Onate et al 2004g - Revision history

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 The <math display="inline">h_i's</math> in above equations are ''characteristic lengths'' of the domain where balance of momentum and mass is enforced. In Eq.([[#eq-9|9]]) these lengths define the domain where equilibrium of boundary tractions is established. Details of the derivation of Eqs.([[#eq-7|7]])&#8211;([[#eq-10|10]]) can be found in [Oñate (1998; 2000) <span id="citeF-27"></span>[[#cite-27|[27]],<span id="citeF-28"></span>[[#cite-28|28]]]; Oñate ''et al.'' (2001) <span id="citeF-32"></span>[[#cite-32|[32]]]].
-Eqs.([[#eq-7|7]])&#8211;([[#eq-10|10]]) are the starting  point for deriving stabilized finite element methods to solve the incompressible Navier-Stokes equations in a Lagrangian frame of reference using equal order interpolation for the velocity and pressure variables [Idelsohn ''et al.'' (2002; 2003a; 2003b; 2004) <span id="citeF-x19"></span>[[#cite-19|[19]]-<span id="citeF-21"></span>[[#cite-21|21]],<span id="citeF-23"></span>[[#cite-23|23]]]; Oñate ''et al.'' (2003) <span id="citeF-33"></span>[[#cite-33|[33]]]; Aubry ''et al.'' (2004) <span id="citeF-1"></span>[[#cite-1|[1]]]]. Application of the FIC formulation to finite element and meshless analysis of fluid flow problems can be found in [García and Oñate (2003 <span id="citeF-12"></span>[[#cite-12|[12]]]); Oñate (2000; 2004) <span id="citeF-28"></span>[[#cite-28|[28]],<span id="citeF-29"></span>[[#cite-29|29]]]; Oñate ''et al.'' (2000; 2004) <span id="citeF-31"></span>[[#cite-31|[31]],<span id="citeF-34"></span>[[#cite-34|34]]]; Oñate and García (2001) <span id="citeF-32"></span>[[#cite-32|[32]]]; Oñate and Idelsohn (1998) <span id="citeF-30"></span>[[#cite-30|[30]]]].
+Eqs.([[#eq-7|7]])&#8211;([[#eq-10|10]]) are the starting  point for deriving stabilized finite element methods to solve the incompressible Navier-Stokes equations in a Lagrangian frame of reference using equal order interpolation for the velocity and pressure variables [Idelsohn ''et al.'' (2002; 2003a; 2003b; 2004) <span id="citeF-19"></span>[[#cite-19|[19]]-<span id="citeF-21"></span>[[#cite-21|21]],<span id="citeF-23"></span>[[#cite-23|23]]]; Oñate ''et al.'' (2003) <span id="citeF-33"></span>[[#cite-33|[33]]]; Aubry ''et al.'' (2004) <span id="citeF-1"></span>[[#cite-1|[1]]]]. Application of the FIC formulation to finite element and meshless analysis of fluid flow problems can be found in [García and Oñate (2003 <span id="citeF-12"></span>[[#cite-12|[12]]]); Oñate (2000; 2004) <span id="citeF-28"></span>[[#cite-28|[28]],<span id="citeF-29"></span>[[#cite-29|29]]]; Oñate ''et al.'' (2000; 2004) <span id="citeF-31"></span>[[#cite-31|[31]],<span id="citeF-34"></span>[[#cite-34|34]]]; Oñate and García (2001) <span id="citeF-32"></span>[[#cite-32|[32]]]; Oñate and Idelsohn (1998) <span id="citeF-30"></span>[[#cite-30|[30]]]].
 ===3.1 Transformation of the mass balance equation. Integral governing equations===

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 Initially (<math display="inline">t = 0.0</math>) the ship is released from a fixed position and the equilibrium configuration found is consistent with Archimedes principle. During the following time steps the external wave hits the ship and both the internal and the external fluids interact with the ship boundaries. At times <math display="inline">t=6.15</math> and 8.55 breaking waves and some water drops slipping along the ship deck can be observed. Figure [[#img-23|23]] shows the pressure pattern at two time steps.
+<div id='img-23'></div>
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 |-

