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Thomas Richter - One of the best experts on this subject based on the ideXlab platform.
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A Posteriori Error Estimation for the Fractional Step Theta Discretization of the Incompressible Navier-Stokes Equations
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Dominik Meidner, Thomas RichterAbstract:Abstract In this work, we derive a goal-oriented a posteriori error estimator for the error due to time discretization. As time discretization scheme we consider the fractional step theta method, that consists of three subsequent steps of the one-step theta method. In every sub-step, the full incompressible system has to be solved (in contrast to time integrators of operator splitting type). The resulting fractional step theta method combines various desirable properties like second order accuracy, strong A-stability and very little numerical dissipation. The derived error estimator is based on a mathematical trick: we define an intermediate time-discretization scheme based on a Petrov–Galerkin Formulation. This method is up to a numerical quadrature error equivalent to the theta time stepping scheme. The error estimator is assembled as one weighted residual term given by the Dual Weighted Residual method measuring the error between real solution and solution to the Petrov–Galerkin Formulation (that at no time has to be calculated) and one additional residual estimating the discrepancy between actual time stepping scheme used for simulation and the intermediate Petrov–Galerkin Formulation.
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Goal-Oriented Error Estimation for the Fractional Step Theta Scheme
Computational Methods in Applied Mathematics, 2014Co-Authors: Dominik Meidner, Thomas RichterAbstract:In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin Formulation that is up to a numerical quadrature error equivalent to the theta timestepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin Formulation. 2010 Mathematical subject classification: 65M60, 65L60, 65L70, 65M15.
M.h. Aliabadi - One of the best experts on this subject based on the ideXlab platform.
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A Galerkin Formulation for shear deformable plate bending dynamics
International Journal for Numerical Methods in Engineering, 2004Co-Authors: J.j. Pérez-gavilán, M.h. AliabadiAbstract:A Galerkin boundary element Formulation for shear deformable plate bending dynamics is developed. The Formulation makes use of the static fundamental solutions for the weighted residual integral equations. The domain integrals carrying the inertia terms and generic static loads are considered as body forces and approximated with boundary values using the dual reciprocity method. The load is modelled as a series of impact loads of time varying intensity and moving in space in a predetermined path. The Formulation was implemented and tested solving a benchmark problem. The results are compared with finite element solutions. Copyright © 2004 John Wiley & Sons, Ltd.
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A symmetric Galerkin Formulation and dual reciprocity for 2D elastostatics
Engineering Analysis with Boundary Elements, 2001Co-Authors: J.j. Pérez-gavilán, M.h. AliabadiAbstract:In this paper, a symmetric Galerkin boundary integral equation including body force terms is presented. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is presented in the context of the Galerkin Formulation. Several numerical examples involving self-weight and centrifugal body forces are studied to demonstrate the efficiency of the method.
Ludovic Noels - One of the best experts on this subject based on the ideXlab platform.
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Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin Formulation
Computer Methods in Applied Mechanics and Engineering, 2013Co-Authors: Van Dung Nguyen, Gauthier Becker, Ludovic NoelsAbstract:Abstract When considering problems of dimensions close to the characteristic length of the material, the size effects can not be neglected and the classical (so-called first-order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second-order multiscale computational homogenization (SMCH) scheme. This second-order scheme uses the classical continuum at the micro-scale while considering a second-order continuum at the macro-scale. Although the theoretical background of the second-order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high-order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro-scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak Formulations allowing for inter-element discontinuities either at the C 0 level or at the C 1 level, and it can thus be used to constrain weakly the C 1 continuity at the macro-scale. The C 0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C 0 displacement-based finite elements. Therefore, two Formulations can be used at the macro-scale: (i) the full-discontinuous Galerkin Formulation (FDG) with weak C 0 and C 1 continuity enforcements, and (ii) the enriched discontinuous Galerkin Formulation (EDG) with high-order term enrichment into the conventional C 0 finite element framework. The micro-problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non-linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way.
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Full discontinuous Galerkin Formulation of shell in large deformations with fracture mechanic applications
2011Co-Authors: Gauthier Becker, Ludovic NoelsAbstract:Different methods have been developed to model tearing prediction, as e.g., the combination between the cohesive principle and the finite element method. Unfortunately, this method has some well known issues that can be fixed by recourse to discontinuous Galerkin Formulation. Such a Formulation allows to insert very easily an extrinsic cohesive element at onset of fracture without any mesh modification. This promising technique has been recently developed by the authors for linear shell. Although promising numerical results were obtained, it is difficult to compare the method with experiments due to the large plastic deformation present in material before the fracture apparition. Thus, the method is extent herein to elasto-plastic finite deformations. The simulations of some benchmarks prove the ability of this new framework to model accurately the continuum part of the deformation and the crack propagation.
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A discontinuous Galerkin Formulation of non-linear Kirchhoff–Love shells
International Journal for Numerical Methods in Engineering, 2009Co-Authors: Ludovic NoelsAbstract:Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown-field derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901–2929) to develop a Formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one-field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG Formulation. It is the purpose of the present paper to extend this Formulation to finite deformations and non-linear elastic behaviors. While the initial linear Formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple-stress from the displacement field at the mid-surface of the shell, the non-linear Formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reFormulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.
