Finite Element Method - Steady - state field problems - heat condution, electric and magnetic potential, fluid flow, etc_07

Finite Element Method - Steady - state field problems - heat condution, electric and magnetic potential, fluid flow, etc_07 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

7

conduction, electric and magnetic

potential, fluid flow, etc

7.1 Introduction

While, in detail, most of the previous chapters dealt with problems of an elastic con- tinuum the general procedures can be applied to a variety of physical problems Indeed, some such possibilities have been indicated in Chapter 3 and here more detailed attention will be given to a particular but wide class of such situations Primarily we shall deal with situations governed by the general ‘quasi-harmonic’ equation, the particular cases of which are the well-known Laplace and Poisson equations.lP6 The range of physical problems falling into this category is large To list but a few frequently encountered in engineering practice we have:

Heat conduction

Seepage through porous media

Irrotational flow of ideal fluids

Distribution of electrical (or magnetic) potential

Torsion of prismatic shafts

Bending of prismatic beams,

Lubrication of pad bearings, etc

The formulation developed in this chapter is equally applicable to all, and hence little reference will be made to the actual physical quantities Isotropic or anisotropic regions can be treated with equal ease

Two-dimensional problems are discussed in the first part of the chapter A

generalization to three dimensions follows It will be observed that the same, Co,

‘shape functions’ as those used previously in two- or three-dimensional formulations

of elasticity problems will again be encountered The main difference will be that now only one unknown scalar quantity (the unknown function) is associated with each point in space Previously, several unknown quantities, represented by the displace- ment vector, were sought

In Chapter 3 we indicated both the ‘weak form’ and a variational principle applic- able to the Poisson and Laplace equations (see Secs 3.2 and 3.8.1) In the following sections we shall apply these approaches to a general, quasi-harmonic equation and

indicate the ranges of applicability of a single, unsed, approach by which one com- puter program can solve a large variety of physical problems

The general quasi-harmonic equation 141

- k <

7.2.1 The general statement

In many physical situations we are concerned with the dzflusion or flow of some

quantity such as heat, mass, or a chemical, etc In such problems the rate of transfer

per unit area, q, can be written in terms of its Cartesian components as

If the rate at which the relevant quantity is generated (or removed) per unit volume

is Q, then for steady-state flow the balance or continuity requirement gives

Generally the rates of flow will be related to gradients of some potential quantity 4

This may be temperature in the case of heat flow, etc A very general linear relation-

ship will be of the form

q = { +

where k is a three by three matrix This is generally of a symmetric form due to energy

arguments and is variously referred to as Fourier’s, Fick’s, or Darcy’s law depending

on the physical problem

is obtained by substitution of

Eq (7.5) into (7.4), leading to

The final governing equation for the ‘potential’

142 Steady-state field problems

which has to be solved in the domain R On the boundaries of such a domain we shall usually encounter one or other of the following conditions:

1 On I'4,

i.e., the potential is specified

2 On rp the normal component of flow, qn, is given as

where n is a vector of direction cosines of the normal to the surface, this condition

can immediately be rewritten as

With respect to such axes we have

k' = [ f, and the governing equation (7.6) can be written (now dropping the prime)

with a suitable change of boundary conditions

Lastly, for an isotropic material we can write

where I is an identity matrix This leads to the simple form of Eq (3.10) which was

discussed in Chapter 3

7.2.3 Weak form of general quasi-harmonic equation [Eq (7.6)]

Following the principles of Chapter 3, Sec 3.2, we can obtain the weak form of

Finite element discretization 143

Eq (7.6) by writing

v ( - V T k V 4 + Q ) d R + v [ ( k V 4 ) T n + q + a 4 ] d r = O (7.11)

for all functions w which are zero on I'd

Integration by parts (see Appendix G) will result in the following weak statement

which is equivalent to satisfying the governing equations and the natural boundary

conditions (7.7b):

(Vv)TkVc$ dR + v(a4 + 4 ) d r = 0 The forced boundary condition (7.7a) still needs to be imposed

7.2.4 The variational principle

(7.12)

We shall leave as an exercise to the reader the verification that the functional

gives on minimization [subject to the constraint of Eq (7.7a)l the satisfaction of the

original problem set in Eqs (7.6) and (7.7)

