tokyo 04 dias

7/21/2019 Tokyo 04 Dias

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Alain Bossavit

The question of forces

Laboratoire de Génie Électrique de Paris (CNRS)

[email protected]

Old controversies,

modern geometrical tools

http://www.lgep.supelec.fr/mse/perso/ab/bossavit.html

in ElectromagneticsOverlays, not part of

presented slides, contain

summary of oral comments

Overlays, not part of presented slides, contain

summary of oral comments

Tokyo,

Friday 11 June 2004

Thanks to organizers and

Prof. Takagi for support

Tokyo,

Friday 11 June 2004

Thanks to organizers and

Prof. Takagi for support

7/21/2019 Tokyo 04 Dias


Magnetoformingform

Conducting plate

Coil, with strong, shorttransient pulse

Countless problems require force computation in moving or deforming conductorsand/or magnets. Most often, the J × B formula suffices ...

Countless problems require force computation in moving or deforming conductorsand/or magnets. Most often, the J × B formula suffices ...

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Magnetoforming

J

sJ

B FJ × B, ↑

uCoenergy:Φ(u) ~ u |J |

s 2

Force: ∂ Φ > 0u

Force is, of course, upwards,

as more involved, but also more

general, rule based on derivation

of coenergy shows. (More on

that later.)

Force is, of course, upwards,

as more involved, but also more

general, rule based on derivation

of coenergy shows. (More on

that later.)

7/21/2019 Tokyo 04 Dias


Melting and stirring of liquid metals

σ > 0

µ >> µ0

J × B also OK for MHD applications, although Eulerian setup makes things more

difficult to handle (term v × B, with no obvious discrete form, to be included inOhm’ s law)

J × B also OK for MHD applications, although Eulerian setup makes things more

difficult to handle (term v × B, with no obvious discrete form, to be included in

Ohm’ s law)

7/21/2019 Tokyo 04 Dias


Rail

Mobile

µ >> µ0

µ >> µ0

B

Maglev

(manip)

In some situations, J × B definitely

not relevant. E.g., with magnets (be it

electromagnets, as here, or permanentones).

In some situations, J × B definitely

not relevant. E.g., with magnets (be it

electromagnets, as here, or permanentones).

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B

HB

– Hc

r B = µ (H + H )c

B = B + µ Hr

0 0

div B = 0

Hc

B = µ H0

rot H = rot H

div B = 0

rot H = 0c

~

~

B = µ (H + H )0c

F = (rot H ) × B

c

F

[1] Suppose hard magnet, with

simple affine constitutive law,

that can be expressed in twoequivalent ways, either this “

(with constant remanent

induction), or ...

[1] Suppose hard magnet, with

simple affine constitutive law,

that can be expressed in twoequivalent ways, either this “

(with constant remanent

induction), or ...

[2] … that “ (with

constant coercive field). If magnet alone in space,

equations are as shown

below, ” ...

[2] … that “ (with

constant coercive field). If magnet alone in space,

equations are as shown

below, ” ...

[3] … which is

equivalent to this “ ,

with amperian currents

as source, ...

[3] … which is

equivalent to this “ ,

with amperian currentsas source, ...

[4] … from which it would seem

that a plausible force formula’ would derive. But ….

[4] … from which it would seem

that a plausible force formula’ would derive. But ….

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B

HB

– Hc

r B = µ (H + H )c

B = B + µ Hr

0 0

div B = 0

Br

B = B + µ H0

rot H = 0

div B = div B

rot H = 0

~

B = µ H0

F = (div B ) H

r

Fr ~

r

– – ––––

+ ++

[5] … this alternative procedure

(take equivalent magnetic chargeas source) is just as reasonable, ...

[5] … this alternative procedure

(take equivalent magnetic chargeas source) is just as reasonable, ...

[6] … and gives a totally different

result for the force field.

[6] … and gives a totally different

result for the force field.

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Which for sure shatters confidence about naive approaches to the

force question.

All the more so that classical treatises are not of great help to the

programmer who is only looking for “some reliable formula to code”.

Indeed, here follows a sample of what the literature offers about

“the” analytical expression of force density:

Which for sure shatters confidence about naive approaches to the

force question.

