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Application of tensor analysis to the finite element method
C. Hernandez, R.B.B. Ovando-Martinez, M.A. Arjona
Instituto Tecnolgico de La Laguna, Divisin de Estudios de Posgrado e Investigacin, Paseo de las Gemas 52, Residencial La Rosa, 27266 Torren, Coah, Mexico
a r t i c l e i n f o
Keywords:
Finite element methodPoisson equation
Tensor analysis
a b s t r a c t
This paper presents the application of tensor analysis to the finite element (FE) method. A FE
formulation is presented where the suffix notation and the Einsteins convention are usedinstead of the matrix algebra notation. The usage of the proposed notation avoids the draw-
backs of matrixalgebra and allowsthe easy implementation of the FE methodin a computer
code. The Dirichlet boundary condition is included in the FE formulation through the use of
suffix constraints. Afterwards, the suffix notation is applied to the FE formulation of the
Poisson equation and it is implemented in a C language. Finally, two numerical examples
are solved to illustrate the advantages of using the proposed notation in the FE method.
2012 Elsevier Inc. All rights reserved.
1. Introduction
The finite element (FE) is a numerical method which was introduced by Courant in 1943 and can be used for solving partial
differential equations (PDE) [15]. The FE method has also been successfully applied in the solution of multiphysics problems[68]. This numerical method assumes a test function of the unknown variable within a discrete domain. The test function can
be linear which is represented by triangular elements[9]. However, some authors have used a high order polynomial approx-
imation in the element representation [10]. The accuracy of the finite element method (FEM) is dependant on the order of the
test functions and the level of the mesh discretization. Nevertheless, a high computation time is needed in a fine discretized
mesh with higher order test functions due the large number of unknown variables. The traditional FE formulations involve
matrix algebra which may be difficult to handle due to the diversity of variables and equations in engineering problems [11].
This paper proposes the usage of the suffix notation and the Einstein summation convention in the FEM. The suffix notation
can handle multiple equations that otherwise must be represented by arrays [12]. The resulting shorthand FE formulation is
more compact, more understandable and easier to program in a computer language than the classical FE matrix formulation.
The introduction of suffixes reduces the number of equations in the mathematical processes without affecting the meaning of
the equations[13]. Some PDEs contain tensors and complex matrix operations due to the large number of resulting discrete
equations [14]. Some examples are found in the NavierStokes and Maxwell equations where the viscous tensor and the pres-
ence of anisotropic materials are modeled[4,15]. The developed shorthand FE formulation uses suffixes to reduce the set ofequations (matrix) into a single equation. Afterwards, the common operations used in algebra, differentiation and integration
are applied to the single equation giving the system solution. The improvement attained by applying the notation in the FEM
allows the search of new algorithms and solution strategies for the PDEs [16]. Wherefore, some investigators use tensors in
their research[17]. The mathematical operations using suffixes could represent an obstacle to its usage, nonetheless there is
literature where a brief introduction to tensor theory can be found[18,19].
0096-3003/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2012.10.074
Corresponding author.
E-mail address: [email protected](M.A. Arjona).
Applied Mathematics and Computation 219 (2013) 46254636
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te / a m c
http://dx.doi.org/10.1016/j.amc.2012.10.074mailto:[email protected]://dx.doi.org/10.1016/j.amc.2012.10.074http://www.sciencedirect.com/science/journal/00963003http://www.elsevier.com/locate/amchttp://www.elsevier.com/locate/amchttp://www.sciencedirect.com/science/journal/00963003http://dx.doi.org/10.1016/j.amc.2012.10.074mailto:[email protected]://dx.doi.org/10.1016/j.amc.2012.10.074 -
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The paper firstly presents the use of the tensor notation in the FE method. Afterwards, the solution of the Poisson equation
is presented to illustrate the advantages of the suffix notation. Finally, the shorthand FE formulation of an example is coded
in C programming language.
