磁头/磁盘界面修正Reynolds方程的新模型及其数值模拟分析
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摘要
近几年,硬盘的数据存储密度越来越大,其主要原因之一就是因为磁头和磁盘之间的间隙(飞行高度)大幅度减小以及复杂滑块设计技术的进步。当磁头与磁盘之间的间隙接近或小于分子平均自由程时,气体的动力学性能就不能直接用连续润滑理论(即连续Reynolds方程)进行描述,而要考虑气体稀薄效应的影响。
基于广泛应用于计算硬盘磁头/磁盘界面压力分布的修正Reynolds方程的FK模型,本文提出了一种考虑气体稀薄效应修正Reynolds方程的新模型——LFR (Linearized Flow Rate)模型,这种模型是通过一个分段的线性函数拟合FK模型的流率函数而实现的。相对于FK模型,LFR模型的数学表达式比较简单,而且计算出来的压力曲线几乎与FK模型得到的结果重合；更重要的是,LFR模型的计算速度比FK模型的快很多。
由于Reynolds方程的非线性和滑块形状的复杂性,必须采用数值方法求解Reynolds方程。因此,本文分别详细阐述了有限差分法(FDM)、最小二乘有限差分法(LSFD)、有限体积法(FVM)、无网格局部Peterov-Galerkin法(MLPG)和基于核重构思想的最小二乘配点法的基本理论,并且在不同离散节点的求解域中把这些方法应用到了修正Reynolds方程。计算结果显示：在相同节点数的条件下,MLPG方法的计算速度最慢,但是,这种方法能在较少离散节点数的情况下,得到较高精度的计算结果；LSFD方法的计算速度较快,计算精度也较高,但是,这种方法需要足够多的离散节点,否则,计算不能进行；FDM方法的计算速度最快,但是,这种方法的计算精度是最差的；基于核重构思想的最小二乘配点法能获得很好的计算精度,然而,这种方法的计算速度稍差些,并且在进行计算的时候,要注意精化参数αx的选择；FVM方法的计算速度很快,并且计算的精度也很高,是一种综合性能最佳的数值方法。
为了进一步检验LFR模型的有效性,分别采用FK模型和LFR模型,对平面倾斜滑块的压力分布进行数值模拟分析。其结果表明,在二维的情况下,LFR模型的计算结果与FK模型相差很小,但其计算速度明显快于FK模型。然后,基于LFR模型,对两轨滑块、三垫滑块以及两个负压滑块的压力分布进行了数值模拟分析,其结果显示：对于同样形状的滑块,在滑块尺寸和飞行参数发生变化时,所得到的压力分布图的形状具有很大差异；对于求解形状复杂的滑块的压力分布,在压力梯度很大(或者滑块形状复杂)区域,必须要采用局部网格加密技术,否则就会导致计算时间过长或计算过程不能进行；随着飞行高度的降低,计算需要越来越多的离散点,并且计算收敛速度越来越慢,计算时间变得更长。
The hard disk drive (HDD) has achieved extraordinary growth in areal density of data storage in the past few years. One of the key factors for this explosive growth in storage capacity is the substantially reduced and tiny spacing (flying height) between the read/write head and recording media and the resulting more complex slider design. When the spacing between the flying head and the rotating disk approximates the molecular mean-free path or less, the gas dynamics cannot be described directly from the continuum transport theory, i.e., continuum Reynolds equation, and the gaseous rarefaction effects must be taken into account in this case.
Based the FK models of corrected Reynolds that has been used widely in calculating pressure profiles of the interface of head/disk in HDD, a new model of corrected Reynolds considered the gaseous rarefaction effects is proposed, i.e., the model of linearized flow rate (LFR). The implementation of this new model is arrived by using a piecewise linear function to fit the flow rate function of the FK model, and the mathematical formation of the FLR model is simpler more than that of FK model. Moreover, the pressure distribution and the computational speed of FLR model are numerically compared with those of FK model, the results show that the pressure profiles of these two models are almost the same but the computational speed of FLR model is much faster than that of FK model.
Due to the non-linearity of Reynolds equation and the complex shape of head slider, numerical methods are used to solve the equations of these two models. Therefore, finite difference method (FDM), least square finite difference (LSFD) method, finite volume method (FVM), meshless local Peterov-Galerkin method (MLPG), and least-square collocation meshless method based reproducing kernel particle idea, are stated detailedly and employed to solve the FLR model of corrected Reynolds equation for different number of discretized nodes, respectively. The resulting pressure distributions and computational speeds are obtained. The pressure profile with a small relative error value can be obtained in the case of the definitive area with a few discretized nodes by using MLPG method, but the computational speed of this method is slow comparing with other several numerical methods. The computational speed and accuracy of LSFD are relatively good in the condition of the definitive area with enough amounts of discretized nodes, unless the computational process is not proceeding. The FDM possesses the fastest computational speed but the accuracy of it is also the worst. The least-square collocation meshless method based reproducing kernel particle idea has a good accuracy with a relatively slow computational speed, noting that the refining parameter should be selected carefully. The FVM is the best numerical method with a fast computational speed and a high accuracy.
To more closely test availability of the LFR model, the pressure distributions of the FK model and the LFR model are numerically obtained for the plane inclined slider in two-dimesional case. The computational time of two models in this case are also compared. The numerical reslults show that the pressure distributions of two models have little difference but the computational time of LFR model is less than that of FK model. Than, based on the LFR model, the two-dimensional pressure distributions of several typical positive pressure slider and two negative pressure slider with complex geometry shapes and great pressure gradients are obtained numerically. Some interesting results are found. For the slider having the same geometry shape, under the changing of its sizes and parameters of flying, it may possess entirely different pressure profiles. To obtain the pressure distributions of slider with complex geometry shape and great pressure gradients, it will result in a long computational time or even the failure of computational proceed, unless techniques of the local refining nodes must be implemented. With the reducing of flying height, more numbers of discretized nodes are used to obtain the pressure profiles of the interface of head/disk.
