Becker et al 2018a - Revision history

Cinmemj at 12:46, 26 March 2019

2019-03-26T12:46:28Z

Cinmemj at 12:39, 26 March 2019

2019-03-26T12:39:03Z

Cinmemj at 13:49, 28 January 2019

2019-01-28T13:49:37Z

Cinmemj at 13:35, 28 January 2019

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Cinmemj: Cinmemj moved page Draft Samper 820077714 to Becker et al 2018a

2018-10-16T11:43:44Z

Cinmemj moved page Draft Samper 820077714 to Becker et al 2018a

Cinmemj at 11:27, 16 October 2018

2018-10-16T11:27:12Z

Cinmemj at 11:20, 16 October 2018

2018-10-16T11:20:53Z

Cinmemj at 11:19, 16 October 2018

2018-10-16T11:19:26Z

Cinmemj at 11:17, 16 October 2018

2018-10-16T11:17:47Z

Cinmemj at 11:16, 16 October 2018

2018-10-16T11:16:59Z

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 Published in ''Computational Mechanics'' (2015). Vol. 55, pp. 1091–1104 <br />
 doi: 10.1007/s00466-014-1107-0
 ==Abstract==

← Older revision		Revision as of 12:39, 26 March 2019
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−	Published in ''Computational Mechanics'' (2015). Vol. 55, pp. 1091–1104 ~~(preprint)~~ <br />	+	Published in ''Computational Mechanics'' (2015). Vol. 55, pp. 1091–1104 <br />
	doi: 10.1007/s00466-014-1107-0		doi: 10.1007/s00466-014-1107-0

← Older revision		Revision as of 13:49, 28 January 2019
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		+	Published in ''Computational Mechanics'' (2015). Vol. 55, pp. 1091–1104 (preprint) <br />
		+	doi: 10.1007/s00466-014-1107-0
		+

	==Abstract==		==Abstract==

@@ Line 1,068: / Line 1,068: @@
 ==8 Acknowledgements==
-This research has been partly funded by  the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n° 289911. This work was  also supported by the ERC Advanced Grant ‘REALTIME’ project (AdG-2009325), the ERC Advanced Grant ‘SAFECON’ project (AdG-26752) and  the HFLUIDS project of the National RTD Plan of the Spanish Ministry of Science and Innovation (BIA2010-15880).
+This research has been partly funded by  the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n° 289911. This work was also supported by the ERC Advanced Grant ‘REALTIME’ project (AdG-2009325), the ERC Advanced Grant ‘SAFECON’ project (AdG-26752), the HFLUIDS project of the National RTD Plan of the Spanish Ministry of Science and Innovation (BIA2010-15880) and the ERC Proof of Concept FORECAST (Ref. 664910) under H2020 Programme.
 The authors would also like to thank Dr. Riccardo Rossi for suggesting the special shape functions used in this work.

@@ Line 87: / Line 87: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\displaystyle \rho \frac{D  ( {\mathbf{V}} )}{D t}  = \nabla \sigma   + \rho \mathbf{g} \qquad \hbox{in}  \Omega   </math>
+| style="text-align: center;" | <math>\displaystyle \rho \frac{D  ( {\mathbf{V}} )}{D t}  = \nabla \sigma   + \rho \mathbf{g} \qquad \hbox{in }  \Omega   </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.a)
 |-
-| style="text-align: center;" | <math> \displaystyle \frac{D  p}{D t} + \kappa \nabla .   {\mathbf{V}} = 0  \qquad  \quad   \hbox{in}  \Omega    </math>
+| style="text-align: center;" | <math> \displaystyle \frac{D  p}{D t} + \kappa \nabla .   {\mathbf{V}} = 0  \qquad  \quad   \hbox{in }  \Omega    </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.b)
 |}

← Older revision	Revision as of 11:43, 16 October 2018
(No difference)

@@ Line 922: / Line 922: @@
 ===6.3 Turek FSI benchmark===
-To test the PFEM-2 algorithm in a fully coupled scenario, the Turek FSI benchmark <span id='citeF-20'></span>[[#cite-20|[20]]], was simulated. This problem is typically used as a reference to validate new FSI strategies and it is particularly challenging for staggered schemes due to the similar properties of the fluid and the solid. On the other hand, the monolithic solver used in this work is well suited for this benchmark. Unlike the previous examples presented in this work (in which the PSPG stabilization was used) in this example the OSS stabilization is employed, with the explicit projection of the pressure gradient in the RHS. This is done to avoid having zeros in the RHS of the pressure equation of Equation [[#eq-23|(23)]] due to lack of gravity, which would provide excessive numerical diffusion.
+To test the PFEM-2 algorithm in a fully coupled scenario, the Turek FSI benchmark <span id='citeF-20'></span>[[#cite-20|[20]]], was simulated. This problem is typically used as a reference to validate new FSI strategies and it is particularly challenging for staggered schemes due to the similar properties of the fluid and the solid. On the other hand, the monolithic solver used in this work is well suited for this benchmark. Unlike the previous examples presented in this work (in which the PSPG stabilization was used) in this example the OSS stabilization is employed, with the explicit projection of the pressure gradient in the RHS. This is done to avoid having zeros in the RHS of the pressure equation of Equation [[#eq-20|(20)]] due to lack of gravity, which would provide excessive numerical diffusion.
 The problem models a 2D rigid cylinder attached to an elastic beam immersed in a fluid stream. The dimensions and problem parameters can be found in <span id='citeF-20'></span>[[#cite-20|[20]]]. In this paper a fixed mesh of 48500 3-noded triangles was used. The frequency and amplitude of the oscillations are analyzed to test the accuracy. For the case studied we use <math display="inline">Re=200</math>, with a parabolic inlet velocity profile and the following parameters:

