非线性有限体积法大旋转问题研究
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (6): 1012-1018  DOI: 10.11990/jheu.201703058 0

### 引用本文

LIU Qi, MING Pingjian, ZHANG Wenping. Research on the nonlinear finite volume numerical method for the large rotating of disk[J]. Journal of Harbin Engineering University, 2018, 39(6), 1012-1018. DOI: 10.11990/jheu.201703058.

### 文章历史

Research on the nonlinear finite volume numerical method for the large rotating of disk
LIU Qi, MING Pingjian, ZHANG Wenping
College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: To study the application of the finite volume method for nonlinear large rotation, a large-strain finite volume method (LS-FVM) using an improved distance-weighted interpolation algorithm was proposed. For the problem of large disk rotation, the effects of the traditional distance-weighted interpolation algorithm, the least square method, and the improved distance-weighted interpolation algorithm on grid movement were compared. Moreover, the effects of the gradient algorithm with different precisions and the load increment on the stress distribution in the large rotation disk were analyzed. Results show that a large numerical error may be introduced at the grid border when the traditional distance-weighted interpolation algorithm is used for moving grids. By introducing the effect of the boundary unit midpoint, the improved distance-weighted interpolation algorithm could effectively increase the numerical precision of moving boundary grids. Large rotation has high precision requirements for the gradient algorithm, and the application of the Gauss gradient algorithm may lead to untrue stress. Therefore, the least square gradient algorithm with enhanced precision was selected to calculate the central gradient of a grid unit. The LS-FVM under the large increment load presented excellent numerical precision.
Key words: large rotating    finite volume method    updated lagrangian formulation    geometrically nonlinear    mesh movement    large strain theory

1 非线性问题控制方程 1.1 线动量守恒方程

 $\oint\limits_{{A_u}} {{\mathit{\boldsymbol{n}}_u} \cdot \left( {\delta {\mathit{\boldsymbol{S}}_{\rm{u}}} + {\mathit{\boldsymbol{S}}_{\rm{u}}} \cdot \delta \mathit{\boldsymbol{F}}_{\rm{u}}^{\rm{T}} + \delta {\mathit{\boldsymbol{S}}_{\rm{u}}} \cdot \delta \mathit{\boldsymbol{F}}_{\rm{u}}^{\rm{T}}} \right){\rm{d}}{A_u}} = 0$ (1)

1.2 本够关系

 $\delta S = 2\mu \left( {\delta E} \right) + \lambda {\rm{tr}}\left( {\delta E} \right)I$ (2)

 $\begin{array}{*{20}{c}} {\delta \mathit{\boldsymbol{E}} = \frac{1}{2}\left[ {\nabla \delta \mathit{\boldsymbol{u}} + {{\left( {\nabla \delta \mathit{\boldsymbol{u}}} \right)}^{\rm{T}}} + \nabla \delta \mathit{\boldsymbol{u}} \cdot {{\left( {\nabla \mathit{\boldsymbol{u}}} \right)}^{\rm{T}}} + } \right.}\\ {\left. {\nabla \mathit{\boldsymbol{u}} \cdot {{\left( {\nabla \delta \mathit{\boldsymbol{u}}} \right)}^{\rm{T}}} + \nabla \delta \mathit{\boldsymbol{u}} \cdot {{\left( {\nabla \delta \mathit{\boldsymbol{u}}} \right)}^{\rm{T}}}} \right]} \end{array}$ (3)

 $\begin{array}{*{20}{c}} {\delta \mathit{\boldsymbol{S}} = \mu \left[ {\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u + }}{{\left( {\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}}} \right)}^{\rm{T}}}} \right] + \lambda {\rm{tr}}\left( {\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}}} \right)\mathit{\boldsymbol{I}} + }\\ {\mu \nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}} \cdot {{\left( {\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}}} \right)}^{\rm{T}}} + \frac{1}{2}\left( {\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}}:\nabla {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{u}}} \right)\mathit{\boldsymbol{I}}} \end{array}$ (4)

 $\mu = \frac{E}{{2\left( {1 + \nu } \right)}}$ (5)
 $\lambda = \left\{ {\begin{array}{*{20}{l}} {\frac{{\mathit{\boldsymbol{\nu }}E}}{{\left( {1 + \mathit{\boldsymbol{\nu }}} \right)\left( {1 - \mathit{\boldsymbol{\nu }}} \right)}},\;\;\;\;\;{\rm{平面应力}}}\\ {\frac{{\mathit{\boldsymbol{\nu }}E}}{{\left( {1 + \mathit{\boldsymbol{\nu }}} \right)\left( {1 - 2\mathit{\boldsymbol{\nu }}} \right)}},\;\;\;{\rm{平面应变和三维问题}}} \end{array}} \right.$ (6)
1.3 应力更新

 ${\mathit{\boldsymbol{S}}_{\rm{u}}} = {J^{ - 1}}\mathit{\boldsymbol{F}} \cdot {\mathit{\boldsymbol{S}}_{\rm{u}}} \cdot {\mathit{\boldsymbol{F}}^{\rm{T}}}$ (7)

 $\mathit{\boldsymbol{F}} = \mathit{\boldsymbol{I}} + {\left( {\nabla \mathit{\boldsymbol{u}}} \right)^{\rm{T}}}$ (8)
1.4 网格移动 1.4.1 距离加权插值法

