CN106168942A

CN106168942A - A kind of fluctuation types dynamic data reconstructing method based on singular boundary method

Info

Publication number: CN106168942A
Application number: CN201610547589.6A
Authority: CN
Inventors: 陈文�; 李珺璞
Original assignee: Hohai University HHU
Current assignee: Hohai University HHU
Priority date: 2016-07-12
Filing date: 2016-07-12
Publication date: 2016-11-30
Anticipated expiration: 2036-07-12
Also published as: CN106168942B

Abstract

The invention discloses a wave type dynamic data reconstruction method based on the singular boundary method, which directly uses the time basic solution of the wave equation as the kernel function, replaces the source point singular term with the source point intensity factor, has no numerical integral, singular integral, and no network Grid division, directly using the basic time solution of the wave equation as the kernel function, without complex mathematical transformation, its simple and efficient characteristics meet the requirements of fast, stable and accurate data reconstruction technology for dynamic data reconstruction of wave types; experimental comparison shows that, Applying the technology proposed by the present invention to deal with the problem of dynamic data reconstruction of fluctuation type, under the condition of obtaining similar precision, the general time-consuming only needs about 10% of the traditional linear boundary element algorithm, which has high precision, fast calculation, simple mathematics, and the program Simple and convenient, it can be applied to engineering technology fields such as sound wave noise reduction, wave energy dissipation, earthquake prediction and other wave data reconstruction, image processing and so on.

Description

A Dynamic Data Reconstruction Method Based on Singular Boundary Method

技术领域technical field

本发明涉及一种波动类型动态数据重构方法，具体涉及一种奇异边界法的波动类型动态数据重构方法。The invention relates to a method for reconstructing dynamic data of a fluctuation type, in particular to a method for reconstructing dynamic data of a fluctuation type based on a singular boundary method.

背景技术Background technique

波动现象作为自然界最常见的自然现象之一，在工程应用领域具有广泛应用与影响，如声波减噪、海浪消能、地震波预测等科研工程领域，均需处理大量波动类型动态数据。由于波动现象基本控制方程的独特性，如波传播的滞后性、三维无后效性等，使波动方程基本解具有与扩散方程、拉普拉斯方程基本解有明显的不同形式，这使得波动类型动态数据的重构极为困难。As one of the most common natural phenomena in nature, fluctuation phenomenon has a wide range of applications and influences in engineering applications. For example, scientific research and engineering fields such as acoustic noise reduction, ocean wave energy dissipation, and seismic wave prediction need to process a large amount of dynamic data of fluctuation types. Due to the uniqueness of the basic governing equations of wave phenomena, such as the hysteresis of wave propagation, three-dimensional no aftereffect, etc., the basic solutions of the wave equation have obviously different forms from the basic solutions of the diffusion equation and the Laplace equation, which makes the wave Refactoring of type dynamic data is extremely difficult.

传统上处理波动类型动态数据重构，一般有有限元方法、边界元方法、有限差分法等。但作为传统区域型网格方法，有限元在处理如声波散射，声波辐射等一系列无限域问题的数据重构时，往往会出现网格划分困难、计算速度慢、计算荷载大等难以克服的问题。而边界元方法作为边界型网格方法，虽然在一定程度上克服了有限元方法需要划分区域网格的缺点，提高了计算效率，但由于在计算过程中需要处理大量奇异与近奇异积分，严重影响了数据的重构速度。基本解方法作为一种无网格方法一定程度上克服了传统网格类算法需要划分网格的缺陷，但由于在基本解方法中为克服奇异性引入了虚假边界，使得该方法在处理一些复杂工况条件下的数据重构问题时，常出现数值不稳定的现象(见文献1.Young DL,Gu M H,Fan C M.The time-marching method of fundamental solutions for waveequations[J].Engineering Analysis with Boundary Elements,2009,33(12):1411-1425.)。Traditionally, there are finite element methods, boundary element methods, and finite difference methods to deal with dynamic data reconstruction of fluctuation types. However, as a traditional regional grid method, when dealing with data reconstruction of a series of infinite domain problems such as acoustic wave scattering and acoustic wave radiation, the finite element often has difficulties in grid division, slow calculation speed, and large calculation load. question. As a boundary-type grid method, the boundary element method overcomes to a certain extent the disadvantage that the finite element method needs to divide the regional grid and improves the calculation efficiency. Affected the reconstruction speed of the data. As a gridless method, the basic solution method overcomes the defect that the traditional grid algorithm needs to divide the grid to a certain extent. Numerical instability often occurs when data is reconstructed under working conditions (see literature 1. Young DL, Gu M H, Fan C M. The time-marching method of fundamental solutions for waveequations[J]. Engineering Analysis with Boundary Elements, 2009, 33(12):1411-1425.).

发明内容Contents of the invention

发明目的：本发明的目的在于针对现有技术的波动类型动态数据重构技术过程复杂、计算荷载大计算时间长的缺陷，提供一种简单、高效、无积分、无网格的基于奇异边界法的波动类型动态数据重构方法，从而达到节约重构时间，提高重构效率的目的。Purpose of the invention: The purpose of the present invention is to provide a simple, efficient, integral-free and grid-free method based on the singular boundary method, aiming at the defects of complex process of dynamic data reconstruction technology of fluctuation type and long calculation time in the prior art. The fluctuation type dynamic data reconstruction method, so as to save reconstruction time and improve reconstruction efficiency.

