Finite Elements Approaches in the Solution of Field Functions in Multidimensional Space: A Case of Boundary Value Problems

Journal of Minerals & Materials Characterization & Engineering, Vol. 9, No.10, pp.929-959, 2010

929

Finite Elements Approaches in the Solution of Field Functions in

Multidimensional Space: A Case of Boundary Value Problems

Chukwutoo Christopher Ihueze*1, Okafor Emeka Christian1, Edelugo Sylvester

Onyemaechi2

1Department of Industrial ⁄ Production Engineering, Nnamdi Azikiwe University Awka.

2Department of Mechanical Engineering University of Nigeria

*Corresponding Author: ihuezechukwutoo @yahoo.com

ABSTRACT

An idealized two dimensional continuum region of GRP composite was used to develop an

efficient method for solving continuum problems formulated for space domains. The continuum

problem is solved by minimization of a functional formulated through a finite element procedure

employing triangular elements and assumption of linear approximation polynomial. The

assemblage of elements functional derivatives system of equations through FEM assembly

procedure made possible the definition of a unique and parametrically defined model from which

the solution of continuum configuration with an arbitrary number of scales is solved. The finite

element method(FEM )developed is recommended to be applied in the evaluation of the function

of functions in irregular shaped continuum whose boundary conditions are specified such as in

the evaluation of displacement in structures and solid mechanics problems, evaluation of

temperature distribution in heat conduction problems, evaluation of displacement potential in

acoustic fluids evaluation of pressure in potential flows, evaluation of velocity in general flows,

evaluation of electric potential in electrostatics, evaluation of magnetic potential in

magnetostatics and in the solution of time dependent field problems. A unified computational

model with standard error of 0.15 and correlation coefficient of 0.72 was developed to aid

analysis and easy prediction of regional function with which the continuum function was

successfully modeled and optimized through gradient search and Lagrange multipliers

approach. Above all the optimization schemes of gradient search and Lagrangian multiplier

confirmed local minimum of function as 0.006-0.00847 to confirm the predictions of FEM and

constraint conditions.

Keywords: finite element, continuum, functional of function, extremum, boundary value

930 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

1. INTRODUCTION

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 [1]. Also in the finite element process an approximate solution is sought

to the problem of minimizing a functional. The concept of the finite element approach to

elasticity as a process in which the total potential energy is minimized with respect to nodal

displacements can obviously be extended to a variety of physical problems in which an

extremum principle exists. The two concepts are combined in this study. Zienkiewicz and

Cheung [2] applied similar approach to solve continuum problem expressed in derivative format

employing the concept of functional minimization with FEM.

Above all, there are many problems encountered in engineering and physics where the

minimization of the integrated quantity usually referred as functional and subject to some

boundary conditions results in the exact solution.This functional may represent a physical

recognizable variable in some instances, for many purposes it is simply a mathematically defined

entity.

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 engineering field continuum problems can be basically in form of wave phenomenon,

diffusion phenomenon and potential phenomenon usually represented by hyperbolic, parabolic

and elliptic differential equations respectively [3].

The objective of this study is therefore to present a methodical approach to solve multiple

dimensional field problems using integrated variational and FEM approach to establish relations

for all elements functional of continuum where the minimization of the elements functionals

system and solution are expected to give the stationary values of the function which extremize

the functional.

2. THEORETICAL BACKGROUND

A finite element model of a two dimensional quadratic function is expected to present a

methodical approach to employ for solution of multidimensional field functions that may have

regular or irregular field regions. Zienkiewicz and Cheung [2] presented Euler theorem to

approximate field functions if the integral or functional of the form

.

I (u) = ∫∫∫ f( x,y,z,u, ∂u

∂x , ∂u

∂y , ∂u

∂z ) dxdydz (1)

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 931

is to be minimized. 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

∂

∂x [∂f

∂(∂u/∂x) ] + ∂

∂y [∂f

∂(∂u/∂y) ] + ∂

∂z [∂f

∂(∂u/∂z) ] - ∂f

∂u = 0 (2)

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

equation governing the behaviour of unknown physical quantity u can generally be expressed as

∂

∂x (kx

∂u

∂x ) + ∂

∂y (ky

∂u

∂y ) + ∂

∂z ( kz ∂u

∂z ) + Q = 0 (3)

where

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

kx, ky, kz , Q = 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

χ =∫∫∫ {1

2 [kx (∂u

∂x ) 2 + ky(∂u

∂y )2 + kz(∂u

∂z )2] - Q u}dxdydz (4)

subject to u obeying the same boundary conditions.

