Finite Element Equations

A 3D finite element analysis is developed in which a provision has been made for time-dependent plasticity and rupturing in steel and cracking in materials such as concrete. The influence of studs, tugs and connectors is included. Concrete steel liners and studs are represented by solid isoparametric elements, shell elements and line elements with or without bond hnkages. To begin with, a displacement finite element is adopted. 

The displacement field within each element is defined in Fig. 3.1 as 

In order to maintain equilibrium with the element, a system of external nodal forces F is applied which will reduce the virtual work (dW) to zero. In the 

The force-displacement relationship for each element is given by 

Equations (3.4) and (3.5) represent the relationships of the nodal loads to the stiffness and displacement of the structure. These equations now require modification to include the influence of the liner and its studs. The material compliance matrices [D] are given in Tables 3.1 and 3.2. The numerical values are given for various materials or their combinations in Tables 3.3,3.4,3.5 and 3.6. These values of the constitutive matrices are recommended in the absence of specific information. 

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Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

The Galerldn finite element method results when the Galerldn method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerldn finite element equations. The element integrals are defined as... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

Before we proceed to our discussion of global stiffness matrix storage schemes, we will discuss the last aspect of the finite element implementation, namely, the application of the boundary conditions. As discussed earlier, the natural boundary conditions are imbedded in the finite element equation system - it is implied that every boundary node without an... 

Following the procedure used with the one-dimensional FEM model and using the constant strain triangle element developed in the previous section, we can now formulate the finite element equations for a transient conduction problem with internal heat generation rate per unit volume of Q. The governing equation is given by... 

The finite element expression for a problem using isoparametric finite elements will be similar to the ones developed in previous examples. For example, a problem governed by Poisson s equation will result in the following finite element equation,... 

The isoparametric element works quite well to formulate the finite element equations for flow problems, such as flows with non-Newtonian shear thinning viscosity. Due to the flexibility that exists to integrate variables throughout the elements, the method lends itself... 

SUPG) developed by Brooks and Hughes [3]. Essentially, the finite element equations remain the same however, as shown here, modified shape functions are introduced on the upwind side of a nodal point. Hence, we have two interpolation, or shape, functions that define the temperature, or convected variable, distribution. One definition uses the conventional shape functions given by... 

What would the constant strain finite element equations look like for the transient heat conduction problem with internal heat generation if you were to use a Crank-Nicholson time stepping scheme ... 

H. Matthies and G. Strang, Int. J. Numer. Methods Engin., 14,1613 (1979). The Solution of Nonlinear Finite Element Equations. 

Only numerical solutions of the VERSE model can be obtained [65]. The partial differential equations are discretized by application of the method of orthogonal collocation on fixed finite elements. Equation 16.59 is divided into 50 or 60 elements, each with four interior collocation points. Legendre polynomials are used for each element. For Eq. 16.62, only one element is required. It is described by a Jacobi polynomial with two interior collocation points. The resulting set of ordinary differential equations, with their initial and boundary conditions and the chemical equations, are solved using a differential algebraic system solver (DASSL) [65,66]. 

Selecting ( ) = / and substituting the approximate solution into the weak formulation, one obtains the local Galerkin finite element equation,... 

The structural analysis can be done using finite element methods. The finite element equation generated from Eq. 6.60 takes the following general form ... 

Some other complications in the course of numerical simulation of rubber and tires are rubber near-incompressibUity and reinforcement in tires. The rubber nearincompressibility makes the system of finite element equations ill conditioned since in this case the volumetric stiffness greatly exceeds the shear stiffness. The nearincompressibility and incompressibility conditions are essentially constraints imposed on the solution, and depending on the ratio of the number of discrete equations and discrete number of constraints solution may or may not exist. Therefore, the design of specific finite elements to satisfy these conditions becomes very important. 

As we have mentioned in the introduction, rubber parts typically experience large displacements and strains during their deformation history and, therefore, linearization based on the theory infinitesimal strains and small displacements that is traditionally employed for steel, reinforced concrete, and so on will produce inaccurate results. In order to retain the accuracy and realistic description of the deformation process in mbber, a fully nonlinear description of the deformation process should be considered. In the following discussion, we will obtain discretized finite element equations and outline their solution methods. 

To outline a finite element analysis approach, we will formulate a boundary value problem, transform it into a weak or variational form, and obtain discretized finite element equations. We begin with the equations of equihhrium that are written in the deformed configuration [1] ... 

Equation (8.6) is now suitable to obtain a set of discrete finite element equations. Specifically, the undeformed domain is discretized by subdividing it into a collection of nodes and elements, and after making an assumption of how nodal variables vary locally within each element, Eq. (8.6) represents a set of discrete equations consisting of a set of primary unknown nodal velocities. 

Since the nodal test functions u>iK are arbitrary, we obtain a set of nonlinear finite element equations ... 

Bathe KJ, Cimento AP (1980) Some practical procedures for the solution of nonlinear finite element equations. Comput Methods Appi Mech Eng 22(l) 59-85... 

Bathe KJ, Dvorkin EN (1983) On the automatic solution of nonlinear finite element equations. Comput Struct 17(5-6) 871-879... 

This finite element is be used to compose a simple substructure as shown in Fig. 26.1 where two elements are used to model a part of a tapered bar. The stiffness matrix given in Eq. 26.1 can be applied to these two elements to formulate the principal finite element equation for both elements as ... 

Assembling of these individual finite element equations gives the global finite element formulation in the general form Ku = F as ... 

The principal finite element equations for each substructure (s = A V B) can be... 

These two equations can be assembled to the global finite element equation Ku = F to obtain... 

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