All the more so that classical treatises are not of great help to the

programmer who is only looking for “some reliable formula to code”.Indeed, here follows a sample of what the literature offers about

“the” analytical expression of force density:

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F = M · ∇B + M × (∇ × B) – ∇B · (v × (M × v))/c 2

P. Penfield, Jr., H.A. Haus: Electrodynamics of Moving Media, The MIT Press (Cambridge, Ma.), 1967, p. 216

(electric field contribution neglected)

F = (J × B) – –– H ∇µ1

2

2

F.N.H. Robinson: Macroscopic Electromagnetism, Pergamon Press (Oxford), 1973

F = – ∇p – –– H ∇µ + –– J × H0

8π

1 2 µ

cL. Landau, E. Lifschitz: lectrodynamique des milieux continus, Mir (Moscou), 1965, p. 190

(for a fluid; p is pressure)0

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Numerical problems with J × B

H = ∑ h We

e ∈ E e => J = rot H

µ H · grad ϕ' = 0 ∀ ϕ'

edge elementB = µH

=> div B ~ 0 (weakly)

F = (rot H) × B

conformal non-conformal

Numerical methods are either

“h-oriented” (rot H = J exactly

enforced, but Faraday

satisfied in only “mesh-weak”form) or “b-oriented” (the other

way round). In both cases,

terms in the J x B formula are

not approximated with the

same accuracy.

Numerical methods are either

“h-oriented” (rot H = J exactly

enforced, but Faraday

satisfied in only “mesh-weak”

form) or “b-oriented” (the other

way round). In both cases,

terms in the J x B formula are

not approximated with the

same accuracy.

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And what about the Maxwell tensor

Ω F(Ω) = FΩ

F(Ω) = [(n · B) H – –– (B · H) n]1

2∂Ω

(flux of M = B H – –– B H δ )

ij i j 1

2

k kij ij

Main drawback of this popular device: it yields force integrals,

i.e., resultants and/or torques, not the force field .

Main drawback of this popular device: it yields force integrals,

i.e., resultants and/or torques, not the force field .

Besides, the expression of M is not a primitive result in

Maxwell’s theory. It is derived from that of local force.

Besides, the expression of M is not a primitive result in

Maxwell’s theory. It is derived from that of local force.

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1976Oxford

1989Tokyo

"Force" papers in the Compumag record 1978

Grenoble1981

Chicago1983Genoa

1985Ft-Collins

1987Graz

1991Sorrento

1993Miami

1995Berlin

1997Rio

1999Sapporo

2001Évian

Bizarre, bizarre.

Reichert

Aldefeld

WeissenburgerAldefeld Coulomb

Müller

Miya Miyata–Miya

Eastham

Reyne

Kabashima

Brauer

Matinescu

Morisue

Kabashima

Tani

Ito–Kanazawa

Kawase-Ito

Müller

Henneberger

Sattler, Shen

Liu, Freeman

Ren

Bossavit

Tsuchimoto

Arturi

Rodger

Kameari

Ren–Razek

Henrotte

JenkinsSmithHowe

Nishiguchi

Richard

NiikuraKameari

Takagi–Tani

Park KimS.-y. Hahn

Copeland KoppO'Handley

BartschWeiland

Ren

TegopoulosWiak

Tsukerman

Takahashi

Im–Kim

Tsuboi

Brauer

Besbes–Ren

Kurz FetzerLehner

TegopoulosWen Yao

McFee

Ito–Kawase

Richard

ReyneMeunier

Hur, Chun,Lee, Hyun

Vandevelde

Medeiros

Divoux, Reyne

SaviniDi BarbaPerugia

Fireteanu

Sebestyen

Hameyer

VandeveldeMelkebeek

Curiously, interest for forces in the Compumag

community has passed its peak, it seems. Yet,

theoretical problems discussed at the very first

Compumag conference (post-talk discussions were

opportunely recorded) seem still to be with us.

Curiously, interest for forces in the Compumag

community has passed its peak, it seems. Yet,

theoretical problems discussed at the very first

Compumag conference (post-talk discussions were

opportunely recorded) seem still to be with us.

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Rail

Mobile

µ >> µ0

µ >> µ0

B

Maglev

(manip)

Was presented, for instance,a 2D modelling of this model

problem for levitation

devices:

Was presented, for instance,

a 2D modelling of this model

problem for levitation

devices:

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The first Compumag paper on forcesK. Reichert, H. Freundl, W. Vogt: "The calculation of forcesand torques within numerical magnetic field calculationmethods", Proc. Compumag, Oxford (1976), pp. 64-73.

Integration

pathWinding

Rail

Core

Winding

Integration

surfaces

Magnetic levitation system

Note use of Maxwell’s tensor integrationNote use of Maxwell’s tensor integration

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Carpenter: "We need an operational definition of 'stress',

which is not a quantity which is directly observable." Authors: "A permanent magnet (...) can be modelled either by surfacecharges or by Ampère's currents. The total force in both cases is the

f = J × (B – M ) – H div M – grad µ H dH

With B = µH + M , they findp

p p0

H

Say "inadvisable" because of non-linearity, "magnetic history should

be known". Use surface integration with surface in µ = const (air).

Hammond: Can the surface traverse iron?

Authors: No, "artificial airgap not allowed",because "it may change the original fielddistribution", unless modification is slight.

Integration

paths

same but the stress inside is different." (We agree that) "the distinction between mag-

netic and mechanical forces inside is 'model depending' and therefore meaningless."

