Modelling of submarine landslides of rock and soil

Mariotti, C.; Heinrich, P.

doi:10.1002/(sici)1096-9853(19990410)23:4<335::aid-nag974>3.0.co;2-i

Cited by 9 publications

(9 citation statements)

References 5 publications

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“…From mechanical and engineering points of view, the great amount of research in Bingham fluid has been done in the following areas: non-Newtonian fluid in porous media [68,67,11,47], heat transfer [65,66], mudflows resulting from submarine slides [52,42,43], chemical processing [15,54,25,62], electrorheological (ER) fluids [51,48] and other mechanical properties [43,64,24,41,49,17]. From mathematical point of view, Duvaut, Lions, Brézis and Glowinski have laid a mathematical foundation of Bingham fluid in the framework of variational inequalities [4,5,16,32,29].…”

Section: Discussionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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The flow of a Bingham fluid in a cylindrical pipe can give rise to free boundary problems. The fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries. Therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods. Considerable progress has been made in the development of numerical algorithms for Bingham fluids [2,20,27,29,30,32]. However, very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution, that is the error estimate of these numerical methods. Error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters. This paper concerns the error estimates of a unsteady Bingham fluid modeled as a variational inequality due to Duvaut-Lions [16] and Glowinski [30]. The difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term. A common technique, called the regularization method, is to replace the non-differentiable term by a perturbed differentiable term which depends on a small regularization parameter . The regularization method effectively reduces the variational inequality to an equation (a regularized problem) which is much easier to cope with. This paper has achieved the following. (1) Error estimates are derived for a continuous time Galerkin method in suitable norms. (2) We give an estimate of the difference between the true solution and the regularized solution in terms of . (3) Some regularity properties for both regularized solution and the true solution are proved. (4) The error 154 Y. Zhangestimates for full discretization of the regularized problem using piecewise linear finite elements in space, and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t. (5) We are able to improve our estimates in the one-dimensional case or under stronger regularity assumptions on the true solution. The estimates for the one-dimensional case are optimal and confirmed by numerical experiment. The estimates from (4) and (5) provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h, t and in order to control the error (see Corollary 4.4). The above estimates extend the error bounds derived in Glowinski, Lions and Trémolières [32] (chapter 5, pp. 348-404) for the stationary Bingham fluid to the time-dependent one, which is the main contribution of this paper. (2000): 35k85, 65M15, 65M60, 76A05, 76M10 Mathematics Subject Classification

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Section: Discussionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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“…Although many non-Newtonian models could reproduce the deformation of landslides very well ( Ataie-Ashtiani and Shobeyri, 2008 ;Rzadkiewicz et al, 1997 ;Capone et al, 2010 ;Mariotti and Heinrich, 1999 ), two significant characteristics of our coupling model can be seen compared with non-Newtonian fluid models. The first one is about the method for choosing the values of parameters.…”

Section: Comparison With a Non-newtonian Fluid Modelmentioning

confidence: 60%

“…For a saturated submerged landslide, like the experiment of submarine landslide mentioned above, a good agreement can be seen between the simulation results of non-Newtonian fluid models and experiments data ( Ataie-Ashtiani and Shobeyri, 2008 ;Rzadkiewicz et al, 1997 ;Capone et al, 2010 ;Mariotti and Heinrich, 1999 ). But for a fast subaerial landslide, like the experiment of Fritz et al, the landslide may behave too softly with the non-Newtonian fluid models compared with the real situations ( Quecedo et al, 2004 ;Cremonesi et al, 2011 ).…”

Section: Comparison With a Non-newtonian Fluid Modelmentioning

confidence: 99%

“…recently. Using rheological theory, a non-Newtonian fluid model was used to describe the deformation and movement of landslides by Quecedo et al (2004) , Cremonesi et al (2011) , Rzadkiewicz et al (1997) , Capone et al (2010) , Ataie-Ashtiani and Shobeyri (2008) , Mariotti and Heinrich (1999) ( Table 2 ), and Manenti et al (2015) . Simple governing equations and fluid-fluid interaction for the coupling process in this model will reduce the simulated difficulties.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation of landslide-generated waves using a soil–water coupling smoothed particle hydrodynamics model

Shi

et al. 2016

Advances in Water Resources

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“…In certain papers, for example, , the soil material is simply modeled as a Newtonian fluid. More complex viscoplastic fluid models are applied in . Furthermore, the inclusion of models for turbidity currents on the surface of submarine landslides is shown in and .…”

Section: Introductionmentioning

confidence: 99%

Extended space–time finite elements for landslide dynamics

Pasenow

Zilian

Dinkler

2012

Numerical Meth Engineering

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SUMMARY The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level‐set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level‐set equation driven by the physical velocity field for all three level‐set functions in a block Jacobi approach. The three boundary‐coupled fluid‐like continua involved are modeled as incompressible, governed by a generalized non‐Newtonian material law taking into account capillary pressure at moving fluid–fluid interfaces. The weighted residual formulation of the level‐set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non‐smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three‐fluid tank sloshing and air–water‐liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. Copyright © 2012 John Wiley & Sons, Ltd.

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Modelling of submarine landslides of rock and soil

Cited by 9 publications

References 5 publications

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Numerical simulation of landslide-generated waves using a soil–water coupling smoothed particle hydrodynamics model

Extended space–time finite elements for landslide dynamics

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