For two dimensional differential equation representing some physical quantities then

χ =∫∫ {1

2 [kx (∂u

∂x ) 2 + ky(∂u

∂y )2] - Q u}dxdy (5)

For the case of our interest, the equivalent functional to be minimized for 2-D Laplace model

reduces to

χ =∫∫ {1

2 [kx (∂u

∂x ) 2 + ky(∂u

∂y )2] }dxdy (6)

The finite element version of an integrated functional is obtained and minimized with respect to

degrees of freedoms of the associated elements. The element functional equations are assembled

and boundary conditions applied, resulting in a system of equations equal to the number of

unconstrained degrees of freedoms of the continuum.

932 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

3. FINITE ELEMENT METHOD (FEM)

Euler variational minimum integral theorem was applied with the procedure of [4] on the

general equation governing the behavior of field functions presented by [2] to develop a finite

element version of elements functions functionals. The elements function functionals are

minimized with respect to degrees of freedoms in the finite element method of assembly are

applied to obtain the system model that is solved for the field of function . Basic approaches to

achieve finite element solouttion of continuum are also available in [5-8}.

3.1 Formulation of Finite Elements Equations

The elements functional of the study are derived for each element and minimized using equation

(6). Minimization of element functional entails finding the partial derivatives of the element

functional at its nodes. The contributions of each element nodes are established and added for all

continuum nodes to obtain the finite element model of the system. The formulation of finite

element model starts by choosing the element type and then choosing the approximation

polynomial coefficients are determined for establishing the element equations from where the

interpolation functions for u are established for all elements. This function u is used then

employed in finding the finite element model of the elements functionals from where the sought

functions are found.

3.1.1 Discritization and element topology description

The region is discretized into 16 triangular elements with 26 degrees of freedom and assuming

displacement in the global system of coordinate (horizontal direction only) only as in Figure 1

elements topologies are described in Table 1 for the establishment of element interpolation

functions for the functional equations for the finite element minimization scheme.

Figure 1: Idealized Finite element Model of two Dimensional Composite Body.

. . .

.

(1)

(2) (3)

(4)

(9)

(6) (7)

(8) (14)(15)

(16)

(13)

(12)

(11) (10)

(5)

10 11 13

5

14 15

3

6

7

8

12

4

9

1

2

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 933

Table 1: Element Topology Description.

Element

Number

Active degrees of freedom

of elements

Element coordinates Element

nodes

1 u5, u4, u14, v5, v4, v14 (0,0), (0,21), (16,16) 14, 5, 4

2 u1, u4, u14, v1, v4, v14 (0,0), (16,16), (21, 0) 14, 4, 1

3 u1, u4, u3, v1, v4, v3 (21, 0), (16, 16), (25, 10) 1, 4, 3

4 u1, u3, u15, v1, v3, v15 (21, 0), (21, 0), (25, 10) 1, 3, 15

5 u5, u10, u9, v5, v10, v9 (0, 21), (0, 37), (10, 25) 5, 10, 9

6 u5, u9, u4, v5, v9, v4 (0, 21), (0, 25), (16, 16) 5, 9, 4

7 u9, u4, u8, v9, v4, v8 (16, 16), (10, 25), (22, 13)4, 9, 8

8 u8, u4, u3, v8, v4, v3 (16, 16), (22, 23), (25, 10)4, 8, 3

9 u15, u3, u2, v15, v3, v2 (25, 10), (35, 10), (35, 0) 15, 3, 2

10 u9, u10, u11, v9, v10, v11 (0, 37), (10, 37), (10, 25) 9, 10, 11

11 u9, u11, u13, v9, v11, v13 (10, 25), (10, 37), (18, 37)9, 11, 13

12 u9, u13, u12, v9, v13, v12 (10, 25), (18, 37), (19, 29)9, 13, 12

13 u9, u12, u8, v9, v12, v8 (10, 25), (19, 29), (22, 23)9, 12, 8

14 u3, u8, u7, v3, v8, v7 (25, 10), (22, 23), (29, 19)3, 8, 7

15 u3, u7, u6, v3, v7, v6 (25, 11), (29, 19), (35, 18)3, 7, 6

16 u3, u6, u12, v3, v6, v12 (25, 10), (35, 18), (35, 10)3, 6, 2

3.2 Determination of FEM Characteristics

3.2.1 Element 1 interpolation and functional equation formulation

By assuming a linear approximation polynomial of the form

yaxaayxu 210

),(



 (1)

