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Convex Analysisの二,三の進展について

Numerical Initial Value Problems in Ordinary Differential Equations

Convex Analysis: (pms-28)

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Higher-order finite element methods.

Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems

http://library.utia.cas.cz/separaty/2014/MTR/valdman-0427638.pdf

A TFETI domain decomposition solver for elastoplastic problems

In this paper we explore a numerical solution to elastoplastic constitutive initial value problems. An improved form of the implicit return-mapping scheme for nonsmooth yield surfaces is proposed that systematically builds upon a subdifferential formulation of the flow rule. The main advantage of this approach is that the treatment of singular points – apices or edges at which the flow direction is multivalued – only involves a uniquely defined set of non-linear equations, similarly to smooth yield surfaces. This paper focuses on isotropic models containing: a) yield surfaces with one or two apices (singular points) on the hydrostatic axis, b) plastic pseudo-potentials that are independent of the Lode angle, and c) possibly nonlinear isotropic hardening. We show that for some models the improved integration scheme also enables us to a priori decide about a type of the return and to investigate the existence, uniqueness, and semismoothness of discretized constitutive operators. The semismooth Newton method is also introduced for solving the incremental boundary-value problems. The paper contains numerical examples related to slope stability with publicly available Matlab implementations.

Subdifferential‐based implicit return‐mapping operators in computational plasticity

Application of a modified semismooth Newton method to some elasto-plastic problems

In the paper, a general constitutive elastoplastic model for associated plasticity is investigated. The model is based on the thermodynamical framework with internal variables and can include basic plastic criteria with a combination of kinematic hardening and non-linear isotropic hardening. The corresponding initial value constitutive elastoplastic problem is discretized by the implicit Euler method. The discretized one-time-step constitutive problem defines the elastoplastic operator, which is formulated by a simple generalization of a projection onto a convex set. Properties of the so-called generalized projection are used for deriving basic properties of the elastoplastic operator like potentiality, monotonicity, Lipschitz continuity and local semismoothness. Further, hardening variables are eliminated from the projective definition of the elastoplastic operators, which yields relations among the models with hardening variables and the perfect plasticity model. Also a simplification of the operator for plastic criteria in eigenvalue forms is introduced. The simplifications are useful for a numerical implementation and can be used for deriving other properties like strong semismoothness of the elastoplastic operators for the classical plastic criteria or strong monotonicity of the stress-strain operator for some models with hardening. The derived properties can be important for convergence analyses of Newton-like methods and other mathematical and numerical analyses.

Stanislav Sysala

Papers

Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems

A TFETI domain decomposition solver for elastoplastic problems

Subdifferential‐based implicit return‐mapping operators in computational plasticity

Application of a modified semismooth Newton method to some elasto-plastic problems

Properties and simplifications of constitutive time-discretized elastoplastic operators