Onate et al 2014c - Revision history

Scipediacontent at 11:00, 14 January 2019

2019-01-14T11:00:18Z

Scipediacontent at 10:58, 14 January 2019

2019-01-14T10:58:33Z

Scipediacontent at 10:57, 14 January 2019

2019-01-14T10:57:36Z

Move page script: Move page script moved page Draft Samper 718004671 to Onate et al 2014c

2018-05-08T13:02:32Z

Move page script moved page Draft Samper 718004671 to Onate et al 2014c

Cinmemj at 10:17, 27 April 2018

2018-04-27T10:17:35Z

Cinmemj at 10:12, 27 April 2018

2018-04-27T10:12:35Z

Cinmemj at 10:09, 27 April 2018

2018-04-27T10:09:57Z

Cinmemj at 10:08, 27 April 2018

2018-04-27T10:08:14Z

Cinmemj at 10:06, 27 April 2018

2018-04-27T10:06:53Z

Cinmemj at 10:00, 27 April 2018

2018-04-27T10:00:00Z

@@ Line 548: / Line 548: @@
 with <math display="inline">{N}_j = N_j {I}_2</math> where <math display="inline">{I}_2</math> is the <math display="inline">2\times 2</math> unit matrix.
 In Eq.([[#eq-34|33]]) vectors <math display="inline">\bar{v}</math>, <math display="inline">\bar {p}</math> and <math display="inline">\bar {T}</math>  contain the nodal velocities, the nodal pressures and the nodal temperatures for the whole mesh, respectively and the upperindex denotes the nodal value for each vector or scalar magnitude.
@@ Line 585: / Line 586: @@
 |}
-where <math display="inline">{\dot{\bar{a}}}</math>  denotes the material time derivative of the components of  a vector <math display="inline">{a}</math>. The element form of the matrices and vectors in Eqs.([[#5 FEM DISCRETIZATION|5]]) is given in Box 1.
+where <math display="inline">{\dot{\bar{a}}}</math>  denotes the material time derivative of the components of  a vector <math display="inline">{a}</math>. The element form of the matrices and vectors in Eqs.([[#eq-34|34]]) is given in Box 1.
 <div class="center" style="font-size: 75%;">
@@ Line 595: / Line 596: @@
-We note that Eqs.([[#5 FEM DISCRETIZATION|34]]) involve the geometry at the updated configuration (<math display="inline">{}^{n+1}V</math>). This geometry is unknown; hence the solution of Eqs.([[#5 FEM DISCRETIZATION|34]]) has to be found iteratively. The iterative solution scheme chosen in this work is presented in the next section.
+We note that Eqs.([[#eq-34|34]]) involve the geometry at the updated configuration (<math display="inline">{}^{n+1}V</math>). This geometry is unknown; hence the solution of Eqs.([[#eq-34|34]]) has to be found iteratively. The iterative solution scheme chosen in this work is presented in the next section.
 <br/>

@@ Line 407: / Line 407: @@
 where <math display="inline">\Delta t</math> is the time step used for the transient solution and <math display="inline">l^e</math> is a characteristic element length computed as <math display="inline">l^e = 2(V^e)^{1/n_s}</math> where <math display="inline">V^e</math> is the element area (for 3-noded triangles) or volume (for 4-noded tetrahedra). For heterogeneous material the values of <math display="inline">\mu </math> and <math display="inline">\rho </math>  are computed at the element center <span id='citeF-38'></span>[[#cite-38|[38]]]. For  a solid material, the viscosity <math display="inline">\mu </math> is substituted by <math display="inline">\bar \mu \Delta t</math> with <math display="inline">\bar \mu </math> defined in Eq.(7).
-We note that <math display="inline">\tau =0</math> for “compressible” type problems not requiring stabilization. In these cases Eq.([[#eq-23|22]]) recovers the standard form for the conservation of mass of the infinitesimal theory (see Eqs.([[#2.3 Mass conservation equation|16]])).
+We note that <math display="inline">\tau =0</math> for “compressible” type problems not requiring stabilization. In these cases Eq.([[#eq-23|22]]) recovers the standard form for the conservation of mass of the infinitesimal theory (see Eqs.([[#eq-16|16]])).
 ==4 VARIATIONAL EQUATIONS==

