Dr. Reddy is internationally known for his contributions to theoretical and SpringerúVerlag, ; The Finite Element Method in Heat Transfer and Fluid. J. N. Reddy, Energy Principles and Variational Methods in Applied J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press. The Finite Element Method (FEM) is a numerical and computer-based technique of J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford.

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Reddy: An Introduction to the Finite Element Method. Rosenberg and .. lumping ), alternative finite element formulations, and nonlinear finite element models. An introduction to nonlinear finite element analysis (J N Reddy) Engineering 12 NONLINEAR FINITE ELEMENT ANALYSIS give the exact solution .. DOWNLOAD FULL PDF EBOOK here { caite.info }. An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics. J. N. Reddy. Abstract.

Classical, Early, and Medieval Prose and Writers: With the advent of computers, there has been a tremendous explosion in the development and use of numerical methods. Like this document? The radiative heat transfer boundary condition which is a nonlinear function of u is not considered here. The assembly procedure described above can be used to assemble elements of any shape and type. This particular choice of weight functions is a natural one when the weight function is viewed as a virtual variation of the dependent unknown i. The mathematical model is developed using laws of physics and assumptions concerning the process behavior.

Such a nonlinearity is known as the softening type. In the present case, the geometric nonlinearity dominates if both nonlinearities are included.

An example of another type of nonlinearity is provided by see Hinton [5] the axial deformation of an isotropic, homogeneous, linear elastic rod with constrained end displacement, as shown in Figure 1. The rod is of length 2L, uniform cross-sectional area of A, and loaded with an axial force P at its midpoint.

The lower end of the rod is constrained so that it can at most have an axial displacement of u0 which is assumed to be very small compared to the length L. Analysis is an aid to design and it involves 1 mathematical model development, 2 data acquisition by measurements, 3 numerical simulation, and 4 evaluation of the results in light of known information and corrections to the mathematical model.

The mathematical model is developed using laws of physics and assumptions concerning the process behavior. The data includes the actual system parameters geometry, loading, and boundary conditions and constitutive properties. The constitutive properties such as the modulus, conductivity, and so on are determined in laboratory experiments.

Nonlinearities in a mathematical model arise from changing geometry or material behavior. It is necessary to employ numerical methods to compute an approximate solution to the mathematical model.

Then we form a set of objectives for the analysis. If the analysis objective is to help develop a preliminary design of the system, the analysis can be very simple. Thus, the objectives will dictate the type of idealization of the system to be adopted; for example, should we model as a two-dimensional or three-dimensional problem?

Once the system idealization is complete i. For example, a mathematical model based on linear elasticity is adequate for determining linear elastic solutions of a solid but inadequate for determining its nonlinear response. The validation exercise allows one to modify the mathematical model to include the missing elements that make the computed response come closer to the physical response.

In fact, a mathematical model can never be validated; it can only be invalidated. Validation is a must when studying new and mulit-physics problems. References 1. Bathe, K.

Belytschko, T. Hinton, E. Oden, J. Owen, D. Reddy, J. Yang, Y. Zienkiewicz, O. Sargent, R. A Review 2. Divide the whole i. Over each representative element, develop the relations among the secondary and primary variables e. Assemble the elements i. Table 2. Since there are n unknown parameters, we need n relations to determine them.

Substituting the approximate solution 2. One way of making the residual zero is in weighted-integral sense Z xb xa we i x Re x, c1, c2,. Equation 2. There are other choices of we i that may be used. This particular choice is known as the Galerkin method.

Figure 2. The resulting integral form is termed the weak form of Eq. A three-step procedure of constructing the weak form of Eq. Step 1. Examining the boundary term appearing in the weak form 2. For the model equation at hand, the primary and secondary variables are Primary variable: Students of engineering who have taken a course in For heat conduction problems, Qe a denotes the heat input at the left end and Qe b the heat output from the right end of the element.

With the notation in 2. Remarks 1. The weak form in 2. The expression containing both we i and ue h is called the bilinear form i. Those who have a background in applied mathematics or solid and structural mechanics will appreciate the fact that the weak form 2.

Thus, a linear polynomial see Figure 2. Of course, we can also carry the nodal values ue a and ue b so that they can be used to join adjacent elements and the parameter ce 3 as the unknowns of the approximation. Identifying the third node at the center of the element [see Figure 2. These are: It should be a complete polynomial, that is, include all lower-order terms up to the highest order term used.