(Yet another expression of

local force field …)

(Yet another expression of

local force field …)

Discussion suggests basic disagreements about the very status of the “localforce” conce t. Some even seem to doubt its obective character.

Discussion suggests basic disagreements about the very status of the “local

force” concept. Some even seem to doubt its objective character.

As we saw← earlier

As we saw← earlier

Maxwell

tensor’s

integration

around a

node to getlocal force

Maxwell

tensor’s

integration

around a

node to getlocal force

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Carpenter : "there must be a definitive answer [to the questionof force density] in every case, and one could use strain gaugesto discover it. There ought, therefore, to be a mathematicalmethod which is physically more correct than the others."

Should we agreeinc(ε) = 0 σ = κε div(σ) =

strainstress force

*

*As quoted by G. Reyne, thesis, 1987.

Moon: "(...) the representation of the magnetic body forceis not unique (...). This raises serious questions regarding

the internal stress state in a magnetized body."

“ incompatibility

tensor

Elasticity equations

(here in simplified,symbolic form), say ε

should satisfy

compatibility conditions,

constitutive relationshould hold, and σ

should balance body

force.

Elasticity equations

(here in simplified,symbolic form), say ε

should satisfy

compatibility conditions,

constitutive relation

should hold, and σ

should balance body

force.

Strain ε is directly

observable, but notσ. Hence room for

disagreement. But i

constitutive law

known, then

cannot fail to beunique.

Strain ε is directly

observable, but notσ. Hence room for

disagreement. But if

constitutive law

known, then f

cannot fail to beunique.

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What I believe:

Force unambiguously derives, viathe virtual work principle, from

the material's constitutive law

Constitutive laws are mixed in magnetoelasticity:

both H and σ depend on both B and ε.Incomplete knowledge won't do.

Conflicting force formulas come from diverging,

unspoken assumptions about constitutive laws

Constitutive laws ought to be

measured, not theorized about

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Let’s proceed. The language of differential forms will be used. Here

follows a short reminder on the differential-geometric approach to

Maxwell’s equations, a dictionary of correspondences with the better

known vector-fields formalism, and a review of discretization

principles which underlie modern “mesh-based” computational

methods (FDM, FEM, FVM).

Let’s proceed. The language of differential forms will be used. Here

follows a short reminder on the differential-geometric approach to

Maxwell’s equations, a dictionary of correspondences with the better

known vector-fields formalism, and a review of discretization

principles which underlie modern “mesh-based” computational

methods (FDM, FEM, FVM).

7/21/2019 Tokyo 04 Dias


The electric field: a mappinge : ORIENTED CURVE → REAL

Unit charge pushedalong curve c ∫∫ ∫ ∫ e

cvolts

with additivity:

and continuity (w.r.t. variations of )c

∫∫ ∫ ∫

e = ∫∫ ∫ ∫

e + ∫∫ ∫ ∫

e,c + c 1 2 c2c1

Work involved:

Most economical

translation of observable

properties of e: as a map

(from curves to reals). This

is the very definition of a

differential form (of degree

1, the dimension of curves).

Most economicaltranslation of observable

properties of e: as a map

(from curves to reals). This

is the very definition of adifferential form (of degree

1, the dimension of

curves).

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E

(the vector field) e(the 1-form)as a proxy fo

Change " ", change E (and τ ), for same e

c

∫ e = Ec

∫ c

The observable is not E but e, the differential form

Standard representation

using vector field E ...

Standard representationusing vector field E ... … involves irrelevant metric

structure (dot product, length …)

… involves irrelevant metric structure (dot product, length …)

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Σ∀- ∂ ∫ d + ∫ h = ∫ jΣ ∂Σ Σt

S∀∂ ∫ b + ∫ e = 0S ∂St

–∂ d + dh = jt

∂ b + de = 0t

b = µh, d = εe source of field

New status for ε and µ, now operators,(Hodge) of type 1-FORM → 2-FORM :

Discretization strategy: Instead of all

S, Σ, enforce this for surfaces spanned

by faces of interlocked meshes:

rima

dua

outer oriented surfaceouter oriented surface

inner oriented surfaceinner oriented surface

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Correspondences: ∫ τ · E

c ∫ e

c

∫ n · JΣ ∫ jΣ

grad ϕ dϕ

rot E dediv J d j

J · E ∧ e ν |B|

2 ν b ∧ b

v × B i bv

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N → E → F → VG R D

grad rot div

here, R = – 1

fe

e, aat edges

h at dual edges(i.e., faces)

d, j at dual faces

h = h : f ∈ F f b = b : f ∈ F f

b at faces

e = e : e ∈ E e d = d : e ∈ E e

νν

εεεε

Set of nodesSet of nodes

Set of facesSet of faces

”

Incidence matrices”