and following the method of Ihueze etal (2009) and Asterly (1992)

5[0, 21]

4[16, 16]

14[0, 0]

1

u

v

u4

v5

1

5

v14

u14

v4

u5

4

14

934 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

Where a0, a1, a2 are called polynomial coefficients or shape constants so that by passing (1)

through the nodes of element1the system of unknown function of the element becomes:

142141014 yaxaau 

424104yaxaau 

525105 yaxaau 

Putting the above polynomial function in matrix form then







































5

4

14

2

1

0

55

44

1414

1

u

a

yx

By applying Crammers rule

 





14441455141454455414

2

1

0yxyxuyxyxuyxyxua A









(2)







414514545414

2

1

1yyuyyuyyuaA









144551444514

2

1

2xxuxxuxxua A





Substituting (2) in (1) then

yaxaau 2101













yxxuxxuxxu

xyyuyyuyyu

yxyxuyxyxuyxyxuu

A

144551444514

2

1

414514545414

2

1

14441455141454455414

2

1















(3)

Recall that the approximation function is given as

55441414 uNuNuNu  (4)

Comparing (5) and (6) we evaluate shape and interpolation function thus

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 935

 





yxxxyyyxyxyxN A45544554

2

1

14 ,





 (5)

 





yxxxyyyxyxyxN A514145514145

2

1

4,







 





yxxxyyyxyxyxN A144414144414

2

1

5,







But A =

 





1444145141454554

2

1yxyxyxyxyxyx





 = 168mm2 (6)

where A= area of triangular element so that



yxN 165336

336

1

14 



xN 21

336

1

4



yxN 1616

336

1

5

Substituting (7) in (4)







5

336

1

4

336

1

14

336

1161621165336 uyxuxuyxu









336

16

336

21

336

5

414 u

uu

x

u







336

16

336

16 5

14 u

u

y

u







By assuming a two dimensional Laplace function for the continuum function of the form

∂2u

∂x2 + ∂2u

∂y2 = 0 (10)

The minimum function integral called functional to be minimized becomes in which case

kx = ky = kz = 1 and Q = 0 (11)

So that (4 ) reduces to





 









 



dxdyx y

u

x

u22

2

1

(9)

(8)

(12)

(7)

936 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

By substitutingg the first partial derivatives of the element 4 interpolation functions in ( 12) with

dxdy = A = 168

2

5

2

454145144

2

14 762.0656.0_524.0313.0418.0 uuuuuuuuux  (13)

By differentiating w.r.t.u14,u4,and u5



5.0*524.0313.0836.0

5414

14

uuu

u

x





5.0*313.0312.1 5144

4

uuu

u

x





5.0*524.0524.1 4145

5

uuu

u

x



3.2.2 Element 2 interpolation and functional equation formulation

By assuming a linear approximation polynomial of the form

yaxaayxu 210, )(  (15)

Passing (15) through the nodes then

142141014 yaxaau

121101 yaxaau

424104 yaxaau 

Putting the system in matrix form then







































4

1

14

2

1

0

44

11

1414

1

u

a

yx

(14)

u

u1

v4

v

[16, 16]

[21, 0]

[0, 0]

2

4

14

2

4

u14

v14

v1

u4

1

14

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 937

By applying Crammers rule,







14144114144114

2

1

0xxuyyuyxyxua A













414114414141441

2

1

1xxuyyuyxyxuaA













14141441411144

2

1

2xxuyyuyxyxua A





 (16)

Substituting (16) in (15)















 14144114144114

2

1xxuyyuyxyxuu A











 414114414141441 xxuyyuyxyxu

(17)









