第二小組邊界單元法

1、1-0Study of the BEM邊界元法學(xué)習(xí)邊界元法學(xué)習(xí)成員：高成路、郭焱旭、梅潔、李銘、金純、 劉克奇、 匡偉、 高松、 李崴1-1巖土工程的數(shù)值方法巖土工程的數(shù)值方法工程問(wèn)題數(shù)學(xué)模型偏微分方程的邊值問(wèn)題或初值問(wèn)題邊界積分方程問(wèn)題解析方法數(shù)值方法解析方法數(shù)值方法FDMFEMEFM其它BEM其它Key words about BEM Character advantage/ disadvantage Application and transformation of the BEM Basic concepts Development of the BEM Basic concepts

2、of the BEM 目錄目錄1-2Study of the BEMKey words1-3applicability 適用性 stress and deformation analysis 應(yīng)力和變形分析 integral statement 功互等定理 kernels 核函數(shù) quadratic elements 二次單元 discretization 離散化 approximation 近似值shape functions 形函數(shù) intrinsic coordinate 本征坐標(biāo)Gaussian quadrature 高斯正交 singularity 奇異性，奇異點(diǎn) Cauchy Pr

3、incipal Value 柯西主值. variational formulation 變分公式化，變分表述1-4numerical integration 數(shù)值積分 sparse and symmetric matrices 稀疏對(duì)稱矩陣 fully populated and asymmetric matrices 全充填非對(duì)稱矩陣Weighted residual principle 加權(quán)余量法 isoparametric elements 等參單元underground excavations 地下開(kāi)挖 fracturing processes 破裂過(guò)程 In-situ stress

4、原位應(yīng)力 permeability measurements 滲透性觀測(cè)coupled thermo-mechanical 熱力耦合material heterogeneity 材料各向異性Somiglianas identity 索米利亞納恒等式hybrid model 混合模型Key words1-5damage evolution processes 損傷演化過(guò)程 homogeneous and linearly elastic bodies. 各向同性線彈性體source densities 原密度 fracture analysis 斷裂分析 field point 場(chǎng)點(diǎn)global

5、 stiffness matrices 整體剛度矩陣 normal derivative 法向?qū)?shù) fracture propagation problems 裂隙傳播問(wèn)題borehole stability 鉆孔穩(wěn)定性 rock spalling 巖石開(kāi)裂 stress intensity factors(SIF)應(yīng)力強(qiáng)度因子 maximum tensile strength 最大抗拉強(qiáng)度microscopic 微觀的Key words1-6heat gradients 熱力梯度 sharp corners 鈍化邊角 degrees of freedom 自由度 potential func

6、tion 勢(shì)函數(shù) meshless technique 無(wú)單元技術(shù) moving least squares 移動(dòng)最小二乘法simplification of the integration 積分簡(jiǎn)化 least square method 最小二乘法analytical integration of domain integrals.積分域的解析解Fourier expansion of integrand functions.被積函數(shù)的傅里葉展開(kāi)higher order fundamental solutions.高階基本解the Dual Reciprocity Method (DRM)

7、.雙重互易法Key wordsKey words about BEM Character advantage/ disadvantage Application and transformation of the BEM Basic concepts Development of the BEM Basic concepts of the BEM 目錄目錄1-7Study of the BEMBasic concepts1-8Unlike the FEM and FDM methods, the BEM approach initially seeks a weak solution at t

8、he global level through an integral statement, based on Bettis reciprocal theorem and Somiglianas identity. For a linear elasticity problem with domain ; boundary of unit outward normal vector n ，and constant body force f , for example, the integral statement is written asi (8)The solution of the in

9、tegral Eq. (8) requires the following steps:1-9(1) Discretization of the boundary with a finite number of boundary elements. Basic concepts (9)1-10(2) Approximation of the solution of functions locally at boundary elements by (trial) shape functions, in a similar way to that used for FEM. The displa

10、cement and traction functions within each element are then expressed as the sum of their nodal values of the element nodes:Basic concepts (10)1-11Substitution of Eqs. (10) into (9) and forEq. (8) can be written in matrix form asBasic concepts (11) (12)1-12(3) Evaluation of the integrals Tij , Uij an

11、d Bi with point collocation method by setting the source point P at all boundary nodes successively.(4) Incorporation of boundary conditions and solution. Incorporation of the boundary conditions into the matrix Eq. (12) will lead to final matrix equationBasic concepts (14)1-13(5) Evaluation of disp

12、lacements and stresses inside the domain. For practical problems, it is often the stresses and displacements at some points inside the domain of interest that have special significance. Unlike the FEM in which the desired data are automatically produced at all interior and boundary nodes, whether so

13、me of them are needed or not, in BEM the displacement and stress values at any interior point, P, must be evaluated separately byBasic concepts (16)(15)Key words about BEM Character advantage/ disadvantage Application and transformation of the BEM Basic concepts Development of the BEM Basic concepts

14、 of the BEM 目錄目錄1-14Study of the BEM1-15The development of BEM In 1963, Jaswon and Symm gave the boundary integral equation method for solving potential problems. In 1967 , Rizzo and Cruse got the breakthrough for stress analysis in solids. In 1978, Cruse studied for fracture mechanics applications,

15、 based on Bettis reciprocal theorem (Betti, 1872) and Somiglianas identity in elasticity theory (Somigliana, 1885). In 1977 , Brebbia and Dominguez written the basic equations using the weighted residual principle. Watson (1976) gave the introduction of isoparametric elements using different orders

16、of shape functions in the same fashion as that in FEM, greatly enhanced the BEMs applicability for stress analysis problems. 1-16 Crouch and Fairhurst (1973), Brady and Bray (1978) taken most notable original developments of BEM application in the field of rock mechanics. In the early 80s, Pan and M

