Chapter 6 Finite Element Method (FEM)

Chapter 6 Finite Element Method (FEM) 6.1 Introduction to FEM

•History and Development

•Basic Finite Element Concepts

•Electromagnetic Analysis

•Program Techniques and Software Package

6.1.1 History and Development

Turner, Clough, Martin, and Topp introduced the finite element concepts 1956. Almost simultaneously, Argyris and Kelsey developed similar concepts in a series of publications on energy theorems. Courant, Hrennikoff and McHencry are also early precursors of the finite element methods.

Starting in 1967, many books have been written on the finite element method. The three editions of the book authored by Prof. Zienkiewicz received worlwide diffusion. Gallagher, Rockey et al., as well as the books written by Absi and Imbert.

The first FEM book (in English) for electrical engineer written by P. P. Silvester is published in 1983.

1. Who and When

Any computer codes in the early 1960s for solving structural analysis problems in a manner similar to what is now called the finite element approach existed only within the aircraft industry, which was one of the earliest motivated to have improved analysis methods. Turner, Clough, Martin, and Topp introduced the finite element concepts 1956. Almost simultaneously, Argyris and Kelsey developed similar concepts in a series of publications on energy theorems. Courant, Hrennikoff and McHencry are also early precursors of the finite element methods.

Starting in1967, many books have been written on the finite element method. The three editions of the book authored by Prof. Zienkiewicz received worldwide diffusion. Gallagher, Rockey et al., as well as the books written by Absi and Imbert.

The first FEM book (in English) for electrical engineer written by P. P. Silvester is published in 1983.

2. What and Why

The principles used to establish valid equations describing the behavior of the engineering problem at hand include equilibrium, Newton's idea regarding force acting on amass,

potential energy, strain energy, conservation of total energy, virtual work, thermo-dynamics, conservation of mass, Maxwell's equations, and many more.

The problem seemed to always be that once all the hard work of formulating the problem was complete, solving the resulting mathematical equations (sometimes linear and often nonlinear partial differential equations), was almost impossible.

Ritz had extended his idea to use different geometric regions, establish separate approximating functions in each region, and then hook them together. This is precisely what finite elements are about.

The idea had to wait until modern digital computers took away the fear of large numbers of algebraic equations. Soon matrices and matix methods of organizing large numbers of algebraic equations where brought into the finite element approach; recall that the word "matrix" was in the title of the important Air Force Conference in 1965.

Now mathematicians reentered the scene to remind the engineers of all the methods of solving matrix equations from linear algebra.

Restated, the finite element method is one wherein the difficulty of mathematically solving large complex geometric problems is transformed from a differential equation approach to an algebraic problem, wherein the building blocks or finite elements have all the complex equations solved for their simple shape (say a triangle, rod, beam, etc.)

The representation of the relationship of the important variables for the little, but not infinitesimal, element is determined through a Rayleigh or Ritz approach just for each element. Once this is done, a matrix of size equal to the number of unknowns for the element can be produced which represents the element.

Infect, the method can be used to solve almost any problem that can be formulated as a field problem. The development of additional software products and use within industry has taken the last 25 or 30 years.

The area still under the early phase of use is probably the electromechanical and electromagnetic area, probably owing to delayed recognition of need and transfer of knowledge from structural to electrical engineers. today's increased world competition, the pressure to design for electromagnetic compatibility, and new electronic and communication devices.

6.1.2 Basic Finite Element Concepts

1. Field Analysis

Engineering design is aided by engineering analysis, the calculation of performance of a trial design. To predict the performance it is often necessary to calculate a field, which is defined as a quantity that varies with position within the device analyzed.

There are many kinds of fields,and each field has a different in fluency on the device performance.

Table 2.1 Various Aspects of Performance

________________________________ Field Potential _________________________________

Heat flux Temperature

Mechanical stress Displacement

Electric field Voltage

Magnetic field Magnetic

vector potential

Fluid velocity Fluid potential

_________________________________

Various Problems in Electromagnetic Fields:

Static Fields

• Electrostatic and magnetostatic field calculations for both linear and non-linear

problems Quasi-Static Fields

• Time-dependent fields, including the transient and steady state behavior of

electromagnetic devices, eddy currents and skin effect Wave Propagation

• Wave propagation problems including microwaves and antennas, Scattering and radiation

Optimization

• Optimization using deterministic and stochastic methods, inverse problems, AI applications, neural networks

Material Modeling

• Modelling of material properties covering superconducting, composite, and microwave absorbing materials and the numerical treatment of anisotropy, semi-conductor, hysteresis, permanent magnets

Coupled Problems:

• Moving boundary problems, as well as electromagnetic fields coupled to mechanical, electronic, thermal and/or flow systems

2. Finite Element Modelling

Calculation of all the above fields and potentials can be performed using finite element analysis. The analysis begins by making a finite elements model of the device. The model is an assemblage of finite elements, which are pieces of various sizes and shapes.