141411441411144 xxuyyuyxyxu





Recalling that the approximation function is

44111414 uNuNuNu  (18)

Comparing (18) and (17) then







yxxxyyyxyxNA14411441

2

1

14













yxxxyyyxyxN A414144414144

2

1













yxxxyyyxyxN A141114141114

2

1

4





 (19)

But A =

 





1411144141441441

2

1yxyxyxyxyxyx





 =168mm2 (20)

so that

 







yxxN 5163362116160336 336

1

336

1

14







 





yxyxN 16161600160 336

1

336

1







 





yyN 2102100336

1

336

1

4 (21)

Substituting (22) into (18)







4

336

1

336

1

14

336

1211616516336uyuyuyxu





 (22)

938 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

2121336

16

336

16 114114 uuuu

x

u











1621336

5

336

21

336

16

336

5411414114 uuuuuu

y

u











(23)

By substitutingg the first partial derivatives of the element 4 interpolarion functions in ( 12) with

dxdy = A = 168mm2





2

441144

2

1141

2

14 328.05.0156.038.03611.0190.0 uuuuuuuuux  (24)

By differentiating w.r.t.u14,u1,and u4

4114

14

156.0261.0418.0 uuu

u

x



5141

1

5.0261.076.0 uuu

u

x



1144

4

5.0156.0656.1 uuu

u

x



(25)

3.2.3 Element 3 interpolation and functional equation formulation

By assuming a linear approximation polynomial of the form



yaxaayxu 210

, (26)

and passing (26) through the nodes then

121101 yaxaau

323103 yaxaau 

424104 yaxaau 

v

[16, 16]

4

3 [25, 10]

[21, 0]

3

u

v1

u1

u3

v3

v4

u4

3

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 939

Putting the above equations in matrix form then,







































4

3

1

2

1

0

44

33

11

1

u

a

yx

(27)

By applying Crammers rule,







34143134431

2

1

0xxuyyuyxyxua A













41314341143

2

1

1xxuyyuyxyxua A













13431413314

2

1

2xxuyyuyxyxua A





 (28)

Substituting (28) into (26)















 34143134431

2

1xxuyyuyxyxuu A











 41314341143 xxuyyuyxyxu













13431113314 xxuyyuyxyxu





(29)

Recall that the approximation function or interpolation function is expressed as:

443311 uNuNuNu  (30)

Comparing (29) and (30) then,







yxxxyyyxyxN A34433443

2

1













yxxxyyyxyxN A41144114

2

1

3













yxxxyyyxyxNA13311331

2

1

4







(31)

But A =









133141143441

2

1yxyxyxyxyxyx





= 57mm2 (32)

then







yxyxN 9624025161610160400 114

1

114

1















yxyxN 51633616210163360114

1

114

1

3

















yxyxN 41052521251000525 114

1

114

1

4







 (33)

940 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

Substituting (30) into (31)







4

114

1

3

114

1

114

141052551633696240 uyxuyxuyxu













114

10

114

16

114

614

3

1u

u

x

u









114

4

114

5

114

94

3

1u

u

y

u







(34)

By substitutingg the first partial derivatives of the element 4 interpolarion functions in ( 12) with

dxdy = A = 57

2

443

2

34131

2

1254.0789.0616.0105.0224.0257.0 uuuuuuuuux  (35)

By differentiating w.r.t.u1,u3, and u4

431

1

105.0224.0514.0 uuu

u

x



413

3

789.0224.0232.0 uuu

u

x



314

4

789.0105.0508.1uuu

u

x



(36)

3.2.4 Element 4 interpolation and functional equation formulation

By substitutingg the first partial derivatives of the element 4 interpolarion function in (12)

2

315331

2

15151

2

135.02.05.0207.0214.0357.0 uuuuuuuuux  (37)

By differentiating w.r.t.u1,u15, and u3

3151

1

5.02144.0714.0 uuu

du

dx 

3115

15

2.0214.0414.0 uuu

du

dx 

1513

3

2.05.07.0 uuu

du

dx 

(38)

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 941

3.2.5 Element 5 interpolation and functional equation formulation

By substitutingg the first partial derivatives of the element 4 interpolarion functions in (12) with

dxdy = A =57mm2

2

10109

2

910595

2

5181.02.04.0163.06.035.0 uuuuuuuuux  (39)

