Liu et al 2023a - Revision history

Rimni at 13:06, 30 May 2023

2023-05-30T13:06:40Z

HaiLiu at 12:04, 30 May 2023

2023-05-30T12:04:46Z

Rimni at 10:57, 30 May 2023

2023-05-30T10:57:19Z

Rimni at 10:55, 30 May 2023

2023-05-30T10:55:58Z

Rimni at 10:53, 30 May 2023

2023-05-30T10:53:06Z

Rimni: /* 2.5 Minimum principle for incremental total energy */

2023-05-29T13:43:58Z

‎2.5 Minimum principle for incremental total energy

Rimni at 13:42, 29 May 2023

2023-05-29T13:42:48Z

Rimni: /* 2.5 Minimum principle for incremental total energy */

2023-05-29T13:38:18Z

‎2.5 Minimum principle for incremental total energy

Rimni at 13:37, 29 May 2023

2023-05-29T13:37:33Z

Rimni at 13:33, 29 May 2023

2023-05-29T13:33:38Z

@@ Line 490: / Line 490: @@
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Yang H.,  Li H.,  Zhang Z.Y., Zhan M.,  Liu J.,  Li G. Advances and trends on tube bending forming technologies. Chinese J. Aeronaut.,  25(1):1-12, 2012.
+[1] Yang H.,  Li H.,  Zhang Z.Y., Zhan M.,  Liu J.,  Li G. Advances and trends on tube bending forming technologies. Chinese J. Aeronaut.,  25(1):1–12, 2012.
 [2]  Teng B.G.,  Hu L.,  Liu G.,  Yuan S.J. Wrinkling behavior of hydro bending of carbon steel Al alloy bilayered tubes. Trans. Nonferrous Met. Soc. China, 22:560−565, 2012.
@@ Line 496: / Line 496: @@
 [3] Xu X.F., Fan Y.B., Wu Y.W., Wu K.W., Xiao J. A novel differential lubrication method for push-bending of L-shaped thin-walled tube with 1D bending radius. International Journal of Material Forming, 14:691–701, 2021.
-[4]  Zeng Y.,  Li Z. Experimental research on the tube push-bending process. Journal of Materials Processing Technology, 122(2-3):237-240, 2002.
+[4]  Zeng Y.,  Li Z. Experimental research on the tube push-bending process. Journal of Materials Processing Technology, 122(2-3):237–240, 2002.
-[5] Oh I.Y., Han S.W., Woo Y.Y., Ra J.H., Moon Y.H. Tubular blank design to fabricate an elbow tube by a push-bending process. Journal of Materials Processing Technology, 260:112-122, 2018.
+[5] Oh I.Y., Han S.W., Woo Y.Y., Ra J.H., Moon Y.H. Tubular blank design to fabricate an elbow tube by a push-bending process. Journal of Materials Processing Technology, 260:112–122, 2018.
-[6] Liu H.,  Zhang S.H.,  Ding Y.P., Shi G.L., Geng Z. A simplified formulation for predicting wrinkling of thin-wall elbow tube in granular media–based push-bending process. International Journal of Advanced Manufacturing Technology, 115:541-549, 2021.
+[6] Liu H.,  Zhang S.H.,  Ding Y.P., Shi G.L., Geng Z. A simplified formulation for predicting wrinkling of thin-wall elbow tube in granular media–based push-bending process. International Journal of Advanced Manufacturing Technology, 115:541–549, 2021.
-[7]  Du B.,  Zhao C.C.,  Dong G.J., Bi J. FEM-DEM coupling analysis for solid granule medium forming new technology. J. Mater. Process. Technol., 249:108-117, 2017.
+[7]  Du B.,  Zhao C.C.,  Dong G.J., Bi J. FEM-DEM coupling analysis for solid granule medium forming new technology. J. Mater. Process. Technol., 249:108–117, 2017.
 [8]  Chen H.,  Güner A.,  Khalifa N.B.,  Tekkaya A.E. Granular media-based tube press hardening. J. Mater. Process. Technol., 228:145–159, 2016.

