General Formulation Of Conventional Numerical Methods
Appendix AGeneral formulation ofconventional numerical methodsA.1IntroductionThe general formulation of continuum and discontinuum methods, as well as the simulation of fracture process using different numerical tecniques, described in this appendix,is based on the previous work made by Jing (2003).A.2Numerical methods in rock engineeringIn rock engineering several modelling methods have been developed for the study anddesign of rock engineering structures. In the literature these approaches are divided intotwo main groups which differ for the type of representation of the problem used.In the discrete representation, engineering problems are represented by an adequatemodel using a finite number of well-defined components. The behaviour of such components is either well known, or can be independently treated mathematically. Theglobal behaviour of the system can be determined through well-defined inter-relationsbetween the individual components (elements). In such problems, termed discrete, thediscrete representation and solution of such systems by numerical methods are usuallystraightforward.In other problems, the definition of such independent components may require an infinite sub-division of the problem domain, and the problem can only be treated using the251
252General formulation of conventional numerical methodsmathematical assumption of an infinitesimal element, implying in theory an infinite number of components. This usually leads to differential equations to describe the systembehaviour at the field points. Such systems are termed continuous representation and haveinfinite degrees of freedom. To solve such a continuous problem by numerical methods,the problem domain is usually subdivided into a finite number of sub-domains (elements) whose behaviour is approximated by simpler mathematical descriptions with finite degrees of freedom. These sub-domains must satisfy both the governing differentialequations of the problem and the continuity condition at their interfaces with adjacentelements. This is the so-called discretization of a continuum. It is an approximation ofa continuous system with infinite degrees of freedom by a discrete system with finitedegrees of freedom. The continuum assumption implies that at all points in a problemdomain, the materials cannot be torn open or broken into pieces. Of course, at the microscopic scale, all materials are discrete systems. However, representing the microscopiccomponents individually is intractable mathematically and unnecessary in practice.The concepts of continuum and discontinuum are therefore not absolute but relative,depending especially on the problem scales. Due to the differences in the underlyingmaterial assumptions, different numerical methods have been developed for continuousand discrete systems. The most commonly applied numerical methods for rock mechanics problems are:continuum methods F INITE D IFFERENCE M ETHOD (FDM). The FDM method is the oldest numericalmethod, and it is the basis of the explicit approach of the DEMs. The FDM is adirect approximation of the governing Partial Differential Equations (PDEs) by replacing partial derivatives with differences at regular or irregular grids imposedover problem domains. The solution of the system equation is obtained after imposing the initial and boundary conditions; F INITE E LEMENT M ETHOD (FEM). The FEM is perhaps the most widely appliednumerical method in engineering because its flexibility in handling material heterogeneity, non-linearity and boundary conditions. It is also the basis of the implicit approach of the DEM. The FEM requires the division of the problem domaininto sub-domains (elements) of standard shapes (triangle, quadrilateral, tetrahedral, etc.) with fixed number of nodes at the vertices and/or on the sides. Polynomial functions are used to approximate the behaviour of PDEs at the elementlevel and generate the local algebraic equations representing the behaviour of theelements. The local elemental equations are then assembled, according to the topologic relations between the nodes and elements, into a global system of algebraic
A.2 Numerical methods in rock engineering253equations whose solution then produces the required information in the solutiondomain, after imposing the properly defined initial and boundary conditions; B OUNDARY E LEMENT M ETHOD (BEM). The BEM requires discretization at theboundary of the solution domains only. The information required in the solutiondomain is separately calculated from the information on the boundary, which isobtained by solution of a boundary integral equation, instead of direct solutionof the PDEs, as in the FDM and FEM. It has greater accuracy over the FDM andFEM at the same level of discretization and is also the most efficient technique forfracture propagation analysis.discontinuum methods D ISCRETE E LEMENT M ETHOD (DEM). The DEM for modelling a discontinuum focuses mostly on applications in the fields of fractured geological media. The DEMrepresents the fractured medium as assemblages of blocks connected by fractures.The equations of motion of these blocks are solved through continuous detectionand treatment of contacts between the blocks. The blocks can be rigid or be deformable with FDM or FEM discretizations. Large displacements caused by rigidbody motion of individual blocks, including block rotation, fracture opening andcomplete detachments is straightforward in the DEM, but impossible in the FDM,FEM or BEM; D ISCRETE F RACTURE N ETWORK (DFN) method. The DFN method is an alternative DEM for fluid flow in fractured rock masses. It simulates fluid flow throughconnected fracture networks, with the matrix permeability either ignored or approximated by simple means. The stress and deformation of the fractures are generally ignored as well. This method is conceptually attractive for simulating fluidflow in fractured rocks when the permeability of the rock matrix is low comparedto that of the fractures.hybrid continuum/discontinuum models Hybrid FEM/BEM; Hybrid DEM/DEM; Hybrid FEM/DEM, and Other hybrid models.
