Finite Difference Methods

Finite difference methods also provide solutions to engineering problems governed by differential equations but approximate to these

in a discrete manner over a typical infinitesimal region.

Various methods of solution exist, the simplest being the explicit method. For a differential equation, say where temperature is

expressed in terms of space and time, as in transient heat flow, and the temperatures are required at a later point in time, a forward value is used for the time derivative but backward known values are used for the calculation of the space derivative. Stability of the

solution however requires that a quantity known as the Fourier number

which is expressed in terms of the space and time step is less than or equal to 0.5. This imposes limitations on the time and space steps that can be used in the construction of the mesh.

More complex fully implicit methods are stable for any value of the

Fourier number. These represent the second order space derivative in 142

terms of forward unknown values at the next time step. Each equation contains three unknowns and the solution is obtained by solving a system of equations. The method has the disadvantage that there is a larger truncation error and the accuracy which increases with the choice of time step is reduced.

General implicit methods resolve these limitations on the accuracy and time step by evaluating the second-order space derivative at a point intermediate between the two time steps, applying a weighting, A and 1 - A to each of these values, respectively. The accuracy of the method decreases with increasing A and is stable for all time steps and Fourier numbers provided A is in the range 0.5 to 1.0. For A = 0.5. The most accurate solution is obtained for a system that is

stable for all time steps and Fourier numbers. The method is then

known as the Crank-Nicolson method. 3.1.3 Temperature Distributions

The PAFEC analysis assumes a temperature distribution {0} governed by the Poisson equation

826 826 826 86

+ --- + --- = a — 5x2 8y2 8z2 8t

A finite element solution can be obtained from the minimum value of the functional.

[S]{0) + [M]{6} = [Q] where,

{0} = the derivatives of temperature with respect to time [S] = the square symmetric thermal conductivity matrix [M] = the square symmetric thermal mass matrix

[Q] = the vector of heat fluxes which enter the structure at the nodes

Knowing {0} at a particular instant in time, PAFEC calculates the temperature gradients at that instant in time and rather than a finite element method uses a finite difference, Crank-Nicolson method to predict the transient temperature distribution at a later point in time, (t + 0t) . Although the Crank-Nicolson method is stable for all time steps and Fourier numbers it does exhibit an oscillatory

behaviour which does not decay. This is most severe at points where a large thermal shock is applied and in such cases it is important that the Fourier number has a value close to unity.

For steady state problems {0} = 0 the Poisson equation reduces to the Laplace equation,

520 828

+ = 0

8x2 8y2

The minimum of the functional, [S]{0} = [Q]

can then be solved provided at any one node either the temperature or

the heat flux at that node is known. For such steady state problems

PAFEC uses a finite element method of solution.

Temperature calculation elements are used in the analysis having one 144

degree of freedom at each node, viz temperature. 3.1.4 Stress Distributions

For stress analyses various variational functions exist depending on which of the displacements, strains and stresses are unknown. The more complex of these have all of the stresses, strains and displacements unknown. By imposing certain conditions simpler variational functions

are obtained. For example the condition that specified stress-strain

relations are always satisfied.

The simplest situations use the principle of minimum potential energy, having only the displacements unknown and the strain displacement relations imposed; for example a linear elastic solid deforming according to Hooke's law, and the principle of minimum complementary energy having only the stresses unknown and the condition that the stresses are in equilibrium imposed.

For the case of minimum potential energy the variational function is given by the strain energy stored internally less the work done by externally applied loads. The structure is divided into a mesh of elements, the strain energy calculated for each element and summed over all elements. One dimensional, linear, two dimensional,

triangular and rectangular and three dimensional, tetrahedral and brick elements exist.

The type of element used in the analysis depends on the deformations permissible, the degree of accuracy and the economy of computing required. Each type of element has a characteristic polynomial shape function, [N] expressed as a function of position and used to

describe the displacements at a general point within the element, {/x} in terms of the nodal displacements of that element, {/* ) • The

resulting pattern of displacements is known as the interpolation function,

(A*> - [N] (/ie )

The shape functions are chosen to give the appropriate nodal displacements when the co-ordinates of the appropriate node are inserted. They have to satisfy certain convergence criteria, one of which is that continuity of the displacements (and their derivatives) occurs across the element boundaries. This ensures that finite

integral values are obtained in the solution.

Shape functions are also used to describe the variation of the unknown function across the element.

The number of degrees of freedom or unknowns in the displacements and unknown function are dependent on the order of the shape function. Obviously the greater the number of nodes on an element the higher the

degrees of freedom and the higher the order of the shape function. For

a given degree of accuracy elements having higher order shape functions will require fewer but more complex elements. The effect this has on computing efficiency is relevant.

This is the situation for simple triangular and rectangular elements.

For elements with more arbitrary shapes perhaps with curved sides it is necessary to define a local co-ordinate system. The elements are transformed into a simple shape in the £, rj, f co-ordinate system using, a geometric shape function [N]. Thus the curvilinear co­

ordinates are expressed in terms of the nodal co-ordinates. The geometric shape function is generally the same as that used to

calculate the displacements and unknown function when the elements are termed isoparametric but can be different as in super- and sub-

parametric elements. These elements are of importance in the modelling of the complex geometries often met with in practice where simple element shapes are impractical.

For conditions of plane stress or plane strain translational

displacements are considered in two mutually perpendicular directions.

A commonly used element is the eight noded isoparametric curvilinear element for plane stress and plane strain.

Displacements are expressed as a quadratic function of the curvilinear co-ordinates £ and rj. Knowing the displacements at each node the

displacements at a general point, (£,*7) can be expressed in terms of the nodal displacements using the shape function. The normal and shear strains are then determined from the derivatives of the displacements and the stresses calculated by multiplying the strain vector by the elasticity matrix [D].

The element strain energy, S.E. given by 1

S.E. = -

2 (e}T {cr) h dA,

can then be expressed in terms of the nodal displacements. The

superscript T indicates the transpose of the strain vector and h and A are the thickness and area of the element, respectively.

Inclusion of rotational degrees of freedom requires the use of other categories of element.