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• Numerical analysis of particulate flows with the finite element method

  G. Casas, E. Oñate, R. Rossi

  Monograph CIMNE (2018). M181

  
  

  Abstract

  In this work we study the numerical simulation of particle-laden fluids, with an emphasis on Newtonian fluids and spherical, rigid particles. Our general strategy consists in using the discrete element method (DEM) to model the particles and the finite element method (FEM) to discretize the continuous phase, such that the fluid is not resolved around the particles, but rather averaged over them. The effect of the particles on the fluid is taken into account by averaging (filtering) their individual volumes and particle-fluid interaction forces. In the first part of the work we study the Maxey–Riley equation of motion for an isolated particle in a nonuniform flow; the equation used to calculate the trajectory of the DEM particles. In particular, we perform a detailed theoretical study of its range of applicability, reviewing the initial effects of breaking its fundamental hypotheses, such as small Reynolds number, sphericity of the particle, isolation etc. The output of this study is a set of tables containing order-of-magnitude inequalities to assess the validity of the method in practice. The second part of the work deals with the numerical discretization of the MRE and, in particular, the study of different techniques for the treatment of the history-dependent term, which is difficult to calculate efficiently. We provide improvements on an existing method, proposed by van Hinsberg et al. (2011), and demonstrate its accuracy and efficiency in a sequence of tests of increasing complexity. In the final part of the work we give three application examples representative of different regimes that may be encountered in the industry, demonstrating the versatility of our numerical tool. For that, we describe necessary generalizations to the MRE to cover problems outside its range of applicability. Furthermore, we give a detailed account of the stabilized FEM algorithm used to discretize the fluid phase and compare several derivative recovery tools necessary to calculate some of the interphase coupling terms. Finally, we generalize the algorithm to include the backward-coupling effects according to the theory of multicomponent continua, allowing the code to deal with arbitrarily dense flow regimes.

  Abstract
  In this work we study the numerical simulation of particle-laden fluids, with an emphasis on Newtonian fluids and spherical, rigid particles. Our general strategy consists [...]

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• Benchmarks for geotechnical modeling

  E. Alarcón, E. Alonso, D. Aubry, K. Axelsson, E. Oñate

  Int. J. Numer. Meth. Engng. (1988). Vol. 26 (11), pp. 2559-2568

  
  

  Abstract

  This letter represents an initiative started by a number of researchers signed below who are working in the field of numerical modelling of soil mechanics problems. We belive that the type of approach outlined is important in a serious numerical study of all problems in geomechanics and indeed is valid in other areas. 

  Abstract
  This letter represents an initiative started by a number of researchers signed below who are working in the field of numerical modelling of soil mechanics problems. We belive [...]

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• A total Lagrangian formulation for the geometrically nonlinear-analysis of structures using finite-elements. Part II: arches, frames and axisymmetric shells

  J. Oliver, E. Oñate

  Int. J. Numer. Meth. Engng. (1986). Vol. 23 (2), pp. 253-274

  
  

  Abstract

  A geometrically nonlinear finite element formulation based on a total Lagrangian approach for axisymmetric shells, arches and frames has been presented. The formulation allows for large displacement-large rotations of the structure. Shear deformation effects have also been taken into account. It has been shown how the formulation can be presented in a unified manner to treat simultaneously the three types of structures.

  Abstract
  A geometrically nonlinear finite element formulation based on a total Lagrangian approach for axisymmetric shells, arches and frames has been presented. The formulation allows [...]

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• A total Lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. Part I. Two‐dimensional problems: Shell and plate structures

  J. Oliver, E. Oñate

  Int. J. Numer. Meth. Engng. (1984). Vol. 20 (12), pp. 2253-2281

  
  

  Abstract

  A total Lagrangian finite element formulation for the geometrically nonlinear analysis (large displacement/large rotations) of shells is presented. Explicit expressions of all relevant finite element matrices are obtained by means of the definition of a local co‐ordinate system, based on the shell principal curvature directions, for the evaluation of strains and stresses. A series of examples of nonlinear analysis of shell and plate structures is given.

  Abstract
  A total Lagrangian finite element formulation for the geometrically nonlinear analysis (large displacement/large rotations) of shells is presented. Explicit expressions of [...]

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• A general formulation for coupled thermal flow of metals using finite elements

  O. Zienkiewicz, E. Oñate, J. Heinrich

  Int. J. Numer. Meth. Engng. (1981). Vol. 17 (10), pp. 1497-1514

  
  

  Abstract

  A finite element formulation to deal with the flow of metals coupled with thermal effects in presented. The deformation process of the metal is treated using the visco‐plastic flow approach and the solution technique for the coupled problem implies a simultaneous solution for velocities and temperatures. Some aspects of the numerical solution of the problem are given and in the last part of the paper some examples of steady‐state extrusion and rolling problems showing the applicability of the method are shown.