General formulation of conventional numerical methods254A.3Continuum methodsA.3.1F INITE D IFFERENCE M ETHOD (FDM)The FDM is the oldest numerical method to obtain approximate solutions to PDEs inengineering. The basic concept of FDM is to replace the partial derivatives of the objective function (e.g. displacement) by differences defined over certain spatial intervals inthe coordinate directions x, y, 4z, which yields a system of algebraic simultaneousequations of the objective functions at a grid (mesh) of nodes over the domain of interest(Figure A.1) (Wheel, 1996). Solution of the simultaneous algebraic system equations,incorporating boundary conditions defined at boundary nodes, will then produce therequired values of the objective function at all nodes, which satisfy both the governingPDFs and specified boundary conditions. The conventional FDM utilizes a regular gridof nodes, such as a rectangular grid as shown in Figure A.1a.Using a standard FDM scheme, the so-called 5-point difference scheme (Figure A.1b),the resultant FDM equation at grid node (i, j) will be expressed as combinations of function values at its four surrounding nodes. For a Navier equation of equilibrium for elasticsolids in 2D, the FDM equation of equilibrium at point (i, j) is given asi,ji 1,ji,ji 1,ju x a1 u xuy b1 uyi,j 1 a2 u xi,j 1 b2 uyi,j 1 a3 u xi,j 1 b3 uyi 1,j a4 u xi 1,j b4 uyi 1,j 1 a5 u xi 1,j 1 b5 uyi,j a6 Fxi,j b6 Fy(A.1)where coefficients ak and bk (k 1, 2, ., 6) are functions of the grid intervals x andi,ji,j y and the elastic properties of the solids, and Fx and Fy are the body forces lumpedat point (i, j), respectively. Assembly of similar equations at all grid points will yield aglobal system of algebraic equations whose solution can be obtained by direct or iterativemethods. FDM schemes can also be applied in the time domain with properly chosentime steps, t, so that function values at time t can be inferred from values at t t.The fundamental nature of FDM is the direct discretization of the governing PDEs byreplacing the partial derivatives with differences defined at neighboring grid points. Thegrid system is only a convenient way of generating objective function values at samplingpoints with small enough intervals between them, so that errors thus introduced aresmall enough to be acceptable. No local trial (or interpolation) functions are employedto approximate the PDE in the neighborhoods of the sampling points, as is done in FEMand BEM.The conventional FDM with regular grid systems does suffer from shortcomings,most of all in its inflexibility in dealing with fractures, complex boundary conditionsand material inhomogeneity. This makes the standard FDM generally unsuitable for
the governing PDFs and specified boundary conditions.The conventional FDM utilizes a regular grid of nodes,such as a rectangular grid as shown in Fig. 8a.Using a standard FDM scheme, the so-called 5-pointdifference scheme (Fig. 8b), the resultant FDM equationat grid node ði; jÞ will be expressed as combinations offunction values at its four surrounding nodes. For aNavier equation of equilibrium for elastic solids in 2-D,the FDM equation of equilibrium at point ði; jÞ is givenasshortcomings, most of all in its inflexibility in dealingwith fractures, complex boundary conditions andmaterial inhomogeneity. This makes the standardFDM generally unsuitable for modelling practical rockmechanics problems. However, significant progress hasbeen made in the FDM so that irregular meshes, such asquadrilateral grids (Perrone and Kao, 1975, [10]) andthe Voronoi grids (Brighi et al., 1998, [11]) can also beused. Although such irregular meshes can enhance theapplicability of the FDM for rock mechanics problems,however, the most significant