The second requirement is necessary in order to capture all possible states, that is, constant, linear and so on, of the actual solution. The third requirement is necessary in order to enforce continuity of the primary variables at the end points where the element is connected to other elements.

The model can be developed using an arbitrary degree of interpolation. If nodes 1 and n denote the end points of the element, then Qe 1 and Qe n represent the unknown point sources, and all other Qe i are the point sources at nodes 2, 3,.

Substituting Eq. Some of these equations are provided by the boundary conditions of the problem and the remaining by the balance of the secondary variables Qe i at nodes common to elements. The balance of equations can be implemented by putting the elements together i. Upon assembly and imposition of boundary conditions, we shall obtain exactly the same number of algebraic equations as the total number of unknown primary ue i and secondary Qe i degrees of freedom.

We will discuss the numerical integration concepts in the sequel.

Suppose that ae, ce, and fe denote the element-wise constant values of a x , c x , and f x. Then the following matrices can be derived by evaluating the integrals exactly. For example, in a heat transfer problem, u denotes temperature T, axx and ayy denote the conductivities, kxx and kyy, and f is the internal heat generation.

The following two types of boundary conditions are assumed: In Eq. The radiative heat transfer boundary condition which is a nonlinear function of u is not considered here.

However, radiation boundary condition will be considered in the nonlinear analysis. We shall discuss simple geometric shapes and orders of approximation shortly. To keep the formulative steps very general i. As we shall see shortly, the interpolation functions depend not only on the number of nodes in the element, but also on the shape of the element.

We shall develop the weak form of Eqs. For n independent choices of w, we obtain a set of n linearly independent algebraic equations. Since we need n independent algebraic equations to solve for the n unknowns, ue 1, ue 2, This particular choice of weight functions is a natural one when the weight function is viewed as a virtual variation of the dependent unknown i.

In matrix notation, Eq. The polynomials used to represent ue x, y must be complete i.

All terms in the polynomial should be linearly independent. The number of linearly independent terms in the representation of ue dictates the shape and number of degrees of freedom of the element. Here we review the interpolation functions of linear triangular and rectangular elements. An examination of the variational form 2. It contains three linearly independent terms, and it is linear in both x and y.

The polynomial is complete because the lower-order term, namely, the constant term, is included. Thus the polynomial in Eq. This polynomial requires an element with four nodes. It is a rectangle with nodes at the four corners of the rectangle.

The interpolation functions for linear triangular and rectangular elements are given below. Higher-order two-dimensional elements i. Linear triangular element The linear interpolation functions for the three-node triangle [see Figure 2.

Note that x, y are the global coordinates used in the governing equation 2. For a right-angled triangular element with base a and height b, and node 1 at the right angle nodes are numbered counterclockwise , [Ke ] takes the form see Reddy [1, p.

It is necessary to compute such integrals only when For additional details, see Reddy [1, pp. When the element is non-rectangular, that is, a quadrilateral, we use coordinate transformations to represent the integrals over a square geometry and then use numerical integration to evaluate them.

Continuity of the primary variable i. Balance of secondary variables i. From the mesh shown in Figure 2. Nodes 1, 2, 3, and 4 of element 2 correspond to global nodes 2, 4, 5, and 3, respectively. Note that the continuity of the primary variables at the inter-element nodes guarantees the continuity of the primary variable along the entire inter-element boundary.

For the two elements shown in Figure 2. Now we are ready to assemble the element equations for the two-element mesh. This is done by means of the connectivity relations, that is, correspondence of the local node number to the global node number. The assembly procedure described above can be used to assemble elements of any shape and type. The procedure can be implemented in a computer with the help of the local-global nodal correspondence.

To include the convective boundary condition 2. The interpolation functions are developed here for regularly shaped elements, called master elements. These elements can be used for numerical This requires a transformation of the geometry from the actual element shape to its associated master element. We will discuss the numerical evaluation of integrals in Section 2.

Higher- order triangular elements i. University Press Scholarship Online. Sign in.

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Reddy Abstract The development of realistic mathematical models that govern the response of systems or processes is strongly connected to the ability to translate them into meaningful discrete models that allow for a systematic evaluation of various parameters of the systems and processes.

More The development of realistic mathematical models that govern the response of systems or processes is strongly connected to the ability to translate them into meaningful discrete models that allow for a systematic evaluation of various parameters of the systems and processes. Bibliographic Information Print publication date: June DOI: Authors Affiliations are at time of print publication. Print Save Cite Email Share.