The algebraic manipulations required to verify the above principle follow precisely

the lines of Sec 3.8 of Chapter 3 and can be carried out as an exercise

7.3 Finite element discretization

This can now proceed on the assumption of a trial function expansion

using either the weak formulation of Eq (7.12) or the variational statement

Eq (7.13) If, in the first, we take

4)

of

according to the Galerkin principle, an identical form will arise with that obtained

from the minimization of the variational principle

Substituting Eq (7.15) into (7.12) we have a typical statement giving

( J o ( V N j ) ' L V N dR + Jrq N a N 1 d r ) a + I N , Q d o + Jr,N,YdT = O

i = I , , n (7.16)

or a set of standard discrete equations of the form

144 Steady-state field problems

with

( V N i ) T k V N j dR + NiaN, d r A = 1 NiQ dR + jrq Niq d r

n

on which prescribed values of 4 have to be imposed on boundaries F4

We note now that an additional ‘stiffness’ is contributed on boundaries for which a radiation constant Q is specified but that otherwise a complete analogy with the elastic structural problem exists

Indeed in a computer program the same standard operations will be followed even including an evaluation of quantities analogous to the stresses These, obviously, are the fluxes

and, as with stresses, the best recovery procedure is discussed in Chapter 14

7.4.1 Anisotropic and non-homogeneous media

Clearly material properties defined by the k matrix can vary from element to element in

a discontinuous manner This is implied in both the weak and variational statements of the problem

The material properties are usually known only with respect to the principal (or sym- metry) axes, and if these directions are constant within the element it is convenient to use them in the formulation of local axes specified within each element, as shown in Fig 7.1

- ,

Fig 7.1 Anisotropic material Local coordinates coincide with the principal directions of stratification

Some economic specializations 145

With respect to such axes only three coefficients k,, k y , and k, need be specified, and

now only a multiplication by a diagonal matrix is needed in formulating the co-

efficients of the matrix H [Eq (7.17)]

It is important to note that as the parameters a correspond to scalar values, no trans-

formation of matrices computed in local coordinates is necessary before assembly of the

global matrices

Thus, in many computer programs only a diagonal specification of the k matrix is

used

7.4.2 Two-dimensional problem

The two-dimensional plane case is obtained by taking the gradient in the form

and taking the flux as

(7.19)

(7.20)

On discretization by Eq (7.16) a slightly simplified form of the matrices will now be

found Dropping the terms with cx and ij we can write

(7.21)

No further discussion at this point appears necessary However, it may be worth-

while to particularize here to the simplest yet still useful triangular element (Fig 7.2)

With

aj + bjx + cjy

N =

2A

as in Eq (4.8) of Chapter 4, we can write down the element 'stiffness' matrix as

bjbj bib; bib, cjcj cjcj cic,

He = & 4A [ bjb, bjbm 1 +$ [ cjc; c j c m ] (7.22)

symmetric bmbm symmetric CmCm

The load matrices follow a similar simple pattern and thus, for instance, the reader

can show that due to Q we have

(7.23)

a very simple (almost 'obvious') result

146 Steady-state field problems

Fig 7.2 Division of a two-dimensional region into triangular elements

Alternatively the formulation may be specialized to cylindrical coordinates and used for the solution of axisymmetric situations by introducing the gradient

and integration carried out as described in Chapter 5, Section 5.2.5

7.5 Examples - an assessment of accuracy

It is very easy to show that by assembling explicitly worked out ‘stiffnesses’ of trian- gular elements for ‘regular’ meshes shown in Fig 7.3a, the discretized plane equations are identical with those that can be derived by well-known finite difference methods.’

Examples - an assessment of accuracy 147

Fig 7.3 ‘Regular’ and ’irregular’ subdivision patterns

Obviously the solutions obtained by the two methods will be identical, and so also

will be the orders of approximati0n.t

If an ‘irregular’ mesh based on a square arrangement of nodes is used a difference

between the two aproaches will be evident [Fig 7.3(b)] This is confined to the ‘load’

vector f‘ The assembled equations will show ‘loads’ which differ by small amounts

from node to node, but the sum of which is still the same as that due to the finite

difference expressions The solutions therefore differ only locally and will represent

the same averages

In Fig 7.4 a test comparing the results obtained on an ‘irregular’ mesh with a

relaxation solution of the lowest order finite difference approximation is shown

Both give results of similar accuracy, as indeed would be anticipated However, it

can be shown that in one-dimensional problems the finite element algorithm gives

exact answers of nodes, while the finite difference method generally does not In

general, therefore, superior accuracy is available with the finite element discretization

Further advantages of the finite element process are:

1 It can deal simply with non-homogeneous and anisotropic situations (particularly

when the direction of anisotropy is variable)

2 The elements can be graded in shape and size to follow arbitrary boundaries and to

allow for regions of rapid variation of the function sought, thus controlling the

errors in a most efficient way (viz Chapters 14 and 15)