@@ Line 494: / Line 494: @@
 [2]  Teng B.G.,  Hu L.,  Liu G.,  Yuan S.J. Wrinkling behavior of hydro bending of carbon steel Al alloy bilayered tubes. Trans. Nonferrous Met. Soc. China, 22:560−565, 2012.
-[3] Yang H.,  Li H.,  Zhang Z.Y.,  Zhan M.,  Liu J.,  Li G. Advances and trends on tube bending forming technologies. J. Plast. Eng., 8:83–85, 2012.
+[3] Xu X.F., Fan Y.B., Wu Y.W., Wu K.W., Xiao J. A novel differential lubrication method for push-bending of L-shaped thin-walled tube with 1D bending radius. International Journal of Material Forming, 14:691–701, 2021.
 [4]  Zeng Y.,  Li Z. Experimental research on the tube push-bending process. Journal of Materials Processing Technology, 122(2-3):237-240, 2002.

@@ Line 458: / Line 458: @@
 In the FEM-DEM coupling program, at the <math display="inline"> i </math>-th increment, the shape of tube is calculated in FEM under the force boundary conditions (contact forces between particles and tube) determined at the (<math display="inline">i-1</math>)th increment in DEM. The contact forces distribution in granular media is calculated in DEM under the displacement boundary conditions (tube shape) determined at the (<math display="inline">i-1</math>)th increment in FEM.
-The PFC<sup>3D</sup>, ABAQUS and MATLAB use different default computational accuracy to complete the simulation or optimization. The default computational accuracy of the solver “quadprog” in MATLAB (tolerance <math display="inline">\le 10^{-6}</math>) is much smaller than that of PFC<sup>3D </sup>(the maximum unbalanced force <math display="inline">\le 10^{-2}</math>). For comparison, the completed condition of PFC<sup>3D</sup> simulation is set to “the maximum unbalanced force <math display="inline">\le 10^{-6}</math>.”
+The PFC<sup>3D</sup>, ABAQUS and MATLAB use different default computational accuracy to complete the simulation or optimization. The default computational accuracy of the solver “quadprog” in MATLAB (tolerance <math display="inline">\le 10^{-6}</math>) is much smaller than that of PFC<sup>3D </sup>(the maximum unbalanced force <math display="inline">\le 10^{-2}</math>). For comparison, the completed condition of PFC<sup>3D</sup> simulation is set to “the maximum unbalanced force <math display="inline">\le 10^{-6}</math>”.
 The time-consuming results of the different FEM-DEM coupling models with no minimum principle, with Eq. (22) and with Eq. (23) under the same simulation condition, are listed in [[#tab-3|Table 3]].

@@ Line 458: / Line 458: @@
 In the FEM-DEM coupling program, at the <math display="inline"> i </math>-th increment, the shape of tube is calculated in FEM under the force boundary conditions (contact forces between particles and tube) determined at the (<math display="inline">i-1</math>)th increment in DEM. The contact forces distribution in granular media is calculated in DEM under the displacement boundary conditions (tube shape) determined at the (<math display="inline">i-1</math>)th increment in FEM.
-The PFC<sup>3D</sup>, ABAQUS and MATLAB use different default computational accuracy to complete the simulation or optimization. The default computational accuracy of the solver “quadprog” in MATLAB (tolerance <math display="inline">\le 10^{-6}</math>) is much smaller than that of PFC<sup>3D </sup>(the maximum unbalanced force <math display="inline">\le 10^{-2}</math>). For comparison, the completed condition of PFC<sup>3D</sup> simulation is set to “the maximum unbalanced force <math display="inline">\le 10^{-6}</math>”.
+The PFC<sup>3D</sup>, ABAQUS and MATLAB use different default computational accuracy to complete the simulation or optimization. The default computational accuracy of the solver “quadprog” in MATLAB (tolerance <math display="inline">\le 10^{-6}</math>) is much smaller than that of PFC<sup>3D </sup>(the maximum unbalanced force <math display="inline">\le 10^{-2}</math>). For comparison, the completed condition of PFC<sup>3D</sup> simulation is set to “the maximum unbalanced force <math display="inline">\le 10^{-6}</math>.”
 The time-consuming results of the different FEM-DEM coupling models with no minimum principle, with Eq. (22) and with Eq. (23) under the same simulation condition, are listed in [[#tab-3|Table 3]].

