1. Dimakopoulos, Y; Makrigiorgos, G; Georgiou, G C; Tsamopoulos, J

The PAL (Penalized Augmented Lagrangian) method for computing viscoplastic flows: A new fast converging scheme Journal Article

In: Journal of Non-Newtonian Fluid Mechanics, 256 , pp. 23-41, 2018, ISSN: 03770257, (cited By 13).

Abstract | Links | BibTeX | Tags: Augmented Lagrangian method, Bubble rise, Filament, Lid-driven cavity, Papanastasiou regularization; Penalty method, stretching, Viscoplastic fluids

@article{Dimakopoulos201823,

title = {The PAL (Penalized Augmented Lagrangian) method for computing viscoplastic flows: A new fast converging scheme},

author = {Y Dimakopoulos and G Makrigiorgos and G C Georgiou and J Tsamopoulos},

url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85044113517&doi=10.1016%2fj.jnnfm.2018.03.009&partnerID=40&md5=46144d6483fc39fdf658de1f3f65dd06},

doi = {10.1016/j.jnnfm.2018.03.009},

issn = {03770257},

year = {2018},

date = {2018-01-01},

journal = {Journal of Non-Newtonian Fluid Mechanics},

volume = {256},

pages = {23-41},

abstract = {Computation of viscoplastic fluid flows has always been a challenging task. Viscoplastic models are intrinsically discontinuous at the yielded-unyielded interface, which leads to numerical difficulties, because of the singularity in the Jacobian matrix of the resulting discretized equations. For this reason, several modeling or numerical approaches have been proposed, the most popular being the Papanastasiou regularization (PR) and the Augmented Lagrangian (AL) methods, respectively. Recently, studies on AL methods have focused on developing accelerated algorithms, since the required computational cost of using AL is extremely high. In the present work, a fast converging and efficient algorithm is proposed for tracking the yield surface and predicting the flow field of viscoplastic fluids accurately. The numerical procedure is the Penalized Augmented Lagrangian (PAL) method, which is based on a monolithic Newton solver for AL, where the governing equations of the Lagrange-multiplier tensor for both the rate-of-strain projection and the extra-stress tensors are penalized. To test the efficiency of our algorithm, five benchmark flow-problems with fixed, free and moving boundaries are studied. First, the problem of the steady rise of a bubble in a viscoplastic medium is addressed validating the new algorithm with the findings by Dimakopoulos et al. (2013). Then the entrance flow in a rectangular channel is solved, where a primary unyielded region is found around the centerline in the developed part of the flow and secondary unyielded regions near the entrance. In addition, the lid-driven cavity problem is solved, which is an often used test for various numerical algorithms and the results are compared to relevant studies for viscoplastic fluids such as those of Syrakos et al. (2013, 2014) and Treskatis et al. (2016). Furthermore, the developed flow in a square duct is examined, similarly to Saramito (2016). Finally, the transient filament stretching of a shear-thinning, yield stress fluid is examined, and the results are compared to those by Balmforth et al. (2010). In all cases, either steady or transient, the algorithm captures the yield surfaces correctly, while maintaining a low computational cost, because the convergence of the method requires only a few (i.e. 5–30) Newton iterations. Based on these extensive tests, PAL is found to be superior combining accuracy and speed to all existing solution methods for viscoplastic fluids. © 2018 Elsevier B.V.},

note = {cited By 13},

keywords = {Augmented Lagrangian method, Bubble rise, Filament, Lid-driven cavity, Papanastasiou regularization; Penalty method, stretching, Viscoplastic fluids},

pubstate = {published},

tppubtype = {article}

}

Computation of viscoplastic fluid flows has always been a challenging task. Viscoplastic models are intrinsically discontinuous at the yielded-unyielded interface, which leads to numerical difficulties, because of the singularity in the Jacobian matrix of the resulting discretized equations. For this reason, several modeling or numerical approaches have been proposed, the most popular being the Papanastasiou regularization (PR) and the Augmented Lagrangian (AL) methods, respectively. Recently, studies on AL methods have focused on developing accelerated algorithms, since the required computational cost of using AL is extremely high. In the present work, a fast converging and efficient algorithm is proposed for tracking the yield surface and predicting the flow field of viscoplastic fluids accurately. The numerical procedure is the Penalized Augmented Lagrangian (PAL) method, which is based on a monolithic Newton solver for AL, where the governing equations of the Lagrange-multiplier tensor for both the rate-of-strain projection and the extra-stress tensors are penalized. To test the efficiency of our algorithm, five benchmark flow-problems with fixed, free and moving boundaries are studied. First, the problem of the steady rise of a bubble in a viscoplastic medium is addressed validating the new algorithm with the findings by Dimakopoulos et al. (2013). Then the entrance flow in a rectangular channel is solved, where a primary unyielded region is found around the centerline in the developed part of the flow and secondary unyielded regions near the entrance. In addition, the lid-driven cavity problem is solved, which is an often used test for various numerical algorithms and the results are compared to relevant studies for viscoplastic fluids such as those of Syrakos et al. (2013, 2014) and Treskatis et al. (2016). Furthermore, the developed flow in a square duct is examined, similarly to Saramito (2016). Finally, the transient filament stretching of a shear-thinning, yield stress fluid is examined, and the results are compared to those by Balmforth et al. (2010). In all cases, either steady or transient, the algorithm captures the yield surfaces correctly, while maintaining a low computational cost, because the convergence of the method requires only a few (i.e. 5–30) Newton iterations. Based on these extensive tests, PAL is found to be superior combining accuracy and speed to all existing solution methods for viscoplastic fluids. © 2018 Elsevier B.V.2. Dimakopoulos, Y; Pavlidis, M; Tsamopoulos, J

In: Journal of Non-Newtonian Fluid Mechanics, 200 , pp. 34-51, 2013, ISSN: 03770257, (cited By 55).

