García-Espinosa Ribó 2001a - Revision history

Julio: Julio moved page Draft García-Espinosa 298267853 to García-Espinosa Ribó 2001a

2018-03-28T08:32:59Z

Julio moved page Draft García-Espinosa 298267853 to García-Espinosa Ribó 2001a

Julio at 08:32, 28 March 2018

2018-03-28T08:32:43Z

Julio at 08:31, 28 March 2018

2018-03-28T08:31:28Z

Julio at 08:26, 28 March 2018

2018-03-28T08:26:56Z

Julio at 08:26, 28 March 2018

2018-03-28T08:26:28Z

Julio at 08:25, 28 March 2018

2018-03-28T08:25:31Z

Show changes

Julio at 08:18, 28 March 2018

2018-03-28T08:18:51Z

Julio at 08:08, 28 March 2018

2018-03-28T08:08:41Z

Julio at 08:00, 28 March 2018

2018-03-28T08:00:50Z

Julio: Created page with "
'''A FINITE ELEMENT METHOD FOR HYDRODYNAMIC TESTING OF SHIP FORMS'''..."

2018-03-28T07:57:14Z

Created page with " <div id="_GoBack" class="center" style="width: auto; margin-left: auto; margin-right: auto;"> <big>'''A FINITE ELEMENT METHOD FOR HYDRODYNAMIC TESTING OF SHIP FORMS'''</big>..."

Show changes

@@ Line 278: / Line 278: @@
 Figure shows the flow chart of the solution algorithm of eqs. (8).
-[[Image:draft_García-Espinosa_298267853-picture-Group 48.svg|center|600px]]
+[[File:Draft_García-Espinosa_298267853_5660_1.jpg|center|600px]]
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 567: / Line 567: @@
 The basic results of these analyses are presented next:
+[[Image:Draft_García-Espinosa_298267853_9225_2.jpg|center|600px]]
-[[Image:draft_García-Espinosa_298267853-picture-Group 43.svg|center|600px]]
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 367: / Line 367: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 41%;"| [[Image:draft_García-Espinosa_298267853-image19.png|600px]]
+|  style="vertical-align: top;width: 41%;"| [[Image:draft_García-Espinosa_298267853-image19.png|center|600px]]
 |-
 |  style="text-align: center;vertical-align: top;"|<span id='_Ref526945909'></span><span style="text-align: center; font-size: 75%;">'''Figure 6. KVLCC2. Hull geometry based on the NURBS entities.'''</span>
@@ Line 375: / Line 375: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image20-c.png|600px]]
+|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image20-c.png|center|600px]]
 |-
 |  style="text-align: center;vertical-align: top;"|<span id='_Ref526945928'></span><span style="text-align: center; font-size: 75%;">'''Figure 7. KVLCC2. Surface mesh on the hull.'''</span>
@@ Line 440: / Line 440: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image23-c.png|600px]]
+|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image23-c.png|center|600px]]
 |-
 |  style="vertical-align: top;"|<span id='_Ref526946547'></span><span style="text-align: center; font-size: 75%;">'''Figure 10. Contours of the X component of the velocity on a plane at 2.71 m from the aft perpendicular. Comparison with experimental data (right).'''</span>
@@ Line 448: / Line 448: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image24-c.png|600px]]
+|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image24-c.png|center|600px]]
 |-
 |  style="vertical-align: top;"|<span id='_Ref526946706'></span><span style="text-align: center; font-size: 75%;">'''Figure 11. Contours of the X component of the velocity on a plane at 2.82 m from the aft perpendicular. Comparison with experimental data (right).'''</span>

@@ Line 293: / Line 293: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 41%;"| [[Image:draft_García-Espinosa_298267853-image15-c.png|600px]]
+|  style="vertical-align: top;width: 41%;"| [[Image:draft_García-Espinosa_298267853-image15-c.png|center|600px]]
 |-
 |  style="text-align: center;vertical-align: top;"|<span id='_Ref526945208'></span><span style="text-align: center; font-size: 75%;">'''Figure 2. DTMB 5415. Hull geometry based on the NURBS entities.'''</span>
@@ Line 301: / Line 301: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image16-c.png|600px]]
+|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image16-c.png|center|600px]]
 |-
 |  style="text-align: center;vertical-align: top;"|<span id='_Ref526945245'></span><span style="text-align: center; font-size: 75%;">'''Figure 3. DTMB 5415. Surface mesh on the hull.'''</span>
@@ Line 311: / Line 311: @@
 {| style="width: 100%;"
 |-
-|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image17.png|600px]]
+|  style="vertical-align: top;width: 100%;"| [[Image:draft_García-Espinosa_298267853-image17.png|center|600px]]
 |-
 |  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Figure 4. DTMB 5415. Wave Profile on hull.'''</span>
@@ Line 360: / Line 360: @@
 |  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Figure 5. DTMB 5415. Wave Profile y/L = 0.082.'''</span>
 |}
 ===Example 2. KVLCC2 Model===