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a discontinuous Galerkin Formulation of non linear kirchhoff love shells
International Journal for Numerical Methods in Engineering, 2009Co-Authors: Ludovic NoelsAbstract:Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown-field derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901–2929) to develop a Formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one-field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG Formulation. It is the purpose of the present paper to extend this Formulation to finite deformations and non-linear elastic behaviors. While the initial linear Formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple-stress from the displacement field at the mid-surface of the shell, the non-linear Formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reFormulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.
Dominik Meidner - One of the best experts on this subject based on the ideXlab platform.
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A Posteriori Error Estimation for the Fractional Step Theta Discretization of the Incompressible Navier-Stokes Equations
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Dominik Meidner, Thomas RichterAbstract:Abstract In this work, we derive a goal-oriented a posteriori error estimator for the error due to time discretization. As time discretization scheme we consider the fractional step theta method, that consists of three subsequent steps of the one-step theta method. In every sub-step, the full incompressible system has to be solved (in contrast to time integrators of operator splitting type). The resulting fractional step theta method combines various desirable properties like second order accuracy, strong A-stability and very little numerical dissipation. The derived error estimator is based on a mathematical trick: we define an intermediate time-discretization scheme based on a Petrov–Galerkin Formulation. This method is up to a numerical quadrature error equivalent to the theta time stepping scheme. The error estimator is assembled as one weighted residual term given by the Dual Weighted Residual method measuring the error between real solution and solution to the Petrov–Galerkin Formulation (that at no time has to be calculated) and one additional residual estimating the discrepancy between actual time stepping scheme used for simulation and the intermediate Petrov–Galerkin Formulation.
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Goal-Oriented Error Estimation for the Fractional Step Theta Scheme
Computational Methods in Applied Mathematics, 2014Co-Authors: Dominik Meidner, Thomas RichterAbstract:In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin Formulation that is up to a numerical quadrature error equivalent to the theta timestepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin Formulation. 2010 Mathematical subject classification: 65M60, 65L60, 65L70, 65M15.
Garth N. Wells - One of the best experts on this subject based on the ideXlab platform.
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A C0 discontinuous Galerkin Formulation for Kirchhoff plates
Computer Methods in Applied Mechanics and Engineering, 2007Co-Authors: Garth N. Wells, N T DungAbstract:Abstract A particular discontinuous Galerkin finite element Formulation for the simulation of Kirchhoff plates is presented. It is rotation-free and utilises standard C 0 Lagrange finite element basis functions, with the required continuity imposed in a weak sense across element boundaries. The implications of the scheme in terms of coercivity and convergence of the Galerkin problem in various norms are studied, with the Formulation shown to be stable for any positive value of a penalty parameter. A priori error estimates are supported by a range of numerical examples. Properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples. Based on the results of the analysis and numerical examples, it is concluded that the Formulation is robust, accurate and relatively simple.
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a discontinuous Galerkin Formulation for a strain gradient dependent damage model
Computer Methods in Applied Mechanics and Engineering, 2004Co-Authors: Garth N. Wells, Krishna Garikipati, Luisa MolariAbstract:Abstract The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand C 1 continuity of the shape functions. Here, recent developments in discontinuous Galerkin methods are explored and exploited for the solution of a prototype non-linear strain gradient-dependent continuum model. A Formulation is developed that allows the rigorous solution of a strain gradient damage model using standard C 0 shape functions. The Formulation is tested in one dimension for the simplest possible finite element Formulation: continuous piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed Formulation.
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A continuous/discontinuous Galerkin Formulation for a strain gradient-dependent damage model
2004Co-Authors: Garth N. Wells, Krishna Garikipati, Luisa MolariAbstract:The numerical solution of strain gradient-dependent continuum problems has been hindered by continuity demands on the basis functions. The presence of terms in constitutive models which involve gradients of the strain eld means that the C 0 conti- nuity of standard nite element shape functions is insucient. In this work, a continu- ous/discontinuous Galerkin Formulation is developed to solve a strain gradient-dependent damage problem in a rigorous manner. Potential discontinuities in the strain eld across element boundaries are incorporated in the weak form of the governing equations. The per- formance of the Formulation is tested in one dimension for various interpolations, which provides guidance for two-dimensional simulations.
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A discontinuous Galerkin Formulation for a strain gradient-dependent damage model
Computer Methods in Applied Mechanics and Engineering, 2004Co-Authors: Garth N. Wells, Krishna Garikipati, Luisa MolariAbstract:The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand $C^{1}$ continuity of the shape functions. Here, recent development in discontinuous Galerkin methods are explored and exploited for the solution of a prototype nonlinear strain gradient dependent continuum model. A Formulation is developed that allows the rigorous solution of a strain gradient damage model using standard $C^{0}$ shape functions. The Formulation is tested in one-dimension for the simplest possible finite element Formulation: piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed Formulation.Comment: 22 pages, 7 figures, submitted, corrected to eliminate multiple appearance of pdf fil