” Incidence matrices ”

S uare matrices built as ex lained below and in com anion slideshowSquare matrices, built as explained below (and in companion slideshow)

Everything lives on the discrete structure (mesh + some dual mesh) from now onEverything lives on the discrete structure (mesh + some dual mesh) from now on

Dual mesh (n – p)-cells matched 1-to-1 with p-cells of primal meshDual mesh (n – p)-cells matched 1-to-1 with p-cells of primal mesh

G, analog of G, analog of R, analog of

R, analog of D, analog of

D, analog of

Di ti ti t lkit il bl ( l b b b t b R b t )

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Then, automatic spatial discretizationof Maxwell's equations:

–∂ d + R h = jt

t

∂ b + Re = 0th = bνννν d = eεεεε

hence a Yee-like scheme ("GFD"):

b – bk + 1/2 k – 1/2

+ Re = 0

k

δt

ε– e – ek + 1 k

δt

+ R b = jννννt k + 1/2k + 1/2

Discretization toolkit now available (replace b by b, rot by R, µ by µµµµ, etc.).Discretization toolkit now available (replace b by b, rot by R, µ by µµµµ, etc.).

More general than what we need. Term in ε (displacement currents) to be ignored, and j replaced by σe + js, (source current).More general than what we need. Term in ε (displacement currents) to be ignored, and j replaced by σe + js, (source current).

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Now, some motivation for “Lagrangian” approach to forces,

with “comoving” mesh.

Now, some motivation for “Lagrangian” approach to forces,

with “comoving” mesh.

Note that neglecting term ε ∂D/∂t amounts to ignore Coulomb

forces (as if ε were 0)

Note that neglecting term ε ∂D/∂t amounts to ignore Coulomb

forces (as if ε were 0)

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s

Bs

(AC)

(steady)

A standard benchmark (pbs. 12 &16 of the "TEAM workshop")

Strong short pulse in the coil

creates equivalent of “kicking blow”

to the clamped elastic plate, which

then deforms in complex (and well-

documented, by lab.measurements) manner.

Strong short pulse in the coilcreates equivalent of “kicking blow”

to the clamped elastic plate, which

then deforms in complex (and well-

documented, by lab.measurements) manner.

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u

Some air around

meshed, too

trajectory: t → u(t) (possibly virtual)

placement u

velocity field: v = ∂ ut (at t = 0, when u(0) = id)

Clearly, having mesh nodes, edges, etc., fixed

w.r.t. matter, has advantages. Hence

Lagrangian setup preferred.

Clearly, having mesh nodes, edges, etc., fixedw.r.t. matter, has advantages. Hence

Lagrangian setup preferred.

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Reference space Physical space

u

air

conductor,magnet, ...

State at time 0 State at time t

But thus having two distinct 3D manifolds mandates some care about which differential forms one speaks about. Centralnotion: placement u, i.e., mapping of material points (including fictitious “air points”) from left to right.

But thus having two distinct 3D manifolds mandates some care about which differential forms one speaks about. Centralnotion: placement u, i.e., mapping of material points (including fictitious “air points”) from left to right.

(or “material manifold”)(or “material manifold”)

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(rot H = σ(E + v × B) + J , ∂ B + rot E = 0)t

s

b = µh,

∫ h = ∫ j∂Σ Σ Σ∀

∂ ∫ b + ∫ e = 0S ∂St S∀

= σ(e – i b)v

dh = j

∂ b + dh = 0t

Equations with movement, in Eulerian

formulation, displacement currents ignored

Note v x B term, here in DF notation, “inner product”,or “contraction”, of 2-form b by velocity field v

Note v x B term, here in DF notation, “inner product”,or “contraction”, of 2-form b by velocity field v

Small case for “Eulerian” objectsSmall case for “Eulerian” objects

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Equations in so-called material form

(Lagrangian formulation):

material surface flux

B : S → ∫ b = ∫ u bu (S)t S t

*

"pull-back" of b

Same for H (m.m.f.), J (current through materialsurface), but E = u*(e – i b). Then,

b

ν

hu*

u*B

H

νu

∂ B + d E = 0, d H = J + Js

t

but now, H = ν B, and energy isu12 ν |B| ≡ νb ∧ b = ν B ∧ B

2u

12

12

t v

SMALL CAPITALS for

“Lagrangian” objects

SMALL CAPITALS for

“Lagrangian” objects

Definition of “material” flux

density B, and of pullback

Definition of “material” fluxdensity B, and of pullback

So B is pull-back

of b via u

So B is pull-back

of b via u

But E is not the pull-back

of e: E meant to represent e.m.f. in moving

conductor (as measured by comoving observer)

But E is not the pull-back of e: E meant to represent e.m.f. in moving

conductor (as measured by comoving observer)

Hodge map on material manifold, defined bycommutative dia ram de ends on lacement

Hodge map on material manifold, defined by

commutative diagram, depends on placement

Define two functionals (here quadratic but more generally conjugate convex functions) called energy and coenergy on material manifold

Define two functionals (here, quadratic, but more generally, conjugate convex functions), called energy and coenergy, on material manifold.