@@ Line 290: / Line 290: @@
 |}
-For a quasi-incompressible fluid material <math display="inline">\kappa =\rho c^2</math> where <math display="inline">c</math> is the speed of sound in the fluid. For a solid material, <math display="inline">\kappa = \bar \kappa </math> with <math display="inline">\bar \kappa </math> given by Eq.([[#eq-14|14]]). Eqs.([[#2.3 Mass conservation equation|16]]) define the constitutive equation for the pressure for both solids and quasi-incompressible fluids as <math display="inline">\Delta p = \kappa  \Delta t \varepsilon _v</math>. This is consisted with the expression obtained in Eq.([[#eq-13|13]]) for a hypo-elastic solid material.
+For a quasi-incompressible fluid material <math display="inline">\kappa =\rho c^2</math> where <math display="inline">c</math> is the speed of sound in the fluid. For a solid material, <math display="inline">\kappa = \bar \kappa </math> with <math display="inline">\bar \kappa </math> given by Eq.([[#eq-14|14]]). Eqs.([[#eq-16|16]]) define the constitutive equation for the pressure for both solids and quasi-incompressible fluids as <math display="inline">\Delta p = \kappa  \Delta t \varepsilon _v</math>. This is consisted with the expression obtained in Eq.([[#eq-13|13]]) for a hypo-elastic solid material.
-For a fully incompressible material <math display="inline">\kappa =\infty </math> and Eq.([[#eq-16|16a]]) simplifies to the standard form, <math display="inline">\varepsilon _v =0</math>. In this case the pressure increment can not be obtained from Eqs.([[#2.3 Mass conservation equation|2.3]]) and then the pressure becomes an independent variable <span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-7|[7,48]]]. In our work we will retain the quasi-incompressible form of <math display="inline">r_v</math>  for convenience.
+For a fully incompressible material <math display="inline">\kappa =\infty </math> and Eq.([[#eq-16|16a]]) simplifies to the standard form, <math display="inline">\varepsilon _v =0</math>. In this case the pressure increment can not be obtained from Eqs.([[#eq-16|16]]) and then the pressure becomes an independent variable <span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-7|[7,48]]]. In our work we will retain the quasi-incompressible form of <math display="inline">r_v</math>  for convenience.
 '''Remark 4.''' Note that for Newtonian fluids the deviatoric stresses are directly related to the rates of deformation by Eq.([[#eq-5|5]]), whereas for solids the ''deviatoric stress increments'' are related to the rates of deformation by Eq.([[#eq-13|13]]). On the other hand, the relationship between the pressure increment and the volumetric strain rate has the form of Eq.([[#eq-13|13]]) for both fluids and solids, with the adequate definition of the bulk modulus for each case.

@@ Line 1,000: / Line 1,000: @@
 {| class="wikitable" style="text-align: left; margin: 1em auto;font-size:85%;"
 |-
-| colspan='2' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;" | ''MATERIAL PROPERTIES'':
+| colspan='2' style="border-top: 2px solid;border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;" | ''MATERIAL PROPERTIES'':
 |-

@@ Line 1,435: / Line 1,435: @@
 |-
 |[[Image:draft_Samper_718004671-ball3DInput.png|400px|]]
-|[[Image:draft_Samper_718004671-ball3DDataGeo.png|400px|]]
+|[[Image:draft_Samper_718004671-ball3DDataGeo.png|150px|]]
 |-
 | colspan="2"|[[Image:draft_Samper_718004671-ball3DDataMat.png|600px|Warming of a solid sphere with initial temperature ^∘T=270 K falling into a cylindrical tank containing a fluid at ^∘T=340 K. Initial geometry and material properties.]]
@@ Line 1,447: / Line 1,447: @@
 |[[Image:draft_Samper_718004671-ball3D8.png|400px|]]
 |[[Image:draft_Samper_718004671-ball3D17.png|400px|]]
 |-
 |[[Image:draft_Samper_718004671-ball3D38.png|400px|]]
 |[[Image:draft_Samper_718004671-tempIce.png|400px|]]
 |-
-|[[Image:draft_Samper_718004671-ball3D53.png|400px|]]
+|t=0.38s
 |[[Image:draft_Samper_718004671-ball3D71.png|400px|]]
 |-
-| colspan="2"|[[Image:draft_Samper_718004671-ball3D300.png|300px|Warming of a solid sphere (^∘T=270 K) falling into a cylindrical tank containing a fluid at ^∘T=340 K. Snapshots of the  motion of the sphere and the temperature contours at different time steps.]]
+|t=0.71s
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure 26:''' Warming of a solid sphere (<math>^\circ </math>T=270 K) falling into a cylindrical tank containing a fluid at <math>^\circ </math>T=340 K. Snapshots of the  motion of the sphere and the temperature contours at different time steps.
+| colspan="3" | '''Figure 26:''' Warming of a solid sphere (<math>^\circ </math>T=270 K) falling into a cylindrical tank containing a fluid at <math>^\circ </math>T=340 K. Snapshots of the  motion of the sphere and the temperature contours at different time steps.
 |}

← Older revision	Revision as of 13:02, 8 May 2018
(No difference)

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 |t=4.55s
+|
 |[[Image:draft_Samper_718004671-Ice1001.png|400px|]]
 |[[Image:draft_Samper_718004671-Ice1600.png|300px]]