2. The finite element method
A dependent variable in a PDE is a function of multiple continuous variables in the space and time domains [9]. The PDE is
solved for the dependent variable using traditional mathematical techniques where the boundary conditions are taken intoaccount. In the FEM, a test function is proposed for the dependent variable without having to solve the PDE [20]. A simple
test function is defined by(1)
/ Xiai 1
where / is the potential, Xrepresents the shape vector, a is a row vector, and i = 1, 2, 3.
A repeated suffix in(1)involves a summation which is known as the Einsteins summation convention [12,15]. The con-
vention is used to achieve a compact notation. The Einsteins summation convention establishes that,
mini m1n
1 m2n2 m3n
3 2
wherem and n are functions of the suffix i. The shape vector contains the space variables and is defined as (3)
X1 1 X2 x X3 y 3
while the values for the row vector are given by(4)
a1 a a2 b a3 c 4
wherea,b, andcare unknown constants.
The constants ai are calculated using the FEM and their magnitudes are determined by the physical problem and the
boundary conditions.
Eq. (1) requires a system ofiequations to obtain theai constants. Wherefore, three distinct potentialsjare obtained once
the linear variation(1)has been considered in a triangular region for the three nodes in the space as it shown in Fig. 1. The
system of equations can be defined by(5)
/j Gjiai 5
wherej = 1, 2, 3, and
Gji
1 x1 y1
1 x2 y21 x3 y3
264 375 6The triangular region is namedelementwhile the vertices are known asnodes(seeFig. 1)[1,5]. The nodal potentials/jare
unknown variables, hence the constants ai may be defined as functions of the nodal potentials and they are given by (7)
ai gij/j 7
where
Fig. 1. A first-order finite element.
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gij G1ji
x2y3x3y2 x3y1x1y3 x1y2x2y1
y2y3 y3y1 y1y2
x3x2 x1x3 x2x1
264
375
2S
8
and Sis the area of an element and it is defined by (9)
Sx2x1y3y1 x3x1y2y1
2 9
In this paper, the tensorgij is defined as the geometrical weight tensor because it contains the geometric information ofthe finite element.
The potential distribution /throughout the element is attained from (7) and (1); (10)describes the linear variation of the
potential / within the triangular region
/ Xigij/j 10
The product Xigij is a set of three lineal equations known as the shape functions and they are given by (11)
aj Xigij 11
The shape functions have the following properties,
Xixk;ykgij
1 k j
0 kj
12
whereXi (xk, yk) represents a shape vector Xiwhich is a function of the space variables x and y.The shape functions are represented by three planes in the space throughout the triangular element and they are shown
inFig. 2. The potential distribution/within the triangular element is attained with the superposition of the shape functions.
3. Discretization of the Poisson equation
The Poisson equation is a PDE with a wide range of applications in electrostatics, magnetostatic, mechanics and theoret-
ical physics problems[2123]. The Poisson equation is used to predict the distribution of the field variable in the space. Due
to broad relevance of this PDE, the discretization of the Poisson equation using the suffix notation and the Einstein conven-
tion are explained in this section. The suffix notation of the Poisson equation is shown by Eq.(13)
C@l@l/ f 13
whereCis a constant related to the material properties, fis the source term,l= 2, 3, and @lis a differentiation operator that is
defined by(14)
@l @
@Xl14
Eq. (13) replaces the classical vector notation of the Poisson equation which is given by (15)
Cr2/ f 15
Afterwards, the potential distribution(10)is used in(13)to obtain the discrete form of the Poisson equation that is given
by(16)
C@l@laj/j f 16
In this paper, the Galerkin Residual method represented by the integral Eq. (17) is used to solve the discrete Eq. (16) [5]
ZX a
k
dX 17
Fig. 2. The shape functions represented by three planes in the space.
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where X represents a surface (triangular area) and a are the weight functions.In the Galerkin method, the weight functionsaare identical to the shape functions[1]. After applying the Galerkin meth-
od to Eq.(16), the expression(18) is obtained
C
ZX
ak@l@laj/jdX
ZX
akfdX 18
wherek = 1, 2, 3.