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Magnetic energy and co-energy

Ψ(u, B)

µ H ∧ Hu1

2ν B ∧ Bu1

2

Φ(u, H)

Theorem: Force = – ∂ Ψ ≡ ∂ Φu u

Proof: d [Ψ(u(t), B(t))] = <∂ Ψ, ∂ u> + ∫ H ∧ ∂ Bt ttu

d [energy] = – <F, ∂ u> – ∫ J ∧ Et

∂ B + d E = 0, d H = J => ∫ H ∧ ∂ B = – ∫ J ∧ Et t

t

Define two functionals (here, quadratic, but more generally, conjugate convex functions), called energy and coenergy, on material manifold.

See ICS News, 11, 1 (2004), pp. 4-12, for detailed proof. The point here is that this is a mathematical result, stemming from

an unassailable postulate (middle line)

See ICS News, 11, 1 (2004), pp. 4-12, for detailed proof. The point here is that this is a mathematical result, stemming from

an unassailable postulate (middle line)

Energy comes out as: work of forces exerted on rest of the world (1), plus Joule losses minus power tapped from supply network (2)Energy comes out as: work of forces exerted on rest of the world (1), plus Joule losses minus power tapped from supply network (2)

(1)(1)(2)

(2)

Algebraic consequences of the equations, plus differentiation of energy, using chain rule, then do the job. (Note that H = ∂BΨ is just the B–Hlaw.) Velocity v in tangent space, and force F in cotangent space at u of “configuration manifold” U, with < , > for duality pairing.

Algebraic consequences of the equations, plus differentiation of energy, using chain rule, then do the job. (Note that H =∂

BΨ

is just the B–Hlaw.) Velocity v in tangent space, and force F in cotangent space at u of “configuration manifold” U, with < , > for duality pairing.

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u

Coenergy: Φ(u, H) ~ u |J|2

Force: ∂ Φ > 0u

Example application. Eddy currents in plate make magnetic barrier, forcing a horizontal and uniform H in airgap. Assume mmf invariable

as u increases, thereby proportional to J, coil current. Then, coenergy proportional to u, hence force is upwards.

Example application. Eddy currents in plate make magnetic barrier, forcing a horizontal and uniform H in airgap. Assume mmf invariable

as u increases, thereby proportional to J, coil current. Then, coenergy proportional to u, hence force is upwards.

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Ψ(u, B) µ H ∧ Hu

1 2ν B ∧ Bu1 2

Φ(u, H) Force = – ∂ Ψ ≡ ∂ Φu u

Since ν u* = u* ν, boils down to differentiate u*u t

i.e., virtual power = lim t → 0 [ν – ν] B ∧ B12t u(t)

General computation, simple case when reluctivity uniform in physical space(all nonmagnetic materials). Then ν independent of u.

General computation, simple case when reluctivity uniform in physical space(all nonmagnetic materials). Then ν independent of u.

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. . . which is what the

Lie derivativeis about.

Coming: definitions of inner product and Lie derivativeComing: definitions of inner product and Lie derivative

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ext(c, v, t)

Extrusion (by the flow of a vector field v):

∫ i b = lim ∫ bvc1tt → 0 ext(c, v, t)

of a point:

c

v

of a p-manifold:

x

d u (x) = v(u (x))t

ext(x, v, t)

Inner product:

u (x)t

t tu (x) = x0

Yields 1-form ivb from 2-form bYields 1-form ivb from 2-form b

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Lie derivative of a differential form

b : S → bS

orientedsurface embracedflux

S

u (S)

ext(S, v, t)

tu*b : S → ∫ b

t u (S)t

L b = lim [ b – b]u (S)tSvS

L b = lim [u*b – b]t1tv

1t ≡ lim [ db – b]1t ext(S, v, t) ext(∂S, v, t)

L = i d + d

iv v vhence Cartan's "magic formula":

← definition← definition

Expression in terms of i and d, using StokesExpression in terms of i and d, using Stokes

← definition← definition

Si * * d d * L

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Since ν u* = u* ν and d u* = L ,t vu

t → 0 ν – ν] = L ν – νL1

t u(t)lim v v

so, when Ψ(u, b) = ∫ ν b ∧ b,u12

= ∫ L b ∧ νb – ∫ v vL h ∧ b = 2 ∫ L b ∧ h,v

virtual power = ∫

(νL – L ν)b ∧ b2

×v v

= 2 ∫ di b ∧ h = 2 ∫ i b ∧ dh ≡ 2 ∫ B × v · Jv v

= 2 ∫ J × B · v

Algebraic manipulations, based on

easy-to-prove integration by parts

formulas, yield expected result.