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 ===10.5 Horizontal motion of a rigid ship in a reservoir===
-In the next example (Figure [[#img-17|17]]) the same ship of the previous example moves now horizontally at a fixed velocity in a water reservoir.   The free-surface, which was initially horizontal, takes the correct position at the ship prow and stern. Again, the complex water-wall contact problem is naturally solved in the curved prow region.
+In the next example (Figure [[#img-17|17]]) the same ship of the previous example moves now horizontally at a fixed velocity in a water reservoir. The free-surface, which was initially horizontal, takes the correct position at the ship prow and stern. Again, the complex water-wall contact problem is naturally solved in the curved prow region.
-<div id='img-18'></div>
+<div id='img-17'></div>
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 |-

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 |-
-|[[Image:Draft_Samper_616988227_5402_monograph-Fig4-c.png|651px|Sloshing: Comparison of the numerical and analytical solutions.]]
+|[[Image:Draft_Samper_616988227_5402_monograph-Fig4-c.png|551px|Sloshing: Comparison of the numerical and analytical solutions.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 10:''' Sloshing: Comparison of the numerical and analytical solutions.
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 |-
-|[[Image:Draft_Samper_616988227_8726_monograph-Fig5-c.png|551px|PFEM results for a large amplitude sloshing problems.]]
+|[[Image:Draft_Samper_616988227_8726_monograph-Fig5-c.png|450px|PFEM results for a large amplitude sloshing problems.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 11:''' PFEM results for a large amplitude sloshing problems.

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 |-
-|[[Image:Draft_Samper_616988227_6009_monograph-Fig6-c.png|551px|3D sloshing problem.]]
+|[[Image:Draft_Samper_616988227_6009_monograph-Fig6-c.png|450px|3D sloshing problem.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 12:''' 3D sloshing problem.

← Older revision	Revision as of 08:56, 10 July 2019
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 The examples chosen show the applicability of the PFEM to solve problems involving large fluid motions and FSI situations. The fractional step algorithm of Section [[#5 Fractional Step Method for Fluid-Structure Interaction Analysis|5]] with <math display="inline">\theta _2 =1</math> and <math display="inline">\alpha =1</math> has been used in all cases.
-In examples [[#10.1 Collapse of a water column|10.1]]-[[#10.10 Rigid cube falling into a water container|10.10]] a value of <math display="inline">\theta _1=1</math> has  been chosen. This basically means that the final configuration <math display="inline">\Omega ^{n+1,j}</math> has been taken as the reference configuration at each iteration. In example [[#10.11 The Rayleigh-B´enard Instability|10.11]] <math display="inline">\theta _1=0</math> has been selected and, hence, the initial configuration <math display="inline">\Omega ^n</math> has been taken as a fixed reference configuration for all the iterations within a time step.
+In examples [[#10.1 Collapse of a water column|10.1]]-[[#10.10 Rigid cube falling into a water container|10.10]] a value of <math display="inline">\theta _1=1</math> has  been chosen. This basically means that the final configuration <math display="inline">\Omega ^{n+1,j}</math> has been taken as the reference configuration at each iteration. In example [[#10.11 The Rayleigh-Bénard Instability|10.11]] <math display="inline">\theta _1=0</math> has been selected and, hence, the initial configuration <math display="inline">\Omega ^n</math> has been taken as a fixed reference configuration for all the iterations within a time step.
 <div id='img-7'></div>

@@ Line 655: / Line 655: @@
 ==7 Generation of a New Mesh==
-One of the key points for the success of the Lagrangian flow formulation  described here is the fast regeneration of a mesh at every time step on the basis of the position of the nodes in the space domain. In our work the mesh is generated using the so called extended Delaunay tesselation (EDT) presented in [Idelsohn ''et al.'' (2003a; 2003c; 2004) <span id="citeF-20"></span>[[#cite-20|[20]]],<span id="citeF-23"></span>[[#cite-23|[23]]],<span id="citeF-24"></span>[[#cite-24|[24]]]]. The EDT allows one to generate non standard meshes combining elements of  arbitrary polyhedrical shapes  (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order <math display="inline">n</math>, where <math display="inline">n</math> is the total number of nodes in the mesh (Figure [[#img-3|3]]). The <math display="inline">C^\circ </math> continuous shape functions of the elements can be simply obtained using the so called meshless finite element interpolation (MFEM). Details of the mesh generation procedure and the derivation of the MFEM shape functions can be found in [Idelsohn ''et al.'' (2003a; 2003c; 2004) <span id="citeF-20"></span>[[#cite-20|[20]]],<span id="citeF-23"></span>[[#cite-23|[23]]],<span id="citeF-24"></span>[[#cite-24|[24]]]].
+One of the key points for the success of the Lagrangian flow formulation  described here is the fast regeneration of a mesh at every time step on the basis of the position of the nodes in the space domain. In our work the mesh is generated using the so called extended Delaunay tesselation (EDT) presented in [Idelsohn ''et al.'' (2003a; 2003c; 2004) <span id="citeF-20"></span>[[#cite-20|[20]],<span id="citeF-23"></span>[[#cite-23|23]],<span id="citeF-24"></span>[[#cite-24|24]]]]. The EDT allows one to generate non standard meshes combining elements of  arbitrary polyhedrical shapes  (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order <math display="inline">n</math>, where <math display="inline">n</math> is the total number of nodes in the mesh (Figure [[#img-3|3]]). The <math display="inline">C^\circ </math> continuous shape functions of the elements can be simply obtained using the so called meshless finite element interpolation (MFEM). Details of the mesh generation procedure and the derivation of the MFEM shape functions can be found in [Idelsohn ''et al.'' (2003a; 2003c; 2004) <span id="citeF-20"></span>[[#cite-20|[20]],<span id="citeF-23"></span>[[#cite-23|23]],<span id="citeF-24"></span>[[#cite-24|24]]]].
 Once the new mesh has been generated  the numerical solution is found at each time step using the fractional step algorithm described in the previous section.