improvement comes fromthe so-called Control Volume or Finite Volumeapproaches.255i 1;jui;jþ a2 ui;j 1þ a3 ui;jþ1þ a4 uiþ1;jx ¼ a1 uxxxxþ a5 uiþ1;jþ1þ a6 Fxi;j ;x A.3 Continuummethods yCell 2Cell 3Cell 4klCell 1Pi,j 1i-1,ji,ji 1,ji,j-1Cell 5jiCell 6(a) xCell 8Cell 7(b)Fig. 8. (a) Regular quadrilateral grid for the FDM and (b) irregular quadrilateral grid for the FVM (after Wheel, 1995 [9]).Figure A.1: (a) Regular quadrilateral grid for the FDM and (b) irregular quadrilateralgrid for the FVM (Wheel, 1996).modeling practical rock mechanics problems. However, significant progress has beenmade in the FDM so that irregular meshes, such as quadrilateral grids (Perrone andKao, 1975) and the Voronoi grids (Brighi et al., 1998) can also be used. Although suchirregular meshes can enhance the applicability of the FDM for rock mechanics problems,however, the most significant improvement comes from the so-called Control Volume orFinite Volume approaches.A.3.1.1Finite volume approach of FDM and its application to stress analysisThe Finite Volume Method (FVM) is also a direct approximation of the PDEs, but inan integral sense. An elastostatic problem with body Ω, is divided into a finite number, N, of internal contiguous cells of arbitrary polyhedral (or polygonal in 2D cases)shape, called Control Volumes (CV), Ωk , with boundary Γk , of unit outward normal vector nik , k 1, 2, ., N. The boundary Γk of Ωk is comprised of a number, Mk , polygonalpside (faces or line segments), Γk , p 1, 2, ., Mk . Assuming isotropic, linear elasticityand using Gauss’ divergence theorem, the Navier–Cauchy equation of equilibrium interms of stress can be rewritten in terms of displacement as
General formulation of conventional numerical methods256N"Mk k 1ˆˆpp 1 Γ ktik dΓ #ΩkN"Mk f i dΩ ˆ#pp 1 Γ kk 1pσijk n j dΓ Fxk 0(A.2)where Fik ρgi V k is the body force vector of the CV of volume V k lumped at its center, ρ is the material density and gi is the body force intensity vector, such as gravityacceleration.The task is to formulate the integrals into algebraic functions of the displacementspat nodes defining the boundary sides Γk of Ωk , which vary with different grid schemes.For an unstructured quadrilateral grid system (Figure A.1b), a typical cell P (CV), withits center at node P, has four sides (ij, jk, kl, li ) and four nodes (i, j, k, l ), surrounded byeight neighboring cells with center nodes I, J, ., O. The integral terms in Equation A.2for the cell P are written in terms of displacement variables at the centers of cells (Wheel,1996), written asA p u x Ar urx B p uy Br ury F KxpprpC p uy rrCr ury pD p ux Dr urx F yK(A.3)where coefficients A p , Ar , B p , Br , C p , Cr , D p , Dr are functions of the cell geometry and theelastic properties of the solids, with r 1, 2, ., 8 running through the eight surroundingcells.The FDM/FVM approach is therefore as flexible as FEM in handling material inhomogeneity and mesh generation. As a branch of the FDM, the FVM can overcome theinflexibility of the grid generation and boundary conditions in the traditional FDM withunstructured grids of arbitrary shape.The FDM/FVM approaches are therefore specially suited to simulate non-linear behavior of solid materials. The reason is its special advantage of no-matrix-equationsolving formulation and data structure, so that integration of non-linear constitutiveequations is a straightforward computer implementation step, rather than iterative predictionmapping integration loops required in FEM.At present, the most well-known computer codes for stress analysis for non-linearrock engineering problems using the FVM/FDM approach is perhaps the FLAC codegroup (ITASCA Consulting Group, 1993b), with a vertex scheme of triangle or quadrilateral grids.
