1

Chapter 1

Introduction

This chapter presents a detailed introduction in the subject of coupled field

analysis, which describes particularly the application of coupled field analysis in

electromagnetic devices, with a section dedicated to the history of electromagnetic

forming.

1.1 History and importance of coupled field analysis

A coupled-field analysis is an analysis that takes into account the interaction

(coupling) between two or more disciplines (fields) of engineering, such as

electromagnetic field and mechanical field in electromechanical devices,

electromagnetic field and thermal field in induction-heating applications etc, and can

be used to investigate many physical problems.

When electricity was first introduced, man discovered the effects of strong magnetic

field on conducting bodies: the conducting bodies can be moved and deformed.

These caused by electromagnetic force and both effects (movement and deformation)

have been studied for their positive and negative aspects, ever since the first electrical

machine was designed and brought into operation.

During operation of electrical machines the electromagnetic force produces

sometimes, besides the motion, the undesired deformation of some parts of the

electrical machines. On the other hand, in other applications certain is the desired

effect of electromagnetic force, accompanying the deformation as in electromagnetic

forming.

In the Fig.(1.1), all the fields, that can exist in a transformer, are shown. The coupling

among these fields is also shown in the same figure. It is useful in analyze the

performance and predicting accurate response of the device in different operating

conditions.

2

Fig. 1.1: Coupling between different fields in transformer

Another typical procedure of coupled field analysis is the electromagnetic forming, a

high-velocity forming process aiming at the shaping of conducting objects (the so-

called workpieces) in strong electromagnetic fields. The electromagnetic forming is

the process of giving shape to a metallic object using electromagnetic forces. A high

frequency damped sinusoidal current obtained by L-C resonant circuit is made pass

through a coil kept in front of a metallic object. This damped sinusoidal current,

flowing through the coil, produces a time-varying magnetic field which induces the

eddy currents in the metallic object. This leads to an electromagnetic force, also

known as the Lorentz force, which is given by J×B, where J is the eddy current

density and B is the magnetic flux density, acting on the eddy current region. This

force gets transferred to the metal object leading to its deformation in the desired

manner as shown in fig. (1.2)

3

Fig. 1.2: Deformed work piece with electromagnetic forming

1.2 Motivation and scope of work

A coupled field analysis is a study of interaction of multiple fields with at least

a fraction of their domain being common. In coupled magneto-structural analysis, the

electromagnetic fields, produced by current or voltage-fed structure in the

electromagnetic devices, interact with the materials present in that field. This

interaction will result in the production of some types of forces, may be magnetization

or Lorentz forces depending on the materials present. The structural field analysis in

the coupled magneto-structural field analysis investigates the effects of these forces

on the mechanics of the materials. Therefore, the final output of a coupled magneto-

structural system will be the deformation and stresses in the structural model. In this

report, the analysis is carried out over the electromagnetic structure involves non-

ferromagnetic materials and various methodologies are discussed for the computation

of forces for different types of excitations. In subsequent structural analysis these

forces are applied as the input and output will be the deformation and stresses in the

structure. The coupled analysis, discussed here, is carried out by using a numerical

technique, known as “Finite Element Method”. The main advantage of the finite

4

element method in the coupled field analysis is that the same geometrical

discretization information can be used in the different subsequent field analyses. The

structural analysis, in this work, is restricted to the linear-elastic range. This analysis

may be used to investigate the deformations and vibrations in the winding of electrical

machines. Noise is also one of the main problems to be considered while designing

electrical machines. The present analysis may also be useful for the noise problems in

the electrical machines. This work describes a way to analyze and understand

deformation phenomena of electrical conductors under an electromagnetic field. A

coupled model of magneto-elastic analysis is presented, in which sequential coupling

is used among the different fields involved. The cases, described above, investigate

the negative aspects of the effects of the electromagnetic forces. The same analysis

can be used for the positive aspects in which the deformation in the metals is desired.

It is known as the electromagnetic forming.

Electromagnetic forming process has many advantages over traditional forming

process like improved surface quality, a noncontact type of process, repeatable and

can be controllable by varying electrical parameters. It is useful if we can estimate the

amount and distribution of electromagnetic force acting on the object to be formed.

This work deals with the computation of electromagnetic forces in different operating

conditions as in case of steady flow of direct current, sinusoidal alternating current

and damped-pulse current. Codes in MATLAB are developed which implement force

computation methods through the application of FEM. Further, estimation of the

forces and their distribution on the metallic object (to be formed using

electromagnetic forming process) is done. The results from the code are compared

with the results obtained using ANSYS (a commercial FEM software)

1.3 Outline of the thesis

In chapter 2, a brief literature survey is given on the electromagnetic field

computation, coupled magneto-structural analysis and electromagnetic forming

process by finite element method.

Chapter 3 presents, a general theory about finite element method and electromagnetic

field theory. This chapter describes the electromagnetic field theory, with potential

5

function formulation, results in boundary value problem. The finite element method is

efficient approximation method to solve such type of problems.

Chapter 4 describes the finite element formulation of electromagnetic fields and it

gives linear system of algebraic equations which can be solved by any standard

method for such systems. In this chapter simulation and results are given for linear

static analysis, harmonic and transient case for electromagnetics field analysis.

Chapter 5, this chapter presents a brief theory about elastic and plastic theory and fem

formulation of elastic and elastoplastic range of material. And the various coupling

schemes for magneto-structural problems are explained. The simulation of coupled

magneto-structural by sequential coupling method is presented in the same chapter.

Chapter 6, in this chapter some conclusions are drawn from the analysis of proposed

problem. Some ideas for future research on coupled field analysis of electromagnetic

devices are also given.

6

Chapter 2

Literature review

The first effort to solve the magnetic field problems by finite element method

were made in the late 1960’s. In the analysis of electrical machines by finite element

method was first applied to synchronous machines in [1] and to dc machines in [2]

because the operation of these machine types can be approximate modeled by

stationary fields. Even nowadays, most of research work concerning the magnetic

field analysis of the electric machines modeling of the synchronous machines. They

have been analyzed by using step by step methods to solve the time dependence in [3]

and by three dimensional finite element formulations in [4].

The field of induction motor has to be solved with a method takes the time-

independence in to account. Probably the difficulties connected with the solution of

time dependent nonlinear fields have postponed the numerical analysis of induction

motors. The first publication dealing with this problem appeared at the beginning of

the 1980s. In the present work, only two-dimensional formulations have been used.

Ito [5] computed the nonlinear field of an induction motor using an eddy current

formulation and assuming sinusoidal time variation.

Bouillaut & Razek [6] and Brunelli [7] used time-stepping methods to calculate the

time variation of magnetic fields in induction motors.

Shen [8] presents a formulation for the sinusoid ally varying field quantities. In the

1990s, as the computing resources progressed, the electromagnetic forming process

could be simulated better. This progress has been followed by a large number of

papers dealing with numerical simulations of electromagnetic forming. In papers by

Bendjima et al. [9, 10], Azzouz et al. [11], and Meriched and Feliachi [12], numerical

results obtained with a finite-element method using simplified models of the process

were presented and compared with the experimental ones. The numerical results

showed quite a good agreement with the experimental ones. The finite element

formulations for electromagnetic devices are described in [13]. In [14], [15] and [16],

7

various numerical techniques, such as finite difference method, finite element method,

are explained for various electromagnetic cases. The main concern of the presented

work in this report is the coupled-magneto structural analysis, in which the

electromagnetic forces play an important role in the analysis. Some papers on

electromagnetic force computations are investigated. In [17], Maxwell stress method

for the magnetic force computation in the magnetized core is described. In [18]

authors described application of the Maxwell stress method in more accurate way in

the magnetized force computation. T. Kabashima

et al. [19] and

G. Henneberger et

al. [20] investigate an alternative method known as equivalent magnetizing current

source for the magnetizing force computation.

Z. Ren [21] compares various

formulations for the magnetized force computation. In [22], some aspects of magnetic

force computation and mechanical behaviors of the ferromagnetic materials. J. L.

Coulomb and G. Meunier [23] describe the Coulomb’s virtual work method for the

force and torque computation for 2-D and 3-D cases. In [24], the method for local

magnetic force distribution by using coulomb’s virtual work method by direct

differentiating the magnetic energy or co-energy with respect to the virtual

displacement of each node.

A. V. Kank and S. V. Kulkarni [25] investigate the

computation of Lorentz forces and magnetic forces by using FEM in the rotting

electrical machines. The magnetic force computation using finite element method

technique in global quantities and local force distribution is investigated in [26].

The mechanical behavior of a metal object, under the applied forces, is described by

elastic, elasto-plastic and visco-elsto-plastic range of the characteristics of the metal

object [27], [28] and [29].

