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To cope with the irregular boundary shape with a higher accuracy in analysis, one of the most common approaches is the use of higher-order element, and the isoparametric formulation is the most commonly used at present. Consider an arbitrary four-node quadrilateral element as an example which is schematically shown in Figure 2.

If we can find the transformation from Figure 2 a to b , then it will become easier to carry numerical integration with complicated shapes for an arbitrary element. In Figure 2 a , we define the Cartesian coordinate system, while in Figure 2 b , we define the local coordinate system or natural coordinate system within a specific domain i. The relation between these two kinds of coordinate system can be described as.

Therefore, the regular element in the natural coordinate system can be transformed to the irregular element in the Cartesian coordinate system. The former element is called the parent element, while the latter is called the subelement. Specifically, Eq. Using the same interpolation functions, the element displacement model can be written as. As mentioned before, during the derivation of the element stiffness matrix and the equivalent load vector, the derivative of the shape function and the integration in element surface or volume in the Cartesian coordinate system are required.

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Since the shape functions adopted herein are expressed in natural coordinates, therefore, derivative and integration transformation relationships are essential when isoparametric element is used. Inverse of Eq. For an infinitely small element, the area under the Cartesian coordinate system and the natural coordinate system are related by. Therefore, element stiffness matrix and equivalent nodal load matrix in Eq. For solving the integral equation, usually the Gaussian integration method is employed. In practice, both two and three integration points along each direction of integration are commonly used.

Since the discretized system is usually overstiff, it is commonly observed that the use of two integration points along each direction of integration will slightly reduce the stiffness of the matrix and give better results as compared with the use of three integration points. The use of exact integration is possible for some elements, but such approaches are usually tedious and are seldom adopted. The advantage in using the exact integration is that the integration is not affected by the shape of the element while the transformation as shown in Eq.

The author has developed many finite element programs for teaching and research purposes which can be obtained at ceymchen polyu.

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In practical applications, a limit equilibrium method based on the method of slices or method of columns and strength reduction method based on the finite element method or finite difference method are used for many types of stability problems. These two major analysis methods take the advantage that the in situ stress field which is usually not known with good accuracy is not required in the analysis. The uncertainties associated with the stress-strain relation can also be avoided by a simple concept of factor of safety or the determination of the ultimate limit state. In general, this approach is sufficient for engineering analysis and design.

If the condition of the system after failure has initiated is required to be assessed, these two methods will not be applicable. Even if the in situ stress field and the stress-strain relation can be defined, the post-failure collapse is difficult to be assessed using the conventional continuum-based numerical method, as sliding, rotation and collapse of the slope involve very large displacement or even separation without the requirement of continuity. The basic assumption adopted in these numerical methods is that the materials concerned are continuous throughout the physical processes.

This assumption of continuity requires that, at all points in a problem domain, the material cannot be torn open or broken into pieces. All material points originally in the neighbourhood of a certain point in the problem domain remain in the same neighbourhood throughout the whole physical process. Some special algorithms have been developed to deal with material fractures in continuum mechanics-based methods, such as the special joint elements by Goodman [ 13 ] and the displacement discontinuity technique in BEM by Crouch and Starfield [ 5 ].

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However, these methods can only be applied with limitations [ 21 ]:. Before a slope starts to collapse, the factor of safety serves as an important index in both the LEM and SRM to assess the stability of the slope.

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The movement and growth after failure have launched which is also important in many cases that cannot be simulated on the continuum model, and this should be analyzed by the distinct element method DEM. In continuum description of soil material, the well-established macro-constitutive equations whose parameters can be measured experimentally are used. On the other hand, a discrete element approach will consider that the material is composed of distinct grains or particles that interact with each other.

The commonly used distinct element method is an explicit method based on the finite difference principles which is originated in the early s by a landmark work on the progressive movements of rock masses as 2D rigid block assemblages [ 6 ]. The method has also been developed for simulating the mechanical behaviour of granular materials [ 8 ], with a typical early code BALL [ 7 ], which later evolved into the codes of the PFC group for 2D and 3D problems of particle systems Itasca, Through continuous developments and extensive applications over the last three decades, there has accumulated a great body of knowledge and a rich field of literature about the distinct element method.

Currently, there are many open source Oval, LIGGGHTS, ESyS, Yade, ppohDEM, Lammps as well as commercial DEM programs, but in general, this method is still limited to basic research instead of practical application as there are many limitations which include: 1 difficult to define and determine the microparameters; 2 there are still many drawbacks in the use of matching with the macro response to determine the microparameters; 3 not easy to set up a computer model; 4 not easy to include structural element or water pressure; 5 extremely time consuming to perform an analysis; and 6 postprocessing is not easy or trivial.

It should also be noted that DEM can be formulated by an energy-based implicit integration scheme which is the discontinuous deformation analysis DDA method. This method is similar in many respect to the force-based explicit integration scheme as mentioned previously. In DEM, the packing of granular material can be defined from statistical distributions of grain size and porosity, and the particles are assigned normal and shear stiffness and friction coefficients in the contact relation.

Two types of bonds can be represented either individually or simultaneously; these bonds are referred to the contact and parallel bonds, respectively Itasca, Although the individual particles are solid, these particles are only partially connected at the contact points which will change at different time step.