Algebraic manipulations, based on

easy-to-prove integration by parts

formulas, yield expected result.

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So one falls back on the familiar formula, by using dictionary of

correspondences. One should do that only after completion of the

differential-form calculus. Insistence on using vector notation in

dealing with Lie derivative may result in hopelessly awkward

computations. A sample (pU denotes p-form with proxy vector U):

So one falls back on the familiar formula, by using dictionary of

correspondences. One should do that only after completion of the

differential-form calculus. Insistence on using vector notation in

dealing with Lie derivative may result in hopelessly awkward

computations. A sample (pU denotes p-form with proxy vector U):

i f h bj fi ld

Th f d d h b f ld

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L B = [v div B + rot(B × v)]v2 2

L E = [– v × rot E + grad(v · E)]v1 1

Exists for other objects, e.g., vector fields.

But proxy of L ω is not L of ω's proxy:v v

L = i d + d iv v v

because d B = (div B), d E = (rot E),

i B = (B × v), i E = (v · E), i q = (q v)

2 2

2 1 0

13

1 3v v v

The notion of Lie derivative extends to other objects, e.g., vector fields

7/21/2019 Tokyo 04 Dias


Having thus verified the validity of the method, and gained familiarity

with manipulation of Lv, we turn to the permanent magnet problem,

to realize that considering the 1-form Hc or the 2-form Br as

invariable are two different assumptions, leading therefore todifferent force formulas.

Next slide: How approximations of “permanent” magnet, according

to either of these assumptions, can be realized. Imagine deformableputty peppered with either (1) small, rigid, hard magnets, or (2) small

coils, bearing constant intensity. Equivalent homogeneous material

behaves as described:

Having thus verified the validity of the method, and gained familiarity

with manipulation of Lv, we turn to the permanent magnet problem,

to realize that considering the 1-form Hc or the 2-form Br as

invariable are two different assumptions, leading therefore to

different force formulas.

Next slide: How approximations of “permanent” magnet, according

to either of these assumptions, can be realized. Imagine deformableputty peppered with either (1) small, rigid, hard magnets, or (2) small

coils, bearing constant intensity. Equivalent homogeneous material

behaves as described:

7/21/2019 Tokyo 04 Dias


Hard ("permanent") magnets

2-form b invariabler

b

hb

– hc

r

Two different ways to make one

small magnets small coils

1-form h invariablec(its flux through any materialsurface stays the same, whatever u)

(its m.m.f. across any material closed curvestays the same, whatever u)

Small magnets Small coils

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b

hb

– hc

rb = b + µ hr b = µ (h + h )c

(µν = 1)

Ψ(u, b) = / ∫ ν b ∧ b – ∫ ν b ∧ b

u1

2ru

u u

Ψ(u, b) = / ∫ ν b ∧ b

– ∫ b ∧ hu

12

cu

Φ(u, h) = / ∫ µ h ∧ h

+ ∫ b ∧ h + / ∫ ν b ∧ bu

12

r u1 2r r

Φ(u, h) = / ∫ µ (h + h) ∧ (h + h)u

1

2

c c

Derivatives ∂ Φ ≡ –∂ Ψ differ on left and rightu u

Non-equivalent ways to write down the

B–H law, because Lagrangian Hodgedepends on u.

Non-equivalent ways to write down the

B–H law, because Lagrangian Hodgedepends on u. Corresponding energy and coenergy,easy to write down, differ accordingly

Corresponding energy and coenergy,easy to write down, differ accordingly

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b

hb

– hc

rb = b + µ hr b = µ (h + h )c

(µν = 1)

u u

The bottom line:

F = (J + J ) × BcF = J × B – (H + H ) div B c r

– (J + J ) × Bc r

… which results in different force fields.… which results in different force fields.(Computation left as an exercise…)

(Computation left as an exercise…)

Jc defined as rot HcJc defined as rot Hc

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Now, which of the two is correct? Wrong question: What precedes only

shows that both “hc constant” and “br constant” (as 1- and 2-form,

respectively) are acceptable models for a permanent magnet. But the

latter’s real behavior (called “local law” in next slide) is a matter for

measurement, not speculation. In such measurements, one compares a

template constitutive law (for instance, some weighted mixture of the abovetwo) with the results of macroscopic readings (whose observed relations

form the “global law” of next slide), and one adjusts parameters till a good

match is achieved..

Now, which of the two is correct? Wrong question: What precedes only

shows that both “hc constant” and “br constant” (as 1- and 2-form,

respectively) are acceptable models for a permanent magnet. But the

latter’s real behavior (called “local law” in next slide) is a matter for

measurement, not speculation. In such measurements, one compares a

template constitutive law (for instance, some weighted mixture of the above

two) with the results of macroscopic readings (whose observed relations

form the “global law” of next slide), and one adjusts parameters till a good

match is achieved..