技术方案：本发明提供了一种基于奇异边界法的波动类型动态数据重构方法，包括以下步骤：Technical solution: The present invention provides a method for reconstructing dynamic data of fluctuation type based on singular boundary method, comprising the following steps:

(1)在所考察物质的内部和边界配置若干测试点，获得这些测试点的初始数据和边界数据；(1) Configure several test points in the interior and boundary of the substance under investigation, and obtain the initial data and boundary data of these test points;

(2)直接采用三维波方程的时间基本解作为核函数，建立波传播问题相应的插值矩阵；(2) Directly use the time basic solution of the three-dimensional wave equation as the kernel function to establish the corresponding interpolation matrix for the wave propagation problem;

(3)利用经验公式计算处理波动数据重构的源点强度因子；(3) Use the empirical formula to calculate the source point strength factor for the reconstruction of fluctuating data;

(4)将源点强度因子代入插值矩阵，计算插值矩阵的未知系数；(4) Substitute the source point intensity factor into the interpolation matrix, and calculate the unknown coefficient of the interpolation matrix;

(5)计算任意时刻任意内点的波动数据值。(5) Calculate the fluctuation data value of any interior point at any time.

进一步，步骤(1)中，初始时刻在所考察物质的内部配置N₁个测试点，获得初始时刻这些测试点的波动数据值u_i,i＝1,...,N₁；在所考察物质的表面配置N₂个测试点，随着波动的进行，获得不同时刻这些点的波动数据值u_i,i＝N₁+1,...,N₁+N₂。Further, in step (1), N ₁ test points are arranged inside the substance under investigation at the initial moment, and the fluctuation data values u _i , i=1,...,N ₁ of these test points at the initial moment are obtained; N ₂ test points are arranged on the surface of the substance, and as the fluctuation proceeds, the fluctuation data values u _i of these points at different times are obtained, i=N ₁ +1,...,N ₁ +N ₂ .

进一步，步骤(2)中采用的波动问题控制方程为：Further, the governing equation of the wave problem adopted in step (2) is:

$u u = = \{\begin{matrix} Δ Δ u u - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} u u}{\partial \partial {t t}^{22}} = = 00,, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ u u {| |}_{Γ Γ} = = \overset{&OverBar; &OverBar;}{u u} \\ u u {| |}_{t t = = 00} = = {u u}_{00},, \frac{\partial \partial u u}{\partial \partial t t} {| |}_{t t = = 00} = = {&upsi; &upsi;}_{11} \end{matrix} - - - - - - ((11))$

其中，Ω表示所考察物质的区域，(x,y,z)为空间坐标，t为(x,y,z)所对应的时刻，c代表波速，u代表势函数，代表边界条件，u₀和υ₁表示初值条件，Δ表示Laplace算子，在声波传播中表示声压值；Among them, Ω represents the area of the substance under investigation, (x, y, z) is the space coordinate, t is the moment corresponding to (x, y, z), c represents the wave velocity, u represents the potential function, Represents the boundary conditions, u ₀ and υ ₁ represent the initial value conditions, Δ represents the Laplace operator, and represents the sound pressure value in the sound wave propagation;

三维波传播问题对应基本解为：The corresponding basic solution of the three-dimensional wave propagation problem is:

$G G ((r r,, t t)) = = \frac{11}{44 π π r r} δ δ [[((t t - - τ τ)) - - r r / / c c]] - - - - - - ((22))$

其中δ表示狄拉克函数，c表示波速，t和τ表示配点和源点的时刻，r表示距离；由于波的传播需要时间，基本解G仅仅用延迟时刻的相应声压值的相关组合便可表示当前时刻的声压值，延迟时刻取决于源点与待测点之间的距离和波的传播速度；因此，对于特定测试点，只要求得关于源点{s_j}的延迟时刻的未知系数{α_j}，则测试点所求时刻的声压值即可求出；Among them, δ represents the Dirac function, c represents the wave velocity, t and τ represent the time of the collocation point and the source point, and r represents the distance; since the propagation of the wave requires time, the basic solution G can only be obtained by using the relevant combination of the corresponding sound pressure values at the delay time Indicates the sound pressure value at the current moment, and the delay time depends on the distance between the source point and the point to be measured and the propagation speed of the wave; therefore, for a specific test point, only the unknown of the delay time of the source point {s _j } is required coefficient {α _j }, then the sound pressure value at the moment of the test point can be obtained;

原方程u可以被看作一个初边值问题，应用叠加原理，将其拆分为u₁和u₂，如式(3)和式(4)所示，即其中u₁可以被看作一个边值问题，u₂可以被看作一个初值问题：The original equation u can be regarded as an initial boundary value problem. Applying the principle of superposition, it can be split into u ₁ and u ₂ , as shown in equations (3) and (4), that is Among them, u ₁ can be regarded as a boundary value problem, and u ₂ can be regarded as an initial value problem:

${u u}_{11}^{* *} = = \{\begin{matrix} {Δu Δu}_{11}^{* *} - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} {u u}_{11}^{* *}}{\partial \partial {t t}^{22}} = = 00,, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ {u u}_{11}^{* *} {| |}_{Γ Γ} = = \overset{&OverBar; &OverBar;}{u u} - - {\overset{&OverBar; &OverBar;}{u u}}_{22}^{* *} = = {\overset{&OverBar; &OverBar;}{u u}}_{11}^{* *} \\ {u u}_{11}^{* *} {| |}_{t t = = 00} = = 00,, \frac{\partial \partial {u u}_{11}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = 00 \end{matrix} - - - - - - ((33))$

