Constitutive relation error estimators for (visco)plastic finite element analysis with softening

Advances in Adaptive Computational Methods in Mechanics

In this paper, adaptive algorithms for time and space discretizations are added to an existing solution method previously applied to the Venice Lagoon Tidal Circulation problem. An analysis of the interactions between space and time discretizations adaptation algorithms is presented. In particular, it turns out that both error estimations in space and time must be present for maintaining the adaptation efficiency. Several advantages, for adaptivity and for time decoupling of the equations, offered by the operator-splitting adopted for shallow water equations solution are presented. Copyright © 1999 John Wiley & Sons, Ltd.

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017

In this article we discuss a priori error estimates for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations and subject to inequality control constraints. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters taking into account the spatial and the temporal regularity of the optimal solution. For the treatment of the control discretization we discuss different approaches extending techniques known from the elliptic case. For detailed proofs and numerical results we refer to [18, 19].KeywordsOptimal Control ProblemSpace DiscretizationDiscontinuous Galerkin MethodControl ConstraintTemporal RegularityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Space–time adaptive splitting scheme for the numerical simulation of polycrystallization

The structural behaviour of a bolted, flanged connection subjected to a cyclic load

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

New investigations on the method of characteristics for the evaluation of line transients

A new numerically stable sequential algorithm for coupled finite-strain elastoplastic geomechanics and flow

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the $\mathbb{P}^k$ (piecewise polynomial) finite element is used. When $k=1$ (piecewise linear finite element), the error estimate is obtained under the usual CFL condition $\dt\leq \beta h$ for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for $\mathbb{P}^k$ finite elements with $k>1$ are obtained under a more restrictive CFL condition.

Quadratic programming method in numerical simulation of metal forming process

ABSTRACTWe study residual-based a posteriori error estimates for both the spatially discrete and the fully discrete lumped mass finite element approximation for parabolic problems in a bounded convex polygonal domain in ℝ2. In particular, the space discretization uses finite element spaces that are assumed to be nested one and the time discretization is based on the backward Euler approximation. The main key features used in the analysis are the reconstruction technique and energy argument combined with the stability of L2 projection in H1(Ω). The a posteriori error estimates we derive are optimal order in both the and -norms.

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).

Intelligent finite element method: An evolutionary approach to constitutive modeling

We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in $${\mathbb {R}}^d (d=1,2,3)$$ . Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when $$u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )$$ . Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function $$u_0$$ is only in $$ L^2(\varOmega )$$ . The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.