The Eq. (18) represents a set of three equations related to the triangular element. Some mathematical properties might be
applied to(18). The application of the properties(19) and (20)transforms Eq. (18) into(21)
ak@l@laj/j @la
k@laj/j @la
k@laj/j 19
ZX
@lak@la
j/jdX
ZC
ak@laj/jdC 20
Eq. (20) is analogous to the Gauss Divergence theorem for open surfaces X of closed perimeter C [22,23]
C
ZC
ak@laj/jdCC
ZX
@lak@la
j/jdX
ZX
akfdX 21
The first integral in Eq. (21) represents a Neumann condition imposed over the contour C; wherefore the term @laj/jis a
known function and the discrete contour integral is given by (22)
CZC
ak@laj/jdC C@la
j/jC2
22
The second integral in(21)is solved throughout the triangular element and the expression (23) is obtained
C
ZX
@lak@la
j/jdX C@lak@la
j/jS 23
Nevertheless, a simplified form of Eq. (23) is achieved when(11)and the Einstein summation convention are applied in
the suffix l resulting in(24)
C@lak@la
j/jS CSglkglj/j 24
Finally, the solution for the integral that contains the source terms in(21)is given by(25)
ZX a
k
fdX
f
S
3 25
Eqs.(21)(25)are used to obtain the shorthand FE formulation of the Poisson equation that is shown in (26)
Rkj/j Bk 26
whereRkj = CSglkglj and Bk = (fS/3) (C@laj/jC/2).
The expression(26)represents three equations for each element; otherwise it would be represented by a large array if
matrix notation is used. The resulting shorthand FE formulation (26)is more compact and understandable than the classical
FE matrix formulation. The process of assembling individual elements to achieve a global system and taking into account the
Dirichlet conditions to(26)is best illustrated with an example. A two element mesh is shown inFig. 3, where Eq. (26) is
applied and two different arraysRmn are attained as R1ij and R2
kl.
Fig. 3. Finite elements with local and global node numbering.
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Fig. 3shows two different node numberings for each element. The node numbering at the right mesh is local while the
numbering at the left mesh is known as global. Local numbering goes from 1 to 3 at each element and counterclockwise to
avoid the presence of negative discrete areas. The local arrayRij, kl is assembled using the local numbering of each element.
The global arrayRmn is assembled by replacing the local node number with its equivalent global node number, i.e., the node 5
of the element 1 has the local number 3 and the same process applies to the rest of the nodes and elements of the mesh. The
global Rmn represents anm n array that contains the relationship between the elements and nodes of the mesh, i.e. the
assembling of the elements is attained by the nodes 5 and 8.Table 1shows the global assembly and the node contributions
achieved for the mesh shown inFig. 3.
The global arrayRmn shown inTable 1is also known as the stiffness matrix. The stiffness matrix is a function of the geo-
metrical weight tensor(8).Table 1shows the diagonal values that are obtained from the node contribution of each element
and the off-diagonal values that are obtained from the contribution of the neighbor nodes. Similarly, the Dirichlet nodes areassembled in(26)by imposing the suffix constraints given by Eq.(27)
dnkdnj
Rkj dnk
dnj
" #Rkj/j B
kdnk dnk d
njdkj R
kj
/n 27
wheren is the known global node, k= 1, 2..nodes number,and j= 1, 2..nodes number. The operatord is namedDelta Kronecker
operatorand it has the properties defined by (28)[12]
dji
1 i j
0 ij
28
and its opposite value is defined by (29)
dji 0 i j
1 ij
29
Eq. (27) demonstrates the advantages of the suffix notation and how it allows handling the Dirichlet boundary conditions
clearer and easier than using the matrix formulation.
4. Numerical examples
Two numerical examples are shown in this section to illustrate the application of the shorthand FE formulation of the
Poisson equation. The first example deals with the suffix manipulation in tensor analysis while in the second example a mag-
netostatic problem is solved.
