Onate et al 2011a - Revision history

Move page script: Move page script moved page Draft Samper 395841693 to Onate et al 2011a

2018-10-11T11:10:08Z

Move page script moved page Draft Samper 395841693 to Onate et al 2011a

Cinmemj at 10:19, 31 August 2018

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Cinmemj at 10:18, 31 August 2018

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Cinmemj at 10:16, 31 August 2018

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Cinmemj at 10:13, 31 August 2018

2018-08-31T10:13:07Z

Cinmemj at 10:11, 31 August 2018

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Cinmemj at 10:11, 31 August 2018

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Cinmemj at 10:10, 31 August 2018

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 The quality of the numerical solution  depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.
 ===2.1 Basic steps of the PFEM===
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 </ol>
 ==3 FIC/FEM formulation for a Lagrangian continuum==

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 where <math display="inline">{S}</math> is the second Piola-Kirchhoff stress tensor, <math display="inline">{F}</math> is the deformation gradient tensor and <math display="inline">J = \det {F}</math> <span id='citeF-22'></span><span id='citeF-47'></span>[[#cite-22|[22,47]]]. Parameters <math display="inline">\mu </math> and <math display="inline">\lambda </math>  take the following values for a fluid or solid material:
-'''Fluid:''' <math display="inline">\mu </math>: viscosity; <math display="inline">\lambda = \Delta t K - \frac{2\mu }{3}</math>      '''Solid:''' <math display="inline">\displaystyle \mu = \frac{\Delta t G}{J}</math>; <math display="inline"> \displaystyle \lambda =\frac{2G \nu \Delta t}{J(1-2\nu )}</math>, where <math display="inline">\nu </math> is the Poisson ration, <math display="inline">G</math> is the shear modulus and <math display="inline">\Delta t</math> the time increment.
 In Eqs.(3) and (4), <math display="inline">s _{ij}</math> are the deviatoric stresses, <math display="inline">\dot \varepsilon _{ij}</math>  is the rate of deformation, <math display="inline">\mu </math> is the viscosity and <math display="inline">\delta _{ij}</math> is the Kronecker delta. <math display="inline">{}^{t} (\cdot )</math> denotes values at time <math display="inline">t</math>.

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 | style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | 2. LOOP OVER NUMBER OF ITERATIONS, <math display="inline">i=1</math>, NITER
 |-
-| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math>  Compute  nodal velocities by solving Eq.(8)
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math>  Compute  nodal velocities by solving Eq.(8)
 |-
 | style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\displaystyle \left[\frac{1}{\Delta t} \mathbf{M}+\mathbf{K}\right]{}^{t+1} \bar{v}^{i+1} ={}^{t+1} \mathbf{f}+G{}^{t+1} \bar{p}^{i} +\frac{1}{\Delta t} \mathbf{M} {}^{t} \bar{v}</math>
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 | style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Next time step <math display="inline">t\to t+1</math>
 |-
-| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math> Identify new analysis domain boundary: <math display="inline">{}^{t+1} V</math>
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math> Identify new analysis domain boundary: <math display="inline">{}^{t+1} V</math>
 |-
-| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math> Generate mesh:<math display="inline">{}^{t+1} M</math>
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  <math display="inline">\bullet </math> Generate mesh:<math display="inline">{}^{t+1} M</math>
 |- style="border-bottom: 2px solid;"
-| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] Go to 1
+| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  Go to 1
 |}

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-<div class="center" style="font-size: 75%;">'''Box I'''. Basic PFEM algorithm for a Lagrangian continuum
+===='''Pressure-velocity relationship'''====
 </div>
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 |- style="border-bottom: 2px solid;"
 | style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] Go to 1
 |}
 ==4 Generation of a new mesh==

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	The condition of prescribed velocities at the fixed boundaries in the PFEM is applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed boundaries is accounted for by the incompressibility condition which ''naturally prevents the fluid nodes to penetrate into the solid boundaries'' <span id='citeF-32'></span><span id='citeF-36'></span>[[#cite-32\|[32,36]]].		The condition of prescribed velocities at the fixed boundaries in the PFEM is applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed boundaries is accounted for by the incompressibility condition which ''naturally prevents the fluid nodes to penetrate into the solid boundaries'' <span id='citeF-32'></span><span id='citeF-36'></span>[[#cite-32\|[32,36]]].
−