Constitutive laws ought to be

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gmeasured, not theorized about

I

F

L

Φ

macroscopic levelmicroscopic level

e.g., condensedmatter physics, etc.

B

H

1

2σ3

global lawslocal lawsquantum, etc., laws

B, H, ε, σ Φ, I, L, F ψ, etc.

mesoscopic level

measurementnum. simulationtheory

some lower levelof description,

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Such a process may require extensive numerical computation, and

so requires a good theory of forces (note that samples do deform, as

a rule, when measurements are taken, which makes what precedes

relevant) and a computational technique which go along well with this

theory. For identification of constitutive laws, static computations

(magnetostatics + elasticity) will suffice, as a rule.

But for applications exemplified by TEAM pbs. 12 & 16, one needs to

take eddy currents into account. So let us review the “generalized

finite differences” technique with that in mind:

Such a process may require extensive numerical computation, and

so requires a good theory of forces (note that samples do deform, as

a rule, when measurements are taken, which makes what precedes

relevant) and a computational technique which go along well with this

theory. For identification of constitutive laws, static computations

(magnetostatics + elasticity) will suffice, as a rule.

But for applications exemplified by TEAM pbs. 12 & 16, one needs to

take eddy currents into account. So let us review the “generalized

finite differences” technique with that in mind:

→ → →G R D

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N →

E →

F →

Vgrad rot div

here, R = – 1fe

e, a

at edges

h at dual edges(i.e., faces)

d, j at dual faces

h = h : f ∈ F f b = b : f ∈ F f

b at faces

e = e : e ∈ E e = j : e ∈ E e

fluxes

emf's

mmf's

currents As we saw, this discretization principle provides a toolkit, from which a discrete eddy-current model can be assembled. Neglecting

displacement currents, and taking as main unknown the time-integrals of edge electromotive forces (so-called “A* formulation”), one gets:

As we saw, this discretization principle provides a toolkit, from which a discrete eddy-current model can be assembled. Neglecting

displacement currents, and taking as main unknown the time-integrals of edge electromotive forces (so-called “A* formulation”), one gets:

ss

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σσσσ ∂ a + R ννννRa = j

s

t

t

νννν , σσσσ : square symmetric matriceswhich depend on placement u

= j – σσσσ ∂ a

h = νννν b

t

s

RRt

Two main ways to get them:Galerkin, with edge or face elements

∂ (µµµµ(h + h )) + R σσσσ Rh = jst

s –1t

FIT, or cell-method, with mutually orthogonal meshes

Bold smallcase for arrays of degrees of freedomb, h, etc., now attached to the comoving mesh

Bold smallcase for arrays of degrees of freedom

b, h, etc., now attached to the comoving mesh

( ( )) ( ( ))) W ( ) W ( ) df'1 1ff'

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ν = ν(u (x)) ; ε(u (x))) W (x) · W (x) dxu uf–1 –1

uff

intrinsic reluctivityof the material

material point

h = ν (ε)bu

Integration here is performed on

physical space. Since elements are

comoving, face elements depend

on u (the Jacobian of which is

therefore involved in the element-assembly process).

Integration here is performed on

physical space. Since elements are

comoving, face elements depend

on u (the Jacobian of which is

therefore involved in the element-assembly process).

Here, ν depends on the nature of

the material point that occupies

position x, but not on its state of strain.

Here, ν depends on the nature of

the material point that occupies

position x, but not on its state of strain.

Yet, the Lagrangian Hodge operator, and hence its discrete version νννν

u, dodepend on strain, via u. This is usually called the “shape effect”.

Yet, the Lagrangian Hodge operator, and hence its discrete version ννννu

, do

depend on strain, via u. This is usually called the “shape effect”.

First way to construct ννννu: Galerkin

method with face elements.

First way to construct ννννu: Galerkin

method with face elements.

[ ]

DoF arrays indexed by facets of comoving mesh

” ”

DoF arrays indexed by facets of comoving mesh

” ”

( ( )) ( ( ))) W ( ) W ( ) df'–1 1ff'

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ν = ν(u (x)) ; ε(u (x))) W (x) · W (x) dxu uf–1 –1

uff

intrinsic reluctivityof the material

material point

h = ν (ε)bu

Now, should ν also depend on strain or stress,

because the nature of the material changes with

its state of stress (think e.g. of terfenol), that

could be handled

Now, should ν also depend on strain or stress,

because the nature of the material changes with

its state of stress (think e.g. of terfenol), that

could be handled

The Hodge map then depends on u for two distinct reasons: shape effect,and (via the dependence onε) whatshould properly be called “magnetostrictive” effects. (Beware: meaning of “magnetostriction” is author-dependent.)

