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## Über dieses Buch

This book covers all basic areas of mechanical engineering, such as fluid mechanics, heat conduction, beams and elasticity with detailed derivations for the mass, stiffness and force matrices. It is especially designed to give physical feeling to the reader for finite element approximation by the introduction of finite elements to the elevation of elastic membrane. A detailed treatment of computer methods with numerical examples are provided. In the fluid mechanics chapter, the conventional and vorticity transport formulations for viscous incompressible fluid flow with discussion on the method of solution are presented. The variational and Galerkin formulations of the heat conduction, beams and elasticity problems are also discussed in detail. Three computer codes are provided to solve the elastic membrane problem. One of them solves the Poisson’s equation. The second computer program handles the two dimensional elasticity problems and the third one presents the three dimensional transient heat conduction problems. The programs are written in C++ environment.

## Inhaltsverzeichnis

### Chapter 1. Introduction and History

Abstract
This section provides a brief history of the development of the finite element method. The general picture of the science of mechanics to formulate the most general rules, variational and boundary value problems, are presented. The approximate solution of both types of formulations are discussed and how these approximate solutions are transformed into the finite element method are presented. The key references in the development of finite element method are cited.
M. Reza Eslami

### Chapter 2. Mathematical Foundations

Abstract
This chapter presents basic mathematical concepts and tools for the development of finite element method. The concept of functional, associated with the variational formulation, and the function, associated with the boundary value problems, is discussed; the method of calculus of variation which transfers the variational formulation into the boundary value problem is also presented. A brief discussion of the numerical solution methods to handle the variational formulation and the boundary value problem is presented. Some numerical examples are given to show the convergence efficiency of the numerical methods.
M. Reza Eslami

### Chapter 3. Finite Element of Elastic Membrane

Abstract
To physically show the finite element approximation, an elastic membrane is selected. By the finite element method, we approximate the height of an elastic membrane. This provides a physical feeling of what the finite element approximation means. In addition, since the governing equation of an elastic membrane is Poisson’s equation, many other mechanical problems governed by Poisson’s equation may be treated by the detailed mathematical derivations of this chapter.
M. Reza Eslami

### Chapter 4. Elements and Local Coordinates

Abstract
The presentation of the finite element method for the elastic membrane problem, a triangular element with straight sides and linear shape function to approximate the elevation of the elastic membrane due to lateral pressure, was employed. In this chapter, we correct ourself that there is no limitation as far as the geometry of the element and the order of approximation of the shape function is concerned. In one, two, and three dimensional problems elements with higher order geometries and approximating shape functions may be used to prepare a finite element model. Also, since it is always more efficient to employ the local coordinates for the integration purpose of the element stiffness and force matrices, the local and global coordinate systems are discussed and the Jacobian matrix is explained.
M. Reza Eslami

### Chapter 5. Field Problems

Abstract
Many problems in mechanics are governed by harmonic and biharmonic partial differential equations. This chapter presents detail derivation of the finite element matrices for harmonic and biharmonic problems. The finite element matrices for harmonic equations in Cartesian and cylindrical coordinates, axisymmetric condition, are derived.
M. Reza Eslami

### Chapter 6. Conduction Heat Transfer in Solids

Abstract
Heat conduction problem in solid continuum is one of the major fields in mechanics. This chapter presents detail derivations of the finite element matrices of one, two, and three-dimensional conduction problems. Both variational and Galerkin methods are employed to derive the finite element formulations. Derivation of the capacitance matrix for the transient heat conduction problems is carried out for the one, two, and three-dimensional cases.
M. Reza Eslami

### Chapter 7. Computer Methods

Abstract
Once the theoretical derivations to obtain the finite element matrices for a base element (e) is completed, the element matrices are assembled to form the finite element equilibrium equation. The computer methods to solve this final equation includes the matrix assembling, bandwidth calculation, application of the boundary conditions, solution algorithm, and preparation of the output results. This chapter briefly explains the computer techniques to perform the required operations and through a number of numerical examples show the details of the mathematical concepts. That is, it puts the mathematical concepts into the computer algorithms. At the end of the chapter, a number of classical methods of solution of dynamic finite element equations are discussed.
M. Reza Eslami

### Chapter 8. Finite Element of Beams

Abstract
The finite element derivations for a base element of beams and bars under different types of behaviors are discussed in this chapter. The static lateral defection and axial, torsional, and lateral vibrations of beams and bars are studied and the members of mass, stiffness, and force matrices are derived. The chapter concludes with a discussion of the Timoshenko beam and the derivations of the element matrices.
M. Reza Eslami

### Chapter 9. Elasticity, Galerkin Formulations

Abstract
The chapter begins with derivation of the equations of motion of an elastic continuum and presentation of the basic equations of theory of elasticity. Employing the Galerkin method, the equations of motion are made orthogonal with respect to the assumed element shape function. A base element is considered and the weak formulation is applied to the terms of higher order derivatives and the finite element equation for the three-dimensional elasticity is obtained. The resulting equations are reduced to the case of two-dimensional elasticity, plane stress, and plane strain conditions. Employing the two-dimensional simplex shape functions, the member of element matrices are obtained.
M. Reza Eslami

