Adaptive timestep control for the contact-stabilized Newmark method

The aim of this paper is to devise an adaptive timestep control in the contact-stabilized Newmark method (ContacX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini’s condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact-stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.     

Robust a posteriori error estimates for subgrid stabilization of non-stationary convection dominated diffusive transport

We derive a robust residual a posteriori error estimator for time-dependent convection-diffusion-reaction problem, stabilized by subgrid viscosity in space and discretized by Crank-Nicolson scheme in time. The estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time. Numerical experiments illustrate the theoretical performance of the error estimator.     

A time‐marching finite element method for an electromagnetic scattering problem

In this paper, Newmark time‐stepping scheme and edge elements are used to numerically solve the time‐dependent scattering problem in a three‐dimensional polyhedral cavity. Finite element methods based on the variational formulation derived in Van and Wood (Adv. Comput. Math., to appear) are considered. Existence and uniqueness of the discrete problem is proved by using Babuska–Brezzi theory. Finite element error estimate and stability of the Newmark scheme are also established. Copyright © 2003 John Wiley & Sons, Ltd.     

AN ADAPTIVE VERSION OF GLIMM＇S SCHEME

This article describes a local error estimator for Glimm＇s scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm＇s scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1 （R） for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad- hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers＇ equation and for Euler＇s system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.     

A unified approach for elasto‐plastic FEM simulations,incorporating adaptive spatial and time discretization

In FEM calculations the discretization should be chosen in a way, that further mesh refinement does not change the results. Otherwise the discretization error might yield unphysically stiff material behavior. If elasto‐plasticity is considered, a second type of error due to the time discretization of the evolution equations has to be taken into account. Due to the non‐linearity of the underlying initial boundary value problem, large time increments often result in non‐converging solutions during equilibrium iteration. In our approach the time integration error resulting from a second order BDF2 time integration method is calculated and utilized in an automatic step size control. In conjunction with a DAE‐treatment of the initial boundary value problem, it allows in the average considerably larger time steps compared to classical ‘elastic predictor – plastic corrector’ schemes. To reduce the discretization error, adaptive mesh‐refinement based on a Z²‐error‐estimator is performed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

AN ADAPTIVE VERSION OF GLIMM'S SCHEME

This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice.As a by-product of the local error estimator,the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1(R) for all times.Although there is partial mathematical evidence for the error estimator proposed,at this stage the error estimator must be considered adhoc.Nonetheless,the error estimator is simple to compute,relatively inexpensive,without adjustable parameters and at least as accurate as other existing error estimators.Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.     

Galerkin‐type space‐time finite elements for volumetrically coupled problems

The study focuses on error estimation techniques for a coupled problem with two constituents based on the Theory of Porous Media. After developing space‐time finite elements for this mixed problem, we extend the numerical scheme to a coupled space‐time adaptive strategy. Therefore, an adjoint or dual problem is formulated and discussed, which is solved lateron numerically. One advantage of the presented technique is the high flexibility of the error indicator with respect to the error measure.     

Efficient Simulation of Glass Cooling Processes

A hybrid method for simulations of cooling processes of high quality glass is investigated. The hybrid algorithm relies on two solvers or the radiative transfer equation and on a global error estimator which allows for an optimal choice of the solver in each frequency band. The global error estimator is formulated in a general Hilbert space setting. Abstract formulations of a Gauss‐like integration‐by‐parts formula and its inversion are required.     

Consistency results on Newmark methods for dynamical contact problems

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the classical Newmark method, the improved energy dissipative version due to Kane et al., and the contact-stabilized Newmark method recently suggested by Deuflhard et al. In the absence of contact, any such variant is equivalent to the Störmer–Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.     

The backward euler anisotropic a posteriori error analysis for parabolic integro‐differential equations

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro‐differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz‐Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016     

James-Stein estimators for time series regression models

The least squares (LS) estimator seems the natural estimator of the coefficients of a Gaussian linear regression model. However, if the dimension of the vector of coefficients is greater than 2 and the residuals are independent and identically distributed, this conventional estimator is not admissible. James and Stein [Estimation with quadratic loss, Proceedings of the Fourth Berkely Symposium vol. 1, 1961, pp. 361-379] proposed a shrinkage estimator (James-Stein estimator) which improves the least squares estimator with respect to the mean squares error loss function. In this paper, we investigate the mean squares error of the James-Stein (JS) estimator for the regression coefficients when the residuals are generated from a Gaussian stationary process. Then, sufficient conditions for the JS to improve the LS are given. It is important to know the influence of the dependence on the JS. Also numerical studies illuminate some interesting features of the improvement. The results have potential applications to economics, engineering, and natural sciences.     

