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

@article{Kimura2019TheGF,
  title={The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme},
  author={Masato Kimura and Hirofumi Notsu and Yoshimi Tanaka and Hiroki Yamamoto},
  journal={Journal of Scientific Computing},
  year={2019},
  volume={78},
  pages={1111-1131}
}
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… 
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References

SHOWING 1-10 OF 14 REFERENCES
MIXED FINITE ELEMENT METHODS FOR LINEAR VISCOELASTICITY USING WEAK SYMMETRY
Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin–Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the
Discontinuous Galerkin Finite Element Approximation of Nonlinear Non-Fickian Diffusion in Viscoelastic Polymers
TLDR
Two means of handling the nonlinearity are discussed: either implicitly, which requires the solution of nonlinear equations at each time level, or through a linearisation based on extrapolating from previous time levels, which are proven to have the same optimal orders of convergence.
Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity
TLDR
A finite-element- in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem and a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation are considered.
Linear Viscoelastic Creep Model for the Contact of Nominal Flat Surfaces Based on Fractal Geometry: Standard Linear Solid (SLS) Material
The objective of this study is to construct a continuous mathematical model that describes the frictionless contact between a nominally flat (rough) viscoelastic punch and a perfectly rigid
Boundary Value Problems in Linear Viscoelasticity
The classical theories of Linear Elasticity and Newtonian Fluids, though trium phantly elegant as mathematical structures, do not adequately describe the defor mation and flow of most real materials.
New development in freefem++
  • F. Hecht
  • Computer Science, Mathematics
    J. Num. Math.
  • 2012
TLDR
First the freefem++ software deals with mesh adaptation for problems in two and three dimension, second, it solves numerically a problem with phase change and natural convection, and finally to show the possibilities for HPC the software solves a Laplace equation by a Schwarz domain decomposition problem on parallel computer.
Viscoelastic properties of polymers
  • J. Ferry
  • Materials Science, Engineering
  • 1961
The Nature of Viscoelastic Behavior. Illustrations of Viscoelastic Behavior of Polymeric Systems. Exact Interrelations among the Viscoelastic Functions. Approximate Interrelations among the Linear
The finite element method for elliptic problems
  • P. Ciarlet
  • Mathematics
    Classics in applied mathematics
  • 2002
From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional
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