4. finite element method

30-JAN-2020EIEN20

Design of Electrical Machines

4. Finite Element Method Formulation & implementation Tools & experience

Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-01-30 2

L4: Finite Element Method•

What shall we know about FE method and formulation as a user

–

Computational electromagnetics•

FE formulation

that is based to the weighted

residual method –

Mathematic form

of the field problem that is

suitable of FE method•

FEMM for magnetic calculations

•

Home assignments: are outcomes (power, losses, ..) from thermal and magnetic analysis comparable?

W

W

https://en.wikipedia.org/wiki/Finite_element_method

https://en.wikipedia.org/wiki/Computational_electromagnetics


Equivalent circuit vs FEM

•

The equivalent circuit method (ECM) is essentially a finite element method (FEM) where the quickness

is set against the accuracy

–

The number of elements

employed by ECM is much less than the usual number of elements in the FEM.

–

The ECM flow is bidirectional

and has to be known in advance when defining elements

•

FE analysis results the potential and flow distribution

itself.•

Ka=f

–

Stiffness matrix, vector of unknowns, load vector + boundary

vector

Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-01-30 4100 101 102 10310-2

10-1

100

relative permeability, [-]

rela

tive

dist

ance

, r [-

]

coupling factor, k=/ [-]

0.5

0.5

0.6

0 .6

0.6

0.7

0.7

0 .7

0.8

0 .8

0.8

0.85

0.85

0.85

0.9

0 .9

0.9

0.95

0.95

0.95

0.97

0.97

0.97

0.98

0.98

0.98

0.99

0.99

0.99

0.995

0.99

50

100 300 1000 3000 10000 30000 10000010-2

10-1

100

101

relative permeability [-]

spac

e fa

ctor

1/k

f [%]

equivalent permeability, eqv [-]

10101010

20202020

404040

40

100100100

100 200200

200

400400400

100010001000

20020002000

40040004000

spacefespaceair

feaireq kk

1

fe

space

air

space kkAl

AlR

11


Follow the magnetic coupling …

•

Focus on Self inductance, Mutual inductance and leakage inductance

•

Keeping eye on flux densities in the gap and in the core•

Calculating & linearizing

magnetic coupling between the coils

2

11

221

121

22

11

III

LMML


… and use it in drive circuits

•

Equation system–

Voltage

u=Ri+L

di/dt+e–

Magnetic flux linkage Ψ=Li-Ψpm

–

Torque or force T= Ψpm x i

–

Mechanics T- Tload=J dω/dt

–

Temperature Pheat=Rth

+Cth

d/dt

dphi/dte

a

i

temperature = f(J)

R

pm flux = f(x)

1/ND

Ac/N

1/m

force=f(x,J)

simout

To WorkspaceJ =f(x,flux)

1/s xo1/s

1/s

[T]

[v][x]

[u]

[F]

[J]

[phi]

[T]

[v][x]

[F][J]

[u]

simin

FromWorkspace

[phi]

Xo

Finite Element Method

How much shall we know FEM?


Finite Element Method•

Robust for general analysis

•

Geometry is broken down into simple pieces – Finite Elements

•

These elements are placed in a large sparse matrix equation

•

The equation system is solved using standard matrix solution techniques

•

Direct solver and error evalustion

often used



If the real model, which is a prototype, contains infinite number of degrees of freedom

(d.o.f.) then the

EC model incorporates too low d.o.f. (i.e. nodes)•

the FE gives an approximate solution

to a physic

problem described by differential or integral equations in a number of small regions –

finite number of d.o.f.