${u u}_{22}^{* *} = = \{\begin{matrix} {Δu Δu}_{22}^{* *} - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} {u u}_{22}^{* *}}{\partial \partial {t t}^{22}} = = 00,, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ {u u}_{22}^{* *} {| |}_{Γ Γ} = = {\overset{&OverBar; &OverBar;}{u u}}_{22}^{* *} \\ {u u}_{22}^{* *} {| |}_{t t = = 00} = = 00,, \frac{\partial \partial {u u}_{22}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = 00,, ((x x,, y the y,, z z)) &NotElement; &NotElement; Ω Ω \\ {u u}_{22}^{* *} {| |}_{t t = = 00} = = {u u}_{00},, \frac{\partial \partial {u u}_{22}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = {&upsi; &upsi;}_{11},, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω \end{matrix} - - - - - - ((44))$

式(4)用三维泊松方程直接求出：Formula (4) Use the three-dimensional Poisson equation to find directly:

${u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((55))$

其中表示半径为ct以M为圆心的球面；在式(4)中，仅需要求出测试点M影响区域中的声压值，所以在影响域中布置初始数据采集点；in Indicates a spherical surface with a radius of ct and M as the center; in formula (4), it is only necessary to obtain the influence area of the test point M The sound pressure value in , so in the domain of influence Arrange the initial data collection points in ;

其后，考虑方程(3)由惠更斯原理“波传播中所到达的每一点都可以作为一个新的次波源，所有这些次波所形成的包络面构成下一时刻的新波面”，假定在边界存在一系列随时间变化的点波源这些点波源所发出的次波的总和形成计算域Ω中其后任意时刻的声压值其中表示声压强度，x_m表示边界点，则计算域Ω中的声压可以被表示为Thereafter, considering equation (3) According to Huygens' principle that "every point reached in the wave propagation can be used as a new secondary wave source, and the envelope surface formed by all these secondary waves constitutes the new wave surface at the next moment", it is assumed that there is a series of changing point source The sum of the secondary waves emitted by these point wave sources forms the sound pressure value at any subsequent time in the calculation domain Ω in represents the sound pressure intensity, x _m represents the boundary point, then the sound pressure in the domain Ω is calculated can be expressed as

${u u}_{11}^{* *} (({x x}_{i i})) = = \underset{Ω Ω}{&Integral; &Integral; &Integral; &Integral; &Integral; &Integral;} \frac{α α ((r r))}{44 π π r r} δ δ ((t t - - \frac{r r}{c c})) d d v v,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((66))$

在边界Γ上布置数据采集点，以Δt为时间间隔，v表示空间积分变量，t_n∈((n-1)Δt,nΔt)，t表示计算时刻，t的下标n表示时间层数，声压可以被基本解G的一系列线性组合来近似：Arrange data collection points on the boundary Γ, with Δt as the time interval, v represents the space integration variable, t _n ∈ ((n-1)Δt,nΔt), t represents the calculation time, and the subscript n of t represents the number of time layers, a can be approximated by a series of linear combinations of the fundamental solutions G:

${u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 3...... 3... - - - - - - ((77))$

其中，α_j(t_R)表示对于源点s_j的在延迟时刻t_R的未知系数，G_ij表示在延迟时刻t_R的三维波方程基本解，Q_i表示在延迟时刻t_R的源点强度因子，x_i表示第i个配点，x_j表示第j个源点，N为数据采集点总数；Among them, α _j (t _R ) represents the unknown coefficient of the source point s _j at the delay time t _R , G _ij represents the basic solution of the three-dimensional wave equation at the delay time t _R , Q _i represents the source point at the delay time t _R Intensity factor, x _i represents the i-th collocation point, x _j represents the j-th source point, and N is the total number of data collection points;

在简谐波动中，假定In simple harmonic fluctuations, it is assumed that

${α α}_{j j} (({t t}_{R R})) = = {α α}_{j j}^{* *} (({t t}_{R R})) {e e}^{- - {iωt iωt}_{R R}} \approx \approx {α α}_{j j}^{* *} (({t t}_{m m Δ Δ t t})) {e e}^{- - {iωt iωt}_{R R}} - - - - - - ((88))$

其中，0≤t_mΔt-t_R＜Δt，α_j(t_R)表示延迟时刻未知系数，表示分离时间变量t后的延迟时刻未知系数，m和n表示时间层数，如果m＜n，则未知系数α_j(t_R)已经被前步求出，如果t_R＜0，则表示波未传至配点处，ω表示波的频率，k＝ω/c表示波数；Among them, _{0≤t mΔt} -t _R <Δt, α _j (t _R ) represents the unknown coefficient of delay time, Indicates the unknown coefficient of the delay time after separating the time variable t, m and n represent the number of time layers, if m<n, the unknown coefficient α _j (t _R ) has been obtained in the previous step, and if t _R <0, it means wave Not sent to the distribution point, ω represents the frequency of the wave, k=ω/c represents the wave number;

式(5)、式(7)和式(8)共同构成了奇异边界法处理波传播问题的差值矩阵。Equation (5), Equation (7) and Equation (8) together constitute the difference matrix for the singular boundary method to deal with wave propagation problems.

进一步，步骤(3)中，计算源点强度因子的经验公式为：Further, in step (3), the empirical formula for calculating the source point strength factor is:

${Q Q}_{i i} = = \frac{11}{44 π π} [[\frac{{π π}^{44}}{2525 \sqrt{{A A}_{j j}}} + + \frac{{((l l n no π π))}^{22}}{S S}]] + + \frac{i i k k}{44 π π} - - - - - - ((99))$

其中A_j表示源点s_j的影响区域，S表示整个计算区域的表面积。where A _j represents the area of influence of the source point s _j , and S represents the surface area of the entire calculation area.