4.1. Example I: Suffix manipulation in tensors
This example shows how to deal with the suffix manipulation to obtain the arrayRmn and the vectorBm in a global node
assembly[1]. The studied mesh has two elements and four nodes as it is shown in Fig. 4. The first element represents the
source withf= 1 while the nodes 1 and 3 are known quantities (Dirichlet nodes). In this example, the constant Cis chosen
to have the value of one at each element. The local assembling Rkj for the first element is given by Eq. (30)
Rkj1
S g2kg2j g3kg3j
30
The valuesgmn are calculated using tensor(8)fork= 1, 2, 3, andj= 1, 2, 3; whereas the areaSis obtained by using(9). The
Rkj and Bk achieved for the first element are shown in(31)
R1
1:236 0:7786 0:4571
0:7786 0:6929 0:0857
0:4571 0:0857 0:3714
264
375; B1 0:35
3
1
1
1
264
375
0:116
0:116
0:116
264
375 31
Table 1
Global assembly and node contributions for the mesh shown in Fig. 3.
Node 5 8 11 23
5 R5 51 R5 52 R
5 81 R
5 82 R
5 111 R
5 232
8 R8 51 R8 52 R
8 81 R
8 82 R
8 111 R
8 232
11 R11 51 R11 81 R
11 111
23 R23 52 R23 82 R
23 232
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Fig. 4. FE mesh used for the example I.
Table 2
Global assembly and node contributions for the mesh shown in Fig. 4.
Node 1 2 3 4
1 R1 11 1:236 R1 21 0:7786 0 R
1 31 0:4571
2 R1 21 0:7786 R2 21 R
1 12 1:25 R
1 22 0:4571 R
2 31 R
1 32 0:0143
3 0 R1 22 0:4571 R2 22 0:8238 R
2 32 0:3667
4 R1 31 0:4571 R2 31 R
1 32 0:0143 R
2 32 0:3667 R
3 31 R
3 32 0:8381
Fig. 5. Domain used in the shorthand FE model.
Fig. 6. FE mesh used in the example II.
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Fig. 7. Flowchart of the shorthand FE model.
Fig. 8. Magnetic vector potential distribution for the magnetostatic problem.
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Likewise, the Rkj and Bk calculated for the second element are shown in (32)
R2
0:5571 0:4571 0:1
0:4571 0:8238 0:3667
0:1 0:3667 0:4667
264
375; B2 0:525
3
0
0
0
264
375
0
0
0
264
375 32
The assembling process of the individual elements shown inFig. 4to obtain a global connection is best illustrated with
the numerical data shown inTable 2.
Whereas the terms in the global array Bm, are:B1 = 0.116, B2 = 0.116, B3 = 0, B4 = 0.116.The assembling process of the Dirichlet nodes in (27)is illustrated for the second row of the global array Rmn. The second
row in the global array Rmn corresponds to the relationship between node 2 and nodes 1, 3, and 4. Hence the second row
(k= 2) is given by (33)
dn2dnj
R2j dn
2dnj
" #R2j/j B
2dn2
dn2
dnjd2jR
2j
/n 33
There are two Dirichlet boundary conditions at nodes 1 (n= 1) and 3 (n= 3). For node 1, the expression (33)is trans-
formed into Eq.(34)
d1jR2j/j B
2 d1jd2jR
2j/1 34
The left hand side of(34) manipulates the global array Rmn at the elements in the second row and first column (j= 1). There-
fore, the second row of the global array Rmn change its values to:R21 = 0,R22 = 1.25,R23 = 0.4571, andR24 = 0.0143. The right
side of(34)is equivalent to have B2 due /1= 0 which means thatB2 = 0.116. For the node 3, the expression(35)is obtained
d3jR2j/j B
2 d3jd2jR
2j/3 35
The left side of(35)manipulates the global array Rmn at the elements in the second row and the third column (j= 3).
Wherefore, the second row of the global Rmn becomes:R21 = 0, R22 = 1.25,R23 = 0, and R24 = 0.0143.