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 The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities <math display="inline">{v}^*</math> in Eq.([[#eq-28a|28a]]). The prescribed velocities at the boundary are applied when solving for <math display="inline">\bar{v}^{n+1,j}</math> in step [[#s-3|3]]. The prescribed pressures at the boundary are imposed by making  the pressure increments zero at the relevant boundary nodes and making <math display="inline">\bar{p}^n</math> equal to the prescribed pressure values.
-Details of the treatment of the contact conditions at the solid-fluid interface are given in Section [[#8 Identification of Boundary Surfaces|8]] [Idelsohn ''et al.'' (2004)<span id="citeF-23"></span>[[#cite-23|[23]]]].
+Details of the treatment of the contact conditions at the solid-fluid interface are given in Section [[#8 Identification of Boundary Surfaces|8]] [Idelsohn ''et al.'' (2004) <span id="citeF-23"></span>[[#cite-23|[23]]]].
 Note that solution of steps [[#s-1|1]], [[#s-3|3]] and [[#s-4|4]] does not require the solution of  a system of equations as a diagonal form is chosen for <math display="inline">M</math> and <math display="inline">\hat {M}</math>. The whole solution process within a time step can be linearized by choosing <math display="inline">\theta _1 = \theta _2 =0</math> and now the iteration loop is no longer necessary. The implicit solution for <math display="inline">\theta _1 = \theta _2 =1</math> is however very effective as larger time steps can be used. This requires  some iterations within steps [[#s-1|1]]-[[#s-8|8]] until converged values for the fluid and solid variables  and the new position of the mesh nodes at time <math display="inline">n+1</math> are found.
@@ Line 647: / Line 647: @@
 '''Remark 2'''
-Although not explicitely mentioned for <math display="inline">\theta _1=1</math> all matrices and vectors in Eqs.([[#eq-28|28]])-([[#eq-32|32]]) are computed at the final configuration <math display="inline">\Omega ^{n+1,j}</math>. This  means that the integration domain changes for each iteration and, hence,  all the terms involving space derivatives  must be updated at each iteration. This problem dissapears if <math display="inline">\Omega ^n</math> is taken as the reference configuration (<math display="inline">\theta _1=0</math>) as this remains fixed during the iteration. The penalty to pay in this case, however, is the evaluation of the Jacobian matrix at each iterations [Aubry ''et al.'' (2004)<span id="citeF-1"></span>[[#cite-1|[1]]]].
+Although not explicitely mentioned for <math display="inline">\theta _1=1</math> all matrices and vectors in Eqs.([[#eq-28|28]])-([[#eq-32|32]]) are computed at the final configuration <math display="inline">\Omega ^{n+1,j}</math>. This  means that the integration domain changes for each iteration and, hence,  all the terms involving space derivatives  must be updated at each iteration. This problem dissapears if <math display="inline">\Omega ^n</math> is taken as the reference configuration (<math display="inline">\theta _1=0</math>) as this remains fixed during the iteration. The penalty to pay in this case, however, is the evaluation of the Jacobian matrix at each iterations [Aubry ''et al.'' (2004) <span id="citeF-1"></span>[[#cite-1|[1]]]].
 ==6 Treatment of Contact Between Fluid and Solid Interfaces==