The finite element model contains the following information about the device to be analyzed:

• geometry, subdivided into finite elements

• materials

• excitations

• constraints

Materials properties, excitations, and constraints can often be expressed quickly and easily, but geometry is usually difficult to describe.

Figure 1 shows a typical engineering problem that happens to be a static thermal or heat transfer problem.

Copper wire Rubber sheath

Fig. 1 Cross-section of current-carrying copper wire with a rubber cover sheath. The wire and its current extend into and out of the page.

To perform finite element thermal analysis, the finite element model shown in Fig.2 was constructed by dividing the device into finite elements.

Fig.2 Two-dimensional finite element model of Fig.1.

3. Energy Functional Minimization and Galerkin's Method

All the desired unknown parameters in the finite element model is by minimizing an energy functional. An energy functional consists of all the energies associated with the particular finite element model.

The law of conservation of energy is that the total energy of a device or system must be zero. Thus, the finite element energy functional must equal zero.

The finite element method obtains the correct solution for any finite element model by minimizing the energy functional. Thus, the solution obtained satisfies the law of conservation of energy.

The minimum of the functional is found by setting the derivative of the functional with respect to the unknown grid point potential to zero.

It is known from calculus that the minimum of any function has a slope or derivative equal to zero. The basic equation for finite element analysis is

0

=

dp

dF

(2.1)

where F is the functional (energy) and p is the unknown grid point potential to be

calculated. The above simple equation is the basis for finite element analysis. The functional F and unknown p vary with the type of problem.

In variational calculus the functional is shown to obey a relationship called Euler's equation. Substitution in the appropriate Euler's equation yields the differential equation of the physical system. Thus, the finite element solution obeys the appropriate differential equation.

The Euler's equation can not be always found in some differential equations. In this case, we have to use Galerkin's method to find the discretized equation.

G_i P N dxdy_i S

*₌

∫∫

_{⋅ =0 (2.2)}

where _{Ni is interpolation function.} (a) one-dimensional elements

Linear(2) Quadratic(3) Cubic(4) (b) Two-dimensional elements Triangular elements Linear(3) Quadratic(6) Cubic(9) Quadrilateral elements Linear(4) Quadratic(8) Cubic(12) (c) There-dimensional elements Tetrahedronal elements

Hexahedron elements Linear(8) Quadratic(20) Cubic(32) Prismatical elements Linear(6) Quadratic(15) Cubic(24)

4. Two Dimensional Finite Elements

Two-dimensional finite elements connect three or more grid points lying in a two-dimensional plane as shown in Fig.2. We will briefly derive the equations for a triangular finite element modeling a physical system that obeys Poisson's differential equation.

Poisson's equation in two dimensions is

d dxk dT dx d dyk dT dy P + = (2.3)

where x and y are two dimensions and k is a material property.

Equation (2.3) governs static tempera-turns T, in which case P is power input per unit volume. It also governs static electric or magnetic fields, in which case T is potential (φ or

A) and P is charge density (Q) or current density (J), respectively. The energy functional for all these physical problems is

F k del T ds PT dS s s = _        −      

∫

1_{2 ( )}2

∫

1_{2 (2.4)}

where S is the surface area in the two dimensions. The first term of Eq. (2.4) is the energy stored in the cases of electric or magnetic fields and is related to power dissipated in the case of thermal fields.

The second term is the input energy in the cases of electric or magnetic fields and is related to power input in the case of thermal fields.

The first term involves the gradient

G =del T= ∇T (6.x 2.5) G x y T xu T yu x ( , )=∂ + _y ∂ ∂ ∂ (6.x 2.6)

where _{ux and uy are unit vectors.}

The simplest type of two-dimensional finite element assumes a linear, or firs-order, variation of the unknown potential T over the element.

y

x

Fig.3 Triangular finite element in the xy plane. Within this first-order element T is related to the three unknown T values at the three triangular grid points according to

T T a_{k k} b x c_{k k} k m n = + + = y

∑

( , , l ) (2.7)

Evaluating Eq. (2.7) at the three vertices gives the solution for the a, b, c coefficients:

a a a b b b c c c x y x y x y l m n l m n l m n l l m m n n           =           − 1 1 1 1 (2.8)

G x y b T u_{k k k} c T u_{k k k} (2.9) k m n ( , ) [ ] , , = + =

∑

l

Thus, the temperature gradient is constant within a particular triangular finite element.