Adaptive mesh for algebraic orthogonal subscale stabilization of convective dispersive transport

We derive a residual a posteriori error estimator for the algebraic orthogonal subscales stabilization of convective dispersive transport equation. The estimator yields upper bound on the error which is global and lower bound that is local. Numerical studies show the behaviour of the error indicator and how it is robust to deal with singularities. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

An improved exponential estimator of finite population mean in simple random sampling using an auxiliary attribute

In this paper, we propose an exponential ratio type estimator of the finite population mean when auxiliary information is qualitative in nature. Under simple random sampling without replacement scheme, the expressions for the bias and the mean square error of the proposed estimator have been obtained, up to first order of approximation. To show that our proposed estimator is more efficient as compared to the existing estimators, we have made a comparative study with respect to their mean square errors. Theoretically and numerically, we have found that our proposed estimator is always more efficient as compared to its competitor estimators including all the estimators of Abd-Elfattah et al. [1] [A.M. Abd-Elfattah, E.A. El-Sherpieny, S.M. Mohamed, and O.F. Abdou. Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute. Applied Mathematics and Computation, 215 (2010), 4198-4202].     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

Enhanced low-order displacement finite elements using incompatible mass scaling and static condensation in explicit time integration

For nonlinear transient problems as well as rather slow quasi-static processes explicit time integration has proven to be a suitable and robust time integration algorithm. As a consequence of the Courant-Friedrichs-Lewy (CFL) criterion small time step sizes are necessary and the overall computation time is dominated by the operations on element level. Privileged use of low order finite elements is common in explicit time integration due to their efficiency and robustness. In order to suppress artificial locking effects the work of Bischoff and Romero [1] is followed where a generalization of the method of incompatible modes (IM) is derived being equivalent to the class of enhanced assumed strain (EAS) elements originally proposed by Simo and Rifai [2]. These concepts are bound to a static elimination procedure of internal parameters which may result in a considerable expansion of the computational costs in the case of an explicit time integration scheme. In order to avoid the static elimination procedure inertia is considered for the incompatible parameters as suggested by Mattern et al. [3, 4]. This step allows integrating the incompatible parameters directly in time, but since rather large stiffnesses are related to the incompatible parameters – particularly for the volumetric mode enhancements – the highest eigenfrequency may be increased which would lead to a decrease of the time step size. This motivates scaling the masses associated to these incompatible parameters to control their effect on the time step size. It is possible to derive analytically based estimates for the scaling of the IM-masses in order to achieve results agreeing very good with the classical EAS approach including large deflections and nonlinear material behaviour with elasto-plasticity. The numerical implementation is performed using AceGen [5]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Posteriori Error Estimates and a Local Refinement Strategy for a Finite Element Method to Solve Structural-Acoustic Vibration Problems

This paper deals with an adaptive technique to compute structural-acoustic vibration modes. It is based on an a posteriori error estimator for a finite element method free of spurious or circulation nonzero-frequency modes. The estimator is shown to be equivalent, up to higher order terms, to the approximate eigenfunction error, measured in a useful norm; moreover, the equivalence constants are independent of the corresponding eigenvalue, the physical parameters, and the mesh size. This a posteriori error estimator yields global upper and local lower bounds for the error and, thus, it may be used to design adaptive algorithms. We propose a local refinement strategy based on this estimator and present a numerical test to assess the efficiency of this technique.     

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

In this work, an effective and fast finite element numerical method with high-order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two-level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H¹-norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.     

Error bounds for the large time step Glimm scheme applied to scalar conservation laws

Summary. In this paper we derive an error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2. Received April 10, 2000 / Revised version received January 16, 2001 / Published online September 19, 2001     

ABSOLUTE STABLE HOMOTOPY FINITE ELEMENT METHODS FOR CIRCULAR ARCH PROBLEM AND ASYMPTOTIC EXACTNESS POSTERIORI ERROR ESTIMATE

In this paper, HFEM is proposed to investigate the circular arch problem. Optimal error estimates are derived, some superconvergence results are established, and an asymptotic exactness posteriori error estimator is presented. In contrast with the classical displacement variational method, the optimal convergence rate for displacement is uniform to the small parameter. In contrast with classical mixed finite element methods, our results are free of the strict restriction on h(the mesh size) which is preserved by all the previous papers. Furtheremore we introduce an asymptotic exactness posteriori error estimator based on a global superconvergence result which is discovered in this kind of problem for the first time.     

A dual graph-norm refinement indicator for finite volume approximations of the Euler equations

Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR--Code which is based on a finite volume method of box type on an unstructured grid and present numerical results. Received May 15, 1995 / Revised version received April 17, 1996