發(fā)布時(shí)間：2018-03-18 05:21

  本文選題：偏微分方程　切入點(diǎn)：曲面重構(gòu)　出處：《成都理工大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：地質(zhì)曲面重構(gòu)是三維地質(zhì)構(gòu)造建模的重要基礎(chǔ),在地質(zhì)礦產(chǎn)資源勘探等領(lǐng)域有著重要的應(yīng)用。常用的地質(zhì)曲面重構(gòu)方法主要分為插值法和擬合法,包括三角剖分法、克里金插值法、加權(quán)最小二乘擬合法等。然而,由于地質(zhì)條件的特殊性,地質(zhì)勘探數(shù)據(jù)比較稀疏,分布不均勻,且包含大量的地質(zhì)斷層,包括正斷層、垂直斷層和逆斷層等,上述方法在重構(gòu)復(fù)雜地質(zhì)曲面方面具有一定的局限性�；诖�,本文對(duì)計(jì)算幾何中的幾何偏微分方程曲面造型方法在復(fù)雜地質(zhì)曲面光滑重構(gòu)中的應(yīng)用展開(kāi)了研究,提出一種基于幾何偏微分方程的復(fù)雜地質(zhì)曲面重構(gòu)方法。該方法首先選取合適的偏微分方程,對(duì)其中涉及的幾何微分算子進(jìn)行離散化,并討論離散格式的穩(wěn)定性,在此基礎(chǔ)上,對(duì)地質(zhì)中某一工區(qū)數(shù)據(jù)構(gòu)建滿足偏微分方程離散格式的空間拓?fù)?并將地質(zhì)斷層作為約束邊界條件,采用演化的思想迭代求解幾何偏微分方程,將離散方程的穩(wěn)態(tài)解近似看作幾何偏微分方程的解去逼近原始曲面,構(gòu)造離散化網(wǎng)格表達(dá)的含復(fù)雜斷層約束的光滑地質(zhì)曲面。本文對(duì)復(fù)雜斷層約束下的地質(zhì)曲面光滑重構(gòu)問(wèn)題展開(kāi)研究,應(yīng)用幾何偏微分方程曲面造型技術(shù),實(shí)現(xiàn)了基于不同形式網(wǎng)格表達(dá)的含復(fù)雜斷層約束的光滑地質(zhì)曲面重構(gòu)。具體工作如下:對(duì)所選擇的幾何偏微分方程在矩形網(wǎng)格上進(jìn)行離散化,構(gòu)造相應(yīng)的差分格式,并分析其穩(wěn)定性,通過(guò)有限差分法迭代求解差分方程,將差分方程的穩(wěn)態(tài)解近似看作對(duì)應(yīng)偏微分方程的解,并將其作為原始曲面的逼近,重構(gòu)了基于矩形網(wǎng)格表達(dá)的復(fù)雜地質(zhì)曲面,該方法具有計(jì)算速度快、易于計(jì)算機(jī)編程實(shí)現(xiàn)等優(yōu)勢(shì),缺點(diǎn)在于矩形網(wǎng)格無(wú)法適應(yīng)地質(zhì)斷層多邊形的任意拓?fù)浣Y(jié)構(gòu),重構(gòu)的復(fù)雜地質(zhì)曲面在斷層約束處處理不夠。為適應(yīng)地質(zhì)斷層多邊形的任意拓?fù)湫螤?對(duì)地質(zhì)采樣數(shù)據(jù)進(jìn)行三角網(wǎng)拓?fù)錁?gòu)建。首先將空間散點(diǎn)數(shù)據(jù)投影到二維平面,加入斷層約束條件后進(jìn)行約束Delaunay三角網(wǎng)剖分,然后將平面三角網(wǎng)映射到三維空間,實(shí)現(xiàn)空間拓?fù)錁?gòu)建。通過(guò)在三角網(wǎng)上構(gòu)建微分算子的離散格式,離散化求解所選取的幾何偏微分方程,重構(gòu)了基于三角網(wǎng)表達(dá)的復(fù)雜多約束地質(zhì)曲面�；谌蔷W(wǎng)表達(dá)的方法解決了斷層約束處的曲面重構(gòu)問(wèn)題,但是微分算子離散格式的穩(wěn)定性條件比較苛刻,且計(jì)算速度比較慢。結(jié)合矩形網(wǎng)格和三角網(wǎng)格上偏微分方程曲面重構(gòu)方法的優(yōu)勢(shì),提出了一種基于混合網(wǎng)格表達(dá)的復(fù)雜地質(zhì)曲面光滑重構(gòu)方法。在非斷層區(qū)域,通過(guò)矩形網(wǎng)格上的有限差分法快速迭代求解幾何偏微分方程,使重構(gòu)曲面迅速達(dá)到指定光滑度;在斷層約束區(qū)域,通過(guò)在三角網(wǎng)上求解幾何偏微分方程,重構(gòu)適應(yīng)斷層拓?fù)涞牡刭|(zhì)曲面。這種方式實(shí)現(xiàn)了含復(fù)雜斷層約束的地質(zhì)曲面的快速光滑重構(gòu),同時(shí)也保持了斷層約束的特性。以實(shí)際勘探數(shù)據(jù)為例,對(duì)本文提出的三種地質(zhì)曲面重構(gòu)方法進(jìn)行了驗(yàn)證和對(duì)比分析,應(yīng)用實(shí)例表明,這三種方法充分考慮了地質(zhì)曲面中各種斷層等特殊情況,可以重構(gòu)含復(fù)雜斷層約束的地質(zhì)曲面,并各有其特點(diǎn)。這種基于偏微分方程的復(fù)雜地質(zhì)曲面光滑重構(gòu)方法有望在地質(zhì)領(lǐng)域展開(kāi)更深入的研究和應(yīng)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.

【學(xué)位授予單位】：成都理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：P628

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本文編號(hào)：1628189