Zhou et al 2022b - Revision history

Rimni at 12:47, 25 November 2022

2022-11-25T12:47:40Z

Rimni: /* 4.2 VI for normal contact conditions */

2022-11-25T12:40:08Z

‎4.2 VI for normal contact conditions

Rimni at 12:37, 25 November 2022

2022-11-25T12:37:30Z

Rimni at 12:36, 25 November 2022

2022-11-25T12:36:25Z

Rimni at 12:35, 25 November 2022

2022-11-25T12:35:39Z

Rimni: /* 4.3 VI for tangential contact conditions */

2022-11-25T12:34:23Z

‎4.3 VI for tangential contact conditions

Rimni at 12:33, 25 November 2022

2022-11-25T12:33:12Z

Rimni at 12:25, 25 November 2022

2022-11-25T12:25:58Z

Rimni at 12:01, 25 November 2022

2022-11-25T12:01:26Z

Rimni at 11:19, 25 November 2022

2022-11-25T11:19:56Z

@@ Line 68: / Line 68: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (1)
 |}
 In Eq. (1), <math display="inline">{\lambda }_1</math>, <math display="inline">{\lambda }_2</math> and <math display="inline">{\lambda }_3</math> denote the elongation of a volume element along the three principle axes, respectively. It is noticed that <math display="inline">I_3</math> identically equals to one for the incompressible material. Therefore, <math display="inline">\bf W</math> is only calculated by using <math display="inline">I_1</math> and <math display="inline">I_2</math>, that is
@@ Line 174: / Line 173: @@
 | colspan="1" style="padding:10px;"| '''Figure 2'''. A quadratic example for the B-spline basis function
 |}
 With the basis functions in hand, a B-spline curve can be constructed as their linear combination:
@@ Line 293: / Line 291: @@
 | colspan="1" style="padding:10px;"| '''Figure 3'''. Known tractions and unknown contact forces at the boundary of the water seal head
 |}
 The updated Kirchhoff stress <math>^{t+\Delta t}S_{ij}</math> is the summation of and the updated Kirchhoff stress increment <math>\Delta S_{ij}</math> and the known Cauchy stress <math>\tau_{ij}</math> at time <math>t</math>,  that is
@@ Line 581: / Line 578: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (36)
 |}
 The normal contact force <math>p_\boldsymbol{n}</math> is directly related to the normal distance <math>d_\boldsymbol{n}</math>. The expansion water seal suffers the back pressure on the back and the reservoir water pressure on the head, which both have specified values. If the expansion water seal contacts the steel plate tightly, in other words, <math>d_\boldsymbol{n}=0</math>, <math>p_\boldsymbol{n}</math> must be greater than the normal force induced by the back pressure or the water pressure. The normal force resulted by the back pressure and the water pressure can be calculated as <math>(\boldsymbol{n}_c)^T\boldsymbol{t}</math>, which is actually the modulus of <math>\boldsymbol{t}</math> because both the back pressure and the water pressure act on the expansion water seal boundary along the inner normal direction. Moreover, if <math>d_\boldsymbol{n}> 0</math>, the normal force applied to
@@ Line 610: / Line 606: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (38)
 |}
 It is noticed that the boundaries <math>\Gamma_t</math> and <math>\Gamma_c</math> are determined when the normal contact conditions are formulated as Eq. (37). If <math>d_\boldsymbol{n}>0</math>, the point <math>\boldsymbol{x}^m</math>  is situated at the boundary <math>\Gamma_t</math> of <math>\Omega^m</math>, and the point <math>\boldsymbol{x}^s</math> is situated at the boundary <math>\Gamma_t</math> of <math>\Omega^s</math>. At this point the normal forces applied to the two points are the known tractions. If <math>d_\boldsymbol{n} =0</math>, the point <math>\boldsymbol{x}^m</math> is situated at the boundary <math>\Gamma_c</math> of <math>\Omega^m</math>, and the point <math>\boldsymbol{x}^s</math> is situated at the boundary <math>\Gamma_c</math> of <math>\Omega^s</math>. At this moment the normal forces applied to the two points are uncertain. Because the two cases are uniformly involved in Eq. (37), it is unnecessary to determine the boundaries <math>\Gamma_t</math> and <math>\Gamma_c</math> in advance.
@@ Line 627: / Line 622: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (39)
 |}
 The relative tangential movement in terms of a contact pair is dominated by the Coulomb friction law. Considering that the direction of the friction acting on the slave body <math>\Omega^s</math> is in agreement with the tangential relative movement, it is denoted as <math>p_{\boldsymbol{\tau }}</math>. If a relative displacement happens, that is, <math>d_{\boldsymbol{\tau }}\not = 0</math>, the magnitude of <math>p_{\boldsymbol{\tau }}</math> will arrive at the maximum value <math>\mu p_\boldsymbol{n}</math> allowed by the Coulomb friction law, in which <math>\mu</math> denotes the Coulomb friction coefficient. Otherwise, if <math>d_{\boldsymbol{\tau }}=0</math>, the magnitude of <math>d_{\boldsymbol{\tau }}</math> is certainly smaller than <math>\mu p_\boldsymbol{n}</math>, but its direction is undetermined.