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 |}
-The <math display="inline">\tau _i</math>'s in Eq.([[#eq-11|11]]), when scaled by the density, are termed  ''intrinsic time parameters''. Similar values for <math display="inline">\tau _i</math> (usually <math display="inline">\tau _i =\tau </math> is taken) are used in other works from ad-hoc extensions of the 1D advective-diffusive problem [Codina ''et al.'' (1998) <span id="citeF-4"></span>[[#cite-4|[4]]];  Codina and Blasco (2000) <span id="citeF-5"></span>[[#cite-5|[5]]]; Codina (2002) <span id="citeF-3"></span>[[#cite-3|[3]]]; Codina and Zienkiewicz (2002) <span id="citeF-6"></span>[[#cite-6|[6]]]; Cruchaga and Oñate (1997; 1999) <span id="citeF-7"></span>[[#cite-7|[7]],<span id="citeF-8"></span>[[#cite-8|8]]]; Donea and Huerta (2003) <span id="citeF-9"></span>[[#cite-9|[9]]]; Franca and Frey (1992) <span id="citeF-11"></span>[[#cite-11|[11]]]; Hansbo and Szepessy (1990) <span id="citeF-15"></span>[[#cite-15|[15]]]; Hughes ''et al.'' (1986; 1989; 1994) <span id="citeF-16"></span>[[#cite-16|[16]]-<span id="citeF-18"></span>[[#cite-18|18]]]; Oñate (1998; 2000; 2004) <span id="citeF-27"></span>[[#cite-27|[27]]-<span id="citeF-29"></span>[[#cite-29|29]]]; Sheng ''et al.'' (1996) <span id="citeF-36"></span>[[#cite-36|[36]]]; Storti ''et al.'' (2004) <span id="citeF-37"></span>[[#cite-37|[37]]]; Tezduyar ''et al.'' (1992) <span id="citeF-41"></span>[[#cite-41|[41]]]; Zienkiewicz and Taylor (2000) <span id="citeF-43"></span>[[#cite-43|[43]]]].
+The <math display="inline">\tau _i</math>'s in Eq.([[#eq-11|11]]), when scaled by the density, are termed  ''intrinsic time parameters''. Similar values for <math display="inline">\tau _i</math> (usually <math display="inline">\tau _i =\tau </math> is taken) are used in other works from ad-hoc extensions of the 1D advective-diffusive problem [Codina ''et al.'' (1998) <span id="citeF-4"></span>[[#cite-4|[4]]];  Codina and Blasco (2000) <span id="citeF-5"></span>[[#cite-5|[5]]]; Codina (2002) <span id="citeF-3"></span>[[#cite-3|[3]]]; Codina and Zienkiewicz (2002) <span id="citeF-6"></span>[[#cite-6|[6]]]; Cruchaga and Oñate (1997; 1999) <span id="citeF-7"></span>[[#cite-7|[7]],<span id="citeF-8"></span>[[#cite-8|8]]]; Donea and Huerta (2003) <span id="citeF-9"></span>[[#cite-9|[9]]]; Franca and Frey (1992) <span id="citeF-11"></span>[[#cite-11|[11]]]; Hansbo and Szepessy (1990) <span id="citeF-15"></span>[[#cite-15|[15]]]; Hughes ''et al.'' (1986; 1989; 1994) <span id="citeF-16"></span>[[#cite-16|[16]]-<span id="citeF-18"></span>[[#cite-18|18]]]; Oñate (1998; 2000; 2004) <span id="citeF-27"></span>[[#cite-27|[27]]-<span id="citeF-29"></span>[[#cite-29|29]]]; Sheng ''et al.'' (1996) <span id="citeF-36"></span>[[#cite-36|[36]]]; Storti ''et al.'' (1995) <span id="citeF-37"></span>[[#cite-37|[37]]]; Tezduyar ''et al.'' (1992) <span id="citeF-41"></span>[[#cite-41|[41]]]; Zienkiewicz and Taylor (2000) <span id="citeF-43"></span>[[#cite-43|[43]]]].
 At this stage it is no longer necessary to retain the stabilization terms in the momentum equations. These terms are critical in Eulerian formulations to stabilize the numerical solution for high values of the convective terms. In the Lagrangian formulation the convective terms dissapear from the momentum equations and the FIC terms in these  equations are just useful to derive the form of the mass balance equation given by Eq.([[#eq-11|11]]) and can be disregarded there onwards. Consistenly, the stabilization terms are also neglected in the Neuman boundary conditions (eqs.([[#eq-9|9]])).