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 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)
 |}
 ===2.4 Incremental work===

← Older revision		Revision as of 13:43, 29 May 2023
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−	where <math display="inline">K=\left[ \begin{matrix}{\bf K}^{n}&{\bf 0}\\{\bf 0}&{\bf K}^{s}\end{matrix}\right]</math> is the stiffness matrix, which is diagonal and definite positive. it can be noted that, <math display="inline">\Psi \left( {\Delta {\bf U}}^{\ast }\right) -</math><math>\Psi \left( \Delta {\bf U}\right)</math> is always positive. So it can be concluded that, The incremental total energy <math display="inline">\Psi \left( {\Delta {\bf U}}^{\ast }\right)</math> attains its minimum if the kinematically admissible system of incremental displacements is such to satisfy the equilibrium equations (13).	+	where <math display="inline">{\bf K}=\left[ \begin{matrix}{\bf K}^{n}&{\bf 0}\\{\bf 0}&{\bf K}^{s}\end{matrix}\right]</math> is the stiffness matrix, which is diagonal and definite positive. it can be noted that, <math display="inline">\Psi \left( {\Delta {\bf U}}^{\ast }\right) -</math><math>\Psi \left( \Delta {\bf U}\right)</math> is always positive. So it can be concluded that, The incremental total energy <math display="inline">\Psi \left( {\Delta {\bf U}}^{\ast }\right)</math> attains its minimum if the kinematically admissible system of incremental displacements is such to satisfy the equilibrium equations (13).

	==3. Numerical model==		==3. Numerical model==

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 {| style="text-align: center; margin:auto;"
 |-
-| <math>\begin{align}\Psi \left( {\Delta {\bf U}}^{\ast }\right) \mathit{\boldsymbol{-}}\Psi \left( \Delta {\bf U}\right) & =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T}{\Delta \delta }^{\ast }\mathit{\boldsymbol{-}}\Delta {\bf F}^{T}{\Delta \boldsymbol{\delta }  }^{\ast }\right) \mathit{\boldsymbol{+}}\frac{1}{2}{\Delta {\bf F}}^{T}{A}^{\mathit{\boldsymbol{T}}}\left( \Delta U\mathit{\boldsymbol{-}}{\Delta {\bf U}}^{\ast }\right) \\\, & =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T}\mathit{\boldsymbol{-}}\Delta {\bf F}^{T}\right) {\Delta \boldsymbol{\delta }  }^{\ast }\mathit{\boldsymbol{+}}\frac{1}{2}{\Delta {\bf F}}^{T}\left( \Delta \boldsymbol{\delta }  \mathit{\boldsymbol{-}}{\Delta \boldsymbol{\delta }  }^{\ast }\right) \\\, & =\frac{1}{2}{\left( {\Delta \boldsymbol{\delta }  }^{\ast }\mathit{\boldsymbol{-}}\Delta \boldsymbol{\delta }  \right) }^{\mathit{\boldsymbol{T}}}{\bf K}\left( {\Delta \boldsymbol{\delta }  }^{\ast }\mathit{\boldsymbol{-}}\Delta \boldsymbol{\delta } \right) \end{align}</math>
+| <math>\begin{align}\Psi \left( {\Delta {\bf U}}^{\ast }\right) -\Psi \left( \Delta {\bf U}\right) & =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T} \Delta \boldsymbol{\delta}^{\ast } - \Delta {\bf F}^{T}\Delta \boldsymbol{\delta}^{\ast}\right) + \frac{1}{2}{\Delta {\bf F}}^{T}{\bf A}^{\mathit{\boldsymbol{T}}}\left( \Delta {\bf U} - {\Delta {\bf U}}^{\ast }\right) \\\, & =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T} - \Delta {\bf F}^{T}\right) {\Delta \boldsymbol{\delta }  }^{\ast } + \frac{1}{2}{\Delta {\bf F}}^{T}\left( \Delta \boldsymbol{\delta }  - {\Delta \boldsymbol{\delta }  }^{\ast }\right) \\\, & =\frac{1}{2}{\left( {\Delta \boldsymbol{\delta }  }^{\ast }- \Delta \boldsymbol{\delta }  \right) }^{\mathit{\boldsymbol{T}}}{\bf K}\left( {\Delta \boldsymbol{\delta }  }^{\ast } - \Delta \boldsymbol{\delta } \right) \end{align}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)