C. Karch and K. Roll [30] investigate the electromagnetic forming process by finite

element method. The numerical model, presented in [30], predicts the electromagnetic

field, temperature, stress, and deformation properties that occur during the forming

process. The numerical results of the tube deformation are compared with available

experimental data.

The finite element formulations for the structural analysis are explained in [31], [32],

and [35]. The electromagnetic forming of aluminum tube in the form of compression

and expansion of the tube is investigated by using a constitutive finite element model

8

in [33] and the simulation of electromagnetic forming is carried out in the commercial

FEM software ANSYS. The simulation results are shown in good agreement with

analytical solutions [34].

The visco-plastic behavior of the metal in metal forming process is analyzed in [36].

The coupled magneto-elastic analysis by finite element method for ferromagnetic

materials is presented in [37]. The magnetostriction effect is also taken in to account

in this analysis.

The mechanical behavior of a solenoidal coil under the electromagnetic forces is

analyzed in [38]. The input data, available in this paper, is also used in the presented

linear sequentially coupled magneto-structural analysis in this report.

9

Chapter 3

FEM: Theory and Application

The finite element method has become a well established method in many

fields of computer aided engineering, such as, electromagnetic field computation,

fluid dynamics and structural analysis.

The finite element method is basically, an efficient approximation method to solve the

partial differential equations or boundary-value problems, which are frequently

occurred in different areas of engineering.

3.1 Finite element method- Basic concept

There are three main steps during the solution of partial differential equation

(PDE) with the finite element method. First the domain on which the PDE should be

solved, descritized into finite elements. Depending on the dimension of the problem,

this can be triangles, squares, rectangles or tetrahedrons, cubes, or hexahedrons. The

solution of PDE is approximated by piecewise continuous polynomials and the PDE

hereby descritized and split into finite number of algebraic equations. Thus, the aim is

to determine the unknown coefficient of these polynomials in such a way, that

distance (which is defined by the norm in a suitable vector space) from the exact

solution becomes a minimum. Therefore, the finite element is essentially a variational

minimization technique.

Since the number of elements is finite, we have reduced the problem of finding a

continuous solution to our PDEs to calculating the finite number of coefficients of the

polynomial. The solution of the Poisson’s equation, which is required to calculate the

magnetic vector potential, has to be solved for a given current density distribution.

We write the Poisson’s equation in more general form

In order to apply the finite element method, we have to find variational formulation.

The Galerkin’s methods leads to the week formulation of the problem: we multiply

Poisson equation by the test function u and integrate over the solution domain

10

2 . . (3.2)u r v r dr f r v r dr

Integration by parts gives

(3.3)nu r v r dr u r v r dr f r v r dr

Where d r

denotes the surface normal on the boundary . If appropriate boundary

conditions defines the value of u (Dirichlet boundary condition) or of its derivative

(Neumann boundary conditions) on the boundary, we can simplify (since v

vanishes, where Dirichlet boundary conditions apply

(3.4)n

nu r v r dr gv r dr f r v r dr

The exact solution ui shall be approximated by a linear combination of trial functions

0

(3.5)n

h i i

i

u r u r

And we use a finite set of test functions

.

If we insert this expansion into the equation (4) and assume only Dirichlet

boundary conditions

0

(3.6)n

i i i i

i

u r v r dr f r v r dr

We get a system of algebraic equations.

This can be solved with any standard method for the solution of a system of

algebraic equations, such as Gauss method, the Cholesky decomposition or iterative

scheme like the conjugate gradient method.

11

3.1.1 Boundary conditions

For the solution of partial differential equation like Maxwell’s equations, we

need boundary conditions to find a unique solution. There are three conditions

3.1.1.1 Dirichlet boundary conditions

The value of the solution is explicitly defined on the boundary (or on a part of

it). The value of solution can be zero or non-zero on the boundary. In

electromagnetics, the magnetic vector potential is usually set to zero along a

boundary, which should not be crossed by magnetic flux.

3.1.1.2 Neumann boundary conditions

The normal derivative of solution is defined on the boundary. If we set the

normal derivative of magnetic vector potential to zero, the boundary can be

interpreted can be interpreted as an interface with a highly permeable metal. Then, the

magnetic flux passes the interface at an angle of 90 degree to the plane of interface. In

order to find a unique solution, a Dirichlet condition must be defined somewhere on

the boundary of the domain.

Fig. 3.1: Problem Domain for boundary value problems

12

3.1.1.3 Mixed boundary condition

A combination of above two boundary conditions is called a mixed boundary

condition or also known as Cauchy’s boundary condition. In this case the normal

derivative of the solution and the value of the solution itself on the boundary are

connected by a function.

3.2 Application of FEM in electromagnetic field computation

The theory of electromagnetics took a long time to be established, and it can

be understood by the fact that the electromagnetics quantities are “abstract” or in

other words, cannot be “seen” or “touched” (such as mechanical and thermal

quantity). Electromagnetics can be described by the Maxwell’s equations and

constitutive equations. Actually, the majority of the electromagnetics phenomena

were established by other scientists before Maxwell, such as Ampere (1775-1836),

Gauss (1777-1855), Faraday (1791-1867), Lenz (1804-1865) among others. However,

there was some incompatibility on the formulation and Maxwell (1831-1879), by

introducing an additional term (displacement current) to Ampere’s law, could

synthesize the electromagnetics in four equations. The genius of this man brought the

electromagnetics to a very simple formalism, kept mainly by only four equations. The

consistency of these equations (along with constitutive ones) is so high that very

distinct phenomena (like microwaves and permanent magnet fields) can be precisely

described by these. While the formalism and the basic concepts of the

electromagnetics relatively simple, realistic problems can be very complicated and

difficult to solve. In fact, when, complicated geometries, non-linearity, many non-

static field sources, etc, appear together (or sometimes even alone) it s virtually

impossible to find analytical solutions for such problems and that is the main reason

why numerical methods have become widely used tools in electrical engineering

nowadays.

In this work, low-frequency phenomena’s are of main interest. Electromagnetic can

be divided in different categories as shown in the fig. 3.2

13

3.2.1 Maxwell’s equations

The general, time dependent, Maxwell’s equations in differential form (also

called point or local form) are

(3.7)t

BE

(3.8)t

DH J

(3.9) D

0 (3.10) B

Equation (3.7) is known as Faraday’s law of induction. This equation

describes the basic law of induction and, is often associated with the eddy current

applications. Equation (3.8) is the Ampere’s law in its general form, in which

displacement current t D is taken into account. Equation (3.9) is the Gauss’s law

which, sometimes associated with electrostatic application but which is also required

in other applications. Equation (3.10) is not associated with a particular law and

simply states the nonexistence of the isolated magnetic pole. It is also known as the

magnetic form of the Gauss’s law [13].

Electromagnetics (Maxwell’s equations)

Electromagnetics Low

Frequency Electromagnetics High

Frequency (Waves)

Electrostatic Magnetics

Magnetostatic Magnetodynamics

Fig. 3.2: Classification of electromagnetics

study

14

The integral form of the Maxwell’s equation is given in following.

(3.11)c s

dl dst

B

E

(3.12)c s

dl dst

DH J

0 (3.13) s

ds B

. (3.14)s V

ds dv D

The differential form of Maxwell’s equation will be extensively in this work. The

differential form is more convenient in calculations using methods such as finite

element method or finite difference method, while integral form is more convenient in

analytic calculation of fields and in various integral methods of numerical calculations

such as the method of moments and boundary element methods [14].

3.2.2 Constitutive relations

The two additional relations needed to complete the system of equations are the

material constitutive relations:

(3.15)B H

(3.16) D E

These two vector equations are equivalent two six scalar equations

In addition, a constitutive relation involving current densities and the electric field

intensity

= E (3.17)J

3.2.3 Potential functions

Potential functions are viewed as alternative representation of the

electromagnetic field. These are the simpler, more useful to describe the field

properties rather than to use an abstract field variable like magnetic flux density,

magnetic field intensity etc.

15

These may allow simplification in the computation of fields, better understanding of

results, which we obtained. Potential functions are defined based on properties of

magnetic fields.

3.2.3.1 Electric scalar potential

This potential function is defined based on the irrotational property of the

static electric field. That is

0 (3.18) E

Any function that is irrotational can be written as the gradient of a scalar function.

(3.19)V E T

his scalar function is called the electric scalar potential function.

2.2.3.2 Magnetic scalar potential

The electric scalar potential is defined based on the irrotational nature of the

electric field. Then it follows that when the magnetic field intensity is irrotational, it

can also be defined in terms of the magnetic scalar potential as

0 (3.20) H

(3.21) H

, is called the magnetic scalar potential.

