Top

Journal of Scientific Computing

Published in:

23-08-2018

The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme

Authors: Masato Kimura, Hirofumi Notsu, Yoshimi Tanaka, Hiroki Yamamoto

Published in: Journal of Scientific Computing | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.

previous article Error Estimates of Spectral Galerkin Methods for a Linear Fractional Reaction–Diffusion Equation

next article Adaptive Modal Filters Based on Artificial and Spectral Viscosity Techniques

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Abuzeid, O.M., Eberhard, P.: Linear viscoelastic creep model for the contact of nominal flat surfaces based on fractal geometry: standard linear solid (SLS) material. J. Tribol. 129, 461–466 (2007)CrossRef

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH

Ferry, J.D.: Viscoelastic Properties of Polymers. Wiley, New York (1970)

Golden, J.M., Graham, G.A.C.: Boundary Value Problems in Linear Viscoelasticity. Springer, Berlin (1988)CrossRefMATH

Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetMATH

Karamanou, M., Shaw, S., Warby, M.K., Whiteman, J.R.: Models, algorithms and error estimation for computational viscoelasticity. Comput. Methods Appl. Mech. Eng. 194(2–5), 245–265 (2005)MathSciNetCrossRefMATH

Kimura, M., Notsu, H., Tanaka, Y., Yamamoto, H.: In preparation

Lockett, F.J.: Nonlinear Viscoelastic Solids. Academic Press, Paris (1972)MATH

Macosko, C.W.: Rheology: Principles, Measurements, and Applications. Wiley-VCH, New York (1994)

10.

Nečas, J.: Les Méthods Directes en Théories des Équations Elliptiques. Masson, Paris (1967)MATH

11.

Rivière, B., Shaw, S.: Discontinuous Galerkin finite element approximation of nonlinear non-Fickian diffusion in viscoelastic polymers. SIAM J. Numer. Anal. 44(6), 2650–2670 (2006)MathSciNetCrossRefMATH

12.

Rivière, B., Shaw, S., Wheeler, M.F., Whiteman, J.R.: Discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numerische Mathematik 95(2), 347–376 (2003)MathSciNetCrossRefMATH

13.

Rognes, M.E., Winther, R.: Mixed finite element methods for linear viscoelasticity using weak symmetry. Math. Models Methods Appl. Sci. 20, 955–985 (2010)MathSciNetCrossRefMATH

14.

Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Springer, Berlin (2005)MATH

15.

Shaw, S., Whiteman, J.R.: A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems. Comput. Methods Appl. Mech. Eng. 193(52), 5551–5572 (2004)MathSciNetCrossRefMATH

Title: The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme
Authors: Masato Kimura
Hirofumi Notsu
Yoshimi Tanaka
Hiroki Yamamoto
Publication date: 23-08-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0799-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 2/2019

An Alternative Formulation of Discontinous Galerkin Schemes for Solving Hamilton–Jacobi Equations

A Posteriori Error Analysis of a Mixed-Primal Finite Element Method for the Boussinesq Problem with Temperature-Dependent Viscosity

A Nonconvex Model with Minimax Concave Penalty for Image Restoration

A Divergence Free Weak Virtual Element Method for the Stokes Problem on Polytopal Meshes

A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems

On Alternative Setups of the Double Mach Reflection Problem

Premium Partner