	===6.2 Contact between solid-solid interfaces===		===6.2 Contact between solid-solid interfaces===

← Older revision	Revision as of 11:10, 11 October 2018
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 The  condition of prescribed velocities  at the fixed boundaries in the PFEM is  applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed  boundaries is accounted for by the incompressibility condition which ''naturally prevents    the fluid nodes to penetrate into the solid boundaries'' <span id='citeF-32'></span><span id='citeF-36'></span>[[#cite-32|[32,36]]].
 <div id='img-2'></div>
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 | colspan="1" | '''Figure 2:''' Modelling of contact conditions at a solid-solid interface with the PFEM
 |}
 This algorithm has proven to be very effective and it allows to identifying and modeling complex frictional contact conditions between two or more interacting bodies moving in water in an extremely simple manner.

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 The algorithm for modeling the  erosion of soil/rock particles at the fluid bed is the following:
 <ol>
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 Figure [[#img-3|3]] shows an schematic view of the bed erosion algorithm described.
 ==8 Examples==

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	The PFEM has been successfully applied to the study of the motion of a 1Tn quasi-spherical rock due to a water stream. The rock lays on a collection of rocks that are kept rigid. Frictional conditions between the analyzed rock and the rest of the rocks have been assumed. Figure [[#img-4\|4]]a shows that a water stream of 1m/s is not able to displace the individual rock. An increase of the water speed to 2m/s induces the motion of the rock as shown in Figure [[#img-4\|4]]b.		The PFEM has been successfully applied to the study of the motion of a 1Tn quasi-spherical rock due to a water stream. The rock lays on a collection of rocks that are kept rigid. Frictional conditions between the analyzed rock and the rest of the rocks have been assumed. Figure [[#img-4\|4]]a shows that a water stream of 1m/s is not able to displace the individual rock. An increase of the water speed to 2m/s induces the motion of the rock as shown in Figure [[#img-4\|4]]b.
		+
		+	<div id='img-4a'></div>
		+	<div id='img-4b'></div>
		+	<div id='img-4'></div>
		+	{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
		+	\|-
		+	\|[[Image:Draft_Samper_395841693-Roca-1ms_1.png\|420px\|Water speed of 1m/s. The individual rock can not be dragged by the stream]]
		+	\|[[Image:Draft_Samper_395841693-Roca-2ms_2.png\|420px\|Water speed of 2m/s. The individual rock is dragged by the stream]]
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| (a) Water speed of 1m/s. The individual rock can not be dragged by the stream
		+	\| (b) Water speed of 2m/s. The individual rock is dragged by the stream
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| colspan="2" \| '''Figure 4:''' Study of the drag of an individual rock of 1Tn under a water stream at speeds of (a) 1m/s and (b) 2m/s
		+	\|}

	===8.2 Impact of sea waves on piers and breakwaters===		===8.2 Impact of sea waves on piers and breakwaters===

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	Figures [[#img-5\|5]] and [[#img-6\|6]] show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of <math display="inline">4\times 4</math> mts). The figures correspond to the study of Langosteira harbour in A Coruña, Spain using PFEM.		Figures [[#img-5\|5]] and [[#img-6\|6]] show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of <math display="inline">4\times 4</math> mts). The figures correspond to the study of Langosteira harbour in A Coruña, Spain using PFEM.

−	~~<div id='img-4a'></div>~~
−	~~<div id='img-4b'></div>~~
−	~~<div id='img-4'></div>~~
−	~~{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"~~
−	\|-
−	~~\|[[Image:Draft_Samper_395841693-Roca-1ms_1.png\|420px\|Water speed of 1m/s. The individual rock can not be dragged by the stream]]~~
−	~~\|[[Image:Draft_Samper_395841693-Roca-2ms_2.png\|420px\|Water speed of 2m/s. The individual rock is dragged by the stream]]~~
−	~~\|- style="text-align: center; font-size: 75%;"~~
−	~~\| (a) Water speed of 1m/s. The individual rock can not be dragged by the stream~~
−	~~\| (b) Water speed of 2m/s. The individual rock is dragged by the stream~~
−	~~\|- style="text-align: center; font-size: 75%;"~~
−	~~\| colspan="2" \| '''Figure 4:''' Study of the drag of an individual rock of 1Tn under a water stream at speeds of (a) 1m/s and (b) 2m/s~~
−	\|}

	<div id='img-5'></div>		<div id='img-5'></div>