3.2.3.3 Magnetic vector potential

Scalar potentials are defined based on the irrotational nature of the electric and

magnetic fields. Another type of potential function can also be defined based on the

nature of solenoidal nature of the electric and magnetic fields.

The magnetic flux density B is solenoidal in nature (i.e. 0 B ), it can be

derived as the curl of another vector

(3.22)B A

Here, A, is called the magnetic vector potential.

16

3.2.3.4 Electric vector potential

Another vector formulation employs the electric vector potential (T) to

describe the current distribution in conducting regions [14]. The electric vector

potential can be regarded as a variation for H as it has same curl relation but,

generally, a different divergence. The magnetic vector potential is defined as

(3.23) T J

Since the divergence of H is zero it can be redefined as

(3.24) H T

Where, , is a magnetic scalar potential.

3.2.4 Gauge conditions

The use of vector potential functions requires the specification of curl and

divergence of the potential functions for unique representation of field quantities. The

curl is normally specified based on the properties of the field (e.g. in the case of

magnetic vector potential the solenoidal nature of B allows the definition of A from

B A ). The divergence must then be specified to be consistent with the field

equations. There are normally two gauge conditions. One is the condition:

0 (3.25) A

is known as Coulomb gauge condition. The Second is

V 0 (3.26)

t

A

is known as the Lorentz gauge condition. This relation is consistent with the field

equations because it leads to continuity equation [15].

3.2.5 Field equations in terms of potential functions

The electromagnetic field equations can be written in terms potential

functions. The potential function formulation of field equations is particularly suited

the numerical methods. For this, the potential functions can be simply substituted in

17

the Maxwell’s equations. Here the magnetic vector potential formulation

(3.27)t

BE

By the definition of magnetic vector potential i.e,

(3.28) B A

Substitute this value in the equation (3.27)

(3.29)

t

AE

And therefore, the field intensity can be written as

(3.30)t

AE

The electric field intensity is only the time dependent part of the electric field

intensity. In addition the static term given

V (3.31) E

Total field intensity is the sum of two electric field intensities

(3.32)Vt

AE

In the case of the magnetic vector potential

1 V (3.33)

t t

S

AA J

Where the relation B Hand D E were used, there scalar form and the current

density J was in SJ (applied or source current density) and eJ (induced or eddy

current density). Assuming linear media, using vector identity

2 (3.34) A A A

Equation (3.33) becomes

2

2

2

V (3.35)

t t

AA A J

In the case of law frequency (static and quasi-static cases ), last term can be neglected

and if there is no electric scalar potential is present, then the resultant equation will

be

2 (3.36) A A J

18

Now, using Coulomb gauge, i.e.

0 (3.37) A

The resultant equation will be partial differential equation in terms of magnetic vector

potential

2 (3.38) A J

this equation can be solved with knowledge of magnetic vector potential on the

boundary of the solution domain [16]. This is now basically a boundary value

problem.

19

Chapter 4

FEM Formulation for Electromagnetic fields and

Force computation

In coupled field analysis for magneto-structural problems, electromagnetic

forming and for other electromagnetic device, the analysis of different fields should

be carried out by an efficient method. The finite element method has been well

established in field computation. In this report, the finite element method is used for

different field analysis. The formulation for electromagnetic field by finite element

method is discussed in following section.

4.1 FEM Formulation for electromagnetic field computation

We start the formulation of magnetic field computation with the governing

partial differential equation in terms of Magnetic vector potential, which is,

2 (4.1) A J

Where, J is the total current, which can be decompose in two components

(4.2)e sJ = J + J

Je , is the eddy current density and J s , is the source current density. Now above

equation can be written as,

2 ( ) (4.3) e sA J + J

Eddy current density can be rewritten as given in equation (3.4) and (3.5)

(4.4)eJ E

V (4.5)t

AE

The above equation will be the result in equation (4.6)

2 V (4.6)t

s

AA J

20

The term, V , in magneto-static and quasi-static cases can be assumed to be zero, so

the final equation

2 (4.7)t

s

AA J

2-D (two dimensional) equation for transient magnetic problems

2 2

2 2

1 1 (4.8)

d d

dx dy t

s

Α A AJ

Let us consider, for an approximated solution A, the residual r is

2 2

2 2

1 1 (4.9)

d dr

dx dy t

s

A A AJ

The weighted residual for element

(4.10)e

iN rdxdy

e

iR

Where, i=1, 2, 3

2 2

2 2 (4.11)

e e e

e e e

i i i

d dN dxdy N dxdy N dxdy

dx dy t

s

A A AJ

Since udv vdu uv

Using 2-D Galerkin’s Finite element formulation, First integral left in Eqn. (4.11) is

1. (4.12)

ee e e

e ee

e e ei ii i i

dN dNd ddxdy N n d N dxdy N dxdy

dx dx dy dy t

e

i s

A A AR D J

Where 1x y

x y

A AD and considering linear triangular interpolation

function

A=a + bx + cy (4.13)

3

1

, , A (4.14)e e

j j

j

x y N x y

eA

21

A

, , A 4.15

A

e

i

e e e e

i j k j

e

k

N N N

eA

Where 1

(4.16)2

e e e e

i i i i

e e e e e

j i j j

e e e e

k k k k

N a b x c y

N a b x c y

N a b x c y

N

A1

, , A (4.17)2

A

i

e e e

i j k j

k

db b b

dx

A

A1

, , A (4.18)2

A

i

e e e

i j k j

k

dc c c

dy

A

1 (4.19)

2

e

i

e

j

e

k

bd

bdx

b

eN

1 (4.20)

2

e

i

e

j

e

k

cd

cdy

c

eN

First integral on the left of equation (3.12) is

1 . (4.21)

ee

e ee

ei ii

dN dNd ddxdy N n d

dx dx dy dy

e

i

A AR D

Line integral is set to zero, which gives natural boundary condition.

22

1 (4.22)

e

e e

i i

e

dN dNd ddxdy

dx dx dy dy

A A

But

(4.23)

e

dxdy

Then by using the derivatives of shape function and interpolation functions, which

will be same in the isoparametric elements, which are very frequently used in the

electromagnetics problems,

2 2

2 2

2 2

A1

A (4.24)4

A

ii i i j i j i k i k

i j i j j j j k j k j

i k i k j k j k k k k

b c bb c c bb c c

bb c c b c b b c c

bb c c b b c c b c

The second integral of equation becomes,

A

, , A (4.25)

Ae e

e

i ie

e e e e e e

j i j k j

ekk

N

N dxdy N N N N dxdyt t

N

e

A

Using formula (4.26), equation (4.25) becomes

1 2 3

2 (4.26)

( 2)e

l m ne e e l m n

N N N dxdyl m n

! ! !

!

A2 1 1

1 2 1 A (4.27)12

1 1 2 A

ie

j

k

t

The forcing function (third term of equation (4.12) becomes)

23

1

= 1 (4.28)3

1e

e

iN dxdy

s

s

JJ

Element level of equations is,

2 2

2 2

2 2

A1

A4

A

ii i i j i j i k i k

i j i j j j j k j k j

i k i k j k j k k k k

b c bb c c bb c c

bb c c b c b b c c

bb c c b b c c b c

-

A2 1 1

1 2 1 A12

1 1 2 A

ie

j

k

t

=

1

1 (4.29)3

1

sJ

Assembling all elements, global set of equation is given as,

(4.30)t

S A T A J

Now, next step is to incorporate the Dirichlet’s boundary conditions in the above

global equation and since, the Neumann’s boundary conditions are implicit in the

formulation so only Dirichlet’s boundary conditions need special treatment [13].

After incorporation of boundary conditions, the resultant algebraic system of equation

can be solved by any standard method of solving linear algebraic equations.

There are generally three types of analysis in the electromagnetics, which are

described below as,

4.1.1 Magneto-static analysis

In the magneto static analysis, the quantities are not time dependent. This

analysis is generally required for the problems of steady flow of dc electric current,

permanent magnets, an applied external field, moving conductors etc. [14].

24

The governing equation in the magneto static case is given as,

2 2

2 2

1 1 (4.31)

d d

dx dy s

A AJ

By using above described finite element formulation, it will result in the algebraic

equation of the form

(4.32)S A J

In this case the time-dependent effects (such as eddy currents) are not considered. It

can also be non linear in the presence of saturable, permanent magnet etc. The flux

density can be calculated as,

(4.33)B A

4.1.1.1 Algorithm for solving magneto-static problems using FEM

1. Prepare the mathematical model of the problems by considering symmetry of

the problems in terms of geometry and loading (if exist).

