基于偏微分方程的复杂地质曲面光滑重构

发布时间：2018-03-18 05:21

本文选题：偏微分方程　切入点：曲面重构　出处：《成都理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：地质曲面重构是三维地质构造建模的重要基础,在地质矿产资源勘探等领域有着重要的应用。常用的地质曲面重构方法主要分为插值法和拟合法,包括三角剖分法、克里金插值法、加权最小二乘拟合法等。然而,由于地质条件的特殊性,地质勘探数据比较稀疏,分布不均匀,且包含大量的地质断层,包括正断层、垂直断层和逆断层等,上述方法在重构复杂地质曲面方面具有一定的局限性。基于此,本文对计算几何中的几何偏微分方程曲面造型方法在复杂地质曲面光滑重构中的应用展开了研究,提出一种基于几何偏微分方程的复杂地质曲面重构方法。该方法首先选取合适的偏微分方程,对其中涉及的几何微分算子进行离散化,并讨论离散格式的稳定性,在此基础上,对地质中某一工区数据构建满足偏微分方程离散格式的空间拓扑,并将地质断层作为约束边界条件,采用演化的思想迭代求解几何偏微分方程,将离散方程的稳态解近似看作几何偏微分方程的解去逼近原始曲面,构造离散化网格表达的含复杂断层约束的光滑地质曲面。本文对复杂断层约束下的地质曲面光滑重构问题展开研究,应用几何偏微分方程曲面造型技术,实现了基于不同形式网格表达的含复杂断层约束的光滑地质曲面重构。具体工作如下:对所选择的几何偏微分方程在矩形网格上进行离散化,构造相应的差分格式,并分析其稳定性,通过有限差分法迭代求解差分方程,将差分方程的稳态解近似看作对应偏微分方程的解,并将其作为原始曲面的逼近,重构了基于矩形网格表达的复杂地质曲面,该方法具有计算速度快、易于计算机编程实现等优势,缺点在于矩形网格无法适应地质断层多边形的任意拓扑结构,重构的复杂地质曲面在断层约束处处理不够。为适应地质断层多边形的任意拓扑形状,对地质采样数据进行三角网拓扑构建。首先将空间散点数据投影到二维平面,加入断层约束条件后进行约束Delaunay三角网剖分,然后将平面三角网映射到三维空间,实现空间拓扑构建。通过在三角网上构建微分算子的离散格式,离散化求解所选取的几何偏微分方程,重构了基于三角网表达的复杂多约束地质曲面。基于三角网表达的方法解决了断层约束处的曲面重构问题,但是微分算子离散格式的稳定性条件比较苛刻,且计算速度比较慢。结合矩形网格和三角网格上偏微分方程曲面重构方法的优势,提出了一种基于混合网格表达的复杂地质曲面光滑重构方法。在非断层区域,通过矩形网格上的有限差分法快速迭代求解几何偏微分方程,使重构曲面迅速达到指定光滑度;在断层约束区域,通过在三角网上求解几何偏微分方程,重构适应断层拓扑的地质曲面。这种方式实现了含复杂断层约束的地质曲面的快速光滑重构,同时也保持了断层约束的特性。以实际勘探数据为例,对本文提出的三种地质曲面重构方法进行了验证和对比分析,应用实例表明,这三种方法充分考虑了地质曲面中各种断层等特殊情况,可以重构含复杂断层约束的地质曲面,并各有其特点。这种基于偏微分方程的复杂地质曲面光滑重构方法有望在地质领域展开更深入的研究和应用。
[Abstract]:Geologic surface reconstruction is an important basis for 3D geological structure modeling, have important applications in the field of Geology and mineral resources exploration. The geological surface reconstruction methods mainly consists of interpolation and fitting, including triangulation method, Kriging method, weighted least square fitting. However, due to the special geological conditions the geological exploration data, relatively sparse, uneven distribution, and contains large geological fault, including normal faults, vertical faults and reverse faults, the method has certain limitations in the reconstruction of complex geological surfaces. Based on this, this paper focuses on the application of computational geometry modeling method in geometric partial differential equation in curved surface complex geological smooth surface reconstruction, proposed a reconstruction method for complex geological surface geometry based on partial differential equations. This method first selects the appropriate partial differential equation of The geometric differential operator involves discretization, and discuss the stability of discrete format, on this basis, a project data in geological construction satisfies the partial differential equation discretization and spatial topology, and geological fault as boundary conditions, the evolution of the thought of iteration for solving geometric partial differential equations, the discrete equations of steady state the approximate solution to the original surface approximation solution as geometric partial differential equations, constructing discrete grids with complex fault restraint smooth geological surface. This paper studies the geological surface of the complex fault under the constraint of smooth reconstruction, geometric partial differential equation of surface modeling technology, to realize the smooth surface reconstruction with complex geological fault constraints the expression of different forms based on grid. The specific work is as follows: the choice of geometric partial differential equation in a rectangular grid of discrete structure Make the corresponding difference scheme, and analyze the stability of differential equations by finite difference iterative method, the differential equations as approximate steady-state points corresponding to the solutions of the partial differential equation, and the approximation of the original surface, complex geological curved rectangular grid based on the expression of the reconstruction, the method of calculation speed fast, easy to program in computer and other advantages, disadvantage is that the rectangular grid cannot adapt to arbitrary topology geological fault polygon, complex geologic surface reconstruction in fault constraint processing is not enough. In order to adapt to arbitrary topology geological fault polygon, triangulation of data sampling. The topology construction of geological spatial data to two-dimensional projection scatter the plane, adding fault constraints after constrained Delaunay triangulation, then the triangulation network is mapped to 3D space, realize spatial topology construction through. Construction of discrete scheme of differential operators in triangular mesh, select the geometric discretization of partial differential equations, based on triangulation reconstruction expression of complex multi constrained geological surface triangulation method. Based on the expression of surface reconstruction to solve the problem of fault constraint, but the differential operator stability conditions for discrete scheme and the calculation speed is relatively harsh. Slow. With rectangular grid and triangular mesh surface reconstruction method of partial differential equation of the advantages of a complex geological surface hybrid grid expression smooth reconstruction method based on non fault region, the finite difference method on rectangular mesh fast iterative geometric partial differential equations, the reconstructed surface reaches the specified smoothness quickly; in fault region, through the triangular mesh for solving geometric partial differential equations, reconstruction of geological surface fault to topology. This way The geological surface with complex fault constrained fast smooth reconstruction, while maintaining the characteristics of fault constraints. In actual exploration data as an example, three kinds of geological surface reconstruction method proposed in this paper are verified and analyzed. The application examples show that the three methods give full consideration to the various faults and other geological surfaces in special circumstances, can the geological surface reconstruction with complex fault constraints, and each has its own characteristics. This is expected to complex geological surface partial differential equation method based on smooth reconstruction in the geological field carried out more in-depth research and application.

【学位授予单位】：成都理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：P628

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