进一步，根据步骤(3)所推得源点强度因子，未知系数由式(7)和(8)求解得出：通过式(7)和(8)，求出在第n时间层，nΔt秒的未知系数进而由式(8)求得延迟时刻的未知系数α_j(t_R)。Further, according to the source point intensity factor deduced in step (3), the unknown coefficient is obtained by solving equations (7) and (8): through equations (7) and (8), find the nth time layer, nΔt seconds The unknown coefficient of Furthermore, the unknown coefficient α _j (t _R ) of the delay time is obtained from formula (8).

进一步，步骤(5)在时刻t任意内点x的声压u通过无积分计算公式求得：Further, the sound pressure u of any internal point x in step (5) at time t is obtained by the non-integral calculation formula:

${u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((1010))$

${u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 33,, ...... … - - - - - - ((1111))$

$u u = = {u u}_{11}^{* *} + + {u u}_{22}^{* *} - - - - - - ((1212))$

有益效果：本发明仅需边界配置边界数据采集点，部分区域配置初始数据采集点，无数值积分、奇异积分，无网格划分，直接应用波动方程时间基本解作为核函数，无复杂的数学变换，其简单高效的特点切合波动类型动态数据重构需要数据重构技术快速、稳定、精确的特征要求；实验对比表明，应用本发明所提出的技术，处理波动类型动态数据重构问题，在取得相似精度条件下，一般耗时只需传统线性边界元算法的10％左右，具有精度高，计算快，数学简单，程序简便的特点，可应用于声波减噪，海浪消能，地震预测等波动型数据重构，图像处理等工程技术领域。Beneficial effects: the present invention only needs to configure border data collection points, some areas to configure initial data collection points, no numerical integration, singular integration, no grid division, directly apply the time basic solution of the wave equation as the kernel function, and no complicated mathematical transformation , its simple and efficient features meet the characteristics of fast, stable and accurate data reconstruction technology for dynamic data reconstruction of fluctuation type; experimental comparison shows that applying the technology proposed by the present invention to deal with the problem of dynamic data reconstruction of fluctuation type can achieve Under the condition of similar accuracy, the general time consumption is only about 10% of the traditional linear boundary element algorithm. It has the characteristics of high accuracy, fast calculation, simple mathematics, and simple program. It can be applied to wave noise reduction, wave energy dissipation, earthquake prediction, etc. Data reconstruction, image processing and other engineering technology fields.

附图说明Description of drawings

图1为本发明波动类型动态数据重构方法流程图；Fig. 1 is a flow chart of the method for reconstructing dynamic data of fluctuation type according to the present invention;

图2(a)为三维波传播问题边界数据采集点布置示意图，图2(b)为区域数据采集点布置示意图；Figure 2(a) is a schematic diagram of the layout of boundary data collection points for the three-dimensional wave propagation problem, and Figure 2(b) is a schematic diagram of the layout of regional data collection points;

图3为边界数据采集点s_j影响区域示意图；Fig. 3 is a schematic diagram of the area affected by boundary data collection point _sj ;

图4为轮胎形区域数据采集点布置示意图；Fig. 4 is a schematic diagram of the arrangement of data collection points in the tire-shaped area;

图5为奇异边界法数据重构结果随边界数据采集点点数增加变化收敛图；Figure 5 is the convergence diagram of the data reconstruction results of the singular boundary method as the number of boundary data collection points increases;

图6为奇异边界法数据重构结果随时间延续变化趋势图；Figure 6 is a trend chart of the data reconstruction results of the singular boundary method over time;

图7为声辐射无量纲实部声压重构数值结果随波数变化图；Fig. 7 is a graph showing the numerical results of sound pressure reconstruction of the dimensionless real part of acoustic radiation versus wave number;

图8为声辐射无量纲虚部声压重构数值结果随波数变化图。Fig. 8 is a graph showing the numerical results of sound pressure reconstruction of the dimensionless imaginary part of acoustic radiation as a function of wave number.

具体实施方式detailed description

下面对本发明技术方案进行详细说明，但是本发明的保护范围不局限于所述实施例。The technical solutions of the present invention will be described in detail below, but the protection scope of the present invention is not limited to the embodiments.

实施例：如图1所示，一种基于无网格奇异边界法重构波动类型动态数据，具体步骤如下：Embodiment: as shown in Fig. 1, a kind of dynamic data of wave type is reconstructed based on gridless singularity boundary method, and concrete steps are as follows:

(1)初始时刻在所考察物质的内部配置N₁个测试点，获得初始时刻这些测试点的波动数据值u_i,i＝1,...,N₁；(1) Configure N ₁ test points inside the substance under investigation at the initial moment, and obtain the fluctuation data values u _i , i=1,...,N ₁ of these test points at the initial moment;

在所考察物质的表面配置N₂个测试点，随着波动的进行，获得不同时刻这些点的波动数据值u_i,i＝N₁+1,...,N₁+N₂。N ₂ test points are arranged on the surface of the substance under investigation, and as the fluctuation proceeds, the fluctuation data values u _i , i=N ₁ +1,...,N ₁ +N ₂ of these points at different times are obtained.

(2)根据奇异边界法的基本思想，直接采用三维波方程的时间基本解作为核函数，建立波传播问题相应的奇异边界法插值矩阵：(2) According to the basic idea of the singular boundary method, the time basic solution of the three-dimensional wave equation is directly used as the kernel function to establish the corresponding singular boundary method interpolation matrix for the wave propagation problem:

应用叠加原理，原数据u可分为初始数据u₂和边界数据u₁，即 Applying the principle of superposition, the original data u can be divided into initial data u ₂ and boundary data u ₁ , namely

初始数据可以用三维泊松方程直接求出Initial data It can be directly obtained by the three-dimensional Poisson equation

${u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((S S .1 .1))$

其中表示半径为ct以M为圆心的球面。在式(S.1)中，奇异边界法仅需要求出测试点M影响区域中的声压值，所以奇异边界法仅需要在影响域中布置初始数据采集点，如图2所示。in Represents a sphere with radius ct centered at M. In formula (S.1), the singular boundary method only needs to find the influence area of the test point M The sound pressure value in , so the singular boundary method only needs to be in the influence domain Arrange the initial data collection points, as shown in Figure 2.