2. Discrete the geometry in terms of the linear elements or quadratic elements.

3. By using governing equation of the field as given in equation (3.31), compute

the elemental equations for each element by using weight functions and shape

functions (using Galerkin’s method or Variational methods) which will be

same in case of isoparametric elements.

4. Assemble all elemental equations in the global system of equation by using

connectivity information which is available from the discretization of the

geometry. It leads to algebraic system of equations.

5. The next step is incorporation of boundary conditions (Dirichlet boundary

condition) in the global system of equations. Neumann boundary conditions

are implicitly used in the formulation hence only Dirichlet boundary

conditions requires special treatment.

6. The resultant algebraic set of equations after incorporation of boundary

conditions is solved using any standard method for solving linear system of

25

equations. The solution of global system of equation gives the values of

unknown variable at each node of the problem domain.

7. The next step is the post processing of the solution. In this phase the secondary

(or derived) quantities can be derived from the solution. These quantities may

be flux density, intensity, magnetic forces etc.

The flow chart for solving magneto-static problems using FEM is given in Fig.

(4.1).

26

4.1.1.2 Flow chart for solving magneto-static problems using FEM

Fig. 4.1: Flow Chart for Magneto – Static Case

Prepare Mathematical Model of the Problem domain

Descritized the problem domain in terms of the finite elements (linear

or quadratic)

Formulate the elemental equations by using Galerkin’s or variational

formulation for each element of the descritized geometry

Assemble all the elemental equations in a global system of equation

using the connectivity information available in the discrete geometry

data and incorporate boundary conditions in the global system of

equation

Compute the Secondary or derived quantities using the solution of

global system of equation

Solve the resultant Global system of equation by using standard method for

solving linear system of equations

Start

End

27

A linear magneto-static case is analyzed using the algorithm discussed above. This

problem is solved by MATLAB and results obtained with MATLAB are Validated by

a commercial FEM software ANSYS.

4.1.1.3 Linear magneto-static case-study

Case study involves: Determination of the magnetic flux densities, force and

deformation in the conductor

4.1.1.3.1 Geometric data and material properties

Outer radius R = 50 mm,

Inner radius r = 40 mm

Height h = 90 mm and

Input current density J = 5×106

amp/m2;

Young Modulus E = 12.2 ×1010

N/m2;

Poisson’s ratio ν = 0.3.

4.1.1.3.2 Assumptions

A long thick solenoid carries a uniform current distribution.

The turns of the solenoid can be modeled as a homogeneous, isotropic

material with modulus of elasticity E and Poisson’s ratio v.

The symmetries of the problem allow us to consider just a one-half or one-

quarter of the domain of analysis.

In this case study, the coil material is the nonmagnetic, only Lorentz force will

exist.

J

r

h

Fig. 4.2: Solenoidal coil

R

28

4.1.1.3.3 Problem model There are two analysis domains in this model.

4.1.1.3.4 Analysis domain of fields

1. EM analysis domain

Coil and air (both) regions

2. Structural analysis domain

The problem domain is descritized with linear triangular elements and by using

Galerkin’s formulation, the linear system of equations is obtained. The solution of

these equations gives magnetic vector potential at all the nodes in the problem

domain. The magnetic vector potential seems to be a bundle of information. Other

interested quantities like magnetic flux density, magnetic intensity, magnetic forces

etc.

The plot of magnetic flux lines are shown in fig. (4.4) and the linkage of this flux

with the coil are shown in the fig. (4.5). Local forces distribution in the coil is shown

in fig. (4.6).

Fig. 4.3: Problem model-1

1. Air region

2. Coil region

29

4.1.1.3.5 Magnetic flux lines plot with ANSYS

The flux lines due to current flow in the coil are shown in the fig. (4.4)

4.1.1.3.6 Flux linkage with coil

The flux produced by the current flowing in the coil is linked with the current

carrying part of the coil. This linkage of flux with the coil is shown in Fig. (4.5)

Fig. 4.4: 2-D flux lines plot with ANSYS

Fig. 4.5: Flux linkage with coil

30

4.1.1.3.7 Force distribution in the coil

The force distribution on the plate, due to flux linkage with the coil, is shown in the

Fig. (4.6).

4.1.1.4 Comparison of Results with ANSYS

The values of maximum flux density and maximum force density is compared

with the values obtained by an FEM software ANSYS shown in the table (4.1)

Table 4.1: Comparison of Results with ANSYS

Maximum flux density (Tesla) Maximum force density (N/m2)

ANSYS 0.0502 2.4918 ×105

MATLAB 0.0501 2.4866 ×105

Fig. 4.6: Force distribution in coil

31

4.1.2 Harmonic analysis

If one is the interesting in knowing the value of peak force in the first cycle,

the problem can be approximated as a time-harmonic one simplifying the

computational burden. In such a case the governing partial governing differential

equation (PDE) is:

2 2

2 2

1 1 j (4.34)

d d

dx dy

s

A AJ A

Where, A has only z-component in case of 2-D problem. The corresponding set of

equations to be solved after FEM discretization is:

(4.35)r i r i r iA jA j A jA J jJ S T

Separating

(4.36)r r

i i

A J

A J

S T

T S

In harmonic analysis, eddy currents can be calculated with the harmonic magnetic

field, which is in terms of real and imaginary terms. The calculated eddy current will

also be in terms of real and imaginary terms such as,

(4.37)j eJ A

eJ and A are the complex quantities.

Magnetic flux density will also be in the complex form, as

(4.38) B A

32

4.1.2.1 Algorithm for time-harmonic analysis

Algorithm for time-harmonic analysis is similar to magneto static analysis as

described in [4.1.1]. In comparison to magneto-static analysis, all the quantities in

time-harmonic analysis are in complex form. Due to presence of eddy currents in

time-harmonic analysis, there will be an additional term in the formulation for time-

harmonic analysis as given in equation (4.35).

A time-harmonic problem is analyzed with formulation and algorithm given above.

This problem is solved by MATLAB and results are verified with commercial FEM

software ANSYS.

4.1.2.2 Electromagnetic Analysis in Time-harmonic case

Involves analysis of determination of the magnetic flux densities, force

Table 4.2 Electrical parameters

Input current (J)

116260sin (Amp.)t

Frequency (ω) 8761.2 (Hz)

Resistivity of workpiece (ρ) 3.333×10

-8 (Ω-m)

Fig .4.7: Electromagnetic forming system

33

Table 4.3 Geometrical parameters of workpiece

Inner radius (rw) 6.5mm

Outer radius (Rw) 7.5mm

Length (hw) 50mm

Material Aluminium

Table 4.4 Geometrical parameters of coil

Inner radius (rc) 9.25mm

Outer radius (Rc) 34.25mm

Length (hc) 21mm

Number of turns (N) 6

Material Copper

4.1.2.3 Assumptions

A long thick solenoid carries a uniform current distribution.

The turns of the solenoid can be modeled as a homogeneous, isotropic

material.

The symmetries of the problem allow us to consider just a one-half or one-

quarter of the domain of analysis.

In this case study, the coil material is the nonmagnetic, only Lorentz force will

exist.

Analysis is carried out by both formulations i.e. by complex and by time-

stepping for same problem.

4.1.2.4 Problem model

There are three analysis domains work piece, coil and air. All the three regions

are shown in fig. (4.8)

34

The problem domain is descritized with 8-node quadratic rectangular

isoperametric elements and by using Galerkin’s formulation, the linear system of

equations is obtained. The solution of these equations gives magnetic vector potential

in complex form at all the nodes in the problem domain. The magnetic vector

potential seems to be a bundle of information. Other interested quantities like

magnetic flux density, magnetic intensity, magnetic forces etc.

4.1.2.5 Simulation with complex Magnetic vector potential

formulation (MATLAB)

The simulation of a time-harmonic problem is carried out with complex vector

potential formulation by developing code in MATLAB. The main advantage of

complex vector potential formulation is that it requires computation in single step in

contrast of time-stepping formulation. The main precaution, which must be taken in

the time-harmonic analysis, is the consideration of the skin effect phenomena. To

consider this effect, the number of element in the one skin depth should be sufficient.

The plots of distribution of various field quantities in analysis domain are given in this

section.