The grid point potentials Tk can be found by minimizing the functional (2.4), where ds=dxdy. Substituting Eq. (2.9) and (2.4) in Eq. (2.1) and considering one triangular finite element yields ∂ ∂T kG PT ds j l m n j s 2 2 − 2 0     = , = ,

∫

, (2.10) ) ) )

Carrying out the integration over the triangle can be shown to yield the 3-by-3 matrix equation

[S][T]=[P] (2.11)

where the "stiffness" matrix is

[ ] (2.12) ( ) ( ) ( ( ) ( ) ( ( ) ( ) ( S k b b c c b b c c b b c c b b c c b b c c b b c c b b c c b b c c b b c c l l l l l m l m l n l n m l m l m m m m m n m n n l n l n m n m n n n n = + + + + + + + + +           ∆

where ∆ is the area of the triangle,

∆ = (2.13)           1 2 1 1 1 x y x y x y l l m m n n

and the right-hand side is the "load vector":

[ ]P (2.14) P P P =           ∆ 3

Equation (2.11) solves for the potential T in a region containing the one triangle with l, m,

n in Figure 3.

For practical problems with N nodes (grid point), the above process is repeated for each finite element, obtaining a stiffness matrix or element coefficient matrix [S] with N rows and N columns. [P] and [T] are then column vectors containing N rows.

All devices designed by engineers are in reality three-dimensional. A special case of a three-dimensional device is one that has axial symmetry. The basic equation can be written as k r r r T r k T z P ∂ ∂ ∂ ∂ ∂ ∂ ( )+ 2₂ = (2.15)

Fig. 4 Axisymmetric finite element.

6. Finite Elements in Three Dimension Three dimensional Poisson's equation:

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ x k T x y k T y z k T z P ( )+ ( )+ ( )= (2.16)

A three-dimensional finite element has at least four grid points, and they do not all lie in one plane. The element forms a solid shape containing a volume of material.

The Two-dimensional and axisymmetric finite elements described above are used whenever possible, because they are simpler to describe and uses than three-dimensional finite elements.

Fig.4 Three-D finite elements:

The derivation of the matrix equation for the tetrahedral finite element is rather similar to that of the triangular element. Thus in the case of quantities obeying Poisson's equation the tetrahedral element is similar to the two-D triangular element.

Extending Eq. (2.7) gives

T T_k a_k b x c y d_{k k k} (2.17) k m n o = + + + =

∑

( , , , l z)

The derivation of the matrix equations proceeds in a fashion similar to that described in 2-D. 7. Example of 2-D FEM y φ=1.0 V x φ=0 V Boundary condition: φ1=φ2=0.0 φ5=φ6=1.0

For the triangular element, the equations that must be satisfied for this linear representation to agree with the functional values at the vertices are

{

}

φ α α α α α α ( )e e e e e e e x y x y = + + =         1 2 3 1 2 3 1 (2.18) In matrix notation, 1 1 1 1 1 2 2 3 3 1 2 3 1 2 3 x y x y x y e e e e e e e e e e e e                   =         α α α φ φ φ (2.19) then α α α φ φ φ 1 2 3 1 1 2 2 3 3 1 1 2 3 1 1 1 e e e e e e e e e e e e x y x y x y         =                   − (2.20)

{

}

φ φ φ φ e e e e e e e e e e x y x y x y x y =                   − 1 1 1 1 1 1 2 2 3 3 1 1 2 3 (2.21)

The expression for φ(x,y) is equivalent to

{

}

φ φ φ φ e e e e e e e e e e e e e e x y b b b c c c d d d =                   − 1 2 1 1 2 3 1 2 3 1 2 3 1 1 2 3 ∆( ) (2.22) where

b

_ie

=

x y

_{je ke}

−

x y

_{ke je}

c

_ie

=

y

_je

−

y

_ke } (2.23)

d

_ie

=

x

_ke

−

x

_je and b b b c c c d d d x y x y x y e e e e e e e e e e e 1 2 3 1 2 3 1 1 2 2 1 2 1 1 1           =           − ∆( ) _(2.24)

∆( )e e e e e e x y x y x y =           1 2 1 1 1 1 1 2 2 3 3 e φ (2.25)

Substitute interpolation functions

(2.26) φ( )e ie ie i N = =

∑

1 3

where Nie are interpolation or shape functions.

N_ie = 1_e b_ie+c x_ie + _ie

2∆( ) ( d y) (2.27)

Substitute Eq.(2.26) to the energy functional corresponding to Laplace's equation, ∇2_φ=0,

W dS x y dxdy = ∇ = _    +           

∫

∫∫

1 2 2 2 2 2 ε φ ε ∂φ_∂ ∂φ_∂ (2.28)

and then Eq.(2.28) can be written as

W N x N x N y N y dxdy i j j i i j i j Se =  +      = =

∑

∑

∫∫

1 2 1 3 1 3 φ φ ∂_∂ ∂_∂ ∂_∂ ∂_∂ ( ) (2.29) where S N x N x N y N y dxdy ij i j i j Se =  +     

∫∫

∂_∂ ∂_∂ ∂_∂ ∂_∂ ( ) (2.30)

∂Nie/∂x and ∂Nie/∂y are given as

∂ ∂ N x c ie ie e = 2∆( ) (2.31) ∂ ∂ N y d ie ie e = 2∆( ) (2.32) dxdy e S e =

∫∫

∆( ) ( ) (2.33)

The energy functional W corresponding to φi,∂W/∂φi=0, we derive

∑

S_{ij j} (i=1,2,...,n) (2.34) j n φ = = 0 1

[S]{φ}={0} (2.35) where (2.36) [ ]S S S S S S_nn =             11 12 21 22 L L M O M { }φ φ φ φ =             1 2 M n (2.37)

where [S]is called element coefficient matrices.

