Onate et al 2011e - Revision history

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 Most of the above problems disappear if a ''Lagrangian description'' is used to formulate the governing equations of both the solid and the fluid domains. In the Lagrangian formulation the motion of the individual particles are followed and, consequently, nodes in a finite element mesh can be viewed as moving material points (hereforth called “particles”). Hence, the motion of the mesh discretizing the total domain (including both the fluid and solid parts) is followed during the transient solution.
-The authors have successfully developed in the past years a particular class of Lagrangian formulation for problems involving complex interactions between fluids and solids. The so called ''particle finite element method'' (PFEM, [[http://www.cimne.com/pfem|PFEM]]), treats the  nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid (or solid) domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM.
+The authors have successfully developed in the past years a particular class of Lagrangian formulation for problems involving complex interactions between fluids and solids. The so called ''particle finite element method'' ([[http://www.cimne.com/pfem   PFEM]]), treats the  nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid (or solid) domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM.
 The FEM solution in the (incompressible) fluid domain implies solving the momentum and incompressibility equations. This is not a simple problem as the incompressibility condition limits the choice of the FE approximations for the velocity and pressure to overcome the well known <math display="inline">div</math>-stability condition <span id='citeF-9'></span><span id='citeF-49'></span>[[#cite-9|[9]],[[#cite-49|49]]]. In our work we use a stabilized mixed FEM based on the Finite Calculus (FIC) approach which allows for a linear approximation for the velocity and pressure variables.

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 Published in ''Particle-Based Methods, Computational Methods in Applied Sciences'', Springer, E. Oñate and R. Owen (Eds.), Vol. 25, pp. 1-48, 2011<br />
-DOI:
+DOI: 10.1007/978-94-007-0735-1_1
 ==Abstract==

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-<!-- metadata commented in wiki content
+Published in ''Particle-Based Methods, Computational Methods in Applied Sciences'', Springer, E. Oñate and R. Owen (Eds.), Vol. 25, pp. 1-48, 2011<br />
-==ADVANCES IN THE PARTICLE FINITE ELEMENT METHOD  (PFEM) FOR SOLVING COUPLED PROBLEMS IN ENGINEERING==
+DOI:
-−
-'''E. Oñate, S.R. Idelsohn<math>^*</math>, M.A. Celigueta,
-−
-R. Rossi J. Marti, J.M. Carbonell, P. Ryzhakov and B. Suárez'''
-−
-{|
-|-
-|International Center for Numerical Methods in Engineering (CIMNE)
-|-
-| Technical University of Catalonia (UPC)
-|-
-| Campus Norte UPC, 08034 Barcelona, Spain
-|-
-| [mailto:onate@cimne.upc.edu onate@cimne.upc.edu], [http://www.cimne.com/eo www.cimne.com/eo]
-|-
-| [http://www.cimne.com/pfem www.cimne.com/pfem]
-|-
-| <math>^*</math> ICREA Research Professor at CIMNE
-|}
--->
 ==Abstract==

@@ Line 1,199: / Line 1,199: @@
 ==APPENDIX==
-The  matrices and vectors in Eqs.(8)-(11) for a 4-noded tetrahedron are:
+The  matrices and vectors in Eqs.([[#eq-8|8]])-([[#cite-11|11]]) for a 4-noded tetrahedron are:
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 197: / Line 197: @@
 ===3.2 Discretization of the equations===
-A key problem in the numerical solution of Eqs.([[#eq-1|1]])-([[#eq-4|4]]) is the satisfaction of the incompressibility condition (Eq.([[#eq-2|2]])). A number of procedures to solve his problem exist in the finite element literature <span id='citeF-9'></span><span id='citeF-49'></span>[[#cite-9|[9]],[[#cite-49|49]]]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure <span id='citeF-27'></span>[[#cite-27|[27]]&#8211;<span id='citeF-29'></span>[[#cite-29|29]]],<span id='citeF-36'></span><span id='citeF-38'></span><span id='citeF-40'></span>[[#cite-36|[36]],[[#cite-38|38]],[[#cite-40|40]]]. The essence of this method is the solution of a ''modified mass balance'' equation which is written  as
+A key problem in the numerical solution of Eqs.([[#eq-1|1]])-([[#eq-4|4]]) is the satisfaction of the incompressibility condition (Eq.([[#eq-2|2]])). A number of procedures to solve his problem exist in the finite element literature <span id='citeF-9'></span><span id='citeF-49'></span>[[#cite-9|[9]],[[#cite-49|49]]]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure <span id='citeF-27'></span>[[#cite-27|[27]]&#8211;<span id='citeF-29'></span>[[#cite-29|29]],<span id='citeF-36'></span><span id='citeF-38'></span><span id='citeF-40'></span>[[#cite-36|36]],[[#cite-38|38]],[[#cite-40|40]]]. The essence of this method is the solution of a ''modified mass balance'' equation which is written  as
 <span id="eq-5"></span>