Fig. 4.8: Problem model-2

35

4.1.2.5.1 Magnetic vector potential plot in work piece

The magnetic vector potential plot in the work piece is shown in fig. (4.9). It is

clear from figure that it has the maximum value at the midpoint of work piece.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.005

0.01

0.015

Element no.across hiegth of workpeice

mag

netic

vec

tor

pote

ntia

l

Plot of magnetic vector potential in work peice

4.1.2.5.2 Axial flux density on the work piece

The axial flux density plot is shown in fig. (4.10). It has the maximum value at

the midpoint in the work piece.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

5

10

15

20

25

Element no. in the plate

Axi

al f

lux

dens

ity in

pla

te

Axial flux density in the plate

Fig. 4.9: Magnetic vector potential on the work piece

Fig. 4.10: Axial magnetic flux density on the work piece

36

4.1.2.5.3 Radial flux density on the work piece

The radial flux density is shown in fig. (4.11). It will be symmetric across the

height of work piece.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-4

-3

-2

-1

0

1

2

3

4

Element no.across the workpeice

radia

l flux d

ensity in w

ork

peic

e

4.1.2.5.4 Eddy current Density on the work piece

As it is time-harmonic case, eddy current will exist in this case. The plot

of eddy current in the work piece is shown in Fig. (4.12).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5x 10

10

Element no. across the hieght of plate

Edd

y cu

rren

t de

nsity

Fig. 4.11: Radial flux density on the work piece

Fig. 4.12: Eddy current density in the work piece

37

4.1.2.5.5 Radial Force density on the work piece

The plot of radial force density is shown in fig. (4.13). It has maximum value

at the mid-height of the work piece.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

0

0.5

1

1.5

2

2.5x 10

11

Element no. on the workpeice

Rad

ial f

orce

den

sity

on

wor

k pe

ice

Radial force density plot on workpeice

4.1.2.5.6 Axial force density on the work piece

The plot axial force density across the height of work piece is shown in Fig.

(4.14). It shows symmetric characteristic across the height of work piece.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

10

Axi

al fo

rce

dens

ity o

n w

ork

peic

e

Element no. on the work peice

Axial force density plot on work peice

Fig. 4.13: Radial Force density on the work piece

Fig. 4.14: axial force density on the work piece

38

4.1.2.6 Simulation with complex Magnetic vector potential

formulation (ANSYS)

The case study of time-harmonic analysis analyzed above by developing own

MATLAB code, here the same case is investigated by using a commercial FEM

software ANSYS. The PLANE 53 elements are used to Descritise the problem

domain in ANSYS. These elements are 8-noded and rectangular in shape. The plots of

distribution of field quantities, obtained with ANSYS, are shown in this section.

4.1.2.6.1 2-D flux lines plot (ANSYS)

2-D lines plot of magnetic flux is shown in fig. (4.15). Flux linkage with the work

piece is also shown in same figure. The flux linkage with work piece is depending on

the input frequency due to skin effect phenomena.

Fig. 4.15: 2-D flux lines plot in electromagnetic domain

39

4.1.2.6.2 Flux density in the work piece

The plot of magnetic flux density in the work piece is shown in the fig. (4.16)

4.1.2.6.3 Nodal force distribution on the work piece

The plot of nodal force in the work piece is shown in fig. (4.17). It shows the

local force distribution on the work piece.

Fig. 4.16: Flux density on the work piece

Fig. 4.17: Nodal force distribution on the

work piece

40

4.1.2.7 Comparison of Results of Time-harmonic analysis with

ANSYS

The results obtained with MATLAB are compared with the ANSYS results, as

shown in table (4.5). The results are shown in good agreement.

Table 4.5: Comparison of Results of Time-harmonic analysis with ANSYS

Max.

Magnetic

vector

potential

(A)

Max. flux

density

(B) 2(wb/m )

Max. Eddy

current

(J )e

2(Amp./m )

Max. force

density

(F) 2(Newton/m )

Max.

Nodal

force

(F)

(Newton)

MATLAB

0.1496

22.1237

102.4063 10

112.1511 10

37.89

ANSYS

0.1498

22.759

102.4675 10

112.3523 10

38.233

4.1.3 Transient magnetic analysis

This type of analysis is required for time-dependent cases, such as short circuit

forces in transformer, pulsed eddy currents, power losses due surges in voltage or

current or pulsed external field.

In this analysis calculating magnetic fields vary with time. A transient analysis can be

either linear or non-linear.

The governing equation (two-dimensional) for transient magnetic problem is given in

equation (4.38), it is

1 (4.39)

t

s

AA J

Using FEM formulation, equation (4.39) will result in the set of algebraic equation as

given as following

(4.40)t t tt

S A T A J

41

Various times stepping schemes can be used to generate linear system from equation

(4.26), the linear system of equations to be solved at each time of step is:

1 1 1 1

J J (4.41)t t t t t t

t t

S T A T S A

Where, S and T are the metrics which depends upon geometry and material

parameters. The value of the constant determine whether the algorithm is of the

forward difference type (=0) backward difference type (=1) or some intermediate

type (01). If =1/2, we have the crank Nicholson method [30]. The linear system

of equation can be solved by using methods such as LU factorization, Cholesky

decomposition, etc.

The eddy current can be calculated with time dependent-magnetic field

( ) (4.42)

t t t

t t

e

A AAJ

Magnetic flux density can be calculated at different time instants but at a particular

instant it will be the function of space only.

( , , ) ( , , ) (4.43) x y t x y tB A

Time-stepping method has some advantages and disadvantages against complex

formulation are

The advantages are

(1) It can handle easily nonlinearities of the problem

(2) Transient problems can only be solved by this method

The disadvantages are

(1) A lot of computation work is required in this method.

(2) If this method is used for analysis for steady-state sinusoidal excitation, it will

introduce some transient effect in the output. Then it is necessary to take sufficient

time steps to disappear this transient effect.

42

4.1.3.1 Algorithm for transient magnetic analysis

1. The steps 1-5 are same as given in algorithm for magneto-static analysis.

2. After incorporation of boundary condition in step no. 5, the resultant equation (4.39)

is solved by Crank-Nicholson method.

3. Initialize t=0, A(t)=0, =0.5, Δt=time-step and n= total no. of time steps

4. Solve equation (4.41) at time instant t+Δt and get A(t+Δt). By using equations (4.42)

and (4.43), calculate the values of Je and B.

5. Replace A(t) in equation (4.41) by A(t+Δt), solve for time instant t+2Δt and get the

values of Je and B.

6. Repeat step no 5 up to time instant t+nΔt.

43

4.1.3.2 Flow chart for solving transient magnetic problems using

FEM

Yes

START

Prepare mathematical model of the problem domain.

Descritise the problem domain in terms of the finite elements

(linear or quadratic)

Formulate the elemental equations by using Galerkin’s or variation

formulation for each element of the descritized geometry.

Assemble all the elemental equations in a global system of equation

using the connectivity information available in the discrete

geometry data and incorporate boundary conditions in the global

system of equation.

Initialize t=0, A(t)=0, =0.5, Δt=time-step and n= total no. of

time steps

Solve equation (4.41) at time instant t+Δt and get A(t+Δt). By

using equations (4.42) and (4.43), calculate the values of Je, B.

Replace A(t) in equation (4.41) by A(t+Δt), solve for time

instant t+kΔt, where k=2,3,4,5….. and get the values of Je, B.

Is k=< n?

END

Fig. 4.18: Flow-chart for linear transient magneto-structural analysis

44

4.1.3.3 Simulation with Time-stepping (MATLAB)

The case study, investigated above with complex magnetic vector potential, is

analyzed with the time stepping formulation. This type of formulation is well suited

for transient problems such as pulsed eddy current problems, electromagnetic forming

etc.

4.1.3.3.1 Max. Magnetic vector potential in time domain

The plot of magnetic vector potential wave form is shown in fig. (4.19). The

excitation, in this case study, is steady state sinusoidal and some transient effects are

appeared as shown in the same figure. To solve a problem, with sinusoidal excitation,

number of time-steps should be sufficient up to disappear this transient phenomenon.

0 50 100 150 200 250 300 350 400 450-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

time steps

Max,

magnetic v

ecto

r pote

ntial

Magnetic vector potential plot

Fig. 4.19: Max. Magnetic Vector potential in time

domain

45

4.1.3.3.2 Max. Axial flux density on the work piece

The plot of axial flux density wave form is shown in fig. (4.20)

0 50 100 150 200 250 300 350 400-25

-20

-15

-10

-5

0

5

10

15

20

25

time steps

flux d

ensity in a

xia

l direction

Axial flux density

4.1.3.3.3 Radial flux density in the work piece

The plot of radial flux density in the work piece is shown in fig. (4.21)

0 2000 4000 6000 8000 10000 12000-4

-3

-2

-1

0

1

2

3

Magnetic f

lux in r

adia

l direction a

t pla

te

Hieght of plate

Radial flux density across plate hight

Fig. 4.20: Max. Axial flux density in time

domain

Fig. 4.21: Radial flux density across the workpiece

46

4.1.3.3.4 Eddy current density on the work piece

The plot of eddy current wave form is shown in the fig. (4.22)

0 50 100 150 200 250 300 350 400-3

-2

-1

0

1

2

3x 10

10

Time steps

Max.e

ddy c

yrr

en d

ensity

Eddy current plot

4.1.3.4 Results of Time stepping formulation with MATLAB

The results of time stepping analysis with MATLAB are given in table 4.6. The

results given in table 4.6 can be compared with results for time-harmonic analysis for

the same case.