By differentiating w.r.t.u5,u9,and u10

1095

5

163.06.070.0 uuu

u

x



1059

9

2.06.080.0 uuu

u

x



9510

10

2.0163.0362.0 uuu

u

x



(40)

3.2.6 Element 6 interpolation and functional equation formulation

By similar procedures as above,

2

595

2

99454

2

4257.0618.0616.0614.0105.0254.0 uuuuuuuuux  (41)

By differentiating w.r.t.u4,u9,and u5

1095

4

508.0 uuu

u

x



559

9

618.0614.0232.1 uuu

u

x



945

5

618.0105.0514.0 uuu

u

x



(42)

3.2.7 Element 7 interpolation and functional equation formulation

By similar procedures as above,

2

998

2

89484

2

4221.0141.0305.0302.0469.0385.0 uuuuuuuuux  (43)

By differentiating w.r.t.u4, u8, and u9

942 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

984

4

302.0469.077.0 uuu

u

x



948

8

141.0469.061.1 uuu

u

x



949

9

141.0302.0442.0 uuu

u

x



(44)

3.2.8 Element 8 interpolation and functional equation formulation

By similar procedures as above,

2

484

2

84383

2

3450.0530.0295.0369.061.0215.0 uuuuuuuuux  (45)

By differentiating w.r.t.u3,u8,and u4

483

3

369.0061.0430.0 uuu

u

x



438

8

530.0061.0590.0 uuu

u

x



934

4

530.0369.090.0 uuu

u

x



(46)

3.2.9 Element 9 interpolation and functional equation formulation

By similar procedures as above,

152

2

15

2

332

2

25.025.025.05.05.0 uuuuuuux  (47)

By differentiating w.r.t.u2,u3,and u15

1532

2

5.05.0 uuu

u

x



23

3

5.05.0 uu

u

x



215

15

5.05.0 uu

u

x



(48)

3.2.10 Element 10 interpolation and functional equation formulation

By similar procedures as above,

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 943

119

2

9

2

101110

2

11 417.0208.03.061.0508.0 uuuuuuux  (49)

By differentiating w.r.t.u11,u10, and u9

91011

11

417.06.0016.1 uuu

u

x



1110

10

6.06.0 uu

u

x



119

9

417.0416.0 uu

u

x



(50)

3.2.11 Element 11 interpolation and functional equation formulation

By similar procedures as above,

119

2

9

2

111311

2

13 333.0167.0542.075.0375.0 uuuuuuux  (51)

By differentiating w.r.t.u9,u13, and u1

119

9

333.0334.0 uu

u

x



1113

13

75.075.0 uu

u

x



91311

11

333.075.0084.1 uuu

u

x



(52)

3.2.12 Element 12 interpolation and functional equation formulation

By similar procedures as above,

2

131312

2

12139129

2

9319.0158.068.0151.068.0214.0 uuuuuuuuux 

By differentiating w.r.t.u9,u12, and u13

13129

9

151.0684.0428.0 uuu

u

x



3912

12

158.0684.0368.1 uuu

u

x



12913

13

158.0151.0638.0 uuu

u

x



(53)

944 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

3.2.13 Element 13 interpolation and functional equation formulation

By similar procedures as above,

2

9129

2

1298128

2

8170.0182.0561.0386.0758.0367.0 uuuuuuuuux  (54)

By differentiating w.r.t.u8, u12, and u9

9128

8

386.0758.0734.0 uuu

u

x



9812

12

182.0758.0122.1 uuu

u

x



1289

9

182.0386.034.0 uuu

u

x



(55)

3.2.14 Element 14 interpolation and functional equation formulation

By similar procedures as above,

2

887

2

78373

2

3307.0665.0563.0051.0462.0206.0 uuuuuuuuux  (56)

By differentiating w.r.t.u3,u7, and u8

873

3

051.0462.0412.0 uuu

u

x



837

7

665.0462.0126.1 uuu

u

x



738

8

665.051.0614.0 uuu

u

x



(57)

