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The Finite Element Method in Engineering is the only book to provide a broad overview of the underlying principles of finite element analysis and where it fits into the larger context of other mathematically based engineering analytical tools. This is an updated and improved version of a finite element text long noted for its practical applications approach, its readability, and ease of use. Students will find in this textbook a thorough grounding of the mathematical principles underlying the popular, analytical methods for setting up a finite element solution based on mathematical equations. The book provides a host of real-world applications of finite element analysis, from structural design to problems in fluid mechanics and thermodynamics. It has added new sections on the assemblage of element equations, as well as an important new comparison between finite element analysis and other analytical methods showing advantages and disadvantages of each. This book will appeal to students in mechanical, structural, electrical, environmental and biomedical engineering.

The uncertain analysis of fixed solar compound parabolic concentrator (CPC) collector system is investigated for use in combination with solar PV cells. Within solar CPC PV collector systems, any radiation within the collector acceptance angle enters through the aperture and finds its way to the absorber surface by multiple internal reflections. It is essential that the design of any solar collector aims to maximize PV performance since this will elicit a higher collection of solar radiation. In order to analyze uncertainty of the solar CPC collector system in the optimization problem formulation, three objectives are outlined. Seasonal demands are considered for maximizing two of these objectives, the annual average incident solar energy and the lowest month incident solar energy during winter; the lowest cost of the CPC collector system is approached as a third objective. This study investigates uncertain analysis of a solar CPC PV collector system using fuzzy set theory. The fuzzy analysis methodology is suitable for ambiguous problems to predict variations. Uncertain parameters are treated as random variables or uncertain inputs to predict performance. The fuzzy membership functions are used for modeling uncertain or imprecise design parameters of a solar PV collector system. Triangular membership functions are used to represent the uncertain parameters as fuzzy quantities. A fuzzy set analysis methodology is used for analyzing the three objective constrained optimization problems.

The Helmholtz equation governs the physical problems, such as the vibration of a membrane, the propagation of electromagnetic waves in a waveguide, the oscillatory or seiche motion of an enclosed mass of water in a lake or harbor, and the acoustic vibrations of a body of fluid enclosed in a room or vehicle. This chapter describes finite element solution for such problems. The chapter illustrates the importance of Helmholtz equation by describing the numerical examples related to the propagation of electromagnetic waves in a waveguide and the problem of noise in the passenger compartment of an automobile. The solution provided by Helmholtz equation for such problems has some physical significance in determining component of the magnetic field strength vector; elevation of free surface measured from the mean water level; transverse displacement of the membrane; and excess pressure above ambient pressure.

The effective lubrication in automobiles requires that the two surfaces should be nearly but not quite parallel. The pressure necessary to separate the two surfaces is developed due to the wedge action. The applied (normal) load is resisted by the fluid pressure, thereby reducing friction and wear. This type of lubrication is called hydrodynamic lubrication and is governed by the Reynolds equation. There are many assumptions made in the derivation of Reynolds equation. Those are as follows: (1) the fluid is incompressible; (2) the fluid is Newtonian; (3) the curvature of the bearing components introduces only second-order negligible effects in journal bearings; (4) the viscosity remains constant throughout the pressure film; (5) the inertia terms governing the motion of fluid are negligible; (6) the pressure gradient through the thickness of the film is zero and (7) the motion of the fluid in a direction to the surface can be neglected compared to the motion parallel to it. This chapter describes the solution procedure of Reynolds equation via finite element method and illustrates the importance of Reynolds equation by considering the numerical examples of step bearing and journal bearing.

This chapter deals with the one-dimensional heat transfer problem. It describes the derivation of the governing differential equation for one-dimensional heat transfer problem under boundary conditions. A fin is a common example of a one-dimensional heat transfer problem. One end of the fin is connected to a heat source (whose temperature is known) and heat is lost to the surroundings through the perimeter surface and the end. The chapter presents the analysis of uniform and tapered fins. The straight uniform fin analysis is a five step procedure. The first step is to idealize the rod into several finite elements. The second step is to assume a linear temperature variation inside any element. The third step involves the derivation of element matrices. The element matrices are then assembled in fourth step to obtain the overall equations and finally, the assembled equations are solved after incorporating the boundary conditions to find the nodal temperatures. A subroutine computer program called HEAT1 is developed for the solution of one-dimensional heat transfer problems. The solution of uniform one-dimensional heat transfer problems is explained by using a quadratic model for the variation of temperature in the element. Time-dependent or unsteady state problems are very common in heat transfer. The chapter also describes the finite element solution of time-dependent heat transfer problems.

This chapter deals with the finite element solution of ideal flow problems of inviscid incompressible flow. The flow around a cylinder, flow out of an orifice, and flow around an airfoil are the typical examples of inviscid incompressible flows. The two dimensional potential flow (irrotational flow) problems are formulated in terms of a velocity potential function (ф) or a stream function (Ψ). In general, the choice between velocity and stream function formulations in the finite element analysis depends on the boundary conditions. If the geometry is simple, no advantage of one over the other can be claimed. If the fluid is ideal, its motion does not penetrate into the surrounding body or separate from the surface of the body and leave empty space. This gives the boundary condition that the component of the fluid velocity normal to the surface must be equal to the component of the velocity of the surface in the same direction. The chapter illustrates the finite element solution of flow problems with reference to the problem of flow over a circular cylinder between two parallel plates considering both potential and stream formulations. The chapter also describes the finite element solution using the Galerkin approach. A subroutine called PHIFLǾ is developed for the solution of the problem of confined flow around a cylinder based on the potential function formulation.

The element matrices and vectors are assembled to obtain the characteristic equations of the entire system of elements. Coordinate transformation is the prerequisite to the assembly of matrices and vectors. The coordinate transformation is necessary when the field variable is a vector quantity such as displacement and velocity. Sometimes, the element matrices and vectors are computed in local coordinate systems suitably oriented for minimizing the computational effort. The local coordinate system may be different for different elements. When a local coordinate system is used, the directions of the nodal degrees of freedom will also be taken in a convenient manner. In such a case, before the element equations can be assembled, it is necessary to transform the element matrices and vectors derived in local coordinate systems so that all the elemental equations are referred to a common global coordinate system. Once the element characteristics, namely, the element matrices and element vectors, are found in a common global coordinate system, the next step is to construct the overall or system equations. The procedure for constructing the system equations from the element characteristics is the same regardless of the type of problem and the number and type of elements used. The assembly procedure is further implemented by computer.

The purpose of this chapter is to derive the finite element equations for the determination of temperature distribution within a conducting body. The basic unknown in heat transfer problems is temperature, similar to displacement in stress analysis problems. The knowledge of temperature distribution within a body is useful in computing the heat added to or removed from a body. Furthermore, if a heated body is not permitted to expand freely in all the directions, some stresses are developed inside the body. The magnitude of these thermal stresses influences the design of devices such as boilers, steam turbines, and jet engines. The first step in calculating the thermal stresses is to determine the temperature distribution within the body. The finite element equations are derived for the determination of temperature distribution within a conducting body either by minimizing a suitable functional using a variational (Rayleigh-Ritz) approach or from the governing differential equation using a weighted residual (Galerkin) approach. The chapter also summarizes the basic equations of heat transfer namely the energy balance and rate equations.