Table 4.6: Results of Time stepping formulation with MATLAB

Max.

Magnetic

vector

potential (A)

Max. flux

density

(B)

2(wb/m )

Max. Eddy

current

(J )e

2(Amp./m )

Max. force

density

(F)

2(Newton/m )

Max. Nodal

force

(F)

(Newton)

0.1499

22.7592

102.4876 10

112.5722 10

38.8933

Fig. 4.22: Max. Eddy current density on the work piece

47

4.2 FEM formulation for force computation

In the analysis of electromagnetic device, electromagnetic force is actually

couple with another field (mechanical field), the accuracy of analysis is strongly

depend upon force computation. There are some methods given in literature for

computation of force field by finite element method. The Lorentz force method is

used for computation of forces in conductive region. For magnetization forces, there

are some methods given in literature. Here in analysis of electromagnetic forming

problem by coupled field theory, the Lorentz forces are essential. Some important

methods, for Lorentz forces and magnetization forces computation, are discussed

following.

The various methods of force computation are-

1-Lorentz force method

2- Maxwell stress tensor method

3-Virtual work method

3-equivalent magnetizing current method

4-equivelent magnetic charges method

Maxwell stress tensor based method and its related aspects such as error of

performance are discussed in [17] and [18]. Theory and implementation of method of

equivalent magnetizing currents for calculation of electromagnetic force acting on a

ferromagnetic material is discussed in [19] and [20]. A comparative study of these

methods of force computation is given [21] and [22].

Practical implementation of virtual work method by determining the derivatives of the

coordinates of the nodes with the respect of the virtual displacement is dealt with [23].

In the virtual work approach, the force is derived at each node by direct differentiation

of the stored energy of the finite elements surrounding the node with respect to virtual

displacements. The magnetic flux distribution should be kept constant while

displacing the node [24].

4.2.1 Lorentz force method

In this method total force on a body is obtained by integrating the forces due

to magnetic field acting on each deferential current element,

48

(4.44)v

v

dv F J B

Where, J×B is the force density in the conducting region.

4.2.2 Maxwell stress tensor method

Maxwell stress tensor is widely used for electromagnetic force computations.

Maxwell stress tensor based method and it is related to aspects such as error

performances are discussed in [18] and [19]. A quantity called stress is defined in this

method whose divergence is actually the force density throughout the volume of the

body on which the force is to be determined. Applying the divergence theorem to the

stress tensor, we can consider Maxwell stress as surface force density which when

integrated over a surface enclosing the body gives total force acting on it. The surface

can be chosen so as to satisfy certain performance criterion and to improve the

accuracy of the result. The expression for stress tensor is,

21 1 B B B (4.45)

2i j ij

ijT

Where ,i j take values (x ,y, z). ij is 1 if i=j, and zero otherwise. The same can be

written in term of force density vector as,

1 1

n n (4.46)2

ds

2F B B B

Where, n is the normal unit vector to surface under consideration. The expression of

stress tensor in equation (4.36) results in to 33 tensor matrix,

2

2

2

1 1 1 1

2

1 1 1 1 (4.47)

2

1 1 1 1

2

x x x y x z

xx xy xz

yx yy yz y x y y y z

zx zy zz

z x z y z z

B B B B B B B

T T T

T T T B B B B B B B

T T T

B B B B B B B

49

In this method body on which the force is calculated, is surrounded by some

imaginary surface. After meshing the region, the edge of the some elements (faces of

the element in case of 3D) will lie on the chosen imaginary surface. Now these tiny

elemental surfaces can be resolved in to three components parallel to three coordinate

surfaces. The x, y and z component of the elemental surface are parallel to y-z, x-z

x-y plans respectively. Now Txx gives surface force density on x component (as i = x)

of the surface in x direction (as j=x), Txy gives surface force density on x- component

of the surface in y direction and so on. Similarly, Tyx,Tyy and Tyz gives surface force

densities on y- component in x, y and z directions respectively.

Tzx,Tzy and Tzz gives surface force densities on z-component of the surface in x, y and

z direction respectively. One care that needs to be taken while using this method is

that the above procedure gives force densities in positive -x or positive-y or positive-z

direction. But, we need to use surface force densities in the direction away from the

enclosed region. Hence all the x values have to be negated if the direction away from

the enclosed region is along negative coordinate axis. The above procedure equivalent

to evaluating the integral, given in Equation (4.45) over the enclosing the Maxwell’s

surface.

The Maxwell stress tensor gives both type of force, viz J×B forces and magnetization

forces. If the Maxwell surface dose not enclosed any current caring part then it gives

the magnetization force acting on the enclosed body. If the surfaces enclose the

current caring part along with the same permeable material then it gives the addition

of the J×B forces acting on the current caring part and the magnetization forces acting

on the permeable material [25].

4.2.3 Virtual work method

The method of virtual work for electromagnetic force calculation based on

generalized principal of virtual displacement. The movable part is assumed to be

displaced and the change of stored magnetic energy divided by the displacement gives

the force acting on the body, as the displacement tends to be infinitesimal. The

displacement is not the actual physical displacement of the body; hence, it is called

virtual displacement. One precaution has to be taken while virtually displacing the

50

body that the flux linkage has to be kept constant throughout the motion. The

implementation can be both at the level of the displacement of nodes. If the nodes are

displaced then the method is called local virtual work method. The expression for the

magnetic energy stored in the field is

0

1 (4.48)

2

B

V

d dV

W H B

Where V is the volume of the field region, B is the flux density and H is the magnetic

flux intensity. The force on the node k, virtually displaced is given by,

(4.49)q

k

WF

4.2.3.1 Derivation and description of local virtual work method

In FEM, total domain is divided in to a number of small elements. Elements

used here are triangular in case of two dimensional analyses and tetrahedral in case of

three dimensional analyses. The total energy content of the domain will be the

addition of energy of its all elements. Hence

1 0

1 (4.50)

2

e

e

BN

e

e V

d dV

e eW H B

Where eV the volume of element e is, eB is the flux density in the element and eH is

the magnetic flux density of that element. N is the total number of elements. If we

want to find the total energy of a particular domain of our problem, the summation

above equation will run over all the elements in that particular domain [26]. If first

order models are used (triangular and tetrahedral are first order models), then the

equation is simplifies to,

2

1

(4.51)2

N

e e

e

v V

B

W

Where ev is the reluctivity of element e. it can be expressed in term of flux density

squared,

51

2 (4.52)e ev v B

Now, differentiating the expression for the energy with respect to virtual displacement

q, we get,

2 2 22

21

(4.53)2 2 2

Me e e e

Kq e e

e

V V v Vv v

q B q q

B B BF Β

In the above expression, the summation is over M elements which surrounds node K.

Here q is the coordinate of the nodes of this particular element. For triangular

elements, q will successively takes the values 1 2 3 1 2 3, , , , ,x x x y y y which are the x and y

coordinates of local nodes 1, 2 and 3 respectively [24].

52

Chapter 5

Structural field analysis

5.1 Elasticity theory

The theories of elasticity and plasticity describe the mechanics of deformation

of engineering solids. Both theories, as applied to metals and alloys, are based on

experimental relations between stress and strain in a polycrystalline aggregate under

simple loading conditions [27].

Strains, as defined by theory of elasticity are given by (U is displacement along i and j

co-ordinates)

1

2 (5.1)

ji

ij

j ix x

UUε

In the simplest form, the stress relationship between strain and stress is governed by a

linear equation known as Hook’s Law.

(5.2)i i j k

E ε σ σ σ

Where, E is Young’s Modulus, σ is the stress, and ν is the Poisson’s ratio. This

relation remains valid while the stresses are below the yield stress and deformation is

in the elastic limit. When, the load exceeds the yield stress, the coefficient of strain in

the above equation (E) no longer remains constant, and becomes a function of strain.

The model is then called a quasi-static (strain-rate independent) elastoplastic model.

Some materials exhibit a behavior where the stress is not just a function of strain but

also of strain-rate, i.e. the rate of deformation. Such behavior can be modeled using a

strain-rate dependent model or by using a state equation such as Johnson-Cook law

[28].

