面向地质条件的贴体网格生成技术
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摘要
模型物理参数的空间网格化是影响正演算法精度的主要因素之一。网格是离散的基础,网格节点是离散化物理量的存储位置。对于网格构造的研究表明,单纯传统意义下的好网格,如大小均匀、长宽适当以及正交等,对某些问题的计算可能并不是最佳的。因为网格构造最终是为数值计算实际问题服务的,网格的好坏应该看它是否与物理问题解的空间分布相适应。
在复杂地质条件下的地球物理研究中,介质的复杂性对网格剖分提出了更高的要求,由于实际模型往往很复杂,具有强反差速度或者不规则边界,如在模拟复杂地质构造和复杂地质体的复杂界面时采用笛卡尔坐标系中的规则网格,必然会出现阶梯状的边界,在这种边界上必然引起人为的虚假绕射波,为了减弱这种虚假绕射波,则必须采用精细网格,而这将导致数值模拟计算储存量的增加和计算量的增大。而不规则网格技术可以部分的解决这一限制,目前涌现出了很多基于不规则网格的差分算法,成为处理复杂地质条件问题的主要技术之一。
本文在对不规则网格技术在地球物理中的应用现状进行分析的基础上,引入计算流体力学(CFD)中常用的微分方程法生成贴体曲网格的技术,根据地球物理学中地质条件的特殊性和流体力学中介质的差别,对地球物理研究中面临的不规则边界和不均匀介质等由复杂地质条件带来的问题,采用合理的贴体网格和自适应网格进行曲坐标剖分。最终生成的网格具有下面三个特点:
1)网格边界线与实际物理域边界线一致,可精确描述模型边界;
2)网格边界处具有良好的正交性,便于边界条件的处理;
3)网格分布可随物理参数变化自由调节,达到与地球物理问题物理解相适应的目的。
在上述网格上做数值计算本文推荐基于坐标变换的差分计算方法,并推导出曲坐标系下的地球物理方程。最后介绍基于上述网格生成算法编制的面向地质条件的曲网格生成系统。
Grid generation is the link between geometric model and numerical algorithm, geometry model has to be divided into a certain standard grid to its numerical solution. On the geophysical studing under complicated geological conditions, the complexity of the medium makes grid generation higher demand. The rectangular grid used in finite difference presents its limitations, irregular grid has emerged as one of the main techniques to solve the problem with complex geological conditions. The current irregular mesh generation methods are interpolation method, mapping method, unstructured grid and variable grid. In the Interpolation method, the broken line is used to approximate the topography, and the value on the free surface is replaced by the value on the regular grid points. So, the method itself leads to the existence of errors; The basic idea of the mapping method is to transform the rough boundary into the horizontal border, while the physical space of the differential equations into differential equations for the calculation of space,in which the calculation of numerical simulation is implemented, The main problem is that the free boundary must be a smooth surface, In addition, the depth of physical space is reseted by division method, which will cause to bring some stability problems (such as zero-denominator problem); the basic idea of non-structural grid metheod is to use split to accommodate the irregular undulating surface and irregular, but it needs a large number of irregular grid computing, meanwhile, finite difference scheme on irregular grid is complex and cumbersome; Variable grid method means changing the grid spacing according to the needs of different regions, the grid spaceing in the adjustment field may lead to instability in numerical calculating.
From the point of numerical grid generation view, this paper is to improve the deficiency of these methods. Grid is the primary problem to be solved, sometimes referred to as pre-treatment in numerical calculation. Quality of the grid is to affect the precision and efficiency of geophysical simulation. Computational domain grid generation is to be divided in accordance with certain principles, so coordinate system on the original problem is transformed to the new coordinate system to solve it. According to the practical problems to design different grid, such as grid should be more intensive in the field where gradient larger, and a new coordinate system should ensure the orthogonality near the boundary. So it’s more effective and convenient, Improve accuracy and ensure the stability of solutions to simplify boundary conditions, thereby reducing the required computing time and computer capacity. Therefore, the grid should be decided by the kind of medium.
Complexity of the medium in different geophysical fields for the physical structure, physical properties and spatial and temporal distribution is different, sometimes even decisive. The medium underground has two intrinsic properties, one is structural form in three-dimensional space; the other is medium in every directions is complexity of variation. Three-dimensional topography model is the approximation of the first properties– tridimensional. while the model parameters are used to describe the second media property - non-uniformity. Current geophysical problems of the complicated geological conditions are also reflected in these two areas, One is topography, the other is complex medium underground.