Under low normal stresses, the strength of the tangential bonds of most granular materials will be weak and the material may flow like a fluid under very small shear stresses. In many particle models for geological materials in practice, the number of particles contained in a typical domain of interest will be very large, similar to the large numbers of molecules.

Compared with a continuum, particles have an additional degree of freedom of rotation which enables them to transmit couple stresses, besides forces through their translational degrees of freedom. At certain moment, the positions and velocities of the particles can be obtained by translational and rotational movement equations and any special physical phenomenon can be traced back from every single particle interactions.

Therefore, it is possible for DEM to analyze large deformation problems and a flow process which will occur after slope failure has initiated. DEM runs according to a time-difference scheme in which calculation includes the repeated application of the law of motion to each particle, a force-displacement law to each contact, and a contact updating scheme. Generally, there are two types of contact in the program which are the ball-wall contact and the ball-ball contact.

Handbook of Differential Equations: Stationary Partial Differential Equations: Volume 5

In each cycle, the set of contacts is updated from the known particles and known wall positions. Force-displacement law is firstly applied on each contact, and new contact force is then calculated according to the relative motion and constitutive relation. Law of motion is then applied to each particle to update the velocity, the direction of travel based on the resultant force, and the moment and contact acting on the particles.

Although every particle is assumed as a rigid material, the behaviour of the contacts is characterized using a soft contact approach in which finite normal stiffness is taken to represent the stiffness which exists at the contact. The soft contact approach allows small overlap between the particles which can be easily observed.

Stress on particles is then determined from this overlapping through the particle interface. The PFC runs according to a time-difference scheme in which calculation includes the repeated application of the law of motion to each particle, a force-displacement law to each contact, and a contact updating a wall position.

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Generally, there are two types of contact exist in the program which are ball-to-wall contact and ball-to-ball contact. In each cycle, the set of contacts is updated from the known particle and the known wall position. The force-displacement law is first applied on each contact. New contact force is calculated and replaces the old contact force. The force calculations are based on preset parameters such as normal stiffness, density, and friction. Next, a law of motion is applied to each particle to update its velocity, direction of travel based on the resultant force, moment and contact acting on particle.

The force-displacement law is then applied to continue the circulation.

Table of Contents

The force-displacement law is described for both the ball-ball and ball-wall contacts. The contact arises from contact occurring at a point. For the ball-ball contact, the normal vector is directed along the line between the ball centres. For the ball-wall contact, the normal vector is directed along the line defining the shortest distance between the ball centre and the wall.

The contact force vector F i is composed of normal and shear component in a single plane surface. The force acting on particle i in contact with particle j at time t is given by. The shear force acting on particle i during a contact with particle j is determined by. The new shear contact force is found by summing the old shear force min F ij t with the shear elastic force.

The motion of the particle is determined by the resultant force and moment acting on it. The motion induced by resultant force is called translational motion. The motion induced by resulting moment is rotational motion. The equations of motion are written in vector form as follows:. I r stands for moment of inertia. F i d and M i d stand for the damping force and damping moment. Unlike finite element formulation, there are now three degree of freedom for 2D problem and six degree of freedom for 3D problems.

This approach also avoids the generation and storage of the large global stiffness matrix that will appear in finite element analysis. On the other hand, the implicit DDA approach will generate a global stiffness matrix which is even larger than that in finite element analysis, as the rotation is involved directly in the stiffness matrix.

In a typical DEM simulation, if there is no yield by contact separation or frictional sliding, the particles will vibrate constantly and the equilibrium is difficult to be achieved. To avoid this phenomenon which is physically incorrect, numerical or artificial damping is usually adopted in many DEM codes, and the two most common approaches to damping are the mass damping and non-viscous damping. This damping is usually applied equally to all the nodes.

As this form of damping introduces body forces, which may not be appropriate in flowing regions, it may influence the mode of failure. Alternatively, Cundall [ 11 ] proposed an alternative method where the damping force at each node is proportional to the magnitude of the out-of-balance-force, with a sign to ensure that the vibrational modes are damped rather than the steady motion.

This form of damping has the advantage that only accelerating motion is damped and no erroneous damping forces will arise from steady-state motion. The damping constant is also non-dimensional and the damping is frequency independent. As suggested by Itasca [ 20 ], an advantage of this approach is that it is similar to the hysteretic damping, as the energy loss per cycle is independent of the rate at which the cycle is executed. While damping is one way to overcome the non-physical nature of the contact constitutive models in DEM simulations, it is quite difficult to select an appropriate and physically meaningful value for the damping.

For many DEM simulations, particles are moving around each other and the dominant form of energy dissipation is for frictional sliding and contact breakages. The choice of damping may affect the results of computations. Currently, most of the DEM codes allow the use of automatic damping or manually prescribed the damping if necessary.

To capture the inherent non-linearity behaviour of the problem with generation and removal of contacts, non-linear contact response and stress-strain behaviour and others , the displacement and contact forces in a given time step must be small enough so that in a single time step, the disturbances cannot propagate from a particle further than its nearest neighbours. For most of the DEM programs, this can be achieved automatically and the default setting is usually good enough for normal cases.