Finite Elements in the Solution of Continuum Field Problems

Journal of Minerals & Materials Characterization & Engineering, Vol. 9, No.5, pp.427-454, 2010

427

Chukwutoo Christopher Ihueze

Department of Industrial/ Production Engineering

Nnamdi Azikiwe University Awka, Nigeria

Email: ihuezechukwutoo @yahoo.com

ABSTRACT

A finite element functional solution procedure was presented employing variational calculus.

The Functionals of field continuum were developed on adoption of Euler minimum integral

theorem and finite element procedures on Laplace model. The elements functionals minimization

resulted to series of partial differential equations describing the variation of the function of

interest at various discrete nodal points. The assembly of the partial differential equations gave

a unifying algebraic system of equation was solved for the unique solutions of the function. To

simulate the finite element model, boundary conditions of temperature field was assumed. The

solution and post processing of FEM of this study showed that once the stiffness matrix of a

continuum is established and the boundary conditions specified the continuum is solved

uniquely. Regression method was used to establish the error associated with FEM results and to

establish a simple prediction model for environmental temperatures. The procedure of this study

presented the basis for insulation design for solid, hollow or shell pipes in fluid transport design

in oil and gas transport system. The finite element method evaluated the temperature distribution

of the region to serve as a guide in quantifying quantity of heat to the environment from the

transit fluid. The error of FEM prediction was estimated at 0.006 and the coefficient of

determination for goodness of regression fit is estimated as 0.99999. This study also presents an

approximate procedure for processing polar systems as rectangular systems by using the

circumference of the circular section as one dimensional independent variable and the difference

between the inner and outer radius (thickness) as the second i n d ependent variable.

Keywords: Calculus of variation, functional, finite element, continuum, field problems, boundary

conditions.

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 428

1. INTRODUCTION

The essence of this work is to use a simple bounded region to present a procedure to solve

functional problems in fluid transport piping system. A finite element functional solution

procedure was presented using variational calculus. Mechanics field problems of heat and mass

transfer and fluid flow problems in oil and gas industries is intended to be solved following the

formulations of this article. Fluid mechanics problems can be studied by isolating a discrete

domain with boundary values such as fluids within shelled pipes in which boundary conditions

are known. The functional approach expects the flow function to be approximate to known field

phenomenon, such as wave phenomenon, , diffusion phenomenon and potential phenomenon,

though, some complex phenomena of engineering sciences may be a combination of the above

phenomena [1].

An immense number of analytical solutions for conduction heat-transfer problems has been

accumulated in the literature over the years past. Even so, in many practical situations the

geometry or boundary conditions are such that an analytical solution has not been obtained at all,

or if the solution has been developed, it involves such a complex series solution that numerical

evaluation becomes exceedingly difficult [2]. However, presently the most fruitful approach to

the problem is one based on finite different techniques, the basic principles of which are outlined

in [3, 4, 5, and 6]. But because of the short falls of finite difference method the major objective

of this article is to present the finite element method, as the most elegant approach of solving

function such as the temperature distribution in a medium. Finite element approach is suited to

solving partial differential equation so that this work used the finite element approach to develop

the partial differential equations of the function at the discrete nodes, and as such the values of

the function at the boundary nodes must be known, may be from the field data. In this study

linear triangular element is used to discretize the field domain or the mathematical model

representing the function, and the following the variational principle and Euler minimum integral

theorem to arrive at the elements equations. The elements equations are then added in a finite

element format called assembly and the system of the equations solved after the application of

boundary conditions to obtain the value of the function that minimizes the functional at the

stationary points. The procedure is analogous to Raleigh – Ritz method for stationary total

potential applied to elastic problems [7].

Above all, Most Quasi – harmonic field phenomenon are represented by either the partial

derivatives of the function or by the well known Laplace and Poisson’s equation [8]. The

velocities, temperatures and densities of fluids in oil and gas pipelines are expected to be known

at various sections of transport line as the mentioned variables determine efficiency of the

transportation unit.

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 429

Also working in polar coordinates for circular or polar systems in engineering and science often

posses some difficulties that this study presents an approximate procedure for processing polar

systems as rectangular systems by using the circumference of the circular section as one

dimensional independent variable and the difference between the inner and outer radius

(thickness) as the second independent variable. The function of interest as the dependent variable

is then analysed by solving an approximate function through FEM approaches, which in this

study is the Laplace steady state 2-D heat equation in cartesian coordinate. The geometry of

field quantities or continuum may be a problem to close form solution of field functions

encountered in engineering and science that appropriate algorithm becomes necessary to obtain

optimum solution, it is then necessary to employ calculus of variation principles and FEM to

obtain optimum continuum field functions whose boundary conditions are specified.