The Hodge map then depends on u for two distinct reasons: shape effect,and (via the dependence on ε) what

should properly be called “magnetostrictive” effects. (Beware: meaning of “magnetostriction” is author-dependent.)

Diagonal discrete HodgeSecond way

t t t νν

Second way

to construct νννν

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Diagonal discrete Hodge for FIT and cell-method

(vectorialarea)

f ~

f~

Here, f // f ~

ν = νf length(f )

area(f )

~depend on u

to construct ννννuto construct ννννu

As material deforms,

keep dual cells

orthogonal to primal

ones. (Not a problem if

deformations are small.)

As material deforms,

keep dual cells

orthogonal to primal

ones. (Not a problem if

deformations are small.)

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What it takes at the numerical level:

u : nodal positions un

Precompute the derivatives ∂ νννν

/ ∂u

ff'

n

at the assembly stage (cf. Coulomb 84)

Galerkin: cf. Z. Ren, 1991

J.L. Coulomb, IEEE Trans., MAG-

19, 6 (1983), pp. 2514-19.

J.L. Coulomb, IEEE Trans., MAG-19, 6 (1983), pp. 2514-19.

Z. Ren, A. Bossavit: "A New Approach to Eddy-Current Problems in Deformable Conductors, and some numericalevidence about its validity", Int. J. Applied Electromagnetics in Materials, 3, 1 (1992), pp. 39-46.

Z. Ren, A. Bossavit: "A New Approach to Eddy-Current Problems in Deformable Conductors, and some numerical

evidence about its validity", Int. J. Applied Electromagnetics in Materials, 3, 1 (1992), pp. 39-46.

Ren's calculations of 1991

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σσσσ ∂ a + R νννν R a = j

f 99

Hollow sphere in transient vertical field (TEAM #6)

iron

x

y

z

zy

x

z

air

iron

20 40 60 90

angle

100

200Normal force along meridian

h-method

a-methodanalytical

∂ (µµµµ (h + h ) + R σσσσ R h = 0uu

s tt

–1

uu

t

t

s

∂ Φ ≡ –∂ Ψu uJ × BRelative error4.97 %

7.17 %

2.12 %

2.10 %

Inferior erformance of J x B due to non-conformal a rox. of B see aboveInferior performance of J x B due to non-conformal approx. of B (see above)

Current challenge: how about FIT

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Current challenge: how about FIT and the cell method?

Only ∂ν / ∂u to care about

f

n

Circumcenters inside tetrahedra. Take Voronoi

cells. Deformed meshes still orthogonal

Not all meshes allow formation of dual with

cells orthogonal to primal ones, as this one does

Not all meshes allow formation of dual with

cells orthogonal to primal ones, as this one doesIn 3D, a convenient simplicial paving exists with this property:

In 3D, a convenient simplicial paving exists with this property:

… the “Sommerville mesh” (Voronoi construction on b.c.c. lattice)… the “Sommerville mesh” (Voronoi construction on b.c.c. lattice)

An exercise in plane geometry

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θ

~

In "FIT", or the cell method,

ν = ν

Move A. What's ∂ ν / ∂θ ?

ff length(f )~

area(f )

A

ff

Proof:

Answer (in 2D): ∂ ν / ∂θ =ff

Coefficients vff depend on placement, but not in so complex a way. Here,

example where parameter u reduces to the sole angle θ.

Coefficients vff depend on placement, but not in so complex a way. Here,

example where parameter u reduces to the sole angle θ.


7/21/2019 Tokyo 04 Dias



θ

~

In "FIT", or the cell method,

ν = ν

Move A. What's ∂ ν / ∂θ ?

ff length(f )~

area(f )

A

ff

Proof: Rθ length(f )

~=

R cos θ

2R sin θ

Answer (in 2D): ∂ ν / ∂θ = – .ff

2 sin θ2

ν

length(f )3D analogue not too difficult3D analogue not too difficult

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"Genuine"

magnetostriction? Bs

b, ε = arginfΨ(u, b) + W(ε(u))

0 = – f + ∂ Wu

h = ν b σ = κ εu

Prospects: A full-fledged theory of magnetoelasticity, fully

accounting for non-linearities and possible dependence of B–H

law on local stress (beyond shape effects)

Prospects: A full-fledged theory of magnetoelasticity, fully

accounting for non-linearities and possible dependence of B–H

law on local stress (beyond shape effects)

Simple basic principle: Just minimize total energy. But ...Simple basic principle: Just minimize total energy. But ...

vector valued~

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h = ν b + c ε

σ = c' b + κ ε

1-formvector-valued

1-form

2-formcovector-valued2-form~

…which differential-geometric entity to use for

ε, still a matter for research.

…which differential-geometric entity to use for

ε, still a matter for research.

tokyo 04 dias

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