For the issue of irregular topography, taking into account the similarity with the fluid medium, this paper introduces to generate body-fitted grid to meet the needs of grid computing. Body-fitted grid is developed in CFD, in which grid lines as irregular borders to guarantee the exact description of the surface, also it allow grid spacing and angle adjustment, the body-fitted grid based on Poisson Equation method is generated with a smooth and inherent flexibility. In addition, Hilgenstock method is used to adjust the grid orthogonality of the boundary nodes to facilitate the realization of free boundary conditions.
For the issue of inhomogeneous media underground, the paper quotes adaptive grid technology. Adaptive grid generation technology to automatically adjust the distribution of media in the region use the dense low grid, high-speed region using the sparse grid, so that we can use the least amount of computation for maximum accuracy.
In summary, the proposed mesh generation algorithm has the following advantages:
1) Body-fitted grid can accurately describe the nature of the surface, so to avoid the error in interpolation method;
2) Grid in the free boundary has good orthogonality to facilitate the realization of free boundary conditions.
3) Adaptive grid generation technology enables to avoid instability in variable step grid;
4) Body-fitted grid of rectangular is a kind of structure grid, so to avoid facing complex difference scheme in the non-structural grid.
After the advisable grid generation of the geophysical model, the application of coordinate transformation method to achieve curved coordinate system is proposed, Numerical simulation of geophysical concept. Coordinate transformation method through the rectangular coordinate system of geophysical equations transform into curvilinear coordinate system grid to solve the rule, not only break the surface mapping method on the shape of the limit, but also avoid the complicated derivation of difference schemes, achieve the numerical calculation really in the curvilinear coordinate system. The uniform transformation relation between the two system make the method more universal, with a wide range of applications.
In this paper, the curvelinear coordinates and numerical simulation of geophysical processes and may experience the issues discussed and the basic potential of a DC field equations and reflection seismic equation gives the song an example coordinate system control equation derived.
Finally, the preparation of a automatic grid generation program, by importing the model parameters (including the model shape, size, distribution and other geophysical parameters data), the model automatically generate body-fitted grid, and Geophysics model gridgeneration is visual to ajust the grid distribution behavior manually, eventually export the grid coordinates and the required data of numerical simulation related with the coordinates.
在复杂地质条件下的地球物理研究中,介质的复杂性对网格剖分提出了更高的要求,由于实际模型往往很复杂,具有强反差速度或者不规则边界,如在模拟复杂地质构造和复杂地质体的复杂界面时采用笛卡尔坐标系中的规则网格,必然会出现阶梯状的边界,在这种边界上必然引起人为的虚假绕射波,为了减弱这种虚假绕射波,则必须采用精细网格,而这将导致数值模拟计算储存量的增加和计算量的增大。而不规则网格技术可以部分的解决这一限制,目前涌现出了很多基于不规则网格的差分算法,成为处理复杂地质条件问题的主要技术之一。
本文在对不规则网格技术在地球物理中的应用现状进行分析的基础上,引入计算流体力学(CFD)中常用的微分方程法生成贴体曲网格的技术,根据地球物理学中地质条件的特殊性和流体力学中介质的差别,对地球物理研究中面临的不规则边界和不均匀介质等由复杂地质条件带来的问题,采用合理的贴体网格和自适应网格进行曲坐标剖分。最终生成的网格具有下面三个特点:
1)网格边界线与实际物理域边界线一致,可精确描述模型边界;
2)网格边界处具有良好的正交性,便于边界条件的处理;
3)网格分布可随物理参数变化自由调节,达到与地球物理问题物理解相适应的目的。
在上述网格上做数值计算本文推荐基于坐标变换的差分计算方法,并推导出曲坐标系下的地球物理方程。最后介绍基于上述网格生成算法编制的面向地质条件的曲网格生成系统。
Grid generation is the link between geometric model and numerical algorithm, geometry model has to be divided into a certain standard grid to its numerical solution. On the geophysical studing under complicated geological conditions, the complexity of the medium makes grid generation higher demand. The rectangular grid used in finite difference presents its limitations, irregular grid has emerged as one of the main techniques to solve the problem with complex geological conditions. The current irregular mesh generation methods are interpolation method, mapping method, unstructured grid and variable grid. In the Interpolation method, the broken line is used to approximate the topography, and the value on the free surface is replaced by the value on the regular grid points. So, the method itself leads to the existence of errors; The basic idea of the mapping method is to transform the rough boundary into the horizontal border, while the physical space of the differential equations into differential equations for the calculation of space,in which the calculation of numerical simulation is implemented, The main problem is that the free boundary must be a smooth surface, In addition, the depth of physical space is reseted by division method, which will cause to bring some stability problems (such as zero-denominator problem); the basic idea of non-structural grid metheod is to use split to accommodate the irregular undulating surface and irregular, but it needs a large number of irregular grid computing, meanwhile, finite difference scheme on irregular grid is complex and cumbersome; Variable grid method means changing the grid spacing according to the needs of different regions, the grid spaceing in the adjustment field may lead to instability in numerical calculating.