•

The system equations are assembled according to the (main) solution methods: a variational

method

(integral kind) and a residual method

(differential kind)



Field solution is based on solving potentials in accordance of the field equation and BC

•

Field solution is based on energy minimization (variational

formulation)

duuW 2

21

02 u

02 gu

dugduuW 2

21


Types of FEM problems•

Closed boundary

problems

•

Equilibrium time-independent

or quasi-static problem in frequency domain –

steady state

analysis, field distribution, stresses and flows•

Eigenvalue

problems –

natural frequencies,

resonance characteristics, mode shapes, stability

•

Propagation time-dependent

problems – transient and dynamics, response, waves


Heat Transfer

x dx

L

H H+dH

Q(x) •

Heat source Q(x) [J/m] per unit of time and length

•

The heat in H [J/s] and outflow H+dH

•

Transfer problem is described by

–

flow balance –

conservation equation

–

Flow potential relation – constitutive relation

•

The stationary time- independent heat problem

QdxdHdHHdxQH

dxd

AHq

0

Q

dxdA

dxd

0 QD


Weak Formulation•

Equation suitable

for a numerical solution

•

To establish the weak of the strong forth the latter is multiplied by an arbitrary weight function

v and this

multiplication is integrated over the pertinent region:

•

The rule of integration by parts

states that

00

dxQ

dxdA

dxdv

l

bab

a

b

a

xxdxdxd

dxddx

dxdy


Solution•

By applying the previous expression and evaluating d/dx

gives

•

By specifying the boundary conditions

as a heat flux (i.e. temperature gradient) the natural boundary conditions are established: q(l)

unknown, q(0)=h,

QdxvdxdAvdx

dxdA

dxdv lll

000

QdxvhvAvAqdxdxdA

dxdv l

xlx

l

00

0


Matrix form•

In the weak forth, however, only the first derivative of the temperature enters i.e. the approximating functions need to be differentiable once.

•

The heat balance equation is expressed through the nodal relation –

stiffness, where the geometry and

material propertied appear, temperature distribution in the node points that the heat flows in our out to the nodes.

•

Flows are initially given as loads, even if they are naturally 0 and unknowns

are calculated in respect to

reference.

loadboundary ffKa


Finite element

•

A geometric object is divided into finite number of simple shape

elements

•

Element joins

geometry and material properties•

Element shape function interpolates the potential values within the element

1D2D

3D


Interpolation of potential•

the ‘potentials’

ae

within the

element bounds is interpolated between its values at the nodal

points according to element shape function Ne

CαCNaN 1ee

j

iej

ei NN

αN 2

121 1

xx

N

3

2

1

321 1 yxyx

CCNaN 1ee

k

j

iek

ej

ei NNN


Element shape functionsl

xx

j

i CC det11

lxi xj

Nje(x)

xNie(x)

Ne i

j

1

0

(x)=α1+ α2x

yxxxyyyxyxN

yxxxyyyxyxN

yxxxyyyxyxN

yxyxyx

ijjiijjiek

kiikkiikej

jkkjjkkjei

C

C

CC

det1

det1

det1

111

33

22

11

ie

j

jei

xxl

N

xxl

N

1

1


From FE to FEM•

FE shape functions are used to derive gradient

of

potential

that is constant

for the 1st order

element•

Weak formulation is used so that the problem becomes differentiable once

•

By solving the weak formulation a weight function is needed for the residual method

•

In Galerkin

weighted residual

method the

weight function is the same as FE shape functions


General form

•

The general form of a differential equation

with its possible coefficients in the particular terms

•

In the coupled field

problems, such coefficients are field dependent and represents the link between the various field types such as magnetic/thermal etc.

1. parabolic, transient term

2. diffusion

3. convection

4. absorption

5. source

gaffvfft

a


2D and 3D•

In the general 3-D case, A is a vector with three components.

•

In the 2-D planar and axisymmetric

cases, two of these three components are zero, leaving just the component in the “out-off the page”

direction.

•

The problem formulation on 3D magnetic field analysis may take advantage either on total scalar potential

or on vector potential

as the unknown in the

node points.


Magnetostatics

•

The problem in electromagnetic analysis on a macroscopic level is the problem of solving Maxwell’s equations

subject to certain boundary conditions.

•

Magnetostatic

problems

are problems in which the fields are time-invariant.

•

The field intensity H

and flux density B

must obey the rules

JH 0 B


Constitutive relation

•

Subject to a constitutive relationship

between B

and H for each material:

•

If a material is nonlinear, the permeability, μ

is actually a function of B:

BHB

HB


Magnetic vector potential•

Finite element magnetic solver calculates a field that satisfies via a magnetic vector potential

approach.