3.2.15 Element 15 interpolation and functional equation formulation

By similar procedures as above,

2

776

2

67363

2

3716.0923.0385.051.0154.0178.0 uuuuuuuuux 

By differentiating w.r.t.u3, u6, and u7

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 945

763

3

51.0154.0356.0 uuu

u

x



763

3

51.0154.0356.0 uuu

u

x



637

7

923.051.0432.1 uuu

u

x



(58)

3.2.16 Element 16 interpolation and functional equation formulation

By similar procedures as above,

2

662

2

232

2

3313.0625.0513.04.02.0 uuuuuuux 

By differentiating w.r.t.u3, u2 and u6

23

3

4.04.0 uu

u

x



632

2

625.04.0026.0 uuu

u

x



26

6

625.0626.0 uu

u

x



(59)

4. SYSTEM ELEMENTS ASSEMBLY ALGORITHMS

The algorithms for element assembly involves the addition of all elements contributing to

minimization dXe

du , this leads to system of equations that equals the degrees of freedoms in the

continuum, the derivatives are then added in a special format called assembly. There are 15

effective degrees of freedoms for the assembly of 16 elements

dXe

ui = 0, i = 1, 2, 3,……, 16

For

i = 1, Xe

u1 = 0 (60)

i = 2, Xe

u2 = 0 (61)

946 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

i = 3, Xe

u3 = 0 (62)

i = 4, Xe

u4 = 0 (63)

i = 5, Xe

u5 = 0 (64)

i = 6, Xe

u6 = 0 (65)

i = 7, Xe

u7 = 0 (66)

i = 8, Xe

u8 = 0 (67)

i = 9, Xe

u9 = 0 (68)

i = 10, Xe

u10 = 0 (69)

i = 11, Xe

u11 = 0 (70)

i = 12, Xe

u12 = 0 (71)

i = 13, Xe

u13 = 0 (72)

i = 14, Xe

u14 = 0 (73)

i = 15, Xe

u15 = 0 (74)

5. ELEMENTS EQUATIONS ASSEMBLY

All the partial derivatives resulting from the minimization scheme with respect to the fifteen (15)

active degrees of freedom (DOF) are added as follows the superscripts on these equations denote

element sources:

X

u1 = Xe

u1 = 0 = X2

u1 + X3

u1 + X4

u1

= 1.988u1 – 0.724u3 - 0.105u4 – 0.5u5 – 0.261u14 – 0.214u15 (75)

X

u2 = Xe

u2 = 0 = X9

u2 + X16

u2

= 1.026u2 – 0.9u3 – 0.625u6 - 0.5u15 (76)

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 947

X

u3 = Xe

u3 = 0 = X16

u3 + X15

u3 + X14

u3 + X9

u3 + X4

u3 + X3

u3 + X8

u3

= - 0.724u1 – 0.9u2 + 3.03u3 – 1.158u4 + 0.154u6 – 0.972u7 – 0.001u8 - 0.2u15 (77)

X

u4 = Xe

u4 = 0 = X1

u4 + X2

u4 + X3

u4 + X6

u4 + X7

u4 + X8

u4

= -1.158u3 + 5.49u4 – 0.395u1 + 0.008u5 – 0.469u8 – 1.832u9 – u10 – 0.001u14 (78)

X

u5 = Xe

u5 = 0 = X1

u5 + X5

u5 + X6

u5

= - 0.409u4 + 1.979u5 – 1.218u9 – 0.163u10 – 0.262u14 (79)

X

u6 = Xe

u6 = 0 = X15

u6 + X16

u6

= 1.396u6 – 0.625u2 + 0.154u3 – 0.923u7 (80)

X

u7 = Xe

u7 = 0 = X15

u7 + X14

u7

= 2.549u7 – 0.972u3 – 0.923u6 – 0.665u9 (81)

X

u8 = Xe

u8 = 0 = X7

u8 + X8

u8 + X13

u8 + X14

u8

= 0.449u3 – 0.999u4 – 0.665u7 + 3.548u8 + 0.245u9 – 0.758u12 (82)

X

u9 = Xe

u9 = 0 = X13

u9 + X12

u9 + X11

u9 + X10

u9 + X7

u9 + X6

u9 + X5

u9

= - 0.302u4 – 1.832u5 + 0.386u8 + 3.851u9 – 0.20u10 – 0.75u11 + 0.502u12 + 0.151u13 (83)