@@ Line 598: / Line 598: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (37)
 |}
 Comparing Eq. (37) with Eq. (35), Eq. (37) is equivalent to a box-constrained variational inequality with <math>a_i =(\boldsymbol{n}_c)^T\boldsymbol{t}</math> and <math>b_i = +\infty</math>; thus, the normal contact force must be the solution of the following ''VI''

@@ Line 668: / Line 668: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>VI\left(K^{\boldsymbol{\tau }},\boldsymbol{F}^{\boldsymbol{\tau }}\right),\mbox{ with }K^{\boldsymbol{\tau }}=</math><math>\left[-\mu p_\boldsymbol{n},\mu p_\boldsymbol{n}\right],\mbox{ and }\boldsymbol{F}^{\boldsymbol{\tau }}=</math><math>-d_{\boldsymbol{\tau }}</math>
+| <math>VI\left(K^{\boldsymbol{\tau }},\boldsymbol{F}^{\boldsymbol{\tau }}\right),\quad\mbox{ with }\quad K^{\boldsymbol{\tau }}=</math><math>\left[-\mu p_\boldsymbol{n},\mu p_\boldsymbol{n}\right],\quad \mbox{ and }\quad \boldsymbol{F}^{\boldsymbol{\tau }}=</math><math>-d_{\boldsymbol{\tau }}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (42)

@@ Line 661: / Line 661: @@
 |}
-Comparing Eqs. (40) and (41) with Eq. (35), the two conditions can be considered as another box-constrained variational inequality with <math>a_i =－\mu p_\boldsymbol{n}</math> and <math>b_i =－\mu p_\boldsymbol{n}</math>; therefore, <math>p_{\boldsymbol{\tau }} </math> must be the solution of the following ''VI''
+Comparing Eqs. (40) and (41) with Eq. (35), the two conditions can be considered as another box-constrained variational inequality with <math>a_i =-\mu p_\boldsymbol{n}</math> and <math>b_i =-\mu p_\boldsymbol{n}</math>; therefore, <math>p_{\boldsymbol{\tau }} </math> must be the solution of the following ''VI''
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 661: / Line 661: @@
 |}
-Comparing Eqs. (40) and (41) with Eq. (35), the two conditions can be considered as another box-constrained variational inequality with <math>a_i =－\mu p_\boldsymbol{n}</math> and <math>b_i =－\mu p_\boldsymbol{n}</math>; therefore, <math>_{\boldsymbol{\tau }} </math> must be the solution of the following ''VI''
+Comparing Eqs. (40) and (41) with Eq. (35), the two conditions can be considered as another box-constrained variational inequality with <math>a_i =－\mu p_\boldsymbol{n}</math> and <math>b_i =－\mu p_\boldsymbol{n}</math>; therefore, <math>p_{\boldsymbol{\tau }} </math> must be the solution of the following ''VI''
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 647: / Line 647: @@
 |}
-The conditions about the tangential relative movement in Eq. (40) are written as a formulation of   <math>d_{\boldsymbol{\tau }</math> with the opposite sign, which aims to provide an intuitive form for the subsequent analysis.
+The conditions about the tangential relative movement in Eq. (40) are written as a formulation of   <math>d_{\boldsymbol{\tau }}</math> with the opposite sign, which aims to provide an intuitive form for the subsequent analysis.
 For the sliding state, we have
@@ Line 672: / Line 672: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (42)
 |}
 For each knot point located on the boundary of the expansion water seal at time <math>t+\Delta t</math>, the normal and tangential contact forces must be the solutions of Eqs. (38) and (42), which are the supplement equations for Eqs. (23) and (32) to estimate the traction on the expansion water seal.