@@ Line 22: / Line 22: @@
 We consider the motion around a body of a viscous incompressible fluid including a free surface. The stabilized finite calculus (FIC) form of the governing differential equations [<span id='cite-11'></span>[[#11|11]]] for the three dimensional (3D) problem can be written as
- <math>\begin{array}{c}
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-r_{m_i}\underline{-\frac{1}{2}h_j\frac{\partial r_{m_i}}{\partial x_j}}=0\begin{array}{cccc}
+|-
   & on & \Omega  & i,j=1,2,3
-\end{array}\\
+\end{array}</math>
-r_d\underline{-\frac{1}{2}h_j\frac{\partial r_d}{\partial x_j}}=0\begin{array}{cccc}
+|}
   & on & \Omega  & j=1,2,3
-\end{array}\\
+\end{array}</math>
-r_{\beta }-\underline{\frac{1}{2}h_{{\beta }_j}\frac{\partial r_{\beta }}{\partial x_j}}=0\begin{array}{cccc}
+|}
   & on & \Omega  & j=1,2
 \end{array}</math>
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+where
-(1a)</div>
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-(1b)</div>
+|-
-−
+|
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+{| style="text-align: left; margin:auto;width: 100%"
-(1c)</div>
+|-
+| style="text-align: center;" | <math>\begin{array}{c}
-<span style="text-align: center; font-size: 75%;"> <math>\begin{array}{c}
 r_{m_i}=\frac{\partial u_i}{\partial t}+\frac{\partial }{\partial x_j}\left(u_iu_j\right)+\frac{\partial p}{\partial x_i}-\frac{\partial {\tau }_{ij}}{\partial x_j}\\
 r_d=\frac{\partial u_i}{\partial x_i}\begin{array}{cc}
@@ Line 51: / Line 71: @@
   & i=1,2
 \end{array},3
-\end{array}</math> </span>where
+\end{array}</math>
 In above ''u<sub>i</sub>'' is the velocity along the i-th global reference axis, ''p'' is the dynamic pressure, ''&#x03b2;'' is the wave elevation and ''&#x03c4;<sub>ij</sub>'' are the viscous stress tensor components.
@@ Line 57: / Line 80: @@
 Let ''n<sub>i</sub>'' be the unit outward normal to the boundary Ω and denoting by an overbar prescribed values, the boundary conditions for the stabilized problem are:
- <math>\begin{array}{c}
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 u=\overline{u}\begin{array}{ccc}
   & on & {\Gamma }_u
@@ Line 76: / Line 104: @@
 \end{array}
 \end{array}</math>
+|}
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
-(2a)</div>
+|}
-−
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
-(2b)</div>
-−
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
-(2c)</div>
 for ''t ''&#x2208; (''t<sub>0</sub>,t<sub>f</sub>''). The boundary Γ has been considered split into three sets of disjoint components Γ''<sub>u</sub>'', Γ''<sub>p</sub>'' and Γ''<sub>&#x03c4;</sub>'', the latter being the part where mixed conditions are prescribed.

← Older revision	Revision as of 08:32, 28 March 2018
(No difference)