Flux density is written in terms of the vector potential, A, as:

•

By rewriting the equation by the Ampere’s circuit lawAB

JMAB

1

PM magnetisation


Time harmonic magnet problem•

Ohm’s Law

•

Cause for electric field intensity

•

Replacing B by A

•

Consider 2D & integration

•

Adding new terms to Amperes Law

•

Fixed frequency and phasor transformation

•

Practical formulation

EJ

tBE

At

AE

VJAAB

1

VAE

tjaetjtaA ResincosRe

VJajaBeff

1

Additional

voltage gradient Complex number

Finite Element Method Magnetics

FEMM for magnetic calculations


Electromagnetic device

•

Symmetry?•

Estimated magnetic flux flux=mmf/(Rg+Ry+Rt)=6e-4 Vs

–

mmf=N*I=180*2=360 A–

Rg=l/(μ0

A)=6e+5 1/H–

Rt=l/(μμ0

A)=2e+4 1/H–

Ry=l/(μμ0

A)=4e+4 1/H

•

Flux density B=flux/A=0.4T•

Force F=0.5B2/ μ0

A=177N


Finite Element Analysis•

Pre-processing

–

Geometry–

Material properties and sources

–

Boundary conditions–

Discretization

–

FE-mesh

•

Processing•

Post-processing

–

Field distribution, flow density, etc

Select appropriate size for the model


Boundary conditions

•

The boundary conditions are used to reduce the size of the FE model

and to

constrain the field that surrounds the electrical device.

–

Dirichlet

(essential) condition -

flux (equi-potential) lines are parallel to the boundary

–

Neumann's (natural) condition (∂/∂n, ∂VM/∂n known) determines the surfaces that the magnetic flux crosses orthogonally

–

Periodicity condition is set to reduce the size of the model according to the periodic structure.


FEMM Pre-Processor•

Drawing the endpoints of the lines and arc segments for a region,

•

Connecting the endpoints with either line segments or arc segments to complete the region,

•

Defining material properties and mesh sizing for each region,

•

Specifying boundary conditions on the outer edges of the geometry.


Coil

•

Wire specification (AWG)•

Coil & circuit specification (number of turns, parallel/series, current)


FEMM Post-Processor•

Flux lines show magnetic coupling between the magnetic conductive parts

•

Flux density indicate magnetic loading

•

Generally the forces are calculated by using weighted Maxwell’s stress tensor

•

Flux linkage can be obtained from circuit-data


Force calculation I


Force calculation II

stator mover

Flux density, B Force integration lines

forces for 1 m long mover


FEMM and LUA Script•

Femm

functions can be called by LUA Script

•

Scripting is similar to Matlab, it is possible to create user-independent calculation loop

•

Matlab

can be used as a design environment that also generates LUA script or calls directly femm

functions

Home assignment A1 and A2

Electromagnetic analysis of a transformer


Methods and implementation

•

FEM and ECM–

Non-linearity

•

Operation points–

No-load,

–

max-load, –

short-circuit

•

Comparability–

Thermal analysis vs

magnetic

analysis–

Models vs

manufacturer

0 2 4 6 80

50

100

150

In/O

utpu

t pow

er, P

i,Pu

[W]

0 2 4 6 80.2

0.4

0.6

0.8

1

effic

ienc

y,

[-]

load current, Iout [A]0 2 4 6 8

0

2

4

6

Cop

per l

osse

s, P

cu [W

]0 2 4 6 8

7

7.5

8

8.5

9

9.5

Cor

e lo

sses

, Pfe

[W]

load current, Iout [A]

0 2 4 6 80

0.5

1

1.5

Inpu

t cur

rent

, Iin

[A]

0 2 4 6 80

10

20

30

40

Out

put v

olta

ge U

2 [V

]

load current, Iout [A]0 2 4 6 8

0

0.5

1

1.5

2

2.5

3

curre

nt d

ensi

ty, J

cm2

[A/m

m2 ]

0 2 4 6 8-20

-15

-10

-5

0

Out

put v

olta

ge re

qula

tion

U

2 [-]

load current, Iout [A]

4. finite element method

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