For the single element, the equation can be written as

∂ ∂φ ∂ ∂φ ∂ ∂φ φ φ φ W W W S S S S S S S S S e e e e e e e e e e e e e e e e e e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 11 12 13 21 22 23 31 32 33 1 2 3                 =                     (2.38) where S_ij e c cie je d die je e ( ) ( ) = + 4∆ (2.39)

Substituting all of Sij and φ in Eq. (2.35), and then system matrix equation can be solved by computer program.

For the four-element example, N=6, so the matrix equati0n s written as

                    =                     Φ Φ Φ Φ Φ Φ                     − − − − − − − − − − − − − − − 0 0 0 0 0 0 2 1 1 0 0 0 1 2 0 1 0 0 1 0 4 2 1 0 0 1 2 4 0 1 0 0 1 0 2 1 0 0 0 1 1 2 2 6 5 4 3 2 1 ε

Transposing all known voltage values to the right-hand side of equation, then the unknown nodal voltages are obtained as the solution of this equation. The results are

V 2 / 1 4 3 =Φ = Φ If ε=1 ) ( ) ( ) ( ) ( 4 4 e je ie x je ie y d ij e je ie je ie e ij d d c c S d d c c S ∆ + = ∆ + = ε ε ε

Supermatrix and band matrix

The matrix can be stored in a rectangular array of dimension n(b+1) as shown below:

                                        0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 99 88 77 66 55 44 33 22 11 99 98 97 96 89 88 87 86 85 79 78 77 76 75 74 69 68 67 66 65 64 63 58 57 56 55 54 53 52 47 46 45 44 43 42 41 36 35 34 33 32 31 25 24 23 22 21 14 13 12 11 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a             =                         4 3 2 1 4 3 2 1 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 y y y y x x x x a a a a a a a a a a a a a a a a

            − − − =                         1 41 4 1 31 3 1 21 2 1 4 3 2 1 44 43 42 34 33 32 24 23 22 ˆ ˆ ˆ ˆ 0 0 0 0 0 0 1 x a y x a y x a y x x x x x a a a a a a a a a Program Techniques

The procedure of numerical techniques: 1. Pre-processor

• Mesh generation and material description by visual techniques

• Dimensions and symmetrical models

• Linearity and nonlinearity 2. Numerical analysis

• FEM program flowchart. See Fig.1 FEM program flowchart.

• Matrix solver. FINITE ELEMENTS

S

EVALUATION OF FLUXES PER ELEMENT

SOLUTION OF THE SYSTEM OF EQUATIONS FOR ALL

ASSEMBLING OF THE TOTAL SYSTEM OF EQUATION

EVALUATION OF ELEMENT MATRICES

INTRODUCTION OF THE ESSENTIAL BOUNDARY CONDITIONS

Electromagnetic Analysis 1. Laplace's Equation

2. Poisson's Equation

A special case of a three-dimensional device is one that has axial symmetry. The basic equation in a cylindrical coordinate system can be written as

1

2 0

r r

r

A

r

z

A

z

J

r z

∂

∂

µ

∂

∂

∂

∂

µ

∂

∂

θ θ

(

)

+

(

)

= −

System matrix equation can be expressed as [S]{A}={K}

For a single element, K has the form of

K_ie J e

= 0

3 θ∆

where J0θ is the current density.

∂

∂

∂

∂

∂

∂

θ θ θ

W

u

W

u

W

u

S

S

S

S

S

S

S

S

S

A

A

A

K

K

K

e e e e e e e e e e e e e e e e e e e e e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

/

/

/

1 2 3 11 12 13 21 22 23 31 32 33 1 2 3 1 2 3

















=





































−

















where Sij can be expressed as

S

_ij e

r

c c

d d

e e ie je ie je ( ) ( ) ( )

=



+







2

4

0

π

ε

∆

where

c

z

z

d

r

r

ie je ke ie ke je

=

−

=

−

(

)

r r r r e e e 0 1 2 3 3 = + + ( ) ( ) ( ) 3. Helments' Equation

Program Techniques and Software Packages

Supermatrix and band matrix

1. Mesh generation and confirmation 2. Developing your own program 3. Post-process