← Older revision	Revision as of 11:34, 27 May 2019
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 <div id="cite-3"></div>
-[3]  K.M. Butler, T.J. Ohlemiller, G.T. Linteris, A Progress Report on Numerical Modeling of Experimental Polymer Melt Flow Behavior, Interflam (2004) 937&#8211;948.
+[[#citeF-3|[3]]] K.M. Butler, T.J. Ohlemiller, G.T. Linteris, A Progress Report on Numerical Modeling of Experimental Polymer Melt Flow Behavior, Interflam (2004) 937&#8211;948.
 <div id="cite-4"></div>
@@ Line 1,057: / Line 1,057: @@
 <div id="cite-7"></div>
-[7]   R. Codina, O.C. Zienkiewicz, CBS versus GLS stabilization of   the incompressible Navier-Stokes equations and the role of the time step as   stabilization parameter, Communications in Numerical Methods in Engineering  (2002)   18(2) (2002) 99&#8211;112.
+[[#citeF-7|[7]]]   R. Codina, O.C. Zienkiewicz, CBS versus GLS stabilization of   the incompressible Navier-Stokes equations and the role of the time step as   stabilization parameter, Communications in Numerical Methods in Engineering  (2002)   18(2) (2002) 99&#8211;112.
 <div id="cite-8"></div>
@@ Line 1,069: / Line 1,069: @@
 <div id="cite-11"></div>
-[11]  J. García, E. Oñate,  An unstructured finite element   solver for ship hydrodynamic problems, J. Appl. Mech.  70 (2003) 18&#8211;26.
+[[#citeF-11|[11]]]  J. García, E. Oñate,  An unstructured finite element   solver for ship hydrodynamic problems, J. Appl. Mech.  70 (2003) 18&#8211;26.
 <div id="cite-12"></div>
-[12]  S.R. Idelsohn, E. Oñate, F. Del Pin, N. Calvo, Lagrangian formulation: the only way to solve some free-surface fluid mechanics problems, Fith World Congress on Computational Mechanics, H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (eds), July 7&#8211;12, Viena, Austria, (2002).
+[[#citeF-11|[11]]]  S.R. Idelsohn, E. Oñate, F. Del Pin, N. Calvo, Lagrangian formulation: the only way to solve some free-surface fluid mechanics problems, Fith World Congress on Computational Mechanics, H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (eds), July 7&#8211;12, Viena, Austria, (2002).
 <div id="cite-13"></div>
@@ Line 1,090: / Line 1,090: @@
 <div id="cite-18"></div>
-[18]  S.R. Idelsohn, J. Marti, A. Limache, E. Oñate, Unified Lagrangian formulation for elastic solids and incompressible fluids: Application to fluid-structure interaction problems via the PFEM. Comput Methods Appl Mech Engrg. (2008) 197 1762&#8211;1776.
+[[#citeF-18|[18]]]  S.R. Idelsohn, J. Marti, A. Limache, E. Oñate, Unified Lagrangian formulation for elastic solids and incompressible fluids: Application to fluid-structure interaction problems via the PFEM. Comput Methods Appl Mech Engrg. (2008) 197 1762&#8211;1776.
 <div id="cite-19"></div>
-[19]  S.R. Idelsohn, M. Mier-Torrecilla, E. Oñate, Multi-fluid flows with the Particle Finite Element Method. Comput Methods Appl Mech Engrg. 198 (2009) 2750&#8211;2767.
+[[#citeF-19|[19]]]  S.R. Idelsohn, M. Mier-Torrecilla, E. Oñate, Multi-fluid flows with the Particle Finite Element Method. Comput Methods Appl Mech Engrg. 198 (2009) 2750&#8211;2767.
 <div id="cite-20"></div>
-[20]  S.R. Idelsohn, M. Mier-Torrecilla, N. Nigro, E. Oñate, On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field. Comput. Mech. (2010) 46 (1) 115&#8211;124.
+[[#citeF-20|[20]]]  S.R. Idelsohn, M. Mier-Torrecilla, N. Nigro, E. Oñate, On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field. Comput. Mech. (2010) 46 (1) 115&#8211;124.
 <div id="cite-21"></div>
@@ Line 1,111: / Line 1,111: @@
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-[25]  M. de Mier Torrecilla, Numerical Simulation of Multi-Fluid Flows with the Particle Finite Element Method. ''Ph.D. Thesis'', Technical  University of Catalonia (UPC), July 2010.
+[[#citeF-25|[25]]]  M. de Mier Torrecilla, Numerical Simulation of Multi-Fluid Flows with the Particle Finite Element Method. ''Ph.D. Thesis'', Technical  University of Catalonia (UPC), July 2010.
 <div id="cite-26"></div>
@@ Line 1,120: / Line 1,120: @@
 <div id="cite-28"></div>
-[28]  E. Oñate, A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comp. Meth. Appl. Mech. Engng. 182(1&#8211;2) (2000) 355&#8211;370.
+[[#citeF-28|[28]]]  E. Oñate, A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comp. Meth. Appl. Mech. Engng. 182(1&#8211;2) (2000) 355&#8211;370.
 <div id="cite-29"></div>
@@ Line 1,126: / Line 1,126: @@
 <div id="cite-30"></div>
-[30]  E. Oñate,  S.R. Idelsohn, A mesh free finite point method for advective-diffusive transport and fluid flow problems, Computational Mechanics 21 (1998) 283&#8211;292.
+[[#citeF-30|[30]]]  E. Oñate,  S.R. Idelsohn, A mesh free finite point method for advective-diffusive transport and fluid flow problems, Computational Mechanics 21 (1998) 283&#8211;292.
 <div id="cite-31"></div>
-[31]   E. Oñate, J. García,  A finite element method for  fluid-structure interaction with surface waves using a finite calculus formulation, Comput. Meth. Appl. Mech. Engrg. 191 (2001) 635&#8211;660.
+[[#citeF-31|[31]]]   E. Oñate, J. García,  A finite element method for  fluid-structure interaction with surface waves using a finite calculus formulation, Comput. Meth. Appl. Mech. Engrg. 191 (2001) 635&#8211;660.
 <div id="cite-32"></div>
@@ Line 1,135: / Line 1,135: @@
 <div id="cite-33"></div>
-[33]  E. Oñate, C. Sacco, S.R. Idelsohn, A finite point method for   incompressible flow problems, Comput. Visual. in Science 2 (2000) 67&#8211;75.
+[[#citeF-33|[33]]]  E. Oñate, C. Sacco, S.R. Idelsohn, A finite point method for   incompressible flow problems, Comput. Visual. in Science 2 (2000) 67&#8211;75.
 <div id="cite-34"></div>
@@ Line 1,147: / Line 1,147: @@
 <div id="cite-37"></div>
-[37]  E. Oñate, A. Valls, J. García,  FIC/FEM formulation   with matrix stabilizing terms for incompressible flows at low and high   Reynold's numbers, Computational Mechanics 38 (4-5) (2006a) 440-455.
+[[#citeF-37|[37]]]  E. Oñate, A. Valls, J. García,  FIC/FEM formulation   with matrix stabilizing terms for incompressible flows at low and high   Reynold's numbers, Computational Mechanics 38 (4-5) (2006a) 440-455.
 <div id="cite-38"></div>