From the point of numerical grid generation view, this paper is to improve the deficiency of these methods. Grid is the primary problem to be solved, sometimes referred to as pre-treatment in numerical calculation. Quality of the grid is to affect the precision and efficiency of geophysical simulation. Computational domain grid generation is to be divided in accordance with certain principles, so coordinate system on the original problem is transformed to the new coordinate system to solve it. According to the practical problems to design different grid, such as grid should be more intensive in the field where gradient larger, and a new coordinate system should ensure the orthogonality near the boundary. So it’s more effective and convenient, Improve accuracy and ensure the stability of solutions to simplify boundary conditions, thereby reducing the required computing time and computer capacity. Therefore, the grid should be decided by the kind of medium.
Complexity of the medium in different geophysical fields for the physical structure, physical properties and spatial and temporal distribution is different, sometimes even decisive. The medium underground has two intrinsic properties, one is structural form in three-dimensional space; the other is medium in every directions is complexity of variation. Three-dimensional topography model is the approximation of the first properties– tridimensional. while the model parameters are used to describe the second media property - non-uniformity. Current geophysical problems of the complicated geological conditions are also reflected in these two areas, One is topography, the other is complex medium underground.
For the issue of irregular topography, taking into account the similarity with the fluid medium, this paper introduces to generate body-fitted grid to meet the needs of grid computing. Body-fitted grid is developed in CFD, in which grid lines as irregular borders to guarantee the exact description of the surface, also it allow grid spacing and angle adjustment, the body-fitted grid based on Poisson Equation method is generated with a smooth and inherent flexibility. In addition, Hilgenstock method is used to adjust the grid orthogonality of the boundary nodes to facilitate the realization of free boundary conditions.
For the issue of inhomogeneous media underground, the paper quotes adaptive grid technology. Adaptive grid generation technology to automatically adjust the distribution of media in the region use the dense low grid, high-speed region using the sparse grid, so that we can use the least amount of computation for maximum accuracy.
In summary, the proposed mesh generation algorithm has the following advantages:
1) Body-fitted grid can accurately describe the nature of the surface, so to avoid the error in interpolation method;
2) Grid in the free boundary has good orthogonality to facilitate the realization of free boundary conditions.
3) Adaptive grid generation technology enables to avoid instability in variable step grid;
4) Body-fitted grid of rectangular is a kind of structure grid, so to avoid facing complex difference scheme in the non-structural grid.
After the advisable grid generation of the geophysical model, the application of coordinate transformation method to achieve curved coordinate system is proposed, Numerical simulation of geophysical concept. Coordinate transformation method through the rectangular coordinate system of geophysical equations transform into curvilinear coordinate system grid to solve the rule, not only break the surface mapping method on the shape of the limit, but also avoid the complicated derivation of difference schemes, achieve the numerical calculation really in the curvilinear coordinate system. The uniform transformation relation between the two system make the method more universal, with a wide range of applications.
In this paper, the curvelinear coordinates and numerical simulation of geophysical processes and may experience the issues discussed and the basic potential of a DC field equations and reflection seismic equation gives the song an example coordinate system control equation derived.
Finally, the preparation of a automatic grid generation program, by importing the model parameters (including the model shape, size, distribution and other geophysical parameters data), the model automatically generate body-fitted grid, and Geophysics model gridgeneration is visual to ajust the grid distribution behavior manually, eventually export the grid coordinates and the required data of numerical simulation related with the coordinates.
引文
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