Journal of Minerals & Materials Characterization & Engineering, Vol. 9, No.10, pp.929-959, 2010
jmmce.org Printed in the USA. All rights reserved
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
1
1
u
u
u
a
a
a
yx
yx
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
A
A
144551444514
2
1
414514545414
2
1
14441455141454455414
2
1
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
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
1
1
u
u
u
a
a
a
yx
yx
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
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
1

 
yyN 2102100336
1
336
1
4 (21)
Substituting (22) into (18)

4
336
1
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
1
1
u
u
u
a
a
a
yx
yx
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
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
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
1
114
141052551633696240 uyxuyxuyxu

114
10
114
16
114
614
3
1u
u
u
x
u

114
4
114
5
114
94
3
1u
u
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)
NodalPoints
Distributionof
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
fx,y0.000042x  0.0017y  0.000059x 0.00034y 0.00015xy 0.00847
subject to the constraint relations
ux,y0.50.0225x 0.5 109
ux,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 xy xu yu 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
N x y u x2 y
2 xy xu yu 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
Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 953
By taking partial derivatives of Lagrange expression
Lx,y,λ,λf
x, yλ
gx,yλ
gx,y 111
0.000042x  0.0017y 0.000059x 0.00034y 0.00015xy 0.00847
λ0.0225x λ
0.0201x  0.0238y
to obtain the following relations
∂L
∂x 0.000042 0.0001x  0.00015y 0.0225λ 0.0201λ0 112
∂L
∂y 0.0017  0.0068y 0.00015x  0.0225λ 0.0238λ0 113
∂L
∂λ0.0225x 0 114
∂L
∂λ0.0201x  0.0238y 0 115
By solving (107)- (110) from(109)
0,λ
0.0356, λ 0.0378
By substituting the variables in (108) the optimum value of the function is obtained as
u(x,y) = f(x,y) = 0.00847
χf
x, y0.000042x  0.0017y  0.000059x 0.00034y 0.00015xy
 0.00847 108
The prediction of functional, χ with (108) are presented in Table 4 using excel package to draw
conclusion with the FEM and multiple linear regression results of Table 3.
954 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10
Table 4: Prediction of Functional with Equation (108).
N x Y Χ
1 21 0 0.035371
2 35 10
0.185715
3 25 10
0.134895
4 16 16
0.176886
5 0 21 0.19411
6 35 18
0.317475
7 29 19
0.296997
8 22 23 0.33281
9 10 25 0.30729
10 0 37 0.53683
11 10 37 0.59865
12 19 29 0.448457
13 18 37 0.656602
14 0 0 0.00847
15 35 0 0.082215
Tables 3 and 4 are compared for u , up and their functional, χ are found approximate.
8.2.1 Extremization by Lagrange gradient search approach
The extremum conditions for continuous and differentiable functions are defined [1]
as follows:
f  ...   
f  ...   
f . 
f  . 
Since fxx and fyy > 0 minimum extremum or local extremum exists.
The extremum at the interior points (x0, y0) is evaluated by solving simultaneous equation
formed by (100) and (101) to obtain x = 4.9767, y = -3.5978. By substituting this value in
equation (108) the function is obtained as 0.006, representing the extrema (minimum) value of
the function u(x, y) within the region.
Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 955
8.2.2 Extremization by Lagra nge multipliers approach
By expressing (108) in the form
fx,y0.000042x 0.0017y  0.000059x 0.00034y 0.00015xy  0.00847
Subject to the constraint relations
ux,y0.5  0.0225x 0.5 120
ux,y 0.0201x  0.0238y0 121
derived for nodes 14 and 10 of elements 1 and 5 at the boundaries.
By taking partial derivatives of Lagrange expression
Lx,y, λ, λx, yλ
gx,yλ
gx,y 122
0.000042x  0.0017y 0.000059x 0.00034y 0.00015xy 0.00847
λ0.0225x λ
0.0201x  0.0238y
to obtain the following relations
∂L
∂x 0.000042 0.0001x  0.00015y 0.0225λ 0.0201λ0 123
∂L
∂y 0.0017  0.0068y 0.00015x  0.0225λ 0.0238λ0 124
∂L
∂λ0.0225x 0 125
∂L
∂λ0.0201x  0.0238y 0 126
By solving (123)- (126) starting from(125)
0,λ
0.0356, λ 0.0378
By substituting the variables in (108) the optimum value of the function is obtained as
u(x,y) = f(x,y) = 0.00847.This value compares favourably with the prediction of 0.006 of
gradient search method showing agreement with the graphics of Figure 2 and Figure 3.
956 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10
Figure 3a, b Distribution of function within the Region.
0.2
0
0.2
0.4
0.6
0.8
1 2 3 4 5 6 7 8 9101112131415
Functtional,χ
Nodenumbers
Functional,χ(x,y)
distributionwithinregi o n
0.035371
0.1857150.134895
0.17886
0.19411
0.317475
0.296997
0.33281
0.30729
0.53683 0.59865
0.448457
0.656602
0.00847 0.082215
0
5
10
15
20
25
30
35
40
0 10203040
Distributionoffunctional,χ
inregi o n(x,y)
b)
a)
Vol.9, No.10 Finite Elements Approaches in the Solution of Field Functions 957
Figure 4a, b and c Versions of 3D Surface Plots of Function.
958 C.C. Ihueze, O.E. Christian, E.S. Onyemaechi Vol.9, No.10
9. CONCLUSSIONS
The methods of this article apply to:
1. Solution of boundary value engineering phenomena whose function can be expressed as
partial differential equation.
2. Solution of of displacement in structures and solid mechanics problems, temperature
distribution in heat conduction problems, displacement potential in acoustic fluids ,
pressure in potential flows , velocity in general flows, electric potential in electrostatics
magnetic potential in magnetostatics , torsion of non – homogenous shaft, flow through
an anisotropic porous foundation, axi – symmetric heat flow, hydrodynamic pressures on
moving surfaces
3. Solution of time dependent field problems such as creep, fracture and fatigue.
4. Equations (97) and (98) are recommended for the prediction of possible values of the
displacement function of GRP composites region from where other properties of the
region could be evaluated.
5. 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.
6. The MatLab 3-D graphics of Figure 4 show potential trend of function within the
regionwith minimum and maximum at the boundaries.
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