X

u10 = Xe

u10 = 0 = X5

u10 + X10

u10

= - 0.163u5 – 0.2u9 + 0.962u10 – 0.6u11 (84)

X

u11 = Xe

u11 = 0 = X11

u11 + X10

u11

= - 0.75u9 – 0.6u10 + 2.1u11 - 0.75u13 (85)

X

u12 = Xe

u12 = 0 = X13

u12 + X12

u12

= - 0.158u13 + 0.758u8 + 0.502u9 + 2.49u12 (86)

948 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

X

u13 = Xe

u13 = 0 = X11

u13 + X12

u13

= - 0.151u9 – 0.75u11 – 0.158u12 + 1.388u13 (87)

X

u14 = Xe

u14 = 0 = X2

u14 + X1

u14

= - 0.261u1 – 0.313u4 – 0.262u5 + 0.836u14 (88)

X

u15 = Xe

u15 = 0 = X4

u15 + X9

u15

= - 0.214u1 – 0.5u2 – 0.2u3 + 0.914u15 (89)

6. APPLICATION OF BOUNDARY CONDITION

In this work a special case where displacements at the boundaries are limited to 0.5mm for an

irregular continuum is considered to predict continuum displacement, strain and stress functions,

while the constrained conditions are taken as zero so that by equating u14 = u15 = 0 and u2 = u5=

u6 = u8 = u10= u13 = 0.50 , (75 - 89) transform to the following:

1.988u1 – 0.724u3 – 0.105u4 = 0.25 (90)

0.900u3 = 0.201 (91)

- 0.724u1 + 3.03u3 – 0.158u4 – 0.972u7 = 0.374 (92)

- 1.158u3 + 5.490u4 – 0.395u1 = 0.731 (93)

- 0.409u4 – 1.218u9 = - 0.907 (94)

0.154u3 – 0.923u7 = - 0.386 (95)

2.549u7 – 0.972u3 – 0.665u9 = 0.462 (96)

0.449u3 – 0.999u4 – 0.665u7 + 0.245u9 – 0.758u12 = - 1.774 (97)

3.851u9 – 0.302u4 – 0.75u11 + 0.502u12 = 0.748 (98)

- 0.200u9 – 0.600u11 = - 0.400 (99)

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 949

2.100u11 – 0.750u9 = 0.675 (100)

- 0.153u3 + 0.502u9 + 2.490u12 = - 0.379 (101)

- 0.151u9 – 0.750u11 – 0.158u12 = - 0.694 (102)

- 0.261u1 – 0.313u4 = 0.131 (103)

- 0.214u1 – 0.200u3 = 0.250 (104)

7. SOLUTION AND POST PROCESSING FOR CONTINUUM FUNCTION

The following nodal displacements in mm are further evaluated by first evaluating u3 = 0.222

from (91) so that other nodal values of the displacement function is as presented in Table 2. The

first partial derivatives of the interpolation function evaluated with active degree of freedom in of

element with respect to the x axis gives the slope of the function and also gives the value of the

strain as presented in Table 2. The computations are achieved with sixteen elements interpolation

functions associated with the elements global coordinate axis. The strains so computed may be

used with Hooke’s law of elasticity to predict the stress distribution function at the respective

nodes when the elastic modulus is known from literature.

Table 2: FEM Results.

n(nodes) u(displacement) 

  (strain)

1 0.210 0.02

2 0.500 0.01

3 0.222 0.02

4 0.059 0.015

5 0.500 0.023

6 0.500 0.015

7 0.455 0.082

8 0.500 0.054

9 0.725 0.028

10 0.500 0.01

11 0.424 0.116

12 0.500 0

13 0.500 0.167

14 0.000 0.026

15 0.000 0.011

950 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

The stress prediction model of a material within the elastic limit is expressed as

σ = Е (105)

where Е = modulus of elasticity

The excel graphics of FEM result using Table 2 of Figure 2 shows a serious indication that the

minimum value of the function is between node 14 and 15 hence another extremization method

is needed to point at which point of the region is this extremum.

Figure 2: Distribution of Function within the Region.

