A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems

Damanik, H.; Hron, J.; Ouazzi, Abderrahim; Turek, Stefan

doi:10.1016/j.jnnfm.2010.05.008

Cited by 50 publications

(88 citation statements)

References 45 publications

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“…In addition to preserving the positive-definiteness, the log-conformation representation improves the representation of large stress gradients, since it linearizes the exponential stress profiles. The log-conformation representation has been implemented in various finite-volume [39][40][41][42] and finite-element [37,[43][44][45][46][47][48] codes. Alternatively, Vaithianathan and Collins [49] have presented two other schemes that guaranty the positive-definiteness of the conformation tensor, based on matrix decompositions.…”

Section: Introductionmentioning

confidence: 99%

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

Comminal

Spangenberg

Hattel

2015

Journal of Non-Newtonian Fluid Mechanics

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Section: Introductionmentioning

confidence: 99%

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

Comminal

Spangenberg

Hattel

2015

Journal of Non-Newtonian Fluid Mechanics

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“…Since then, many computational techniques based upon this matrix transformation have been published, invariably showing good stability properties for solving challenging problems in viscoelastic flows (e.g. [5][6][7][8][9][10][11][12]). By taking the advantage that the conformation tensor A is a symmetric positive definite tensor, Balci et al [13] proposed a simple and interesting square root matrix transformation.…”

Section: Introductionmentioning

confidence: 99%

A numerical study of the Kernel-conformation transformation for transient viscoelastic fluid flows

Martins

Oishi

Afonso

et al. 2015

Journal of Computational Physics

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This work presents a numerical application of a generic conformation tensor transformation for simulating highly elastic flows of non-Newtonian fluids typically observed in computational rheology. In the Kernel-conformation framework [14], the conformation tensor constitutive law for a viscoelastic fluid is transformed introducing a generic tensor transformation function. The numerical stability of the application of the Kernel-conformation for highly elastic flows is ultimately related with the specific kernel function used in the matrix transformation, but also to the existence of singularities introduced either by flow geometry or by the characteristics of the constitutive equation. In this work, we implement this methodology in a free-surface Marker-And-Cell discretization methodology implemented in a finite differences method. The main contributions of this work are two fold: on one hand, we demonstrate the accuracy of this Kernel-conformation formulation using a finite differences method and free surfaces; on the other hand, we assess the numerical efficiency of specific kernel functions at high-Weissenberg number flows. The numerical study considers different viscoelastic fluid flow problems, including the Poiseuille flow in a channel, the lid-driven cavity flow and the die-swell free surface flow. The numerical results demonstrate the adequacy of this methodology for high Weissenberg number flows using the Oldroyd-B model.

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“…It is clear that this formula can be evaluated along the same lines as the evaluation of the double integral itself (11). Similar considerations also yield the different derivatives of the exponential mapping as involved in the discretized weak form (9)…”

Section: Linearization and Evaluationmentioning

confidence: 79%

“…Nevertheless, considering the derivatives of eigenvectors, it is known that they become singular in the case of degenerate eigenvalues due to the ambiguity in the eigenvectors. As a remedy for this and for the difficulty of taking the derivative of the matrix-exponential function, first attempts resorted to the approximation of the Jacobian matrix by difference quotients [10,11].…”

Section: Introductionmentioning

confidence: 99%

The fully-implicit log-conformation formulation and its application to three-dimensional flows

Knechtges¹

2015

Journal of Non-Newtonian Fluid Mechanics

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The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation method as proposed by Fattal and Kupferman, the application of the efficient Newton-Raphson algorithm to the full monolithic system of governing equations, consisting of the log-conformation equations and the Navier-Stokes equations, has always posed a problem. In particular, it is the formulation of the constitutive equations by means of the spectral decomposition that hinders the application of further analytical tools. Therefore, up to now, a fully monolithic approach could only be achieved in two dimensions, as, e.g., recently shown in [P. Knechtges, M. Behr, S. Elgeti, Fully-implicit log-conformation formulation of constitutive laws, J. Non-Newtonian Fluid Mech. 214 (2014) 78-87].The aim of this paper is to find a generalization of the previously made considerations to three dimensions, such that a monolithic Newton-Raphson solver based on the log-conformation formulation can be implemented also in this case. The underlying idea is analogous to the two-dimensional case, to replace the eigenvalue decomposition in the constitutive equation by an analytically more "well-behaved" term and to rely on the eigenvalue decomposition only for the actual computation. Furthermore, in order to demonstrate the practicality of the proposed method, numerical results of the newly derived formulation are presented in the case of the sedimenting sphere and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found that the expected quadratic convergence of Newton's method can be achieved.

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A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems

Cited by 50 publications

References 45 publications

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

A numerical study of the Kernel-conformation transformation for transient viscoelastic fluid flows

The fully-implicit log-conformation formulation and its application to three-dimensional flows

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