第二小组边界单元法

1、1-0Study of the BEM边界元法学习边界元法学习成员：高成路、郭焱旭、梅洁、李铭、金纯、 刘克奇、 匡伟、 高松、 李崴1-1岩土工程的数值方法岩土工程的数值方法工程问题数学模型偏微分方程的边值问题或初值问题边界积分方程问题解析方法数值方法解析方法数值方法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 目录目录1-2Study of the BEMKey words1-3applicability 适用性 stress and deformation analysis 应力和变形分析 integral statement 功互等定理 kernels 核函数 quadratic elements 二次单元 discretization 离散化 approximation 近似值shape functions 形函数 intrinsic coordinate 本征坐标Gaussian quadrature 高斯正交 singularity 奇异性，奇异点 Cauchy Pr

3、incipal Value 柯西主值. variational formulation 变分公式化，变分表述1-4numerical integration 数值积分 sparse and symmetric matrices 稀疏对称矩阵 fully populated and asymmetric matrices 全充填非对称矩阵Weighted residual principle 加权余量法 isoparametric elements 等参单元underground excavations 地下开挖 fracturing processes 破裂过程 In-situ stress

4、原位应力 permeability measurements 渗透性观测coupled thermo-mechanical 热力耦合material heterogeneity 材料各向异性Somiglianas identity 索米利亚纳恒等式hybrid model 混合模型Key words1-5damage evolution processes 损伤演化过程 homogeneous and linearly elastic bodies. 各向同性线弹性体source densities 原密度 fracture analysis 断裂分析 field point 场点global

5、 stiffness matrices 整体刚度矩阵 normal derivative 法向导数 fracture propagation problems 裂隙传播问题borehole stability 钻孔稳定性 rock spalling 岩石开裂 stress intensity factors(SIF)应力强度因子 maximum tensile strength 最大抗拉强度microscopic 微观的Key words1-6heat gradients 热力梯度 sharp corners 钝化边角 degrees of freedom 自由度 potential func

6、tion 势函数 meshless technique 无单元技术 moving least squares 移动最小二乘法simplification of the integration 积分简化 least square method 最小二乘法analytical integration of domain integrals.积分域的解析解Fourier expansion of integrand functions.被积函数的傅里叶展开higher order fundamental solutions.高阶基本解the Dual Reciprocity Method (DRM)

7、.双重互易法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 目录目录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 目录目录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 目录目录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.优点

20、: 降低求解问题的维数, 3D问题变为2D问题, 2D变为1D问题. 具有较高的精度, 原因: 仅仅对边界进行离散, 域内点的值采用边界上的已知量计算得到.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 目录目录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 目录目录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