@@ Line 98: / Line 98: @@
 ==Implicit Fractional Step Formulation==
-Let us discretize in time the stabilized momentum equation (1a) using the trapezoidal
+Let us discretize in time the stabilized momentum equation (1a) using the trapezoidal rule (or ''θ'' method) as
-rule (or ''θ'' method) as
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
- <math>\frac{u_i^{n+1}-u_i^n}{\delta t}+\frac{\partial }{\partial x_j}\left(u_i^{n+\theta }u_j^{n+\theta }\right)+</math><math>\frac{\partial p^{n+\theta }}{\partial x_i}-\frac{\partial {\tau }_{ij}^{n+\theta }}{\partial x_j}-</math><math>\frac{1}{2}h_j\frac{\partial r_{m_i}^{n+\theta }}{\partial x_j}=</math><math>0</math>
+{| style="text-align: left; margin:auto;width: 100%"
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|-
-(3)</div>
+| style="text-align: center;" | <math>\frac{u_i^{n+1}-u_i^n}{\delta t}+\frac{\partial }{\partial x_j}\left(u_i^{n+\theta }u_j^{n+\theta }\right)+</math><math>\frac{\partial p^{n+\theta }}{\partial x_i}-\frac{\partial {\tau }_{ij}^{n+\theta }}{\partial x_j}-</math><math>\frac{1}{2}h_j\frac{\partial r_{m_i}^{n+\theta }}{\partial x_j}=0</math>
 the superscripts n and ''θ'' refer to the time step and to the trapezoidal rule discretization parameter, respectively. For ''θ'' = 1 the standard backward Euler scheme is obtained, which has a temporal error of 0(t). The value ''θ'' = 0.5 gives the standard Crank Nicholson scheme, which is second order accurate in time 0(t<sup>2</sup>).
@@ Line 111: / Line 114: @@
 A classical implicit fractional step method can be simply derived by splitting eq. (3). The resulting continuous problem, omitting the boundary and initial conditions for brevity, is as follows:
- <math>\begin{array}{c}
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-\frac{{\overline{u}}_i^{n+1}-u_i^n}{\delta t}+\frac{\partial }{\partial x_j}\left({\overline{u}}_i^{n+\theta }{\overline{u}}_j^{n+\theta }\right)+\frac{\partial p^n}{\partial x_i}-\frac{\partial {\overline{\tau }}_{ij}^{n+\theta }}{\partial x_j}-\frac{1}{2}h_j\frac{\partial {\overline{r}}_{m_i}^{n+\theta }}{\partial x_j}=0\\
+|-
-\delta t\frac{{\partial }^2}{\partial x_i\partial x_i}\left(p^{n+1}-p^n\right)=\frac{\partial u_i^\ast }{\partial x_i}+\frac{1}{2}h_j\frac{\partial r_d}{\partial x_j}\\
-u_i^{n+1}={\overline{u}}_i-\delta t\frac{\partial }{\partial x_i}\left(p^{n+1}-p^n\right)
+{| style="text-align: left; margin:auto;width: 100%"
-\end{array}</math>
+|-
+| style="text-align: center;" | <math>\frac{{\overline{u}}_i^{n+1}-u_i^n}{\delta t}+\frac{\partial }{\partial x_j}\left({\overline{u}}_i^{n+\theta }{\overline{u}}_j^{n+\theta }\right)+\frac{\partial p^n}{\partial x_i}-\frac{\partial {\overline{\tau }}_{ij}^{n+\theta }}{\partial x_j}-\frac{1}{2}h_j\frac{\partial {\overline{r}}_{m_i}^{n+\theta }}{\partial x_j}=0</math>
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+|}
-(4a)</div>
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4a)
+|}
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+{| class="formulaSCP" style="width: 100%; text-align: left;"
-(4b)</div>
+|-
-−
+|
-<div style="text-align: right; direction: ltr; margin-left: 1em;">
+{| style="text-align: left; margin:auto;width: 100%"
-(4c)</div>
+|-
 where the terms denoted by an overbar are those calculated with the intermediate velocity ''ū<sub>i</sub>'', which is introduced to allow the momentum splitting.

← Older revision		Revision as of 08:00, 28 March 2018
Line 1:		Line 1:
−
−
−	~~<div id="_GoBack" class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''A FINITE ELEMENT METHOD FOR HYDRODYNAMIC TESTING OF SHIP FORMS'''</big></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''J. García <sup>(1)</sup>, R. Ribó <sup>(2)</sup> y M. López <sup>(3)</sup>'''</big></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>''' <sup>(1)</sup>'''</big>Aula FNB-CIMNE</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~Universidad Politécnica de Cataluña</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~Pla de Palau 18, 08003 Barcelona, España</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~e-mail: [mailto:julio@cimne.upc.es julio@cimne.upc.es], web page: [http://www.upc.e http://www.upc.e]s/fnb</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''<sup>(2)</sup>'''</big>Compass Ingeniería y Sistemas S.A.</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~C/ Manuel Girona, 61 Bajos. 08034 Barcelona, España</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~[mailto:e-mail:ramsan@compassis.com e-mail:ramsan@compassis.com], web page: [http://www.compassis.com http://www.compassis.com]</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''<sup>(3)</sup>'''</big>Nautatec, Ingeniería Naval Diseño de Yates S.L.</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~C\ Arquitecto Gaudí,11 Bajo Exterior. 28016 Madrid, España</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~e-mail: [mailto:nautatec@nautatec.com nautatec@nautatec.com], web page: [http://www.nautatec.com http://www.nautatec.com]</div>~~
−
−	~~'''Key words''': Finite element, naval hydrodynamics, incompressible ﬂows, free surface, ﬁnite calculus, fractional step method.~~

	==Introduction==		==Introduction==

García-Espinosa Ribó 2001a - Revision history

Julio: Julio moved page Draft García-Espinosa 298267853 to García-Espinosa Ribó 2001a

Julio at 08:32, 28 March 2018

Julio at 08:31, 28 March 2018

Julio at 08:26, 28 March 2018

Julio at 08:26, 28 March 2018

Julio at 08:25, 28 March 2018

Julio at 08:18, 28 March 2018

Julio at 08:08, 28 March 2018

Julio at 08:00, 28 March 2018

Julio: Created page with " '''A FINITE ELEMENT METHOD FOR HYDRODYNAMIC TESTING OF SHIP FORMS'''..."

Julio: Created page with "
'''A FINITE ELEMENT METHOD FOR HYDRODYNAMIC TESTING OF SHIP FORMS'''..."