其后，考虑边界数据奇异边界法在边界Γ上布置边界数据采集点，如图2所示，以Δt为时间间隔，t的下标表示时间层数。声压t_n∈((n-1)Δt,nΔt)，可以被基本解G的一系列线性组合来近似，如式(S.2)所示：Afterwards, consider the boundary data The singular boundary method arranges boundary data collection points on the boundary Γ, as shown in Figure 2, with Δt as the time interval, and the subscript of t indicates the number of time layers. a t _n ∈ ((n-1)Δt,nΔt), can be approximated by a series of linear combinations of the basic solution G, as shown in formula (S.2):

${u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 3...... 3... - - - - - - ((S S .2 .2))$

其中α_j(t_R)表示对于源点s_j的在延迟时刻t_R的未知系数，G_ij表示在延迟时刻t_R的三维波方程基本解，Q_i表示在延迟时刻t_R的源点强度因子。where α _j (t _R ) represents the unknown coefficient of the source point s _j at the delay time t _R , G _ij represents the basic solution of the three-dimensional wave equation at the delay time t _R , Q _i represents the source point intensity at the delay time t _R factor.

在简谐波动中，我们假定In simple harmonic motion, we assume

${α α}_{j j} (({t t}_{R R})) = = {α α}_{j j}^{* *} (({t t}_{R R})) {e e}^{- - {iωt iωt}_{R R}} \approx \approx {α α}_{j j}^{* *} (({t t}_{m m Δ Δ t t})) {e e}^{- - {iωt iωt}_{R R}} - - - - - - ((S S .3 .3))$

其中0≤t_mΔt-t_R＜Δt，如果m＜n，则未知系数α_j(t_R)已经被前步求出，如果t_R＜0，则表示波未传至配点处，ω表示波的频率，k＝ω/c表示波数。Among them, _{0≤t mΔt} -t _R <Δt, if m<n, the unknown coefficient α _j (t _R ) has been obtained in the previous step, if t _R <0, it means that the wave has not reached the collocation point, ω represents the frequency of the wave, and k=ω/c represents the wave number.

式(S.1)，式(S.2)和式(S.3)共同构成了奇异边界法处理波传播问题的插值矩阵。Equation (S.1), Equation (S.2) and Equation (S.3) together constitute the interpolation matrix for the singular boundary method to deal with wave propagation problems.

(3)利用经验公式计算奇异边界法的源点强度因子：三维波动方程的源点强度因子计算公式如式(S.4)所示。(3) Use the empirical formula to calculate the source point strength factor of the singular boundary method: the calculation formula of the source point strength factor for the three-dimensional wave equation is shown in formula (S.4).

${Q Q}_{i i} = = \frac{11}{44 π π} [[\frac{{π π}^{44}}{2525 \sqrt{{A A}_{j j}}} + + \frac{{((l l n no π π))}^{22}}{S S}]] + + \frac{i i k k}{44 π π} - - - - - - ((S S .4 .4))$

其中A_j表示源点s_j的影响区域，如图3所示，S表示整个计算区域的表面积。where A _j represents the influence area of the source point s _j , as shown in Figure 3, and S represents the surface area of the entire calculation area.

(4)将源点强度因子代入奇异边界插值矩阵，计算奇异边界法求解方程的未知系数。应用步骤(3)所推得源点强度因子，步骤(4)中奇异边界法的未知系数可由式(S.2)和(S.3)求解得出。通过式(S.2)和(S.3)，求出在第n时间层，nΔt秒的未知系数进而由式(S.3)求得延迟时刻t_R的未知系数α_j(t_R)。(4) Substitute the source point intensity factor into the singular boundary interpolation matrix, and calculate the unknown coefficients of the singular boundary method to solve the equation. Using the source point intensity factor derived in step (3), the unknown coefficients of the singular boundary method in step (4) can be obtained by solving equations (S.2) and (S.3). Through the formulas (S.2) and (S.3), find the unknown coefficient of the nth time layer, nΔt seconds Furthermore, the unknown coefficient α _j (t _R ) of the delay time t _R is obtained from formula (S.3).

(5)根据奇异边界法公式，计算任意时刻任意内点的波动数据值。步骤(5)中，通过步骤(4)求得的未知系数，在时刻t任意内点x的声压u可通过下面奇异边界法公式求得：(5) Calculate the fluctuation data value of any interior point at any time according to the singular boundary method formula. In step (5), through the unknown coefficient obtained in step (4), the sound pressure u of any interior point x at time t can be obtained by the following singular boundary method formula:

${u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((S S .5 .5))$

${u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 33 ... ... ... ... ((S S .6 .6))$

$u u = = {u u}_{11}^{* *} + + {u u}_{22}^{* *} - - - - - - ((S S .7 .7))$

实施例1：考虑如图4所示的轮胎形区域波动数据重构，区域方程为Embodiment 1: Consider the tire-shaped regional fluctuation data reconstruction as shown in Figure 4, the regional equation is

其中R＝0.8，r＝0.2，边界数据采集点如图3所示。Among them, R=0.8, r=0.2, and the boundary data collection points are shown in Figure 3.

