Leonidas Xanthis

A posteriori error estimation for elasto-plastic problems based on duality theory

Abstract In this paper we introduce a new approach to a posteriori error estimation for elasto-plastic problems based on the duality theory of the calculus of variations. We show that, in spite of the prevailing view, duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem. Rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts: the error in the constitutive law and the error in the equilibrium equations. This representation is both physically meaningful and computationally important. The duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the Galerkin orthogonality condition. In particular, they encompass the familiar smoothening or gradient averaging techniques commonly used in practice. We prove that the duality error estimators are ‘equivalent to the error’ when the approximate solution is the exact solution of the discretised problem. Moreover, in the case of linear elasticity, the known residual type error estimators can be obtained from the duality error estimators. Numerical results for a model elasto-plastic problem show the accuracy of the duality error estimators.

On Fast Domain Decomposition Solving Procedures for hp-Discretizations of 3-D Elliptic Problems

Abstract A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by Kinterpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently.

Iterative subspace correction methods for thin elastic structures and Korn's type inequality in subspaces

We present a methodology for the efficient solution of three–dimensional problems for thin elastic structures, such as shells, plates, rods, arches and beams, based on the synergy of two fundamental concepts: the subspace correction and the Korns type inequality in subspaces. The former provides the theoretical background which enables the development of modern iterative methods for large–scale problems, such as domain decomposition and multilevel methods, which are sine qua non for high–performance scientific and engineering computing, and the latter is responsible for the design of iterative methods for thin elastic structures with convergence which is uniform with respect to the thickness. The subspace correction methods are based on the decomposition of the space where the solution is sought into the sum of subspaces. In this paper we show that using the Korns type inequality in subspaces we can introduce subspace decompositions for which the convergence rate of the corresponding subspace correction methods is independent of both the thickness and the discretization parameters.

A new Korn's type inequality for thin domains and its application to iterative methods

We present a new Korns type inequality which estimates the ratio of the energy norm to the Sobolev norm in a given subspace in terms of the angle it forms with an explicitly extracted finite dimensional subspace. This inequality provides crucial information for improving the convergence of various iterative algorithms for elasticity problems in thin domains. This is demonstrated in the case of the semi-discrete iterative algorithm EDRA (see E.E. Ovtchinnikov and L.S. Xanthis, Effective dimensional reduction for elliptic problems, C.R. Acad. Sci. Paris, Series I, 320: 879–884, 1995). We show both theoretically and numerically that the convergence of the modified EDRA is independent of the thickness of the domain and of the semi-discretisation parameters.

A posteriori error estimation for nonlinear variational problems

Abstract We introduce a new modus operandi for a posteriori error estimation for nonlinear (and linear) variational problems based on the duality theory of the calculus of variations. We derive what we call duality error estimates and show that they yield computable a posteriori error estimates without directly solving the dual problem.

Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates

We present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem Au =λBu for symmetric positive definite operators A and B. We call this method successive eigenvalue relaxation, or the SER method (homoechon of the classical successive over‐relaxation, or SOR method for linear systems). In particular, there are two significant features of SER which render it computationally attractive: (i) it can effectively deal with preconditioned large‐scale eigenvalue problems, and (ii) its practical implementation does not require any information about the preconditioner used: it can routinely accommodate sophisticated preconditioners designed to meet more exacting requirements (e.g. three‐dimensional elasticity problems with small thickness parameters). We endow SER with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a discretized partial diwfferential equation exhibiting clusters of eigenvalues.

The Korn's type inequality in subspaces and thin elastic structures

We present the concept of the Korns type inequality in subspaces generalized to embrace thin elastic structures of various kinds such as shells, plates, rods, arches and beams of arbitrary geometries which are viewed and analysed as three-dimensional solids. We show that based on this inequality we can eliminate the deterioration of the convergence from which iterative algorithms are known to suffer when applied to thin elastic structures. As a paradigm we consider a class of iterative algorithms and show that a simple modification based on the Korns type inequality in subspaces yields a radical improvement of their convergence.

A new Korn's type inequality for thin elastic structures

Abstract We introduce a new Korns type inequality and show its usefulness in making the convergence of iterative algorithms for thin elastic structures independent of their thickness.

The method of arbitrary lines in optimal shape design: problems with an elliptic state equation☆

Abstract Optimal shape design problems with elliptic state equations are defined. The shape of an optimised domain is controlled by a vector of design variables uniquely denning a Bezier curve. The state equations are semi-discretised by the h - p method of arbitrary lines (MAL) and approximate shape design problems are derived. The existence and convergence of both semi-discrete state solutions and approximate optimal shapes is proved. In shape design sensitivity analysis, the material derivative method is used. The resulting formulae are expressed by boundary integrals which contain the normal derivative of the state solution. The derivative is approximated by means of the MAL semi-discrete state solution and the convergence properties of the approximation are analysed. Finally, numerical examples are given which demonstrate the performance of MAL solutions as compared with other approaches based on (conforming and nonconforming) finite element methods.

The discrete Korn's type inequality in subspaces and iterative methods for thin elastic structures

Abstract When iterative methods are applied to thin elastic structures, such as shells, plates, rods, arches and beams, they suffer from slow convergence with diminishing thickness. To overcome this difficulty the authors have introduced the fundamental concept of the Korns type inequality in subspaces (see Proc. R. Soc. Lond. A 453 (1997) 2003–2016). Here, we present the discrete Korns type inequality in subspaces which, when applied to discrete methods, such as the finite element method, enables the design of iterative algorithms for thin elastic structures with convergence rate independent of both the thickness and the discretisation parameters. As a paradigm we consider the steepest descent method with the subspace correction preconditioning and present p-version finite element results for a model shell problem which show that a simple modification of this method based on the discrete Korns type inequality in subspaces yields a radical improvement in the convergence.