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运用有限元法求解问题的途径是将待求解域分解成有限个无穷小的单元,设待求解函数为X, 则可以得到:
$ X \approx X' = \sum\limits_{i = 1}^n {{N_i}{Y_i}} $
(1) 式中, X′是X的近似值,Ni是每个小单元的个数,Yi是每个小单元的计算结果。将(1)式代入以积分形式定义的标量ω泛函中,可以得到:
$ {\rm{ \mathsf{ δ} }}\omega = \frac{{{\rm{ \mathsf{ δ} }}\omega }}{{{\rm{ \mathsf{ δ} }}{a_1}}}{\rm{ \mathsf{ δ} }}{a_1} + \frac{{{\rm{ \mathsf{ δ} }}\omega }}{{{\rm{ \mathsf{ δ} }}{a_2}}}{\rm{ \mathsf{ δ} }}{a_2} + \cdots + \frac{{{\rm{ \mathsf{ δ} }}\omega }}{{{\rm{ \mathsf{ δ} }}{a_n}}}{\rm{ \mathsf{ δ} }}{a_n} = 0 $
(2) 式中,an为ω的元素函数。在此情况下,如果能确定泛函ω的具体形式,则能通过(2)式对规定的函数进行求导,进而对所需的问题进行求解。
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出于对光的波粒二象性的考虑,在本次的建模过程中将更多地考虑光的波动性带来的影响。首先考虑光的电场及磁场分布的偏微分方程, 并通过设置限制条件可得如下形式的偏微分方程:
$ \nabla \times \left( {\nabla {\mu _{\rm{r}}}^{ - 1} \times {E_z}} \right) - \left[ {{\varepsilon _{\rm{r}}} - \mathit{\boldsymbol{j}}\sigma /\left( {\omega {\varepsilon _0}} \right)} \right]\mathit{\boldsymbol{k}}_0^2{E_z} = 0 $
(3) 式中,Ez为电场强度,μr为介质磁导率,εr和ε0分别为介质和真空中的电容率,σ为电荷密度,j, k0均为单位矢量,▽为哈密顿算子。此方程即为此次求解过程中需要使用到的散射边界条件,反映了电磁波在某一介质内的强度分布情况。使用有限元方法求解问题的过程如图 1所示。
考虑到模型的复杂程度,在此次的计算中采用三角形剖分的方式分解求解域,假设三角形的3个顶点在平面坐标系中的坐标位置分别为A(x1, y1), B(x2, y2), C(x3, y3),可得公式:
$ U\left( {{x_1}, {y_1}} \right) = a{x_1} + b{y_1} + c = {\mu _1} $
(4) $ U\left( {{x_2}, {y_2}} \right) = a{x_2} + b{y_2} + c = {\mu _2} $
(5) $ U\left( {{x_3}, {y_3}} \right) = a{x_3} + b{y_3} + c = {\mu _3} $
(6) 式中,a, b, c为线性系数; μ1, μ2, μ3为3个顶点的线性函数。设置插值算子ξi, ηi, ωi如下:
$ {\xi _1} = {x_2} - {x_3}, {\xi _2} = {x_3} - {x_1}, {\xi _3} = {x_1} - {x_2} $
(7) $ {\eta _1} = {y_2} - {y_3}, {\eta _2} = {y_3} - {y_1}, {\eta _3} = {y_1} - {y_2} $
(8) $ \begin{array}{l} {\omega _1} = {x_2}{y_3} - {x_3}{y_2}, {\omega _2} = {x_3}{y_1} - {x_1}{y_3}, \\ \;\;\;\;\;\;\;\;\;\;\;{\omega _3} = {x_1}{y_2} - {x_2}{y_1} \end{array} $
(9) 将球体和立方体三角剖分以后的结果如图 2和图 3所示。从剖分的结果看来,由于立方体的形状结构较球体更加复杂,所以在立方体结构复杂的部分需增加更多的剖分单元,以期获得更多更精确的结果。从获得的剖分图也可以看到,在两个微粒杂质边界接触的区域增加了剖分的力度,这部分区域也是本次计算过程中的重点区域。最后可以得到差值方程如下:
$ U\left( {x, y} \right) = \frac{1}{D}\left( {x\sum\limits_{i = 1}^3 {{\eta _i}{\mu _i}} - y\sum\limits_{i = 1}^3 {{\xi _i}{\mu _i}} + \sum\limits_{i = 1}^3 {{\omega _i}{\mu _i}} } \right) $
(10) 式中, D为差值系数。
空气中多杂质颗粒对光传输的影响特性研究
Influence of multiple impurity particles in air on light propagation
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摘要: 为了研究空气中杂质颗粒对光传输的影响,采用有限元法分析了有多个杂质微粒同时存在的情况下,颗粒物连接处的光强分布情况,取得了不同颗粒物之间的强度差异数据,并给出了一种解决复杂多杂质微粒存在情况下的通用解决方案。结果表明,对于球体颗粒物杂质,当同时有两个杂质微粒存在时,穿过杂质的微粒呈现出从低到高的趋势,在杂质微粒互相接触的区域光强达到最大值;对于立方体杂质微粒,光强分布呈现出较强的波动特性,且光强强度比球体杂质微粒的光强强度在数量级上多了100倍。该研究模型可移植性强,能推广应用到多个领域,这一结果对后续开展光在气体中传播的理论是有帮助的。Abstract: To study the influence of impurity particles on light propagation in air, a finite element method was used to analyze the distribution of light intensity at the boundary of impurity particles at the presence of multiple impurity particles. The intensity difference data between different particles was acquired. A general solution was provided to solve the problem of complex impurity particles. After theoretical analysis and experimental verification, the results show that, for sphere particles, while there are two impurity particles at the same time, the particles passing through the impurity present a trend from low to high, and the intensity reaches the maximum in the area where the impurity particles are in contact with each other. For cube impurity particles, the intensity distribution exhibits a strong fluctuation property, and the intensity of light is 100 times more than that of sphere impurity particles. The research model has strong portability and can be widely applied to lots of fields. This result is helpful for the subsequent development of light propagation in air.
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Key words:
- optical communication /
- light scattering /
- finite element method /
- particulate matter
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