Abstract | Links | BibTeX | Tags: Augmented Lagrangian method, bubble, Free surface flows, Herschel-Bulkley fluids, Papanastasiou model, viscoplastic material

@article{Dimakopoulos201334,

title = {Steady bubble rise in Herschel-Bulkley fluids and comparison of predictions via the Augmented Lagrangian Method with those via the Papanastasiou model},

author = {Y Dimakopoulos and M Pavlidis and J Tsamopoulos},

url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-84881025361&doi=10.1016%2fj.jnnfm.2012.10.012&partnerID=40&md5=3ee9dc3fd031d766237c1052ec38fa3b},

doi = {10.1016/j.jnnfm.2012.10.012},

issn = {03770257},

year = {2013},

date = {2013-01-01},

journal = {Journal of Non-Newtonian Fluid Mechanics},

volume = {200},

pages = {34-51},

abstract = {The steady, buoyancy-driven rise of a bubble in a Herschel-Bulkley fluid is examined assuming axial symmetry. The variation of the rate-of-strain tensor around a rising bubble necessitates the coexistence of fluid and solid regions in this fluid. In general, a viscoplastic fluid will not be deforming beyond a finite region around the bubble and, under certain conditions, it will not be deforming either just behind it or around its equatorial plane. The accurate determination of these regions is achieved by introducing a Lagrange multiplier and a quadratic term in the corresponding variational inequality, resulting in the so-called Augmented Lagrangian Method (ALM). Additionally here, the augmentation parameters are determined following a non-linear conjugate gradient procedure. The new predictions are compared against those obtained by the much simpler Papanastasiou model, which uses a continuous constitutive equation throughout the material, irrespective of its state, but does not determine the boundary between solid and liquid along with the flow field. The flow equations are solved numerically using the mixed finite-element/Galerkin method on a mesh generated by solving a set of quasi-elliptic differential equations. The accuracy of solutions is ascertained by mesh refinement and comparison with our earlier and new predictions for a bubble rising in a Newtonian and a Bingham fluid. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham, Bn, Bond, and Archimedes numbers. As Bn increases, the bubble decelerates, the yield surfaces at its equatorial plane and away from it approach each other and eventually merge immobilizing the bubble. For small and moderate Bingham numbers, the predictions using the Papanastasiou model satisfactorily approximate those of the discontinuous Herschel-Bulkley model for sufficiently large values of the normalization exponent (≥104). On the contrary, as Bn increases and the rate-of-strain approaches zero almost throughout the fluid-like region, much larger values of the exponent are required to accurately compute the yield surfaces. Bubble entrapment does not depend on the power law index, i.e. a bubble in a Herschel-Bulkley fluid is entrapped under the same conditions as in a Bingham fluid. © 2012 Elsevier B.V.},

note = {cited By 55},

keywords = {Augmented Lagrangian method, bubble, Free surface flows, Herschel-Bulkley fluids, Papanastasiou model, viscoplastic material},

pubstate = {published},

tppubtype = {article}

}

The steady, buoyancy-driven rise of a bubble in a Herschel-Bulkley fluid is examined assuming axial symmetry. The variation of the rate-of-strain tensor around a rising bubble necessitates the coexistence of fluid and solid regions in this fluid. In general, a viscoplastic fluid will not be deforming beyond a finite region around the bubble and, under certain conditions, it will not be deforming either just behind it or around its equatorial plane. The accurate determination of these regions is achieved by introducing a Lagrange multiplier and a quadratic term in the corresponding variational inequality, resulting in the so-called Augmented Lagrangian Method (ALM). Additionally here, the augmentation parameters are determined following a non-linear conjugate gradient procedure. The new predictions are compared against those obtained by the much simpler Papanastasiou model, which uses a continuous constitutive equation throughout the material, irrespective of its state, but does not determine the boundary between solid and liquid along with the flow field. The flow equations are solved numerically using the mixed finite-element/Galerkin method on a mesh generated by solving a set of quasi-elliptic differential equations. The accuracy of solutions is ascertained by mesh refinement and comparison with our earlier and new predictions for a bubble rising in a Newtonian and a Bingham fluid. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham, Bn, Bond, and Archimedes numbers. As Bn increases, the bubble decelerates, the yield surfaces at its equatorial plane and away from it approach each other and eventually merge immobilizing the bubble. For small and moderate Bingham numbers, the predictions using the Papanastasiou model satisfactorily approximate those of the discontinuous Herschel-Bulkley model for sufficiently large values of the normalization exponent (≥104). On the contrary, as Bn increases and the rate-of-strain approaches zero almost throughout the fluid-like region, much larger values of the exponent are required to accurately compute the yield surfaces. Bubble entrapment does not depend on the power law index, i.e. a bubble in a Herschel-Bulkley fluid is entrapped under the same conditions as in a Bingham fluid. © 2012 Elsevier B.V.