The methodology of this work is also supported by the fact that in calculus of variations, instead

of attempting to locate points that extremize function of one or more variables that extremize

quantities called functional, functions of functions that extremize the functional are found [9].

Also in the finite element process an approximate solution is sought to the problem of

minimizing a functional.

2. THEORETICAL BACKGROUND: BASIC FINITE ELEMENT FUNCTIONS

Figure 1 shows the typical triangular element considered, with nodes I,j,m numbered in

anticlockwise order.

Figure 1. An Element of Continuum for deriving Element Equation.

i

m

j

xi

yi



vi

y

x

ui

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 430

2.1 Approximation Function Determination

Linear Polynomials of the form

(1)

(2)

are usually chosen for horizontal and vertical responses or degrees of freedom(DOF).

By passing these polynomials in turn through nodes i,j,m ,the system of the function is obtained

for the element following the method of [7] as

(3)

(4)

(5)

2.2 Shape Function and Interpolation Function Definition

By employing Crammer’s rule for the polynomial coefficients, α0, α1 α2, also called the shape

constants, the interpolation function is established as

(6)

where

(7)

(8)

(9)

and

(10)

So that the shape functions are expressed at the nodes as follows

(11)

(12)

(13)

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 431

and interpolation function expressed as

(14)

2.3 Basic Field Equations and Approximating Functionals

[8] Presented the general equation governing quasi-harmonic field functions as

+ (15)

While the theorem of Euler presented by [10] states that if the integral

(16)

is to be minimized, then the necessary and sufficient condition for this minimum to be reached is

that the unknown function u (x, y, z) should satisfy the following differential equation

+ + (17)

within the same region, provided u satisfies the same boundary conditions in both cases,

where

u = unknown function assumed to be single valued within the region

= specified functions of x, y, z

x, y, z = space variables

The equivalent formulation to that of equation (3) is the requirement that the volume integral

given below and taken over the whole region, should be,

(18)

3. FORMATION OF FUNCTIONAL OF TWO DIMENSIONAL FIELD FUNCTIONS

The functions that can be approximated by two dimensional Laplace equations such as the two

dimensional steady state heat transfer problem with constant thermal conductivity are said to be

governed by the equation [2]

(19)

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 432

The functional of this model is established considering a 2-D case of (15)

expressed as

+ (20)

So that by (18) the equivalent functional to be minimized becomes

(21)

The functional equation (21) is transformed to a more useful form by using the interpolation

functions of elements to obtain each element functional.

3.1 Derivation and Description of Element Characteristics and Functional

This process is accomplished by evaluating the contributions to each differential, such as

from a typical element then adding such contribution and equating to zero. Only the elements

adjacent to node i will contribute to .

3.2 Element Characteristics and Fun c ti o na l M in im i zation Scheme

The contribution of elements to the differential is evaluated as follows:

If the value of functional associated with the element is expressed from (21) as

(22)

Then by partially differentiating χ e ,

= (23)

With the interpolation function, defined as

u = (Ni, Nj, Nm)e (24)

and the shape function defined as

Ni = (25)

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(23) becomes

(26)

An element contributes three differentials associated to its nodes. The contributions of the three

nodes are listed

= (27)

Equation (27) can be written in the form

(28)

The matrix is easily written if are taken as constant within the element, also by nothing

that over the area of the element

(29)

+ (30)

The vector can be found as follows: If Q is assumed constant within an element then the

integral

(31)

can be evaluated

(32)

Where and are the coordinates of the centroid ie

(33)

Vol.9, No.5 Finite Elements in the Solution of Continuum Field Problems 434

so the integral expressed by (31) becomes

(34)

3.3 Assembly Procedure for Functional Minimization Scheme

The final equations of the minimization procedure require the assembly of all the differentials of

and equating of these to zero, expressed typically as

(35)

Equation (35) says that the functionals of all elements are established and partially differentiated

with respect to the associated nodal degrees of freedoms [11].

4. METHODOLOGY

4.1 Finite Element Modeling of 2-D Steady State Regional Function Distribution

Working in polar coordinates for circular or polar systems in engineering and science often