Irrespective of the model used to represent the behavior of metal, the laws of

equilibrium must be obeyed at all points of time [29].

53

0 (5.3)ij

i

ix

σ

f

Where, fi are the body forces per unit mass, and ρ is the mass density.

There are three equilibrium equations (equation 5.3), six stress-train relations, one

hydrostatic stress-strain relation and one yield criterion (Tresca’s or Von mises’) in the

elasticity theory. With the help of above 11 equations, all the unknowns can be

simultaneously solved.

Using equations 5.1, 5.2 and 5.3, a second order partial differential equation is

obtained which relates the applied forces to the displacement of the body. Such an

equation is then numerically descritized and solved. In this report finite element

method is used for elastic analysis in the coupled field analysis of magnet-forming

system

Fig. 5.1: Stress-strain curve

54

5.2 Plasticity theory

In the case of plastic deformation, the strains are not proportional to stress as

in the case of elastic deformation. The stress may vary exponentially or by any other

function as shown in Fig. (5.1). The plastic flow rule gives the relation between

incremental stress and incremental strain is

( ) (5.4)ij pd d ijS

Where, dλ = proportionality constant, Sij = Deviatoric stresses and (dεij)p = Plastic

strain increments.

The proportionality constant dλ is a scalar quantity and known as “plasticity

modulus”. By differentiating equation (5.4) with respect to time, we get

(5.5)ijd d

dt dt

ijS

Or it can be written as

(5.6)ij ijS

Fig. 5.2: Relation between incremental stress and incremental strain

55

Equations (5.4) and (5.5) are referred to as differential and finite forms of Levy-

Mises’ equations. The constant

in equation (5.6) is the function of material

constant and strain rate. There are ten equations in the plastic range as given

following:

(1) One continuity equation: 0ijε

(2) Three equilibrium equation: 0ij

ix

σ

(3) Six Levy-Mises’ equations: ( ) ij pd d ijS

Using above ten equations and the ten equations can be solved simultaneously.

Invariably solving these 10 equations is not easy and several methods are therefore

resorted to. The method of solving to these 10 unknowns is referred as the theory of

plasticity [29].

5.3 Elastoplastic Analysis

The elastoplastic properties of a material can be described by its mass density,

Young’s modulus, Poisson’s ratio, Lame’s coefficient, yield strength, ultimate tensile

strength, rupture stress and maximum elongation or maximum strain. Besides, various

non-linear models of stress and strain relations in the plastic deformation use specific

coefficient to account for temperature rise and for the deformation velocity.

The elastoplastic properties of materials are mostly given as experimental data of the

material and depend on the working methods (annealing, heat treatment etc.). The

influence of the deformation velocity on the elastoplastic is taken into account via the

model of elastoplastic behavior.

The temperature rise has a certain influence on the elastoplastic properties of the of

materials, since each material has its own so-called critical temperature, obtained at

the intersection of curves of the yielding and fracture, but the influence of

temperature rise is not taken into account in the analysis reported here.

The pure elastic and pure plastic condition are described above. In some cases the

elastic and plastic deformations are of comparable magnitude, like example of temper

56

rolling of sheets. In such cases elastic deformation cannot be neglected. Prandtle –

Reuss equations and Hencky’s equation describe the eleastoplastic deformation [29].

In the elastoplastic range, it is assumed that the total deformation is the sum of elastic

and plastic deformation, i. e.

N

(5.7)iy y i

uy

(5.8)tot el plε ε +ε

Differentiating with respect to time, gives

(5.9) tot el plε ε ε

(5.10)2

ij

G

ij

ij

Se S

And according to Hencky, in this region total strain is given by

1 ( ) (5.11)

2G ij ij ijε e S

Here is the comparable strain.

5.4 FEM Formulation for linear structural analysis

For modeling and numerical analysis of electromagnetic forming, finite

element method is extensively used worldwide due to its flexibility and ease for use

for various conditions. Finite element analysis is a numerical tool for piecewise

solution of partial or ordinary differential equations. In the simplest form of finite

element method, the entire domain is split in to a finite number of entities, called

elements defined by points in a space or area, called nodes. The differential equations

are reduced to simultaneous linear equations in each element and all the simultaneous

equation across all elements are solved at common nodes, to give approximation

solutions.

Shape functions, which interpolate a given quantity in terms of nodal values of that

quantity, are used to descritized the specific quantity, such as displacement. For a 2-D

case, using Einstein’s notation (varies from 1to number of node in an element)

57

(5.12)x i x iu N u

(5.13)y i y iu N u

The normal strains and shear strain due to displacement at a given point are,

(5.14)xx

u

x

(5.15)ix x i

Nu

x

(5.16)y

y

u

y

(5.17)iy y i

Nu

y

(5.18)yx

xy

uu

x y

(5.19)i ixy x yi i

N Nu u

x y

Therefore the relation between the nodal displacements and strain at any point, are

related by the derivatives of the shape function in respective direction for a triangular

element the above relations in the matrix form are expressed as follows

31 2

31 2

31 1 2 2

0 0 0

0 0 0

x

x

y

y

xyyx

u NN N

x x x x

u NN N

y y y y

u NN N N Nu

y x y x yy x

ε

1

1

2

2

3 3

3

(5.20)

x

y

x

y

x

y

u

u

u

u

N u

x u

(5.21)ε = B U

The equation which relates the stress to strain or strain rate to any factors on which

the stress is dependent is called constitutive relation. For linear elastic behavior the

58

equation which relates stress to strain is given by the following equation,

(5.22)0 0σ = D ε -ε + σ

Where 0 and 0 are initial strain and stress respectively.

For plane stress condition,

0 02

1 0

1 0 (5.23)1

10 0

2

x x

y y

xy xy

E

Where E is the Young’s modulus and is Poisson’s ratio. In the above equation

2

1 0

1 0 (5.24)1

10 0

2

E

D

D is a matrix of constant values.

For plastic condition, that is, the behavior of material after yielding, the relation

depend on the strain hence [D] dose not remain constant. The constitutive relations

are mostly based on extensive experimentation of material behavior before and after

yielding. Many constitutive equations have suggested in recent years, some of them

generally applicable, while other for specific uses. One of the most extensively used

constitutive relations is modified Johnson cook law, which suggests that the stress is a

function of strain, strain rate of the temperature [30].

To obtain the integral statement and nodal equivalent forces, the principal of the

virtual work is used in [31]. Consider an infinitesimal virtual nodal displacement.

After the equating the external work done by applied surface tractions and point

forces, and internal work done against stress to causes further strain i,e, strain energy,

we get a statement relating external force (q) to internal body forces (b) and stress (σ

=[D] [B] {u}) due to deformation.

59

(5.25)e e

Te ed d

q B σ N b

This is valid for any stress-strain relationship. Using the linear stress strain

relationship in eqn. (5.23)

(5.26)ee e e

q K u f

Where the first term of R.H.S represents the forces developed due to displacement,

while the second term stand for the body forces and initial stress and strain if present.

B D B (5.27)e

Te ed

K

0 (5.28)e e e

T T Te e e

ed d d

e

f N b B D B

5.5 Quasi-static Elastoplastic Analysis

In quasi-static elastoplastic analysis of the work piece, the strain rate

hardening effects are neglected, and so are the temperature effects. The material is

assumed to be linear elastic till the effective stress or Von Mises stress reaches the

uniaxial yield stress of the material. Thereafter, the material is considered plastic and

linear hardening with the increase in strain. A tangent modulus is calculated at each

iteration defining the relation between Von-mises effective stress and strain

increment. The Von mises stress criterion is used for further yielding.

It is observed through the experimentation, that the volume of workpiece remains

constant while deforming plastically [28],

0 (5.29)x y z

Hence all further calculation is done in term of Deviatoric stresses and the normal

component of stresses is neglected, as it does not affect the plastic behavior. Isotropic

60

hardening is assumed, since the bauschinger effect is not significant. The elastic-

plastic stress strain relations, after yielding, are given by prandtl-reuss equations as

given by equation (4.4) in term of the plastic strain increments and Deviatoric stresses

[27]

( ) (5.30)ij pd d ijS

Where, dλ is the proportionality constant which changes in the course of iterations

and ijσ = Deviatoric stresses and ( )ij pd =plastic strain increment

The algorithm can be summarized as [32]

1. Get the data defining geometry and material properties

2. Evaluate the equivalent nodal forces for applied body forces and surface

traction

3. Load increment loop begins

A. Increments the applied loads according to specified load factors

B. The inner loop begins

a. Calculate element stiffness for elastic and elastoplastic material

behavior

b. Solve the simultaneous equations. calculate if the elements is

-yielding the previous load increment and

-the load is increasing

-yielding in the current load increment

- Not yielding

c. Determine the flow vector and elastoplastic matrix

C. See if the solution coverage’s

If yes, exit the inner loop

If no, continue the loop

4. Continue the load increment loop until the load reaches final value.

61

5.6 High strain- Rate analysis

While in quasi-static analysis, the strain-rate hardening effects in material

behavior are ignored, in high strain-rate formulation, efforts are made to incorporate

this behavior is obtain a much more realistic simulation of the electroforming

phenomenon as mentioned in [34] introducing mass matrix in transient dynamic

analysis [35].

