IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
Not submitted to IEEE Trans. Appl. Supercond.
Recent Developments in Finite Element Methods for Electromagne8c Problems Christophe Geuzaine University of Liège, Belgium EUCAS 2015 -­‐ September 9th 2015
1
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
Not submitted to IEEE Trans. Appl. Supercond.
A (subjec8ve) selec8on
• High-­‐order and isogeometric methods — for fast convergence
• Domain decomposiJon techniques — for large scale computa2ons • MulJscale methods — for microstructured, hystere2c materials • StochasJc solvers — for material/geometric uncertain2es • Homology & cohomology solvers — for field-­‐poten2al problems • … 2
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
Not submitted to IEEE Trans. Appl. Supercond.
• High-­‐order and isogeometric methods • Domain decomposiJon techniques • MulJscale methods • StochasJc solvers • Homology & cohomology solvers
3
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Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
Not submitted to IEEE Trans. Appl. Supercond.
• High-­‐order polynomial approximaJons: curved mesh generaJon; provable validity X3
x2
X5
X(x)
x5
(a) Straight sided mesh
x3
y
• Isogeometric analysis: directly based on CAD (e.g. T-­‐splines)
X(⇠)
x(⇠)
X4
⇠3
x6
x
X2
X6
⌘
x4
⇠5
⇠6
X1
Y
X
x1
⇠
⇠1
⇠4
⇠2
Figure 1: Reference unit triangle in local coordinates ⇠ = (⇠, ⌘) and the mappings x(⇠),
X(⇠) and X(x).
Geometrical Validity of High-Order Triangular Finite Elements
13
(b) Raw curvilinear mesh
positive. In all what follows we will always assume that the straight-sided
mesh is composed of well-shaped elements, so that the positivity of det x,⇠
is guaranteed. This standard setting is presented on Figure 1 (left) for the
quadratic triangle.
Let us now consider a curved element obtained after application of the
curvilinear meshing procedure, i.e., after moving some or all of the nodes of
the straight-sided element. The nodes of the deformed element are called
Xi , i = 1 . . . Np , and we have
(a) Straight sided mesh
(c) After elastic analogy
Np
X
(p)
X(⇠)
Li fourth-order
(⇠) Xi . curved triangles.
Fig. 6. Curvilinear mesh
of a=rotor using
i=1
(2)
4
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
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• High-­‐order and isogeometric methods • Domain decomposiJon techniques • MulJscale methods • StochasJc solvers • Homology & cohomology solvers
5
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• Split computaJonal domain into subdomains, and iterate • Scalability requires “coarse grid” (long-­‐distance informaJon exchange); open problem for high-­‐frequency wave propagaJon
• Time-­‐domain approaches gaining tracJon (Waveform relaxaJon, Parareal, …)
Figure 10: Electromagnetic scattering by a Falcon jet.
6
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
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• High-­‐order and isogeometric methods • Domain decomposiJon techniques • MulJscale methods • StochasJc solvers • Homology & cohomology solvers
7
6
84
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CHAPTER 5. NUMERICAL TESTS
• Modern computer architectures enable fully computaJonal homogenizaJon • Possibly heterogeneous models • Convergence of local (in the microstructure) and global quanJJes
CHAPTER 4. COMPUTATIONAL MULTISCALE METHODS
downscaling
upscaling
Figure 5.7: SMC problem, b-conform formulations, nonlinear case. Spatial cuts
8
of the z-component of the eddy currents j (top) and of the x-component of the
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
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• High-­‐order and isogeometric methods • Domain decomposiJon techniques • MulJscale methods • StochasJc solvers • Homology & cohomology solvers
9
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
ensity
452 of
(red),
at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
in the grey matter forInvited keynote presentationNot3A-LS-O1.3
Fig.given
3.IEEE
Probability
density
of
in48,the
matter
TRANSACTIONS
ON MAGNETICS,
VOL.
NO.white
2, FEBRUARY
2012for
submitted
to IEEE
Trans.
Appl. Supercond.
(green).
and
(black),
(red),
(green),
(blue).
II. RESULTS AND DISCUSSION
sive method is governed by three parameters:
is linked to the precision on the approxithe input random variables
and
of truncation of the studied global quantities
452
and the corresponding quantities for IEEE
theTRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012
d is the number of quadratureFig.points.
1. Mesh Herein,
used for the numerical experiments [20]: (a) full head (300 000 nodes, 27 tissues); (b) grey matter; (c) white matter.
, while
and vary. For the sake of
. 1. Mesh used for the numerical experiments [20]: (a) full head (300 000 nodes, 27 tissues); (b) grey matter; (c) white matter.
deal with the white and grey matter
(though the
nervous system (CNS) recommended by ICNIRP are
problem, we use an approach based on a p
ndle other tissues during the same
computation
composition of both the conductivity and th
mA/m and
mV/m.
rvous system (CNS) recommended by ICNIRP are
A/m and
mV/m.