@@ Line 982: / Line 982: @@
 For cases in which the density of the fluid is negligible, it is possible to calculate the analytical vibration frequency in vacuum with the  following expression:
-<span id="eq-68"></span>
+<span id="eq-42"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 990: / Line 990: @@
 | style="text-align: center;" | <math>\omega _{vac} = \sqrt{\frac{C_n^4  E  I}{L^4  \rho _q}}   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (68)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (42)
 |}
 where the subscript <math display="inline">n</math> denotes the mode and <math display="inline">C_n</math> are the positive roots of the following equation
-<span id="eq-69"></span>
+<span id="eq-43"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,003: / Line 1,003: @@
 | style="text-align: center;" | <math>1 + cos (C_n)  cosh (C_n) = 0.   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (69)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (43)
 |}
 Taking the first vibration mode, <math display="inline">C_1 = 1.8751</math> and the frequency of the beam becomes:
-<span id="eq-70"></span>
+<span id="eq-44"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,016: / Line 1,016: @@
 | style="text-align: center;" | <math>\omega _{vac} = \sqrt{\frac{C_1^4  E  I}{L^4  \rho _q}} = 32.097  rad/s  \rightarrow  5.108  Hz   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (70)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (44)
 |}
@@ Line 1,023: / Line 1,023: @@
 To test the influence of the fluid density, according to Van Eysden <span id='citeF-21'></span>[[#cite-21|[21]]] it is possible to estimate the lower vibration modes for rectangular section beams immersed in a fluid using the following formula:
-<span id="eq-71"></span>
+<span id="eq-45"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,031: / Line 1,031: @@
 | style="text-align: center;" | <math>\omega _{fluid}=\omega _{vac} \left(1 + \frac{ \pi \rho _p b }{4 \rho _q H } \Gamma _f \right)^{-1/2}    </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (71)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (45)
 |}

@@ Line 885: / Line 885: @@
 The beam dimensions are <math display="inline">L=10</math> and <math display="inline">H=1</math>. A vertical mass force of <math display="inline">q = 1[m/s^2]</math> is applied to the whole domain. According to Timoshenko beam theory and correcting the force to account for the buoyancy effect, the vertical displacement <math display="inline">w</math> at the tip should be
-<span id="eq-67"></span>
+<span id="eq-41"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 893: / Line 893: @@
 | style="text-align: center;" | <math>w_{max} =  (\rho _q - \rho _p) \left(\frac{q  L^4}{8  E  I} + \frac{1}{2} \frac{q L^2}{10/12   h  \mu } \right) = 0.001511 </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (67)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (41)
 |}

