徑向基函數(shù)無網(wǎng)格配點法及其在巖石力學中的應(yīng)用研究
發(fā)布時間：2019-01-27 12:21
【摘要】：由于巖石天然狀態(tài)的復(fù)雜性，人們面對實際工程問題時很難得到解析解，，需要借助數(shù)值方法進行求解。無網(wǎng)格法作為一種新興的數(shù)值方法，在處理復(fù)雜問題時，能避免傳統(tǒng)有限元方法難以解決的許多難題。徑向基函數(shù)法作為無網(wǎng)格法的一種，因其具有指數(shù)級收斂速度、形式簡單、各向同性等優(yōu)點備受矚目。其形函數(shù)本身具備無窮階可導(dǎo)且連續(xù)的性質(zhì)，在求解偏微分方程時非常適合結(jié)合配點法等強形式算法進行計算，且不需要背景網(wǎng)格進行區(qū)域積分，能大大降低計算時間。 然而，現(xiàn)有的關(guān)于徑向基函數(shù)配點法的研究工作主要集中在算法本身的收斂性和求解邊界值問題等方面，對于其求解動力問題的穩(wěn)定性分析和非連續(xù)介質(zhì)問題的應(yīng)用研究較少。本論文基于von Neumann法提出了一種新的徑向基函數(shù)配點法求解動力問題的穩(wěn)定性評估算法，并將徑向基函數(shù)配點法應(yīng)用于非連續(xù)巖石結(jié)構(gòu)承受靜力和動力荷載問題。 本文的主要研究工作如下： 1.推導(dǎo)了基于von Neumann法的徑向基函數(shù)配點法求解動力問題的穩(wěn)定性分析算法，定義了具體的穩(wěn)定性參數(shù)來定量地對實際計算時如何選擇合適的時間步長進行指導(dǎo)，并通過該參數(shù)對影響徑向基函數(shù)配點法穩(wěn)定性的各個因素進行了詳細分析和討論，對實際算例中出現(xiàn)的無條件不穩(wěn)定情形的原因進行了分析，研究了徑向基函數(shù)配點法求解動力問題穩(wěn)定性的主要影響因素并給出了如何合理地選取徑向基函數(shù)形函數(shù)形狀參數(shù)及點距以提高計算穩(wěn)定性的結(jié)論。 2.將徑向基函數(shù)配點法應(yīng)用于裂紋結(jié)構(gòu)靜力問題的求解，推導(dǎo)了徑向基函數(shù)通過強形式配點法用于求解任意分布多裂紋結(jié)構(gòu)承受復(fù)雜應(yīng)力作用下裂紋結(jié)構(gòu)的算法流程，建立了求解方程組，編寫了FORTRAN靜力計算程序,并對其計算結(jié)果進行了驗證。 3.將徑向基函數(shù)配點法應(yīng)用于裂紋結(jié)構(gòu)承受動力荷載問題的求解，推導(dǎo)了徑向基函數(shù)配點法求解動力荷載作用下裂紋結(jié)構(gòu)的算法流程，建立了求解方程組，編寫了FORTRAN動力計算程序，并對其計算結(jié)果進行了驗證。 4.通過應(yīng)力外推法計算了徑向基函數(shù)配點法數(shù)值解的應(yīng)力強度因子，并以應(yīng)力強度因子為指標，定量分析了靜力問題中不同裂紋長度對應(yīng)力強度因子的影響以及動力問題中不同外荷載頻率對應(yīng)力強度因子放大率的影響，其結(jié)論對實際工程施工時的結(jié)構(gòu)安全評估具有一定的參考價值。
[Abstract]:Because of the complexity of the natural state of rock, it is difficult to obtain an analytical solution when people are faced with practical engineering problems, which need to be solved by numerical method. As a new numerical method, meshless method can avoid many difficult problems which are difficult to solve by traditional finite element method in dealing with complex problems. As a meshless method, radial basis function (RBF) method has attracted much attention for its advantages of exponential convergence rate, simple form and isotropy. The shape function itself has infinitely differentiable and continuous properties, so it is very suitable to solve partial differential equations with collocation method and other strong form algorithms, and it does not need background mesh for domain integration, which can greatly reduce the computation time. However, the existing researches on the radial basis function collocation method mainly focus on the convergence of the algorithm itself and the solution of boundary value problems, etc. There are few researches on the stability analysis of the radial basis function collocation method and the application of the discontinuous medium problem to the solution of the dynamic problem. In this paper, a new radial basis function collocation method is proposed to evaluate the stability of dynamic problems based on the von Neumann method, and the radial basis function collocation method is applied to the static and dynamic load problems of discontinuous rock structures. The main work of this paper is as follows: 1. The stability analysis algorithm of radial basis function collocation method based on von Neumann method for solving dynamic problems is derived, and specific stability parameters are defined to guide how to choose appropriate time step in practical calculation. The factors affecting the stability of radial basis function collocation method are analyzed and discussed in detail, and the causes of unconditional instability in practical examples are analyzed. The main factors affecting the stability of the radial basis function collocation method for solving dynamic problems are studied and the conclusion of how to select the radial basis function shape parameters and the distance between points to improve the stability is given. 2. The radial basis function collocation method is applied to solve the static problem of cracked structure. The algorithm flow of radial basis function (RBF) is derived by using the strong form collocation method to solve the crack structure with arbitrary distribution and multiple cracks subjected to complex stress. The equations are solved, the FORTRAN static calculation program is written, and the calculation results are verified. 3. The radial basis function collocation method is applied to solve the problem of crack structure subjected to dynamic load. The algorithm flow of radial basis function collocation method for solving crack structure under dynamic load is derived, and the solving equations are established. The FORTRAN dynamic calculation program is written and the calculation results are verified. 4. The stress intensity factor of the radial basis function collocation method is calculated by the stress extrapolation method, and the stress intensity factor is taken as the index. The influence of crack length on stress intensity factor in static problem and the effect of different frequency of external load on stress intensity factor in dynamic problem are analyzed quantitatively. The conclusion has certain reference value for the structural safety evaluation in actual engineering construction.