A.3 Continuum methodsA.3.2257F INITE E LEMENT M ETHOD (FEM)The FEM requires the division of the problem domain into a collection of sub-domains(elements) of smaller sizes and standard shapes (triangle, quadrilateral, tetrahedral, etc.)with fixed number of nodes at the vertices and/or on the sides, the discretization. Trialfunctions, usually polynomial, are used to approximate the behavior of PDEs at the element level and generate the local algebraic equations representing the behavior of theelements. The local elemental equations are then assembled, according to the topologicrelations between the nodes and elements, into a global system of algebraic equationswhose solution then produces the required information in the solution domain, afterimposing the properly defined initial and boundary conditions.The FEM has been the most popular numerical method in engineering sciences, including rock mechanics and rock engineering. Its popularity is largely due to its flexibility in handling material inhomogeneity and anisotropy, complex boundary conditionsand dynamic problems, together with moderate efficiency in dealing with complex constitutive models and fractures. All these merits were very appealing to researchers andpracticing engineers alike during early development in the 1960s and 1970s when themain numerical method in engineering analysis was the FDM with regular grids. Sincethen, the FEM method has been extended in many directions.Basically, three steps are required to complete an FEM analysis:1. domain discretization;2. local approximation;3. assemblage and solution of the global matrix equation.The domain discretization involves dividing the domain into a finite number of internalcontiguous elements of regular shapes defined by a fixed number of nodes (e.g., triangleelements with three nodes in 2D and brick elements with eight nodes in 3D). A basicassumption in the FEM is that the unknown function, uie over each element, can be apjproximated through a trial function of its nodal values of the system unknowns, ui , in apolynomial form. The trial function must satisfy the governing PDF and is given byuie M Nij uij(A.4)j 1where the Nij are often called the shape functions (or interpolation functions) defined inintrinsic coordinates in order to use Gaussian quadrature integration, and M is the order
General formulation of conventional numerical methods258of the elements. Using the shape functions, the original PDF of the problem is replacedby an algebraic system of equations written asN hKijein ouej i 1N ( fie )orKu F(A.5)i 1h in owhere matrix Kije is the coefficient matrix, vector uej is the nodal value vector of the unknown variables, and vector f ie is comprised of contributions from body forceterms and initial/boundary conditions.h iFor elasticity problems, the matrix Kije is called the element stiffness matrix given byh i ˆ Kije (A.6)([ Bi ] [ Ni ])T [ Di ] Bj dΩΩiwhere matrix [ Di ] is the elasticity matrix and matrix [ Bi ] is the geometry matrix determined by the relation between the displacement and strain. The global stiffness matrixK is banded and symmetric because the matrices [ Di ] are symmetric. Material inhomogeneity in FEM is most straightforwardly incorporated by assigning different materialproperties to different elements (or regions). To enforce the displacement compatibilitycondition, the order of shape functions along a common edge shared by two elementsmust be the same, so that no displacement discontinuity occurs along and across theedge.“Infinite elements” have also been developed in FEM to consider the effects of aninfinite far-field domain on the near-field behavior, most notably the “infinite domainelements” of Beer and Meek (1981) and the “mapped infinite elements” of Zienkiewiczet al. (1983), with focus on geomechanical applications. The mapped infinite elementsare simply implemented using special shape functions that project boundary nodes atinfinite distances in one or two directions, where the displacements are either zero orhave prescribed values. Additional nodes are needed at the imaginary infinite locations.The infinite domain element technique does not require additional infinite nodes, butrequires a “decay function” to describe the manner in which the displacements vary frommesh boundary to infinity. The shape functions used in the infinite element formulationsare singular at the “infinite” nodes.A.3.3B OUNDARY E LEMENT M ETHOD (BEM)The BEM requires discretization at the boundary of the solution domains only, thus reducing the problem dimensions by one and greatly simplifying the input requirements. The
A.3 Continuum methods259information required in the solution domain is separately calculated from the information on the boundary, which is obtained by solution of a boundary integral equation,instead of direct solution of the PDEs, as in the FDM and FEM.Unlike the FEM and FDM methods, the BEM approach initially seeks a weak solutionat the global level through an integral statement, based on Betti’s reciprocal theorem andSomigliana’s identity. For a linear elasticity problem with domain Ω boundary Γ of unitoutward normal vector ni and constant body force f i for example, the integral statementis written asˆcij u j ˆΓtij u j dΓ Γuij t j dΓ ˆ u ijΓ nf j dΓ(A.7)where u j and t j are the displacement and traction vectors on the boundary Γ the terms uij and tij are called displacement and traction kernels. The term cij is called the free termdetermined by the local geometry of the boundary surfaces, cij 1 when the field pointis inside the domain Ω. The solution of the integral Equation A.7 requires the followingsteps:1. Discretization of the boundary Γ with a finite number of boundary elements. For 2D problems, the elements are 1D line segments which may have one node at the center ofthe element (constant element), two nodes at the two ends of the line segment (linear elements) or three nodes with two end nodes and one central node (quadraticelements). Let N denote the total number of boundary elements. The boundaryintegral equation then is re-arranged into a sum of local integrals over all elementsNcij u j ˆk 1 Γ ktij u j dΓN ˆk 1 Γ kuij t j dΓ N ˆk 1 Γ k uij nf j dΓ(A.8)2. Appr
General formulation of conventional numerical methods A.1 Introduction The general formulation of continuum and discontinuum methods, as well as the simula-tion of fracture process using different numerical tecniques, described in this appendix, is based on the previous work made by Jing (2003). A.2 Numerical methods in rock engineering
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