@@ Line 649: / Line 649: @@
 The solution strategy described in the previous section requires an accurate and robust tool to account for the convective term. Since the aim of the PFEM-2 is to achieve large time steps with small errors, it is not enough to simply move the particles with the last velocity multiplied by the time step (<math display="inline">\delta \mathbf{x} =  {V}^n \cdot \Delta _t</math>).
-The main goal is to obtain as much information as possible from the previous time step to calculate the new position of the particles. The streamline integration method used in this work follows the same strategy of <span id='citeF-1'></span>[[#cite-1|[1]]] for single fluids. For a particle <math display="inline">p</math>, its exact position at time <math display="inline">n+1</math> is computed as [[#eq-59|(59)]]:
+The main goal is to obtain as much information as possible from the previous time step to calculate the new position of the particles. The streamline integration method used in this work follows the same strategy of <span id='citeF-1'></span>[[#cite-1|[1]]] for single fluids. For a particle <math display="inline">p</math>, its exact position at time <math display="inline">n+1</math> is computed as [[#eq-34|(34)]]:
-<span id="eq-59"></span>
+<span id="eq-34"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 659: / Line 659: @@
 | style="text-align: center;" | <math>\mathbf{x}^{n+1}_p = \mathbf{x}^{n}_p + \int _{n}^{n+1} \mathbf{V^t}   dt </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (59)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
 |}
 Unfortunately, the velocity is not known continuously in time but only at discrete time steps( <math display="inline">n-1</math>,<math display="inline">n</math>, <math display="inline">n+1</math> ... ). Approximating the particle trajectory using the velocity of the previous time step to obtain an explicit scheme, the new position of the particle becomes:
-<span id="eq-60"></span>
+<span id="eq-35"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 672: / Line 672: @@
 | style="text-align: center;" | <math>\mathbf{x}^{n+1}_p \approx \mathbf{x}^{n}_p + \int _{n}^{n+1} \mathbf{V^n}   dt </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (60)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
 |}
@@ Line 687: / Line 687: @@
 To overcome this problem, the streamline integration must be deactivated when a particle of the heavier fluid transits across a region of the lighter fluid. In this case, the gravity will be taken as the only acting force, leading to a parabolic (projectile-type) motion, where the last known velocity of the particle is taken as the starting data. Setting the distance function <math display="inline">\varphi \le 0</math> for the denser fluid regions and <math display="inline">\varphi > 0 </math> for the lighter fluid regions, the updated position of the denser fluid particles becomes
-<span id="eq-61"></span>
+<span id="eq-36"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 695: / Line 695: @@
 | style="text-align: center;" | <math>\mathbf{x}^{n+1}_p \approx \Bigg\{{ \begin{matrix}   \mathbf{x}^{n}_p + \int _{n}^{n+1} \mathbf{V^n}   dt \qquad \qquad \hbox{if}   \varphi _{x_p^t} \le 0  \\   \mathbf{x}^{n}_p + \int _{n}^{n+1}  \mathbf{V^n_p} + g\cdot t   dt \quad \hbox{if}   \varphi _{x_p^t} > 0 \end{matrix}  } </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (62)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
 |}
@@ Line 706: / Line 706: @@
 Each particle has an associated material and the interface should be located where the material properties change. The distribution of particles inside each element can define complicated curves that become impossible to manage with simple shape functions due to the large number of particles. As an example, in Figure [[#img-8|8]] some particles are in the "wrong" side of the interface. For this reason the instantaneous local interface inside each element is simply defined by a line (or a plane in 3D) taking into account a weighted average using the shape functions. An instantaneous , pseudo level-set function <math display="inline">\varphi </math> is created that defines the interface where it is valued zero. To calculate the value of <math display="inline">\varphi </math> at each <math display="inline">j</math>th node, all the <math display="inline">i</math> particles in the neighbour elements to <math display="inline">j</math> are used. The weighting function  is defined as the standard shape function of the element for the <math display="inline">j</math>th node in the position of each <math display="inline">i</math>th  particle. Once the contribution of all the particles has been added, the location of the interface is computed as
-<span id="eq-63"></span>
+<span id="eq-37"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 714: / Line 714: @@
 | style="text-align: center;" | <math>\varphi _ j = \frac{ \sum _{i}^{n} {sign}_i N_i^j}{ \sum _{i}^{n} N_i^j} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (63)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (37)
 |}
@@ Line 729: / Line 729: @@
 The same procedure is  for all the variables, replacing the <math display="inline">sign_i</math> by the desired variable of the particle, either scalar, vectorial or tensorial. Using this kernel, the projected velocity can be written as
-<span id="eq-64"></span>
+<span id="eq-38"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 737: / Line 737: @@
 | style="text-align: center;" | <math>\hat { \hat { \mathbf{V}  } } _ j ^{n+1} = \frac{ \sum _{i}^{n} \mathbf{V}_i N_i^j}{ \sum _{i}^{n} N_i^j} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (64)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (38)
 |}
@@ Line 744: / Line 744: @@
 It is worth noting that the element shape functions <math display="inline">N_i</math> are not the only possible kernel. As an example, the function proposed in <span id='citeF-16'></span>[[#cite-16|[16]]] for the  Smoothed Particle Hydrodynamics method assigns a higher weight to those particles that are closer to the node. Setting <math display="inline">h</math> as the element size and <math display="inline">u=R/h</math>, where <math display="inline">R</math> is the distance from the particle to the node, the Wendland kernel is written as
-<span id="eq-65"></span>
+<span id="eq-39"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 752: / Line 752: @@
 | style="text-align: center;" | <math>W_i^j = \frac{7}{4 \pi h^2}(1+2u)(1-u/2)^4; </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (65)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (39)
 |}
 After the information of the Lagrangian domain has been projected and the mesh computations have been performed, the velocity of the particles has to be updated. At this stage it is important not to replace the particle velocity with the updated velocity but only a correction. The reason for this is simple; no matter how good the kernel algorithm may be, it is never exact. Replacing the unknowns in the particles would destroy valuable data, leading to excessive artificial diffusion in the strategy. Based on this fact, the update stage on the particle will only transfer the variation in the velocity (or any other variable) calculated in the mesh, that is:
-<span id="eq-66"></span>
+<span id="eq-40"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 765: / Line 765: @@
 | style="text-align: center;" | <math>\delta \mathbf{V}  _ j ^{n+1} =   \mathbf{V}  _ j ^{n+1}  - \hat { \hat { \mathbf{V}  } } _ j ^{n+1} </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (66)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (40)
 |}