  Abstract
  A finite element formulation to deal with the flow of metals coupled with thermal effects in presented. The deformation process of the metal is treated using the visco‐plastic [...]

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• A simple and efficient element for axisymmetric shells

  O. Zienkiewicz, J. Bauer, K. Morgan, E. Oñate

  International Journal for Numerical Methods in Engineering (1977). Vol. 11 (10), pp. 1545-1558

  
  

  Abstract

  A two noded, straight element which includes shear deformation effects is presented and shown to be extremely efficient in the analysis of axisymmetric shells. A single point of numerical integration is essential for its success when applied to thin shells where the results compare favourably with those achieved with more complex curved elements.

  Abstract
  A two noded, straight element which includes shear deformation effects is presented and shown to be extremely efficient in the analysis of axisymmetric shells. A single point [...]

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• Multi-fluid flows with the particle finite element method

  S. Idelsohn, M. Mier-Torrecilla, E. Oñate

  Comput. Methods Appl. Mech. Engrg., (2009). vol. 198, pp. 2750–2767

  
  

  Abstract

  Particle methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied on it. In this paper the Particle Finite Element Method based on finite element shape functions is used to solve the continuous fluid mechanics equations in the case of heterogeneous density. To evaluate the external applied forces to each particle, the incompressible Navier–Stokes equations are solved at each time step using a Lagrangian formulation. All the information in the fluid is transmitted via the particles. All kinds of density heterogeneous fluids and multiphase flows with internal interfaces including or not free-surfaces, breaking waves and fluid separations may be easily solved with this methodology. 

  Abstract
  Particle methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal [...]

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• New rotation-free finite element shell triangle accurately using geometrical data

  P. Fuentes, E. Oñate

  Comput. Methods Appl. Mech. Engrg., (2010). vol. 199, pp. 383–391

  
  

  Abstract

  A new triangle shell element is presented. The advantages of this element are threefold: simplicity, generality and geometrical accuracy. The formulation is free from rotation degrees of freedom. The triangle here presented can be used regardless of the mesh topology, thus generality is conserved for any meshrepresented surface. From an original first order approach we evolve to a third order geometric description. The higher degree geometric description is based on the Bézier triangles concept, a very well known geometry in the domain of CAGD [G. Farin, Curves and Surfaces for CAGD. A Practical Guide, fifth ed., Morgan Kaufmann Publishers, San Francisco, CA, 2002]. Using this concept we show the path to reconstruct a general third order interpolating surface using only the three coordinates at each node. This work takes as starting point the nodal implementation of a basic triangle shell element [E. Oñate, F. Zárate, Rotation-free triangular plate and shell elements, Int. J. Numer. Methods Engrg. 47 (2000) 557– 603]. In order to use an exact formula for the curvature, the normal directions at each node and the way to characterize them are proposed. Then, the geometrical properties and the mechanical behavior of the surface created are introduced. Finally, different examples are presented to depict the versatility and accuracy of the element.

  Abstract
  A new triangle shell element is presented. The advantages of this element are threefold: simplicity, generality and geometrical accuracy. The formulation is free from rotation [...]

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• A high-resolution Petrov-Galerkin method for the convection-diffusion-reaction problem. Part II-A multidimensional extension

  P. Nadukandi, E. Oñate, J. García-Espinosa

  Comput. Methods Appl. Mech. Engrg., (2012). Vol. 213-216, pp. 327-352

  
  

  Abstract

  A multidimensional extension of the HRPG method using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction–diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors li and the stabilization parameters αi, βi are calculated along these li. Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of αi, βi are a direct extension of their counterparts in 1D. Using αi, βiand li, objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective—stabilization with high-resolution

  Abstract
  A multidimensional extension of the HRPG method using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies [...]

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• Large time-step explicit integration method for solving problems with dominant convection

  S. Idelsohn, N. Nigro, A. Limache, E. Oñate

  Comput. Methods Appl. Mech. Engrg., (2012). Vol. 217-220, pp. 168-185

  
  

  Abstract

  An explicit time integrator without the CFL < 1 restriction for the momentum equation is presented. This allows stable large time-steps in problems dominated by convection, independently of the spatial discretization. The idea is to use the information existing at time t=tn in the velocity streamlines as well as in the acceleration streamlines to update the particle position as well as the velocity in an updated Lagrangian frame. The method may be used with moving or fixed meshes.

  Abstract
  An explicit time integrator without the CFL < 1 restriction for the momentum equation is presented. This allows stable large time-steps in problems dominated [...]

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