@@ Line 989: / Line 989: @@
 The results are shown in Figure [[#img-42|42]]. A rotation and a displacement have been imposed to the roadheader. Note that contact elements only appear in the contact zone. The cone that models the roadheader loses material at the tip due to wear. Ground geometry suffers big changes during the simulation. Remeshing and detection of the boundary via the alpha-shape technique are crucial for capturing the fast and drastic changes of the domain boundary.
 <div id='img-42'></div>
@@ Line 1,001: / Line 997: @@
 | colspan="1" | '''Figure 42:'''Simulation of an excavation with a roadheader using the PFEM. Note the geometry change in the roadheader tip due the wear
 |}
 <div id='img-43'></div>
@@ Line 1,025: / Line 1,025: @@
 | colspan="1" | '''Figure 45:''' Wear of the rock cutting discs in a TBM during the simulation of a tunneling operation using the PFEM. Circles denote worn cutting discs.
 |}
 The previous examples illustrate the good  capabilities of the PFEM for modelling ground excavation processes.

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 The results are shown in Figure [[#img-42|42]]. A rotation and a displacement have been imposed to the roadheader. Note that contact elements only appear in the contact zone. The cone that models the roadheader loses material at the tip due to wear. Ground geometry suffers big changes during the simulation. Remeshing and detection of the boundary via the alpha-shape technique are crucial for capturing the fast and drastic changes of the domain boundary.
-===''Simulation of an excavation with a TBM''===
+''Simulation of an excavation with a TBM''
 Figures [[#img-42|42]]&#8211;[[#img-44|44]] show a simulation of a tunneling process with a TMB (Tunnel Boring Machine) acting on a 3D soil/rock domain. This  example evidences the capability of the PFEM  to model complex excavation settings. The discretization of the TMB  and the soil/rock region is displayed in Figure [[#img-42|42]]. Figure [[#img-43|43]] shows an overview of the simulation as the tunneling process advance and the stress contour lines and Figure [[#img-44|44]] shows the wear of the rock cutting discs in the TBM induced by the excavation forces. Far away from the rotating axis the displacement is bigger for the same rotation velocity and it generates larger friction forces at the edges of the tunneling head.

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 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 40:''' Simulation of the excavation of a soft soil mass with a rotating gear disc with the PFEM. Contour of the modulus of the acceleration vector in the soil at two instances
 |}