17、aier (1997), Elzein (2000) and Ghassemi started to concern BEM formulations for coupled thermo-mechanical and hydro-mechanical processes. Kuriyama and Mizuta (1993), Kuriyama (1995) and Cayol and Cornet (1997) reported 3-D applications due to the BEMs advantage in reducing model dimensions, especial

18、ly using DDM for stress and deformation analysis.The development of BEMKey words about BEM Character advantage/ disadvantage Application and transformation of the BEM Basic concepts Development of the BEM Basic concepts of the BEM 目錄目錄1-17Study of the BEM1-18advantageThe main advantage of the BEM is

19、 the reduction of the computational model dimension by one, with much simpler mesh generation and therefore input data preparation, compared with full domain discretization methods such as the FEM and FDM. The BEM is often more accurate than the FEM and FDM, due to its direct integral formulation.優(yōu)點(diǎn)

20、: 降低求解問(wèn)題的維數(shù), 3D問(wèn)題變?yōu)?D問(wèn)題, 2D變?yōu)?D問(wèn)題. 具有較高的精度, 原因: 僅僅對(duì)邊界進(jìn)行離散, 域內(nèi)點(diǎn)的值采用邊界上的已知量計(jì)算得到.1-19disadvantagethe BEM is not as efficient as the FEM in dealing with material heterogeneity, because it cannot have as many sub-domains as elements in the FEM. The BEM is also not as efficient as the FEM in simulating no

21、n-linear material behaviour, such as plasticity and damage evolution processes, because domain integrals are often presented in these problems. Key words about BEM Character advantage/ disadvantage Application and alternative formulation of the BEM Basic concepts Development of the BEM Basic concept

22、s of the BEM 目錄目錄1-20Study of the BEM1-21Application of BEM Fracture analysis with BEMTo apply standard direct BEM for fracture analysis, the fractures must be assumed to have two opposite surfaces, except at the apex of the fracture tip where special singular tip elements must be used. Denote c as

23、the path of the fractures in the domain with its two opposite surfaces represented by c+ and c- , respectively, Somiglianas identity (when the field point is on the boundary) can be written as (17)1-22Two new techniques were proposed for fracture analysis with BEM. The first one is Dual Boundary Ele

24、ment Method (DBEM), which was first presented by Portela (1992) ,and was extended to 3-D crack growth problems by Mi and Aliabadi(1992, 1994). The essence of this technique is to apply displacement boundary equations at one surface of a fracture element and traction boundary equations at its opposit

25、e surface, although the two opposing surfaces occupy practically the same space in the model. The general mixed mode fracture analysis can be performed naturally in a single domain.DBEMFracture analysis with BEM1-23The second one is DDM. The DDM has been widely applied to simulate fracturing process

26、es in fracture mechanics in general and in rock fracture propagation problems in particular due to the advantage that the fractures can be represented by single fracture elements without need for separate representation of their two opposite surfaces, as should be done in the direct BEM solutions. D

27、DMFracture analysis with BEM1-24Application of BEM Fracture analysis with BEMBut there are still great boundedness in analyzing fracturing processes using BEM, especially for rock mechanics problems. On the one hand, what happens exactly at the fracture tips in rocks still remains to be adequately u

28、nderstood, On the other hand, complex numerical manipulations are still needed for re-meshing following the fracture growth process so that the tip elements are added to where new fracture tips are predicted. Due to the above difficulties, fracture growth analyses in rock mechanics have not been wid

29、ely applied.Key words about BEM Character advantage/ disadvantage Alternative formulation of the BEM Basic concepts Development of the BEM Basic concepts of the BEM 目錄目錄1-25Study of the BEM1-26Alternative formulations associated with BEMThe standard BEM, DBEM and DDM as presented above have a common

30、 feature: the final coefficient matrices of the equations are fully populated and asymmetric, due to the traditional nodal collocation technique. This makes the storage of the global coefficient matrix and solution of the final equation system less efficient, compared with FEM. And this method needs

31、 special treatment for the problem with sharp corners on the boundary surfaces (curves) or at the fracture intersections, and artificial corner smoothing, additional nodes or special corner elements are usually the techniques applied to solve this particular difficulty.1-27Galerkin Boundary Element

32、MethodThe GBEM produces a symmetric coefficient matrix by multiplying the traditional boundary integral by a weighted trail function and integrates it with respect to the source point on the boundary for a second time, in a Galerkin sense of weighted residual formulation. (19)1-28The GBEM is an attr

33、active approach due to the symmetry of its final system equation, which paves the way for the variational formulation of BEM for solving non-linear problems.Galerkin Boundary Element Method1-29Boundary Contour MethodThe Boundary Contour Method (BCM) involves rearranging the standard BEM integral Eq.

34、 (8) so that the difference of the two integrals appearing on the right-hand side of Eq. (8) can be represented by a vector function Fi= Uij*tj tij*uj which is divergence free (8)(22)1-30The BCM approach is attractive mainly because of its further reduction of computational model dimensions and simp

35、lification of the integration. The savings in preprocessing of the simulations are clear. Treatment of fractures and material non-homogeneity has not been studied in BCM; these may limit its applications to rock mechanics problems considering the present state-of the-art.Boundary Contour Method1-31B

36、oundary Node MethodThe method is a combination of traditional BEM with a meshless technique using the moving least squares for establishing trial functions without an explicit mesh of boundary elements. It further simplifies the mesh generation tasks. Its applications concentrate on shape sensitivity analysis at present and solution of potential problems, but can be extended to general geomechanics problems, especially ground