B. Uncertainties
problem, we use an approach based on a polynomial chaos deWe assume the conductivities
to (green)
be of finite
(blue),
(red) and
in
Fig. 4. Probability density of
composition of both the conductivity and the induced fields [18].
). sumption on the shape of the probabilistic
the grey matter (
We assume the conductivities to be of finite variance, with no asIEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012
The values of the induced fields—t
he
Input Parameters
Let us focus on the fields
inducedoninthe
theshape
brain.of
The
Uncertainties
sumption
theconductivprobabilistic distribution.
in the brain
ities of the white matter
of theof
grey
are density
The and
values
thematter
induced fields—the
average current
Let
us
focus
on
the
fields
induced
in
the
brain.
The
conductivy density (PD) of
inmodeled
the grey
matter
finite element method from any
within
a probabilistic
framework, as functions of the by the
—are computed
es of the white matter
and of the grey matter
are density in the brain
. The average densi
and different
values
of(300. is
Fig. 1. Mesh used for the numerical
experiments
[20]:
(a) variable
full head
000
nodes, 27 tissues); (b) grey matter; (c) white matter. ,
random
Therefore
couple of values
odeled within a probabilistic framework, as functions of the by the finite element method from any
thatbelongs
can betospanned
are random
the maxThe average
density
a space by the polynomials
The
the PD obtained and
with
and, as well. In particular, by. using
ndomcurves
variable of. Therefore
imum
entropy
principle
[22]
(arbitrarily)
and written as a truncated
that we
canmodel
be spanned
by the polynomials
and thus series to an order
d
are
random
as
well.
In
particular,
by
using
the
maxy superposed,
which
proves the
convergence
of problem,
nervous system (CNS)
recommended
by ICNIRP
we use uniformly
an approachdistributed
based on a polynomial chaos deasare
independent
written variables,
as a truncated
series to an order
um entropy principle [22] we model (arbitrarily)
and random
• Uncertain geometry and/or material laws • Efficient calculaJon of probability distribuJon of quanJJes of interest
composition of both the conductivity and the induced fields [18].
mA/m and values of . mV/m.
increasing
as independent random variables, uniformly distributed
We assume the conductivities to be of(1)
finite variance, with no asspersion
parameters as the mean and the stan- sumption on the shape of the probabilistic
B. Uncertainties
(5)
distribution.
(2)
(1)
The values of the induced fields—the
average current
heFig.convergence
reached
as
soon
as
To compute
the value of the unknown real
1.
for the
experiments
[20]: (a)
head
(300 000
nodes,
27 tissues); (b) grey matter; (c) white matter.
LetMesh
us used
focus
onnumerical
theisfields
induced
infull
the
brain.
The
conductivdensity
in
the
brain
—are
computed
(2)
wecoefficients
use the orthogonality
of the whitedeviation:
matter
and ofwhen
the grey matter conare To compute the value of the unknown real
/m,itiesstandard
0.07
, properties 10of the H
by the finite element method from any couple of values
modeled within a probabilistic framework, as functions of the
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
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• High-­‐order and isogeometric methods
• Domain decomposiJon techniques
• MulJscale methods
• StochasJc solvers
• Homology & cohomology solvers
11
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
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Homology & cohomology solvers
Consider the model problem of an inductor around a hollow piece of conductor: ⌦c
⌦C
c
• ConducJng domain ⌦c
• Non-­‐conducJng domain ⌦
C
c
• Overall domain ⌦ = ⌦c [ ⌦C
c
• Complementary parts of boundary of ⌦
: S 1 and S2
12
ve equations
ive equations
b = µh,
b = µh,
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
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e = ⇢j.
(4)
Homology & cohomology solvers
e = ⇢j.
• In ⌦
w
e want curl
h =tjo solve curlhe =
= j @t b
curl
div b
= 0 @t b
curl
e=
div b = 0
ve equations
together with consJtuJve laws ive equations
b = µh
(4)
(5)
(6)
(5)
(7)
(6)
(8)
(7)
(8)
(9)
(10)
(9)
= µ⇢ h
j
be =
(10)
and appropriate boundary condiJons
treat the domain
Me =
as ⇢aj circuit
element,
we apply magnetic
boundary
@M . This
is achieved
by the boundary
treat the domain
M as
a circuit element,
we applycondition
magnetic
=0
• @M
In ⌦
. C
, j =
is
0 achieved
and thus boundary
bycurl
the h
boundary
condition
c This
b · n = 0 on @M.
b·n=
on have
@M. no currents:
conducting domain
Ma0 we
(11)
(11)
13
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
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m.
