Application Tensor Analysis Fem

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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.

4626 C. Hernandez et al./ Applied Mathematics and Computation 219 (2013) 46254636


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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.

C. Hernandez et al. / Applied Mathematics and Computation 219 (2013) 46254636 4627


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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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