波动数据控制方程及精确解为：The governing equation and exact solution of the fluctuation data are:

$\{\begin{matrix} {u u}_{t t t t} = = {c c}^{22} (({u u}_{x x x x} + + {u u}_{y the y y the y} + + {u u}_{z z z z})),, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω \\ u u {| |}_{t t = = 00} = = 00 \\ {u u}_{t t} {| |}_{t t = = 00} = = c c k k ((cos cos ((k k x x)) + + c c o o s the s ((k k y the y)) + + cos cos ((k k z z)))) \\ {u u}_{t t} {| |}_{t t = = 00} = = ((cos cos ((k k r r)) + + cos cos ((k k y the y)) + + cos cos ((k k z z)))) sin sin ((c c k k t t)),, ((x x,, y the y,, z z)) &Element; &Element; Γ Γ \end{matrix}$

本例中，我们应用时间步长Δt＝2×10^-1，波速c＝10，对每个测试点布置区域数据采集点N_f＝1255，测试点被布置在半径为0.8，t＝1s的时间层上，图5给出了奇异边界法随边界采集数据点点数增加的数值精度收敛图，其中波数分别为(k₁＝0.5，k₂＝5，k₃＝10)。可以发现奇异边界法以2阶收敛速度快速收敛，当k＝5时，收敛速度甚至达到了C＝5.0。In this example, we apply time step size Δt=2×10 ^-1 , wave velocity c=10, arrange regional data collection points N _f =1255 for each test point, and the test points are arranged in a radius of 0.8, t=1s On the time layer, Fig. 5 shows the numerical accuracy convergence diagram of the singular boundary method with the increase of the number of boundary collection data points, where the wave numbers are (k ₁ =0.5, k ₂ =5, k ₃ =10). It can be found that the singular boundary method converges rapidly at the second-order convergence rate, and when k=5, the convergence rate even reaches C=5.0.

其后，应用时间步长Δt＝2×10^-1，波速c＝10，布置边界数据采集点N_s＝972，对每个测试点布置区域数据采集点N_f＝1255，图6给出了在不同波数情况下(k₁＝0.5，k₂＝5，k₃＝10)奇异边界法数据重构最大绝对误差随时间变化情况，可以看到随时间延续，奇异边界法最大相对误差仍与精确解保持一致，并未随时间变化出现离散情况。Afterwards, apply time step Δt=2×10 ^-1 , wave velocity c=10, arrange boundary data collection points N _s =972, and arrange regional data collection points N _f =1255 for each test point, Fig. 6 shows In the case of different wave numbers (k ₁ = 0.5, k ₂ = 5, k ₃ = 10), the maximum absolute error of the data reconstruction of the singular boundary method changes with time. It can be seen that the maximum relative error of the singular boundary method is still the same as that of The exact solution remains consistent and does not diverge over time.

实施例2：考虑在单位球声压辐射数据重构，其中半径a＝1，波速c＝v₀，精确解可以被表示为：Embodiment 2: Considering the reconstruction of sound pressure radiation data in a unit sphere, where radius a=1, wave velocity c=v ₀ , the exact solution can be expressed as:

$u u ((r r,, θ θ,, t t)) = = \frac{a a}{r r} ((\frac{{ikaz ikaz}_{00}}{i i k k a a - - 11})) {v v}_{00} {e e}^{i i k k ((r r - - a a)) + + i i w w t t},,$

其中z₀＝ρ₀c，表示介质阻抗，ρ₀为介质密度，c为波速，ω＝kc为波频。图7和图8给出了无量纲声压实部解和虚部解在边界数据采集点N_s＝400，时间步长Δt＝0.5s条件下的数值结重构结果。可以发现奇异边界法重构结果和精确解精确拟合，相比之下当虚边界d＝0.5时，基本解方法重构结果和精确解拟合良好，但是当虚边界取为d＝0.9时，数据重构结果发生了明显离散。同时，需要指明的是，对于声辐射，声散射等外域问题的波动数据重构，本发明方法无需采集区域波动数据，大大提高了波动类型数据的重构效率。Among them, z ₀ =ρ ₀ c represents the medium impedance, ρ ₀ is the medium density, c is the wave velocity, and ω=kc is the wave frequency. Fig. 7 and Fig. 8 show the solution of dimensionless acoustic compaction and imaginary solution The numerical structure reconstruction results under the conditions of boundary data collection points N _s =400 and time step Δt=0.5s. It can be found that the reconstruction results of the singular boundary method and the exact solution fit accurately. In contrast, when the virtual boundary d=0.5, the reconstruction results of the basic solution method and the exact solution fit well, but when the virtual boundary d=0.9 , the data reconstruction results are obviously discrete. At the same time, it should be pointed out that for the reconstruction of fluctuation data of external domain problems such as acoustic radiation and sound scattering, the method of the present invention does not need to collect regional fluctuation data, which greatly improves the reconstruction efficiency of fluctuation type data.

综上，本发明波动类型动态数据重构方法以奇异边界法为基础，直接应用波动方程时间基本解作为核函数，用源点强度因子代替源点奇异项，避免了数值积分和网格，仅需边界布点，无网格、无数值积分和数学变换，提高了数据重构效率。相较于现有技术，无需做复杂的快速傅里叶变换或繁琐的多级波形迭代求解，直接在时域上运用时间依赖基本解作为插值基函数，通过应用源点强度因子重构波动类型数据，其简单高效的特点切合波动类型动态数据重构需要计算工具快速、稳定、精确的特征要求，适用于声波，水波，地震波等标量波传播动态数据的重构问题。To sum up, the wave type dynamic data reconstruction method of the present invention is based on the singular boundary method, directly applies the time basic solution of the wave equation as the kernel function, replaces the source point singular term with the source point intensity factor, avoids numerical integration and grid, and only Need boundary layout, no grid, no numerical integration and mathematical transformation, which improves the efficiency of data reconstruction. Compared with the existing technology, there is no need to do complex fast Fourier transform or cumbersome multi-level waveform iterative solution, directly use the time-dependent basic solution as the interpolation basis function in the time domain, and reconstruct the fluctuation type by applying the source point intensity factor Data, its simple and efficient characteristics meet the requirements of fast, stable and accurate characteristics of computing tools for dynamic data reconstruction of wave types, and are suitable for the reconstruction of scalar wave propagation dynamic data such as sound waves, water waves, and seismic waves.

