Equation stiffness

Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenient. Tliis is readily ascertained considering that shape functions defined in the global system have different coefficients in each element. For example... 

Siep 6 assembly of the elemental stiffness equations into a global system of algebraic equations... 

Elemental stiffness equations are assembled over their common nodes to yield... 

After the substitution for T from Equation (2.68), dx from Equation (2.70) and global derivatives of shape functions from Equation (2.71) into the elemental stiffness equation (2,.55) we obtain, for the equation corresponding to N[... 

Note that in contrast to the example shown in Section 2.2,2 the element stiffness equation obtained for this problem is not symmetric. After the substitution for the shape functions and algebraic manipulations... 

After the assembly and insertion of the boundai conditions the following set of global stiffness equations is derived... 

The general elemental stiffness equation can thus be written as... 

As an example we eonAsider the solution of Equation (2.80) with a value of a = 50, in this case the general form of the elemental stiffness equation is written as... 

Therefore the second-order derivative of/ appearing in the original form of / is replaced by a term involving first-order derivatives of w and/plus a boundary term. The boundary terms are, normally, cancelled out through the assembly of the elemental stiffness equations over the common nodes on the shared interior element sides and only appear on the outside boundaries of the solution domain. However, as is shown later in this chapter, the appropriate treatment of these integrals along the outside boundaries of the flow domain depends on the prescribed boundary conditions. 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

Elemental stiffness equations (i.e. the working equations) resulting from the described discredzations are in general written as... 

The main subroutine for evaluation of the elemental stiffness equations and load vectors. 

Let us first consider the assembly of elemental stiffness equations in the simple example shown in Figure 6.4. 

With respect to the selected elemental and global orders of node numbering the elemental stiffness equations for elements ei, eu and em in Figure 6.4 are expressed as... 

Frontal solution requires very intricate bookkeeping for tracking coefficients and making sure that all of the stiffness equations have been assembled and fully reduced. The process time requirement in frontal solvers is hence larger than a straightforward band solver for equal size problems. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

GAUSSP. Gives Gauss point coordinates and weights required in the numerical integration of the members of the elemental stiffness equations. 

SOLVER Assembles elemental stiffness equations into a banded global matrix, imposes boundary conditions and solves the set of banded equations using the LU decomposition method (Gerald and Wheatley, 1984). SOLVER calls the following 4 subroutines. ... 

ASEMB - reads and assembles the elemental stiffness equations in a banded form, as is illustrated in Figure 7.1 to minimize computer memory requirements. 

THIS SUBROUTINE ASSEIIBLES AND SOLVES GLOBAL STIFFNESS EQUATIONS... 

The best packages for stiff equations (see below) use Gear s back-... 

Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

For laminates that are symmetric in both geometry and material properties about the middle surface, the general stiffness equations. Equation (4.24), simplify considerably. That symmetry has the form such that for each pair of equal-thickness laminae (1) both laminae are of the same material properties and principal material direction orientations, i.e., both laminae have the same (Qjjlk and (2) if one lamina is a certain distance above the middle surface, then the other lamina is the same distance below the middle surface. A single layer that straddles the middle surface can be considered a pair of half-thickness laminae that satisfies the symmetry requirement (note that such a lamina is inherently symmetric about the middle surface). ... 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that VJJ) becomes very small near the tube waU. There are also software packages that will handle the discretization automatically. 

The best packages for stiff equations (see below) use Gear s backward difference formulas. The formulas of various orders are [Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971)]... 

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