### Chapter 10. Elasticity, Variational Formulations

Abstract
The derivation of finite element equation of motion based on the variational formulation is presented in this chapter. The Hamilton principle for elastic continuum is derived and basic relations for the linear elasticity, the constitutive law, and the kinematical relations, are presented. Employing the linear shape functions for three displacement components, the elements of the mass, stiffness, and force matrices are derived. The two-dimensional plane stress and plane strain elasticity are discussed and the axisymmetric elasticity formulation follows.
M. Reza Eslami

### Chapter 11. Torsion of Prismatic Bars

Abstract
This chapter deals with the torsion of prismatic bars. Bars of general cross-sectional area are considered and the equilibrium equation of the bars under torsional couple is derived, using the concept of stress function. The final form of the equilibrium equation is that of the Poisson equation, where its functional is similar to that of the elastic membrane. Employing a base element (e), the Ritz method is used to derive the stiffness and force matrices of the finite element equation.
M. Reza Eslami

### Chapter 12. Thermoelasticity

Abstract
The chapter begins with an explanation of the basic governing equations of linear thermoelasticity, including the equations of motion, the linear thermoelastic constitutive law, and the kinematical relations. Using the governing equations, the Navier equations of motion in terms of the displacement components are derived. The condition for the case of zero thermal stresses is then discussed. The displacement-based finite element equations in conjunction with the heat conduction equation are derived employing the Galerkin method. The final of the finite element equation of motion is reduced to the case of two-dimensional thermoelasticity problems, where the element matrices for the base element (e) are derived. The dynamic finite element equation is further reduced to the one-dimensional case, where for a one-dimensional simplex element the detail of element matrices are calculated.
M. Reza Eslami

### Chapter 13. Incompressible Viscous Fluid Flow

Abstract
The flow of incompressible and viscous fluid is considered in this chapter. The continuity condition and the equations of motion, the Navier-Stokes equations for the Newtonian fluid, are derived. The Stokes equation for the low Reynolds number flow is discussed and its associated functional expression is presented. The governing equations are then made dimensionless and employing the Galerkin method, the finite element equation is derived, using two different shape functions for the velocity components and pressure. The general finite element equations are then reduced to the case of two-dimensional fluid flow and the elements of mass, stiffness, and force matrices are calculated. The general form of boundary conditions are discussed and the selection/limitation of the shape functions for the velocity components and pressure is explained. As another method of fluid flow problems, the vorticity transport technique is presented and the finite element modeling of the fluid flow in terms of the vorticity and stream function is presented and the element matrices are derived. Since the governing finite element equations are nonlinear, the method of linearization technique is discussed. The finite element matrices for two-dimensional flow of an incompressible viscous fluid employing two-dimensional simplex elements are derived.
M. Reza Eslami

### Chapter 14. One-Dimensional Higher Order Elements

Abstract
The chapter begins with the definition of straight one-dimensional quadratic element, where the natural coordinate and the Jacobian matrix is then calculated. As an application, the field problem is selected and the stiffness and force matrices are calculated. The cubic element in the general and local coordinates are discussed the Jacobian of transformation is calculated. The layer-wise theory is then described and it is applied to a composite beam under static and dynamic loading conditions.
M. Reza Eslami

### Chapter 15. Two-Dimensional Higher Order Elements

Abstract
The triangular elements with quadratic and cubic shape functions in terms of the local area coordinates are presented and coordinate transformation law with the associated Jacobian matrix calculations are given in the chapter. As an application, the Field problem in two dimensions is considered and employing the quadratic triangular element, the stiffness and force matrices are calculated. The quadrilateral element is discussed in the following and the shape functions in the global and local coordinates are obtained. The field problem is reconsidered and the element matrices are calculated employing the bilinear quadrilateral element.
M. Reza Eslami

### Chapter 16. Coupled Thermoelasticity

Abstract
The problems of coupled linear thermoelasticity are among those classes of mechanics which seldom have analytical solution even for simple structures such as beams and rods. Therefore, finite element method is one of the most reliable numerical methods to handle the solution of structural members. The chapter begins with the Galerkin method to obtain the finite element equations of the coupled problems for general three-dimensional case. The members of each related matrice in the resulting finite element equations are calculated and given. The method is then applied to a number of problems. The function- ally graded layer under thermal shock load is analyzed in the next section. Thick spherical vessels under radially symmetric thermal shock load applied to its inside surface is dis- cussed in the next section. The coupled thermoelastic equations for an axisymmetrically loaded disk with different approximation orders is presented in the last section. Elements with various orders are employed to investigate the effects of the number of nodes in an element
M. Reza Eslami

### Chapter 17. Computer Programs

Abstract
In this chapter, three computer codes are presented. These codes are to solve the elevation of elastic membrane under static load, the static elasticity, and the three-dimensional transient heat conduction problems. The descriptions and de- tails of the processor and postprocessor for each computer program is presented in this chapter.
M. Reza Eslami
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