@@ Line 366: / Line 366: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (22)
 |}
 in which, <math>^t\boldsymbol{u}^e</math> and <math>^{t}{x}^e</math> are the arrays containing the displacements and the coordinates of <math>n^e</math> number of control points in an element at time <math>t</math>, <math>\Delta u^e</math> is the array containing the displacement incremental of control points in an element within the time interval <math>\Delta t</math>, <math>\boldsymbol{R}\left(\xi ,\eta \right) = [R_1 (\xi , \eta ){\bf I}, R_1 (\xi , \eta ){\bf I}, \ldots , R_{ne} (\xi , \eta ){\bf I}]</math> is a matrix of size <math>2 \times  2n^e</math> with identity matrix <math>{\bf I}</math> in <math>R^2</math>. Here, <math>R(\xi , \eta )</math> is a simplified notation of <math>R_{ij,pq}(\xi , \eta )</math> in Eq. (13), and <math>n^e</math> denotes the total number of control points whose <math>R(\xi , \eta )</math> has local support in an element <math>\Omega^e</math>.
@@ Line 485: / Line 484: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (31)
 |}
 here, <math>{\bf t} = (t_x, t_y)^T</math>, <math>{\bf p} = (p_x, p_y)^T</math> and <math>{\bf f} = (f_x, f_y)^T</math> are the vector expressions for <math>t_k</math>, <math>p_k</math> and <math>f_k</math>, respectively. There is no barrier to obtain the value of the body force <math>{\bf f}</math>. However, the value of the contact force <math>{\bf b}</math> is temporarily unknown. Furthermore, the boundaries <math>{\Gamma }_t</math> and <math>{\Gamma }_c</math> are uncertain, so the contribution of the known tractions to <math>Q^e</math> is also an issue. The issues will be settled by introducing VI theory, which is demonstrated in the next section.
@@ Line 496: / Line 494: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{K}^\mbox{p}\Delta u^\mbox{p}=\boldsymbol{Q}^\mbox{p}-</math><math>\boldsymbol{F}^\mbox{p}\mbox{  for }{\Omega }^\mbox{e}\in {\Omega }_{\mbox{TP}}\mbox{, }{\Omega }_{\mbox{LP}},\mbox{ }{\Omega }_{\mbox{RP}},\mbox{ }{\Omega }_{\mbox{BP}}</math>
+| <math>\boldsymbol{K}^\mbox{p}\Delta u^\mbox{p}=\boldsymbol{Q}^\mbox{p}-</math><math>\boldsymbol{F}^\mbox{p}\quad \mbox{  for }\quad {\Omega }^\mbox{e}\in {\Omega }_{\mbox{TP}}\mbox{, }{\Omega }_{\mbox{LP}},\mbox{ }{\Omega }_{\mbox{RP}},\mbox{ }{\Omega }_{\mbox{BP}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (32)

@@ Line 211: / Line 211: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>R_{i,p}(\xi )=\frac{N_{i,p}(\xi )w_i}{\sum_{\overset{\mbox{ˆ}}{i}=1}^mN_{\overset{\mbox{ˆ}}{i},p}(\xi )w_{\overset{\mbox{ˆ}}{i}}}</math>
+| <math>R_{i,p}(\xi )=\frac{N_{i,p}(\xi )w_i}{\displaystyle\sum_{\overset{\mbox{ˆ}}{i}=1}^mN_{\overset{\mbox{ˆ}}{i},p}(\xi )w_{\overset{\mbox{ˆ}}{i}}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)
@@ Line 235: / Line 235: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>R_{ij,pq}\left(\xi ,\eta \right)=\frac{N_{i,p}\left(\xi \right)M_{j,q}\left(\eta \right)w_{ij}}{\sum_{\overset{\mbox{ˆ}}{i}=1}^m\sum_{\overset{\mbox{ˆ}}{j}=1}^nN_{\overset{\mbox{ˆ}}{i},p}\left(\xi \right)M_{\overset{\mbox{ˆ}}{j},q}\left(\eta \right)w_{\overset{\mbox{ˆ}}{i}\overset{\mbox{ˆ}}{j}}}</math>
+| <math>R_{ij,pq}\left(\xi ,\eta \right)=\frac{N_{i,p}\left(\xi \right)M_{j,q}\left(\eta \right)w_{ij}}{\displaystyle\sum_{\overset{\mbox{ˆ}}{i}=1}^m\displaystyle\sum_{\overset{\mbox{ˆ}}{j}=1}^nN_{\overset{\mbox{ˆ}}{i},p}\left(\xi \right)M_{\overset{\mbox{ˆ}}{j},q}\left(\eta \right)w_{\overset{\mbox{ˆ}}{i}\overset{\mbox{ˆ}}{j}}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (13)

← Older revision		Revision as of 11:19, 25 November 2022
Line 751:		Line 751:


−	In this study, the parameters in the Extra-Gradient Method algorithm are set as <math>\varepsilon =0.00001</math>, <math>\eta_1=0.4</math> and <math>\eta_2=0.9</math>. More details of the Extra-gradient method can be referred to the monograph~~[20]~~ by Facchinei and Pang and the article~~[21]~~ by Jiang et al.	+	In this study, the parameters in the Extra-Gradient Method algorithm are set as <math>\varepsilon =0.00001</math>, <math>\eta_1=0.4</math> and <math>\eta_2=0.9</math>. More details of the Extra-gradient method can be referred to the monograph by Facchinei and Pang [20] and the article by Jiang et al. [21].

	==6. Performance analysis for the expansion water seal using the proposed method==		==6. Performance analysis for the expansion water seal using the proposed method==