# Manual Introduction to Partial Diff. Eqns. With Applns.

Introduction to partial differential equations with applications. Front Cover. E. C. Zachmanoglou, Dale W. Thoe. Williams & Wilkins, - Mathematics -

This includes an introduction to Hilbert space methods as well as an introductory treatment of distribution theory and its applications to solving problems in partial differential equations. We do not require a specific course prerequisite as the results will be developed as we go along. However, a reasonable amount of mathematical maturity is necessary. M Syllabus. The Transport Equation.

Supplement: Additional techniques. Problem A. Problem A solution. Maple examples. For an FEM problem, the total potential energy is the summation of that from all the elements. Therefore, substituting Eqs. From Eq. Specifically, in order to solve Eq. Without the provision of adequate boundary condition, the system is singular as rigid body motion will produce no stress in the system and such mode will be present in the SEE. For a three-node triangular element, the element strain matrix B is constant, thus Eq.

K i j indicates the i th nodal force along the x- and y -directions in the Cartesian coordinate system when the displacement of the j th node is unit along the x- and y -directions, which can be easily obtained. Moreover, the element stiffness matrix is symmetric, and the computational memory required in an FEM program can be reduced by using this property. It should be noted that for a higher order triangular element e. Towards this, numerical integration methods such as the Gaussian integration or the Newton-Cotes integration can be utilized.

For an FEM process, we need to solve Eq. Most of the elements in the matrix K are 0 simply because each node is only shared by a few surrounding elements. In view of that, a rectangular matrix can represent the global stiffness matrix which is a square matrix , and the half bandwidth D can be defined as. In conclusion, the properties of the global stiffness matrix can be summarized as: symmetric, banded distribution, singularity and sparsity.

Among all the properties, singularity will vanish by introducing appropriate boundary conditions to Eq. Also, other properties like banded distribution should be fully taken into consideration to reduce the computational memory and enhance the computation efficiency. Most of the engineering structure is not regular in shape, and some of them even have very complicated boundary shapes.

Although the use of triangular element can always fit a complicated boundary, the accuracy of this element is low in general. 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. 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.

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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. 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.

## Introduction to Partial Differential Equations with Applications by E.C. Zachmanoglou

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 ]. 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. 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.

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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.

## MA250 Introduction to Partial Differential Equations

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. The force-displacement law is described for both the ball-ball and ball-wall contacts.

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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.

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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. It is, however, sometimes necessary to manually adjust the time step in some special cases when the input parameters are unreasonably high or low. Most of the DEM codes use the central difference time integration algorithm which is a second-order scheme in time step. If the local results in DEM are analyzed, it is found that there will be large fluctuations with respect to both locations and time. Such results are not surprising, as the results are highly sensitive to the interaction between particles and hence the time step under which the results are monitored. It can be viewed that such local results can be meaningless unless the results are monitored over a long time span or region. A number of quantities in a DEM model are defined with respect to a specified measurement circle. These quantities include coordinate number, porosity, sliding fraction, stress and strain rate.

The coordination number and stress are defined as the average number of contacts per particle. Only particles with centroids that are contained within the measurement circle are considered in computation. In order to account for the additional area of particles that is being neglected, a corrector factor based on the porosity is applied to the computed value of stress. Since measurement circle is used, stress in particle is described as the two in-plane force acting on each particle per volume of particle.

Average stress is defined as the total stress in particle divided by the volume of measurement circle. Thus, shape of particle is regardless of the average stress measurement because the reported stress is easily scaled by volume unity. The reported stress is interpreted as the stress per volume of measurement circle. There are also various publications on the numerical solutions of differential equations, and the readers are suggested to the works of Lee and Schiesser [ 24 ], Jovanoic and Suli [ 22 ], Veiga et al. It is impossible for the author to cover every available analytical or numerical method; hence, the author has chosen some methods that are actually used for teaching and research.

The readers are strongly encouraged to consult the numerous resources available in various books and publications. There are still new developments available for the solutions of specific differential equations in large-scale problems, and this is also the current trend in the development of differential equation solution. Due to the importance of the solution of differential equations, there are other important numerical methods that are used by different researchers but are not discussed here, which include the finite difference and boundary element methods computer codes for learning can also be obtained from the author.

This will result in a system of algebraic equations that can be solved implicitly or explicitly. There are various ways to form the derivatives, and the most common methods are the forward difference, backward difference and the central difference schemes. While the finite difference methods may be more suitable for different types of differential equations, this method is less convenient to deal with irregular boundary conditions as compared with the finite element method.

For highly irregular domain where it is not easy to form a nice discretization, the finite element method will also be much easier and natural to deal with for such condition. In this respect, it is not surprising that many engineering programs are written by the use of the finite element method than the finite difference method. The boundary element method BEM is another numerical method for solving linear partial differential equations which can be formulated as integral equations.

The boundary element method uses the given boundary conditions to fit boundary values into the integral equation. In the post-processing stage, the integral equation will be used to calculate the solution directly at any given point inside the solution domain numerically. The dimension of the problem will then be reduced by one. For example, two-dimensional problem will be effectively reduced to one-dimensional problem along the boundary, and this will greatly improve the efficiency of computation.

The requirement from the boundary element method imposes considerable restrictions on the range and generality of problems to which the boundary element method can usefully be applied. There are some new developments to the boundary element method so that it can be used for non-linear problem or problems with several major materials problems with random distribution of material properties are still not applicable. The fundamental solutions are often difficult to integrate. For complicated problems, the boundary element will lose its advantage as compared with other numerical methods.

Due to the various limitations, there are only limited boundary element programs available to the researchers. Interested readers can consult the works of Banerjee [ 1 ], Brebbia et al. It appears that there are less interest in the use and development of the boundary element method in the recent years, due to the various limitations of this method in general non-linear non-homogeneous problem. In history, various techniques have been developed for ordinary differential equations and partial differential equations under different boundary conditions.

## Introduction to Partial Differential Equations with Applications

While these tricks appear to be elegant, they are not readily adopted for normal engineering use due to various limitations. Being an engineer, the author seldom adopted the methods as outlined in this chapter in actual applications but do adopt for teaching , except the numerical methods as outlined in this chapter. At present, there are many proprietary or open source finite elements or distinct element codes being used for many complicated real problems.

Introducing Green's Functions for Partial Differential Equations (PDEs)