【學位授予單位】：上海交通大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：TU45
本文編號：2416254
[Abstract]:Because of the complexity of the natural state of rock, it is difficult to obtain an analytical solution when people are faced with practical engineering problems, which need to be solved by numerical method. As a new numerical method, meshless method can avoid many difficult problems which are difficult to solve by traditional finite element method in dealing with complex problems. As a meshless method, radial basis function (RBF) method has attracted much attention for its advantages of exponential convergence rate, simple form and isotropy. The shape function itself has infinitely differentiable and continuous properties, so it is very suitable to solve partial differential equations with collocation method and other strong form algorithms, and it does not need background mesh for domain integration, which can greatly reduce the computation time. However, the existing researches on the radial basis function collocation method mainly focus on the convergence of the algorithm itself and the solution of boundary value problems, etc. There are few researches on the stability analysis of the radial basis function collocation method and the application of the discontinuous medium problem to the solution of the dynamic problem. In this paper, a new radial basis function collocation method is proposed to evaluate the stability of dynamic problems based on the von Neumann method, and the radial basis function collocation method is applied to the static and dynamic load problems of discontinuous rock structures. The main work of this paper is as follows: 1. The stability analysis algorithm of radial basis function collocation method based on von Neumann method for solving dynamic problems is derived, and specific stability parameters are defined to guide how to choose appropriate time step in practical calculation. The factors affecting the stability of radial basis function collocation method are analyzed and discussed in detail, and the causes of unconditional instability in practical examples are analyzed. The main factors affecting the stability of the radial basis function collocation method for solving dynamic problems are studied and the conclusion of how to select the radial basis function shape parameters and the distance between points to improve the stability is given. 2. The radial basis function collocation method is applied to solve the static problem of cracked structure. The algorithm flow of radial basis function (RBF) is derived by using the strong form collocation method to solve the crack structure with arbitrary distribution and multiple cracks subjected to complex stress. The equations are solved, the FORTRAN static calculation program is written, and the calculation results are verified. 3. The radial basis function collocation method is applied to solve the problem of crack structure subjected to dynamic load. The algorithm flow of radial basis function collocation method for solving crack structure under dynamic load is derived, and the solving equations are established. The FORTRAN dynamic calculation program is written and the calculation results are verified. 4. The stress intensity factor of the radial basis function collocation method is calculated by the stress extrapolation method, and the stress intensity factor is taken as the index. The influence of crack length on stress intensity factor in static problem and the effect of different frequency of external load on stress intensity factor in dynamic problem are analyzed quantitatively. The conclusion has certain reference value for the structural safety evaluation in actual engineering construction.
【學位授予單位】：上海交通大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：TU45
【參考文獻】
相關(guān)期刊論文 前10條
1 苗紅宇,張雄,陸明萬;分階擬合直接配點無網(wǎng)格法[J];工程力學;2003年05期
2 李光耀,鐘志華,韓旭;固體力學問題數(shù)值解的一種驗證方法[J];工程力學;2004年03期
3 葛東云,陸明萬;波在各向異性介質(zhì)中傳播規(guī)律的無網(wǎng)格法數(shù)值模擬[J];工程力學;2004年05期
4 周小平,周瑞忠;無單元法研究現(xiàn)狀及展望[J];工程力學;2005年01期
5 龍述堯,胡德安,熊淵博;用無單元伽遼金法求解幾何非線性問題[J];工程力學;2005年03期
6 張希;姚振漢;;無網(wǎng)格彼得洛夫伽遼金法在大變形問題中的應(yīng)用[J];工程力學;2006年S1期
7 ;ELASTO-PLASTICITY ANALYSIS BASED ON COLLOCATION WITH THE MOVING LEAST SQUARE METHOD[J];Acta Mechanica Solida Sinica;2003年02期
8 李臥東,王元漢,譚國煥;無網(wǎng)格法在彈塑性問題中的應(yīng)用[J];固體力學學報;2001年04期
9 陳文;;奇異邊界法:一個新的、簡單、無網(wǎng)格、邊界配點數(shù)值方法[J];固體力學學報;2009年06期
10 龐作會,朱岳明;無網(wǎng)格伽遼金法(EFGM)求解接觸問題[J];河海大學學報(自然科學版);2000年04期
相關(guān)會議論文 前1條
1 陳勇;黃建明;陸明萬;;地震波傳播計算中的無網(wǎng)格方法[A];1999年中國地球物理學會年刊——中國地球物理學會第十五屆年會論文集[C];1999年
相關(guān)博士學位論文 前1條
1 王飛;位移不連續(xù)法及其在巖體工程中的應(yīng)用[D];上海交通大學;2010年
相關(guān)碩士學位論文 前2條
1 吳少平;固體力學中單位分解方法的研究[D];浙江大學;2007年
2 馮亭;固體力學中的自適應(yīng)無網(wǎng)格方法計算研究[D];浙江大學;2007年
本文編號：2416254
本文鏈接：http://sikaile.net/kejilunwen/sgjslw/2416254.html