8. DISCUSSION AND VALIDATIO N OF RESULTS

Regression analysis was carried out on FEM results to obtain a unified model for elements

function interpolation. The regression model so obtained is further used to transform the element

functional equation to aid extremization of FEM results.

8.1 Regression Analysis

Multiple linear regression analysis was carried out on finite element results to obtain the

following model for the region. By employing the classical multiple linear regression equation of

the form

u(x, y) = ao + a1 x+ a2y (106)

‐0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9101112131415

Function,u(x,y)

NodalPoints

Distributionof

function,u(x,y)within

reg i on

Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 951

a regression model for the FEM is obtained with Table 3 and expressed as (107).

u(x,y) = 0.065 + 0.0036x + 0.0130y (107)

The goodness of fit of regression was evaluated to obtain: Coefficient of determination, r2 = 0.52,

correlation coefficient, r = 0.72, standard error, se = 0.1

where u = field function evaluated through FEM

u1 = average of FEM function

up = field function predicted with regression model

Table 2 and Figure 2 show the variation of the function within the region. Continuum fluid

elements in heat and mass transfer operations associated with pipeline transportation can

elegantly be analyzed following the procedure of this work. The FEM developed can be applied

in the evaluation of the stress distribution in irregular shaped continuum whose boundary

conditions are specified such as in the evaluation of displacement in structures and solid

mechanics problems, evaluation of temperature distribution in heat conduction problems,

evaluation of displacement potential in acoustic fluids ,evaluation of pressure in potential flows

,evaluation of velocity in general flows, evaluation of electric potential in electrostatics and in

evaluation of magnetic potential in magnetostatics.

8.2 Extremization of Functional: Extremization by Lagrange Multipliers Approach

In order to further analyse the FEM results, the functional, of any element is transformed to a

function of (x, y) using the regression model of (107) to obtain:

χf

x, y0.000042x  0.0017y  0.000059x 0.00034y 0.00015xy  0.00847

 108

Figure 4a,b and c show versions of 3D plots of function using Matlab for (108)

The objective function

fx,y0.000042x  0.0017y  0.000059x 0.00034y 0.00015xy 0.00847

subject to the constraint relations

ux,y0.50.0225x 0.5 109

ux,y 0.0201x  0.0238y0 110

derived for nodes 14 and 10 of elements 1 and 5 at the boundaries.

952 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10

Table 3 Computations For Regression and Error Analysis of FEM Results.

N x y u x2 y

2 xy xu yu up (u-u1)2 (u-up)2

1 21 0.0 0.2100 441 0.0000 0.0000 4.4100 0.0000 0.1406 0.0266 0.004816

2 35 10 0.5000 1225 100 350 17.5000 5.0000 0.321 0.0161 0.032041

3 25 10 0.2220 625 100 250 5.5500 2.2200 0.285 0.0228 0.003969

4 16 16 0.0590 256 256 256 0.944 0.9440 0.3306 0.0986 0.073767

5 0.0 21 0.5000 0.0000 441 0.0000 0.0000 10.5000 0.338 0.0161 0.026244

6 35 18 0.5000 1225 324 630 17.5000 9.0000 0.425 0.0161 0.005625

7 29 19 0.4550 841 361 551 13.195 8.6450 0.4164 0.0067 0.00149

8 22 23 0.5000 484 529 506 11.000 11.5000 0.4432 0.0161 0.003226

9 10 25 0.7250 100 625 250 7.2500 18.1250 0.426 0.1239 0.089401

10 0.0 37 0.5000 0.0000 1369 0.0000 0.0000 18.5000 0.546 0.0161 0.002116

11 10 37 0.424 100 1369 370 4.2400 15.6880 0.582 0.0026 0.024964

12 19 29 0.5000 361 841 551 9.5000 14.5000 0.5104 0.0161 0.000108

13 18 37 0.5000 324 1369 666 9.0000 18.5000 0.6108 0.0161 0.012277

14 0.0 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.065 0.1391 0.004225

15 35 0.0 0.0000 1225 0.0000 0.0000 0.0000 0.0000 0.191 0.1391 0.036481

sum 275 282 5.595 7207 7684 4380 100.089 133.122 5.631 0.6721 0.32075