@@ Line 1,393: / Line 1,393: @@
 |[[Image:draft_Samper_718004671-Ice1001.png|400px|]]
 |[[Image:draft_Samper_718004671-Ice1600.png|300px]]
 |-
 |t=10.00s
@@ Line 1,404: / Line 1,405: @@
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:draft_Samper_718004671-iceZoom281.png|400px]]
+|style="padding-right:10px;|[[Image:draft_Samper_718004671-iceZoom281.png|400px]]
-|[[Image:draft_Samper_718004671-iceZoom470.png|400px]]
+|style="padding-right:10px;|[[Image:draft_Samper_718004671-iceZoom470.png|400px]]
 |[[Image:draft_Samper_718004671-iceZoom977.png|400px]]
 |-
@@ Line 1,412: / Line 1,413: @@
 |(c) t=9.77s
 |-
-|[[Image:draft_Samper_718004671-iceZoom1274.png|400px]]
+|style="padding-right:10px;|[[Image:draft_Samper_718004671-iceZoom1274.png|400px]]
-|[[Image:draft_Samper_718004671-iceZoom1469.png|400px]]
+|style="padding-right:10px;|[[Image:draft_Samper_718004671-iceZoom1469.png|400px]]
 |[[Image:draft_Samper_718004671-iceZoom1547.png|400px]]
 |-

@@ Line 1,360: / Line 1,360: @@
 |-
 |[[Image:draft_Samper_718004671-iceInput2.png|400px|]]
-|[[Image:draft_Samper_718004671-iceDataGeo.png|400px|]]
+|[[Image:draft_Samper_718004671-iceDataGeo.png|150px|]]
 |-
 | colspan="2"|[[Image:draft_Samper_718004671-iceDataMat.png|600px|Melting of an ice cylinder. Geometry, material data and initial thermal conditions.]]
@@ Line 1,378: / Line 1,378: @@
 |[[Image:draft_Samper_718004671-Ice44.png|400px|]]
 |[[Image:draft_Samper_718004671-Ice110.png|400px|]]
 |-
-|[[Image:draft_Samper_718004671-Ice263.png|400px|]]
+|t=0.44s
-|[[Image:draft_Samper_718004671-tempIce.png|400px|]]
+|t=1.10s
 |-
 |[[Image:draft_Samper_718004671-Ice455.png|400px|]]
 |[[Image:draft_Samper_718004671-Ice1001.png|400px|]]
 |-
-| colspan="2"|[[Image:draft_Samper_718004671-Ice1600.png|300px|Melting of an ice cylinder. Snapshots of the melting process with temperature contours at different time steps.]]
+|t=10.00s
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 23:''' Melting of an ice cylinder. Snapshots of the melting process with temperature contours at different time steps.
@@ Line 1,393: / Line 1,404: @@
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:draft_Samper_718004671-iceZoom281.png|400px|]]
+|[[Image:draft_Samper_718004671-iceZoom281.png|400px]]
-|[[Image:draft_Samper_718004671-iceZoom470.png|400px|]]
+|[[Image:draft_Samper_718004671-iceZoom470.png|400px]]
 |-
-|[[Image:draft_Samper_718004671-iceZoom977.png|400px|]]
+|[[Image:draft_Samper_718004671-iceZoom1274.png|400px]]
-|[[Image:draft_Samper_718004671-iceZoom1274.png|400px|]]
+|[[Image:draft_Samper_718004671-iceZoom1469.png|400px]]
 |-
-|[[Image:draft_Samper_718004671-iceZoom1469.png|400px|]]
+|(d) t=12.74s
-|[[Image:draft_Samper_718004671-iceZoom1547.png|400px|Melting of an ice cylinder. Zoom of the ice domain at different time instants.]]
+|(e) t=14.69s
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure 24:''' Melting of an ice cylinder. Zoom of the ice domain at different time instants.
+| colspan="3" | '''Figure 24:''' Melting of an ice cylinder. Zoom of the ice domain at different time instants.
 |}

@@ Line 1,317: / Line 1,317: @@
 |[[Image:draft_Samper_718004671-Balls664.png|400px|]]
 |[[Image:draft_Samper_718004671-Balls1816.png|400px|]]
 |-
-|[[Image:draft_Samper_718004671-Balls2656.png|400px|]]
+|t=0.67s
-|[[Image:draft_Samper_718004671-tempIce.png|400px|]]
+|t=1.82s
 |-
 |[[Image:draft_Samper_718004671-Balls4500.png|400px|]]
 |[[Image:draft_Samper_718004671-Balls7000.png|400px|]]
 |-
-| colspan="2"|[[Image:draft_Samper_718004671-Balls8000.png|300px|Falling of three solid objects in a heated tank filled with a fluid. Snapshots with temperature contours at different time steps.]]
+|t=7.00s
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" | '''Figure 20:''' Falling of three solid objects in a heated tank filled with a fluid. Snapshots with temperature contours at different time steps.
+| colspan="3" | '''Figure 20:''' Falling of three solid objects in a heated tank filled with a fluid. Snapshots with temperature contours at different time steps.
 |}