The right hand side of expression(35)is re-evaluated due to theR23 and/3quantities and it gives B2 = 4.455. By apply-
ing the same process to the rest of the rows, the final system is obtained as(36)
1 0 0 0
0 1:25 0 0:0143
0 0 1 0
0 0:0143 0 0:8381
26664
37775
/1
/2
/3
/4
26664
37775
0
4:455
10
3:541
26664
37775 36
The solution of(36)gives a result of [/1/2/3/4]T
= [0 3.612 10 4.297]T
.The employment of the shorthand FE formulation allows a clearer and easier programming of the Dirichlet boundary con-
ditions than the matrix based formulation. Usually little attention in the FE programming is given to this point and it is as-
sumed that the reader understands how to program the FEM into a computer program [24]. The programming stage is more
difficult to grasp when it does not exist a clear relationship between the matrix formulation and the program syntax and
structure. The next subsection shows the application of the shorthand FE formulation to solve a physical problem.
4.2. Example II: A shorthand FE model for solving a magnetostatic problem
In electromagnetics, the Poisson equation is used to describe the magnetic field distribution produced by a current density
and/or a constant magnetic flux[2123]. The magnetostatic field can be obtained by using the relationships (37)in Eq.(13)
C 1=l0lr
f
J
/ Az
9>=>; 37Therefore the Poisson equation(38)is obtained in suffix notation
1
l0lr
@l@lAz J 38
whereJ(A/m2) is the current density,Az is the magnetic vector potential, l0(Wb/(Am)) is the free-space magnetic perme-ability andlr the relative magnetic permeability.
Eq. (38) is solved to find the magnetic vector potential distribution over the problem domain shown in Fig. 5where an
infinitely long square wire with a current density Jhas been placed in the air. The corresponding FE mesh is seen in Fig. 6
[25,26].
Unlike Example I,Fig. 6contains a larger number of finite elements (146 elements). However, the solution of this problem
without a computer program would be very difficult. The flowchart of the developed FE computer program is shown inFig. 7.
A brief description of each stage is explained next.
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Step 1: Input data. Relevant information is input to the program, i.e., the number of nodes, number of elements, region
labels, material properties and boundary conditions.
Step 2:Element global connection. The FE global assembling is achieved with the help of the local and global node num-
bering. The system is assembled by using(26)with the corrections given by(37)for the magnetic vector potential distribu-
tion analysis. Eq. (26) requires a memory allocation to store the data. At this step, it is necessary to take care of the suffix
dynamics. An erroneous suffix numeration will generate an error in the final global assembly.
Step 3: Boundary conditions. Eq. (27) can be programmed using if-then conditions and forloops. Although the Dirichlet
boundary conditions can be explained with matrix algebra, Eq. (27) represents a more general function that is easier to pro-
gram than using the matrix notation. As it was demonstrated in the first numerical example, the Dirichlet boundary condi-
tions modify the global assembling by applying Eq.(27).
Step 4:Solution. The solution of the system can be achieved by solving the set of equations with a direct or iterative solver.
Step 5:Post-processing. Due the computer program gives only numerical results, it is necessary to have a post-processing
stage to plot the magnetic vector potential distribution[27]. The magnetic vector potential distribution for this example is
shown inFig. 8.
Appendix Apresents the C program developed for solving the proposed shorthand FE model defined by (27). The suffix
dynamics and the Delta Kronecker operator were implemented usingif-thenconditions andforloops in the computer code.
The mesh file used in the C program is shown in the appendix. The mesh file must be named mesh_1.txt and be written with
the following data format: (1) number of nodes, (2) number of elements, (3)x ycoordinates, (4) nodes per element,fand C
values, (5) number of boundary condition nodes, and (6) boundary node number and its value.
5. Conclusions
This paper has demonstrated the application of tensor analysis to the finite element method. It was found that the suffix
notation improves the FE formulation. The proposed shorthand FE formulation is more compact and understandable than the
classical FE matrix formulation. The developed shorthand FE formulation allows a clearer and easier programming of the FE
and the Dirichlet conditions compared with the programming derived from the matrix formulation. In addition, the compu-
tational programming is simpler due the similarities between the suffix dynamics and the program syntax and structure. To
illustrate the application of this formulation, a C language program was presented.
Acknowledgements
The authors would like to thank to CONACYT, DGEST and Instituto Tecnolgico de La Laguna for their financial support.
Appendix A
A computer program developed in C programming language is given in this appendix. It illustrates the application of ten-
sor analysis to the solution of the Poisson equation with the FEM.
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