基于广泛应用于计算硬盘磁头/磁盘界面压力分布的修正Reynolds方程的FK模型,本文提出了一种考虑气体稀薄效应修正Reynolds方程的新模型——LFR (Linearized Flow Rate)模型,这种模型是通过一个分段的线性函数拟合FK模型的流率函数而实现的。相对于FK模型,LFR模型的数学表达式比较简单,而且计算出来的压力曲线几乎与FK模型得到的结果重合；更重要的是,LFR模型的计算速度比FK模型的快很多。
由于Reynolds方程的非线性和滑块形状的复杂性,必须采用数值方法求解Reynolds方程。因此,本文分别详细阐述了有限差分法(FDM)、最小二乘有限差分法(LSFD)、有限体积法(FVM)、无网格局部Peterov-Galerkin法(MLPG)和基于核重构思想的最小二乘配点法的基本理论,并且在不同离散节点的求解域中把这些方法应用到了修正Reynolds方程。计算结果显示：在相同节点数的条件下,MLPG方法的计算速度最慢,但是,这种方法能在较少离散节点数的情况下,得到较高精度的计算结果；LSFD方法的计算速度较快,计算精度也较高,但是,这种方法需要足够多的离散节点,否则,计算不能进行；FDM方法的计算速度最快,但是,这种方法的计算精度是最差的；基于核重构思想的最小二乘配点法能获得很好的计算精度,然而,这种方法的计算速度稍差些,并且在进行计算的时候,要注意精化参数αx的选择；FVM方法的计算速度很快,并且计算的精度也很高,是一种综合性能最佳的数值方法。
为了进一步检验LFR模型的有效性,分别采用FK模型和LFR模型,对平面倾斜滑块的压力分布进行数值模拟分析。其结果表明,在二维的情况下,LFR模型的计算结果与FK模型相差很小,但其计算速度明显快于FK模型。然后,基于LFR模型,对两轨滑块、三垫滑块以及两个负压滑块的压力分布进行了数值模拟分析,其结果显示：对于同样形状的滑块,在滑块尺寸和飞行参数发生变化时,所得到的压力分布图的形状具有很大差异；对于求解形状复杂的滑块的压力分布,在压力梯度很大(或者滑块形状复杂)区域,必须要采用局部网格加密技术,否则就会导致计算时间过长或计算过程不能进行；随着飞行高度的降低,计算需要越来越多的离散点,并且计算收敛速度越来越慢,计算时间变得更长。
The hard disk drive (HDD) has achieved extraordinary growth in areal density of data storage in the past few years. One of the key factors for this explosive growth in storage capacity is the substantially reduced and tiny spacing (flying height) between the read/write head and recording media and the resulting more complex slider design. When the spacing between the flying head and the rotating disk approximates the molecular mean-free path or less, the gas dynamics cannot be described directly from the continuum transport theory, i.e., continuum Reynolds equation, and the gaseous rarefaction effects must be taken into account in this case.
Based the FK models of corrected Reynolds that has been used widely in calculating pressure profiles of the interface of head/disk in HDD, a new model of corrected Reynolds considered the gaseous rarefaction effects is proposed, i.e., the model of linearized flow rate (LFR). The implementation of this new model is arrived by using a piecewise linear function to fit the flow rate function of the FK model, and the mathematical formation of the FLR model is simpler more than that of FK model. Moreover, the pressure distribution and the computational speed of FLR model are numerically compared with those of FK model, the results show that the pressure profiles of these two models are almost the same but the computational speed of FLR model is much faster than that of FK model.
Due to the non-linearity of Reynolds equation and the complex shape of head slider, numerical methods are used to solve the equations of these two models. Therefore, finite difference method (FDM), least square finite difference (LSFD) method, finite volume method (FVM), meshless local Peterov-Galerkin method (MLPG), and least-square collocation meshless method based reproducing kernel particle idea, are stated detailedly and employed to solve the FLR model of corrected Reynolds equation for different number of discretized nodes, respectively. The resulting pressure distributions and computational speeds are obtained. The pressure profile with a small relative error value can be obtained in the case of the definitive area with a few discretized nodes by using MLPG method, but the computational speed of this method is slow comparing with other several numerical methods. The computational speed and accuracy of LSFD are relatively good in the condition of the definitive area with enough amounts of discretized nodes, unless the computational process is not proceeding. The FDM possesses the fastest computational speed but the accuracy of it is also the worst. The least-square collocation meshless method based reproducing kernel particle idea has a good accuracy with a relatively slow computational speed, noting that the refining parameter should be selected carefully. The FVM is the best numerical method with a fast computational speed and a high accuracy.
To more closely test availability of the LFR model, the pressure distributions of the FK model and the LFR model are numerically obtained for the plane inclined slider in two-dimesional case. The computational time of two models in this case are also compared. The numerical reslults show that the pressure distributions of two models have little difference but the computational time of LFR model is less than that of FK model. Than, based on the LFR model, the two-dimensional pressure distributions of several typical positive pressure slider and two negative pressure slider with complex geometry shapes and great pressure gradients are obtained numerically. Some interesting results are found. For the slider having the same geometry shape, under the changing of its sizes and parameters of flying, it may possess entirely different pressure profiles. To obtain the pressure distributions of slider with complex geometry shape and great pressure gradients, it will result in a long computational time or even the failure of computational proceed, unless techniques of the local refining nodes must be implemented. With the reducing of flying height, more numbers of discretized nodes are used to obtain the pressure profiles of the interface of head/disk.
引文
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