实验对比表明，应用本发明方法处理波动类型动态数据重构问题，在取得相似精度条件下，一般耗时只需传统线性边界元算法的10％左右，具有精度高，计算快，数学简单，程序简便的特点。该技术为波动型动态数据的重构提供了新的，简单高效的技术路线。Experimental comparison shows that the application of the method of the present invention to deal with the problem of dynamic data reconstruction of fluctuation type, under the condition of obtaining similar precision, generally takes only about 10% of the traditional linear boundary element algorithm, and has high precision, fast calculation, simple mathematics, and Easy features. This technology provides a new, simple and efficient technical route for the reconstruction of fluctuating dynamic data.

Claims

1. A method for reconstructing dynamic data of fluctuation type based on singular boundary method, characterized in that: comprising the following steps:

(1) Configure several test points inside and on the boundary of the substance under investigation, and obtain the fluctuation data values of these test points;

(2) Directly use the time basic solution of the three-dimensional wave equation as the kernel function to establish the corresponding interpolation matrix for the wave propagation problem;

(3) Use the empirical formula to calculate the source strength factor for the reconstruction of fluctuating data;

(4) Substitute the source point intensity factor into the interpolation matrix, and calculate the unknown coefficient of the interpolation matrix;

(5) Calculate the fluctuation data value of any interior point at any time.

2. The wave type dynamic data reconstruction method based on the singular boundary method according to claim 1, characterized in that: in the step (1), at the initial moment, N ₁ test points are configured inside the substance under investigation to obtain the initial moment The fluctuation data values of these test points u _i , i=1,...,N ₁ ; N ₂ test points are arranged on the surface of the substance under investigation, and as the fluctuation proceeds, the fluctuation data values u of these points at different times are obtained _i , i=N ₁ +1,...,N ₁ +N ₂ .

3. the wave type dynamic data reconstruction method based on singular boundary method according to claim 1, is characterized in that: the wave problem governing equation that adopts in the step (2) is:

u u = = \{\begin{matrix} Δ Δ u u - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} u u}{\partial \partial {t t}^{22}} = = 00,, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ u u {| |}_{Γ Γ} = = \overset{&OverBar; &OverBar;}{u u} \\ u u {| |}_{t t = = 00} = = {u u}_{00},, \frac{\partial \partial u u}{\partial \partial t t} {| |}_{t t = = 00} = = {&upsi; &upsi;}_{11} \end{matrix} - - - - - - ((11))

Among them, Ω represents the area of the substance under investigation, (x, y, z) is the space coordinate, t is the moment corresponding to (x, y, z), c represents the wave velocity, u represents the potential function, Represents the boundary conditions, u ₀ and υ ₁ represent the initial value conditions, Δ represents the Laplace operator, and represents the sound pressure value in the sound wave propagation;

The corresponding basic solution of the three-dimensional wave propagation problem is:

G G ((r r,, t t)) = = \frac{11}{44 π π r r} δ δ [[((t t - - τ τ)) - - r r / / c c]] - - - - - - ((22))

Among them, δ represents the Dirac function, c represents the wave velocity, t and τ represent the time of the collocation point and the source point, and r represents the distance; since the propagation of the wave requires time, the basic solution G can only be obtained by using the relevant combination of the corresponding sound pressure values at the delay time Indicates the sound pressure value at the current moment, and the delay time depends on the distance between the source point and the point to be measured and the propagation speed of the wave; therefore, for a specific test point, only the unknown of the delay time of the source point {s _j } is required coefficient {α _j }, then the sound pressure value at the moment of the test point can be obtained;

The original equation u can be regarded as an initial boundary value problem. Applying the principle of superposition, it can be split into u ₁ and u ₂ , as shown in equations (3) and (4), that is Among them, u ₁ can be regarded as a boundary value problem, and u ₂ can be regarded as an initial value problem:

{u u}_{11}^{* *} = = \{\begin{matrix} {Δu Δu}_{11}^{* *} - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} {u u}_{11}^{* *}}{\partial \partial {t t}^{22}} = = 00,, ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ {u u}_{11}^{* *} {| |}_{Γ Γ} = = \overset{&OverBar; &OverBar;}{u u} - - {\overset{&OverBar; &OverBar;}{u u}}_{22}^{* *} = = {\overset{&OverBar; &OverBar;}{u u}}_{11}^{* *} \\ {u u}_{11}^{* *} {| |}_{t t = = 00} = = 00,, \frac{\partial \partial {u u}_{11}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = 00 \end{matrix} - - - - - - ((33))