 ${f_V} = \frac{{\sum\limits_{i = 1}^P {{\omega _{{P_i}}}{f_{{P_i}}}} }}{{\sum\limits_{i = 1}^P {{\omega _{{P_i}}}} }}$ (9)

 ${\omega _{{P_i}}} = \frac{1}{{\left| {{\mathit{\boldsymbol{r}}_V} - {\mathit{\boldsymbol{r}}_{{P_i}}}} \right|}}$ (10)

1.4.2 改进的距离加权插值法

1.4.3 最小二乘插值法

 $f\left( {x,y,z} \right) = ax + by + cz + d$ (11)

1.5 边界条件

 $\delta {u_i} = \delta {u_{iB}}$ (12)
2 非线性问题求解方法

 Download: 图 3 计算大应变程序流程图 Fig. 3 Computation procedure for large strain problems

 ${a_P}{\phi _P} = \sum\limits_{nf} {{a_N}{\phi _N} + \mathit{\boldsymbol{R}}}$ (13)

 $\left\{ \begin{array}{l} \phi = \delta \mathit{\boldsymbol{u}}\\ {a_P} = \frac{{{{\left( {{\rho _u}{V_u}} \right)}_p}}}{{{{\left( \mathit{\Delta t} \right)}^2}}} + \sum\limits_{f = 1}^{{n_i}} {{a_N}} \\ {a_N} = \left( {2{\mu _f} + {\lambda _f}} \right) \cdot \left( {\frac{{{\mathit{\boldsymbol{A}}_{fu}} \cdot {\mathit{\boldsymbol{A}}_{fu}}}}{{{\mathit{\boldsymbol{A}}_{fu}} \cdot {\mathit{\boldsymbol{d}}_f}}}} \right)\\ {\mathit{\boldsymbol{R}} = \sum\limits_{f = 1}^{{n_i}} {\left( {2{\mu _f} + {\lambda _f}} \right)\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right) \cdot \left( {{\mathit{\boldsymbol{A}}_{fu}} - {\mathit{\boldsymbol{d}}_f}\frac{{{\mathit{\boldsymbol{A}}_{fu}} \cdot {\mathit{\boldsymbol{A}}_{fu}}}}{{{\mathit{\boldsymbol{A}}_{fu}} \cdot {\mathit{\boldsymbol{d}}_f}}}} \right)} + }\\ \;\;\;\;\sum\limits_{f = 1}^{{n_i}} {{\mathit{\boldsymbol{A}}_{fu}} \cdot \left[ {{\mu _f}\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)_f^{\rm{T}} + {\lambda _f}{\rm{tr}}{{\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)}_f}\mathit{\boldsymbol{I}} - } \right.} \\ {\;\;\;\;{\left( {\mu + \lambda } \right)_f}\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)_f^{\rm{T}} + \mu {\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)_f} \cdot \left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)_f^{\rm{T}} + }\\ {\;\;\;\;0.5 \cdot {\lambda _f}{\left( {\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}:\nabla \delta {\mathit{\boldsymbol{u}}^{t - 1}}} \right)_f}\mathit{\boldsymbol{I }} + }\\ \;\;\;\;\left. {\left( {\mathit{\boldsymbol{S}}_{fu}^{t - 1} + \delta \mathit{\boldsymbol{S}}_{fu}^{t - 1}} \right) \cdot {{\left( {\delta \mathit{\boldsymbol{F}}_{fu}^T} \right)}^{t - 1}}} \right] \end{array} \right.$ (14)
3 圆盘大旋转算例

 Download: 图 4 大旋转圆盘示意图以及网格划分 Fig. 4 Sketch up of large rotating disk and mesh

 $\left\{ \begin{array}{l} \delta {u_x} = \cos \theta \cdot x + \sin \theta \cdot y - x\\ \delta {u_y} = \sin \theta \cdot x - \cos \theta \cdot y - y \end{array} \right.$ (15)

3.1 不同网格移动插值方法对圆盘旋转的影响

 Download: 图 5 应用最小二乘法插值网格移动 Fig. 5 Mesh movement using least squares interpolation
 Download: 图 6 采用最小二乘法圆盘应力分布 Fig. 6 True stress distributions based on least square method

 Download: 图 7 采用距离加权法圆盘应力分布 Fig. 7 True stress distributions based on inverse distance interpolation method

3.2 应用Gauss法求解单元中心梯度对大旋转问题的影响

 Download: 图 10 应用高斯法计算单元中心梯度圆盘应力分布 Fig. 10 Results of the rotating disk using Gauss method to calculate the cell central gradient
3.3 不同增量载荷对圆盘旋转的影响

 Download: 图 11 不同增量载荷应力分布对比 Fig. 11 Comparisons of the true stress under different increment sizes
3.4 大应变理论以及小应变理论在求解大旋转问题的对比

 Download: 图 12 小应变理论求解大旋转问题 Fig. 12 Results of the rotating disk using small theory
4 结论

1) 采用传统距离加权插值法的网格移动质量最差，当构形移动较大时，优先采用最小二乘法或改进的距离加权法。

2) 大旋转问题对单元中心梯度的计算精度要求较高，采用Gauss梯度算法会产生不合理的应力分布，应选择具有更高精度的最小二乘梯度算法。

3) 应用小应变理论求解大旋转问题时，在很小的增量载荷下，仍会产生错误的应力分布，而LS-FVM在较大的载荷增量步下仍可以得到较好的收敛解，但载荷不宜过大，过大的增量载荷会导致病态矩阵的出现，导致发散。

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