(5.31)M U + C U + K U = F

And visco-plastic formulation by Perzyna model [36]

(5.32)e νpε = ε +ε

1 1- 2 (5.33)

2G Eij kk ij

ε τ σ

Where, G is shear modulus and E is the Young’s modulus.

(5.34)p

ij

FF

Where, F is the flow function, γ is the material constant and (F) is a function which

can be experimentally obtain for each material. Again, the two main loops used in

quasi-static formulation are used here, but incorporating the inertia force and perzyna

model.

5.7 Coupling schemes

The procedure of coupled field analysis depends on which fields are being

coupled, but two distinct methods can be identified in the finite element analysis of

coupled field: sequential coupling (weak coupling) and direct coupling (strong

coupling).

62

5.7.1 Strong or Direct coupling scheme

In a direct coupling, the two field analyses are coupled in such a way that they

result in a single system of equations. The solution of this system gives all the degree

of freedoms. This is useful in the analysis of highly non-linear coupled fields. The

block diagram of direct or strong coupling is given in fig. (5.3).

5.7.2 Sequential or weak coupling scheme

In the sequential coupling, the two fields are solved in sequence and the result

of one analysis is given to subsequent field analysis as an input (fig.5.4). The

sequential coupling is useful in the analysis of linear fields. In the present report, the

sequential coupled model is used in magneto-structural analysis. Firstly, an

electromagnetic analysis is carried out, and subsequently a structural analysis is

performed. A fraction of analysis domain must be common for various fields in

coupled field analysis. Then the same geometrical information can be used in the

various analyses of fields in the coupled field problem. Such an analysis is a step

towards more complex non-linear analyses [15].

Fig. 5.3: Strong coupling scheme

63

In next section, simulation of magneto-structural coupled analysis is carried out

in the following way; firstly the simulation of electromagnetic part of coupled field is

presented for linear static case and then the structural part of coupled field problem is

simulated.

5.8 Coupled linear magneto-structural case

Case study involves: Determination of the magnetic flux densities, force and

deformation in the conductor

5.8.1 Geometric data and material properties

Outer radius R = 50 mm,

Inner radius r = 40 mm

Height h = 90 mm and

Input current density J = 5×106

amp/m2;

Young Modulus E = 12.2 ×1010

N/m2;

Poisson’s ratio ν = 0.3.

Field No.1

Analysis

Geometry Information

Field No .2

Analysis

Analysis # 1

Analysis #2

J R

r

h

Fig. 5.5: Solenoidal coil

Fig. 5.4: Weak coupling scheme

64

5.8.2 Assumptions

A long thick solenoid carries a uniform current distribution.

The turns of the solenoid can be modeled as a homogeneous, isotropic

material with modulus of elasticity E and Poisson’s ratio v.

The symmetries of the problem allow us to consider just a one-half or one-

quarter of the domain of analysis.

In this case study, the coil material is the nonmagnetic, only Lorentz force will

exist.

5.8.3 Problem model

There are two analysis domains in this model.

5.8.4 Analysis domain of fields

1. EM analysis domain

Coil and air (both) regions

2. Structural analysis domain

Coil region

5.8.5 Coupling scheme:

Sequential coupled model

5.8.6 Electromagnetic analysis

Fig. 5.6: Problem model-1

1. Air region

2. Coil region

coil region

65

5.8.6.1 Magnetic flux lines plot with ANSYS

The flux lines plot due to current flow in the coil is shown in fig. (5.7).

5.8.6.2 Flux linkage with coil

The flux linkage with current-carrying coil is shown in fig.(5.8)

5.8.6.3 Force distribution in the coil

Fig. 5.7: 2-D flux lines plot with ANSYS

Fig. 5.8: Flux linkage with coil

66

The local force distribution in the coil region is shown in fig. (5.9)

5.8.7 Structural analysis for Linear Elastic case

The linear structural analysis is carried out in the elastic region of the stress-strain

curve of the material. The input for structural analysis is the local force distribution

obtained from the electromagnetic part of this coupled problem. The analysis domain

for the structural analysis is only the coil region.

5.8.7.1 Deformation in the coil:

The resultant deformation in the coil region is shown in the fig. (5.10). The

deformation shown in this fig (5.10) is obtained with ANSYS.

Fig. 5.9: Force distribution in coil

67

5.8.7.2 Deformation plot across the height of the coil (MATLAB)

The plot of deformation across the height of plate is shown in fig. (5.11). The plot of

deformation is obtained with MATLAB.

Fig. 5.11: Plot of deformation across the height of coil

Fig .5.10: Deformation in the coil

68

5.8.8 Results of coupled field analysis

The results obtained with the MATLAB are compared with results obtained with

ANSYS.

Table 5.1: Comparison of Results of Coupled magneto-structural analysis with

ANSYS

Maximum flux

density (Tesla)

Maximum force

density (N/m2)

Maximum

deformation (m)

ANSYS 0.0502 2.4918 ×105 1.6623× 10

-9

MATLAB 0.0501 2.4866 ×105 1.6251× 10

-9

Ref. [38] 0.04877 1.946× 105 1.923× 10

-9

69

Chapter 6

Conclusion and Future work

6.1 Summary of work done There are two cases have been analyzed in this report.

1. In first case, the sequential coupled linear magneto-structural analysis has

been done by developing MATLAB code. The analysis electromagnetic part

of the coupled magneto-structural analysis is investigated for the static

electromagnetic analysis due static excitation in the solenoidal coil. The

results obtained by developing MATLAB code are verified by the results

obtained with a commercial FEM software ANSYS. This analysis is carried

out further for coupled magneto-structural analysis in which the subsequent

structural analysis is coupled with electromagnetic analysis by means of

electromagnetic forces. These electromagnetic forces are applied as input in

the subsequent structural analysis which gives the final deformations, stresses

in the solenoidal coil. The coupled magneto-structural analysis is also carried

out by developing MATLAB code. These results are validated with

commercial FEM package ANSYS.

2. In the second case, various formulations for force computation in the

conductive region have been done. In first, it is analyzed by complex magnetic

vector formulation and in second, same case analyzed by time-stepping

formulation. The time-stepping formulation is particularly suited for transient

and non-linear problems. The time-stepping formulation is used for analyzed

for the same problem as that for in the complex vector potential formulation.

70

Therefore, the results obtained with time-stepping formulation may be

compared with the results obtained with complex vector potential formulation.

All results are validated by commercial FEM package ANSYS.

6.2 Summary of Results

1. In first case, the coupled linear magneto-structural analysis describes the

deformation in the conductor. Results obtained by MATLAB code are in good

agreement with commercial FEM package ANSYS. This analysis may be

useful in the analysis of deformation and vibration problem in electrical

machines. The analysis can be used to explain the deformation in the rotor

conductor in electrical machines. The result obtained with coupled magneto-

structural analysis shows the deformation in the current carrying conductor. In

the case described above can also be used in the calculation of deformation in

the winding conductor under the short-circuit conditions by using the peak

value of the short-circuit current as static excitation.

2. The accuracy of the coupled field analysis is strongly depends on the accuracy

of computation of the electromagnetic forces. The deformation and stress field

patterns in the metal-forming are depend also on the patterns of applied forces

on the metal object. In the second case, the force computation by various

formulations is analyzed. These formulations can be useful for the analysis for

steady-state excitation (sinusoidal input), particularly time-harmonic or

complex magnetic vector potential formulation, and time-stepping formulation

is particularly for transient excitation. Force computation with time-stepping

formulation is the first step for electromagnetic forming case.

71

6.3 Future work

The linear coupled magneto-structural analysis may be useful in the analysis of

deformation and vibration analysis in electrical machines. This analysis can extend to

nonlinear structural analysis, which can be much close to real world values. The

thermal field can also be taken into account to approaching accurate analysis. The

coupled analysis can extend for magnetic materials by computing force with one of

the magnetization force technique. In case of ferromagnetic materials, there is only

modification that can be made in the analysis presented above is in force computation

technique.

72

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