{u u}_{22}^{* *} = = \{\begin{matrix} {Δu Δ u}_{22}^{* *} - - \frac{11}{{c c}^{22}} \frac{{\partial \partial}^{22} {u u}_{22}^{* *}}{\partial \partial {t t}^{22}} = = 00,, & ((x x,, y the y,, z z)) &Element; &Element; Ω Ω,, t t > > 00 \\ {u u}_{22}^{* *} {| |}_{Γ Γ} = = {\overset{&OverBar; &OverBar;}{u u}}_{22}^{* *} \\ {u u}_{22}^{* *} {| |}_{t t = = 00} = = 00,, \frac{\partial \partial {u u}_{22}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = 00 & ((x x,, y the y,, z z)) &NotElement; &NotElement; Ω Ω \\ {u u}_{22}^{* *} {| |}_{t t = = 00} = = {u u}_{00},, \frac{\partial \partial {u u}_{22}^{* *}}{\partial \partial t t} {| |}_{t t = = 00} = = {&upsi; &upsi;}_{11} & ((x x,, y the y,, z z)) &Element; &Element; Ω Ω \end{matrix} - - - - - - ((44))

Formula (4) Use the three-dimensional Poisson equation to find directly:

{u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((55))

in Indicates a spherical surface with a radius of ct and M as the center; in formula (4), it is only necessary to obtain the influence area of the test point M The sound pressure value in , so in the domain of influence Arrange the initial data collection points in ;

Then, considering equation (3) According to Huygens' principle that "every point reached in the wave propagation can be used as a new secondary wave source, and the envelope surface formed by all these secondary waves constitutes the new wave surface at the next moment", it is assumed that there is a series of changing point source The sum of the secondary waves emitted by these point wave sources forms the sound pressure value at any subsequent time in the calculation domain Ω in represents the sound pressure intensity, x _m represents the boundary point, then the sound pressure in the domain Ω is calculated can be expressed as

{u u}_{11}^{* *} (({x x}_{i i})) = = \underset{Ω Ω}{&Integral; &Integral; &Integral; &Integral; &Integral; &Integral;} \frac{α α ((r r))}{44 π π r r} δ δ ((t t - - \frac{r r}{c c})) d d v v,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((66))

Arrange data collection points on the boundary Γ, with Δt as the time interval, v represents the space integration variable, t _n ∈ ((n-1)Δt,nΔt), t represents the calculation time, and the subscript n of t represents the number of time layers, a can be approximated by a series of linear combinations of the fundamental solutions G:

{u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 33 ... ... ... ... ((77))

Among them, α _j (t _R ) represents the unknown coefficient of the source point s _j at the delay time t _R , G _ij represents the basic solution of the three-dimensional wave equation at the delay time t _R , Q _i represents the source point at the delay time t _R Intensity factor, x _i represents the i-th collocation point, x _j represents the j-th source point, and N is the total number of data collection points;

In simple harmonic fluctuations, it is assumed that

{α α}_{j j} (({t t}_{R R})) = = {α α}_{j j}^{* *} (({t t}_{R R})) {e e}^{- - {iωt iωt}_{R R}} \approx \approx {α α}_{j j}^{* *} (({t t}_{m m Δ Δ t t})) {e e}^{- - {iωt iωt}_{R R}} - - - - - - ((88))

Among them, _{0≤t mΔt} -t _R <Δt, α _j (t _R ) represents the unknown coefficient of delay time, Indicates the unknown coefficient of the delay time after separating the time variable t, m and n represent the number of time layers, if m<n, the unknown coefficient α _j (t _R ) has been obtained in the previous step, and if t _R <0, it means wave Not sent to the distribution point, ω represents the frequency of the wave, k=ω/c represents the wave number;

Equation (5), Equation (7) and Equation (8) together constitute the difference matrix for the singular boundary method to deal with wave propagation problems.

4. the wave type dynamic data reconstruction method based on singular boundary method according to claim 3, is characterized in that: in step (3), the empirical formula of calculating source point intensity factor is:

{Q Q}_{i i} = = \frac{11}{44 π π} [[\frac{{π π}^{44}}{2525 \sqrt{{A A}_{j j}}} + + \frac{{((l l n no π π))}^{22}}{S S}]] + + \frac{i i k k}{44 π π} - - - - - - ((99))

where A _j represents the area of influence of the source point s _j , and S represents the surface area of the entire calculation area.

5. The method for reconstructing dynamic data of fluctuation type based on singular boundary method according to claim 4, characterized in that: according to the source point intensity factor derived from step (3), the unknown coefficient is determined by formula (7) and (8) Solve and get: through equations (7) and (8), find the unknown coefficient of the nth time layer, nΔt seconds Furthermore, the unknown coefficient α _j (t _R ) of the delay time is obtained from formula (8).

6. the wave type dynamic data reconstruction method based on singular boundary method according to claim 5, is characterized in that: step (5) obtains by non-integral calculation formula at the sound pressure u of any internal point x at time t:

{u u}_{22}^{* *} (({x x}_{i i})) = = \frac{11}{44 π π} \frac{\partial \partial}{\partial \partial r r} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{u u}_{00}}{r r} d d s the s + + \frac{11}{44 c c π π} \underset{{S S}_{c c t t}^{M m}}{&Integral; &Integral; &Integral; &Integral;} \frac{{&upsi; &upsi;}_{11}}{r r} d d s the s,, {x x}_{i i} &Element; &Element; Ω Ω - - - - - - ((1010))

{u u}_{11}^{* *} (({x x}_{i i})) = = u u (({x x}_{i i})) - - {u u}_{22}^{* *} (({x x}_{i i})) = = {Σ Σ}_{j j = = 11,, j j &NotEqual; &NotEqual; i i}^{N N} {α α}_{j j} (({t t}_{R R})) {G G}_{i i j j} + + {α α}_{i i} (({t t}_{R R})) {Q Q}_{i i},, {t t}_{R R} = = {t t}_{n no} - - \frac{r r}{c c} > > 00,, i i = = 11,, 22,, 33 ... ... ... ... ((1111))

u u = = {u u}_{11}^{* *} + + {u u}_{22}^{* *} - - - - - - ((1212)) . .