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 {| style="text-align: center; margin:auto;"
 |-
-| <math>\Psi \left( {\Delta {\bf U}}^{\ast }\right) -</math><math>\Psi \left( \Delta {\bf U}\right) =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T}{\Delta \boldsymbol{\delta }  }^{\ast }-\right. </math><math>\left. \Delta {\bf F}^{T}{\Delta \boldsymbol{\delta }  }^{\ast }\right) +\frac{1}{2}\Delta {T}^{T}\left( \Delta {\bf U}-\right. </math><math>\left. {\Delta {\bf U}}^{\ast }\right)</math>
+| <math>\Psi \left( {\Delta {\bf U}}^{\ast }\right) -</math><math>\Psi \left( \Delta {\bf U}\right) =\frac{1}{2}\left( {\Delta {\bf F}}^{\ast T}{\Delta \boldsymbol{\delta }  }^{\ast }-\right. </math><math>\left. \Delta {\bf F}^{T}{\Delta \boldsymbol{\delta }  }^{\ast }\right) +\frac{1}{2}\Delta {\bf T}^{T}\left( \Delta {\bf U}-\right. </math><math>\left. {\Delta {\bf U}}^{\ast }\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)

@@ Line 263: / Line 263: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\int_{\overline{\mathit{\boldsymbol{\delta }}}}^{\overline{\mathit{\boldsymbol{\delta }}}+\Delta \delta }{{\bf F}\left( \boldsymbol{\delta } \right) }^{T}d\boldsymbol{\delta } =</math><math>{\overline{\bf F}}^{T}\Delta \delta +\frac{1}{2}\Delta {\bf F}^{T}\Delta \boldsymbol{\delta }</math>
+| <math>\int_{\overline{\mathit{\boldsymbol{\delta }}}}^{\overline{\mathit{\boldsymbol{\delta }}}+\Delta \delta }{{\bf F}\left( \boldsymbol{\delta } \right) }^{T}d\boldsymbol{\delta } =</math><math>{\overline{\bf F}}^{T}\Delta \boldsymbol{\delta } +\frac{1}{2}\Delta {\bf F}^{T}\Delta \boldsymbol{\delta }</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (16)

@@ Line 202: / Line 202: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\left\{ \begin{align}\Delta \boldsymbol{\delta} & =\left[ \begin{matrix}-\Delta \alpha \\\Delta \gamma \end{matrix}\right] ={\bf A}^{T}\Delta U\\\, \Delta \boldsymbol{\gamma } & =\tilde{\Upsilon }\Delta \boldsymbol{\alpha } \\\, \left[ \tilde{\Upsilon }\right] & =\left[ {\tilde{\boldsymbol{\gamma }}}_{1}\quad {\tilde{\boldsymbol{\gamma }}}_{2}\, \, \ldots \, \, \ldots \quad {\tilde{\boldsymbol{\gamma }}}_{{M}_{c}+{M}_{b}}\right] \end{align}\right.</math>
+| <math>\left\{ \begin{align}\Delta \boldsymbol{\delta} & =\left[ \begin{matrix}-\Delta \boldsymbol{\alpha} \\\Delta \boldsymbol{\gamma} \end{matrix}\right] ={\bf A}^{T}\Delta U\\\, \Delta \boldsymbol{\gamma } & =\tilde{\Upsilon }\Delta \boldsymbol{\alpha } \\\, \left[ \tilde{\Upsilon }\right] & =\left[ {\tilde{\boldsymbol{\gamma }}}_{1}\quad {\tilde{\boldsymbol{\gamma }}}_{2}\, \, \ldots \, \, \ldots \quad {\tilde{\boldsymbol{\gamma }}}_{{M}_{c}+{M}_{b}}\right] \end{align}\right.</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)