. We propose a minimization algorithm based on the reduction of the functional to a quadratic functional in the displacement and the plastic strain increment omitting a certain nonlinear dependency. The algorithm also allows for an easy extension to higher order finite elements. A numerical example in 2D reports on first results for uniform h-and p-mesh refinements.
On the Adaptive Coupling of Finite Elements and Boundary Elements for ElastoPlastic Analysis
The purpose of this paper is to present an adaptive FEM-BEM coupling method that is valid for both two- and three-dimensional e lasto-plastic analyses. The method takes care of the evolution of the elastic a nd plastic regions. It eliminates the cumbersome of a trial and error process in the identification of the FEM and BEM sub-domains in the standard FEM-BEM coupling approaches. The method estimates the FEM and BEM sub-domains and automatically generates/adapts the FEM and BEM meshes/sub-domains, according to the state of computation. The results for two- and three- dimensional application s in elasto-plasticity show the practicality and the efficiency of the adaptive FEM -BEM coupling method.
Computers & Structures, 2012
A three field variational framework is used to formulate a new class of mixed finite elements for the elastoplastic analysis of 2D problems. The proposed finite elements are based on the independent interpolation of the stress, displacement and plastic multiplier fields. In particular new and richer interpolating patterns are proposed for the plastic multiplier in order to go beyond the oversimplifying constant assumption used in Bilotta and Casciaro [1]. As will be shown, this last choice is useful to simplify the return mapping algorithm but adversely affect the accuracy of the finite element. More articulated interpolations transform the return mapping process into a convex nonlinear optimization problem with few variables and constraints, a problem that can be efficiently solved using optimization algorithms, without penalizing overall computational efficiency. Several kinds of interpolations are proposed and compared with respect to accuracy and efficiency by performing a series of numerical tests on plane stress/strain problems modeled on the basis of the von Mises and Drucker-Prager yield functions. The reliability and good performance of the proposed elements are evident.