3 Specified gradient or ‘radiation’ boundary conditions are introduced naturally and

with a better accuracy than in standard finite difference procedures

148 Steady-state field problems

Fig 7.4 Torsion of a rectangular shaft Numbers in parentheses show a more accurate solution due to South- well using a 12 x 16 mesh (values of 4/GOL2)

4 Higher order elements can be readily used to improve accuracy without complicat- ing boundary conditions - a difficulty always arising with finite difference approx- imations of a higher order

5 Finally, but of considerable importance in the computer age, standard programs may be used for assembly and solution

Two more realistic examples are given at this stage to illustrate the accuracy attain- able in practice The first is the problem of pure torsion of a non-homogeneous shaft illustrated in Fig 7.5 The basic differential equation here is

Some practical applications 149

in which q5 is the stress function, G is the shear modulus, and 0 the angle of twist per

unit length of the shaft

In the finite element solution presented, the hollow section was represented by a

material for which G has a value of the order of lop3 compared with the other

materiakt The results compare well with the contours derived from an accurate

finite difference solution.8

An example concerning flow through an anisotropic porous foundation is shown in

Fig 7.6

Here the governing equation is

(7.27)

in which H is the hydraulic head and k, and ky represent the permeability coefficients

in the direction of the (inclined) principal axes The answers are here compared

against contours derived by an exact solution The possibilities of the use of a

graded size of subdivision are evident in this example

7.6.1 Anisotropic seepage

The first of the problems is concerned with the flow through highly non-homo-

geneous, anisotropic, and contorted strata The basic governing equation is still

Eq (7.27) However, a special feature has to be incorporated to allow for changes

of x’ and y’ principal directions from element to element

No difficulties are encountered in computation, and the problem together with its

solution is given in Fig 7.7.3

7.6.2 Axisymmetric heat flow

The axisymmetric heat flow equation results by using (7.24) and (7.25) with q5 replaced

by T Now T is the temperature and k the conductivity

In Fig 7.8 the temperature distribution in a nuclear reactor pressure vessel’ is

shown for steady-state heat conduction when a uniform temperature increase is

applied on the inside

7.6.3 Hydrodynamic pressures on moving surfaces

If a submerged surface moves in a fluid with prescribed accelerations and a small

amplitude of movement, then it can be shown’ that if compressibility is ignored the

standard program

Some practical applications 151

Fig 7.7 Flow under a dam through a highly non-homogeneous and contorted foundation

excess pressures that are developed obey the Laplace equation

V ’ p = 0

On moving (or stationary) boundaries the boundary condition is of type 2 [see

Eq (7.7b)l and is given by

As an example, let us consider the case of a vertical wall in a reservoir, shown in

Fig 7.9, and determine the pressure distribution at points along the surface of the

wall and at the bottom of the reservoir for any prescribed motion of the boundary

points 1 to 7

The division of the region into elements (42 in number) is shown Here elements of

rectangular shape are used (see Sect 3.3) and combined with quadrilaterals composed

152 Steady-state field problems

Fig 7.8 Temperature distribution in steady-state conduction for an axisymmetrical pressure vessel

Fig 7.9 Problem of a wall moving horizontally in a reservoir

Some practical applications 153

Fig 7.10 Pressure distribution on a moving wall and reservoir bottom

of two triangles near the sloping boundary The pressure distribution on the wall and

the bottom of the reservoir for a constant acceleration of the wall is shown in Fig 7.10

The results for the pressures on the wall agree to within 1 per cent with the well-

known, exact solution derived by Westergaard."

For the wall hinged at the base and oscillating around this point with the top

(point 1) accelerating by ao, the pressure distribution obtained is also plotted in

Fig 7.10

In the study of vibration problems the interaction of the fluid pressure with

structural accelerations may be determined using Eq (7.28) and the formulation

given above This and related problems will be discussed in more detail in Chapter 19

In Fig 7.11 the solution of a similar problem in three dimensions is shown.4 Here

simple tetrahedral elements combined as bricks as described in Chapter 6 were used

and very good accuracy obtained

In many practical problems the computation of such simplified 'added' masses

is sufficient, and the process described here has become widely used in this

context I ] - I 3

7.6.4 Electrostatic and magnetostatic problems

In this area of activity frequent need arises to determine appropriate field strengths

and the governing equations are usually of the standard quasi-harmonic type

discussed here Thus the formulations are directly transferable One of the first

applications made as early as 1 9674 was to fully three-dimensional electrostatic

field distributions governed by simple Laplace equations (Fig 7.12)

In Fig 7.13 a similar use of triangular elements was made in the context of

magnetic two-dimensional fields by Winslow6 in 1966 These early works stimulated

considerable activity in this area and much work has now been p ~ b l i s h e d ' ~ - ' ~