ctwo
ohomology solvers
te that the boundary @M is Homology now decomposed&
on
parts according
boundary condition applied on it: @M = S1 \ S2 . On S1 = Mc \ @M
ve the condition e ⇥ n = 0 and on S2 = Ma \ @M we have the condition
0. We can either drive the problem by the net current Is through the
nals, or byWe thecnet
di↵erence
Vs between
an dvoltage
rive the problem either bthe
y terminals:
the net current through the Z
terminals e · dl = Vs ,
y @M isZ now decomposed
on two parts
Z
Z ⇣ according
applied onj ·it:
@M
S2· .n On
1 \h
n da
==S
curl
da =S1 = hM· cdl\=@M
Is ,
@⌃ condition
n = 0 and⌃on S2 = Ma⌃ \ @M we have the
rive the problem by the net current Is through the
or by the et voltage ifference oltage di↵erence
Vsnbetween
thedterminals:
(15)
(16)
(17)
ere ⇣ is abetween curve on tShe the terminals on S1 and ⌃ ⇢ M is a
2 between
terminals
e that isolates the terminals. The boundary @⌃ also lies on the part
Z
the boundary of the domain. In addition, the fields e and j are linked
· dlAll
= these
Vs , considerations
(15) hint a duality
constitutive equation e =e⇢j.
Z ⇣the current density j and between the voltage
enZthe electric field e and
he
= current
curl conditions.
h · n da =
h · dl = Is ,
(16)
⌃
equivalence
class [⇣] of @⌃
curves on S2 between the terminals on S1 beo so-called relative homology space H1 (S2 , @S2 ), while
(17)the class [@⌃] of
belong to a homology space H1 (S2 ). The integral conditions on fields
14
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Homology & cohomology solvers
• The curves [⇣]
belong to the relaJve homology space H1 (S2 , @S2 )
• The curves [@⌃]
belong to the homology space H1 (S2 )
• The integral condiJons on fields e and h
fixing V
s and I s are called cohomology condiJons (they consider integrals over an element of a homology space)
15
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where
Z
(19)
(
Vi when h0 = Ei
h 2 {gradHomology ni , ei , E1 , E2 } and
e ⇥ h · n dasolvers
=
& cohomology 0
otherwise.
S2
(20)
0
0
• Finite element H-­‐formulaJon: look for the magneJc field h
such that NEW:
Z
Z
Z
0
· h0 dV coe
+ cient
⇢ curlvector
h · curl
+
e ⇥by
h0 the
· n dacohomology
= 0 (21)
@t theµ hinteger
where
zihisdVproduced
⌦
⌦c
S2
solver of Gmsh.
0 one
plugging
in the
expression
hof
construct a linear system
Now,Bythe
magnetic
field for
h ish
thecan
form
holds for aunknown
ppropriate test funcJons
from whichX
the unknown coe cients X
i , ti and Vi or Ii can be solved. Note
• At the hd=
iscrete level, if N
n ia+
nd E
denote the stets of In1 Eodes nd i grad
i ei +
1 + Ia
2E
2 , edges on 4
i2N a
(M
i2E(M
))
a ) n
c \(@Ma \@M
in the mesh, nd i and e the associated ncodal and edge shape i
funcJons: NEW:
h=
X
i2N (⌦C
c )
i
grad ni +
X
h i ei + I 1 E 1 + I 2 E 2
i2E(⌦c \(@⌦C
c \@⌦c ))
C
where E
cohomology
basiscohomology functions E1basis and fE
where the
i are the (discrete) uncJons of H1 (⌦by
2 are produced
c )the
computation of the cohomology space H 1 (Ma ) in Gmsh. That is, the function
space from which we look for the approximate solution of h is spanned by
16
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Homology & cohomology solvers
In pracJce:
17
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Homology & cohomology solvers
E1 =
X
z1j ej
j
18
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Homology & cohomology solvers
E2 =
X
z2j ej
j
19
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Example: superconduc8ng wire
• Parametric 3D geometry (in “helix.geo” file): • Number and layers of superconducJng filaments • Twist pitch, radius of surrounding air box, conducJng matrix and filaments • Nonlinear H-­‐formulaJon, with imposed Jme-­‐varying (total) current (in “helix.pro” file): • No ficJJous conducJvity in the air, current imposed through cohomology basis funcJon Ec kjk n 1
(
)
• Nonlinear resisJvity: ⇢ =
Jc Jc
• Implicit Euler Jme-­‐stepping
20
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Example: superconduc8ng wire
• You can give it a try now: • Download the code from hfp://onelab.info • Uncompress the archive and launch Gmsh ( icon) • Open the file “models/superconductors/helix.pro” with the “File>Open” menu • Change some parameters and click on “Run” • The code combines the mesh generator Gmsh (hfp://gmsh.info) and the finite element solver GetDP (hfp://getdp.info). Both are free soiware, released under GNU GPL. 21
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
Invited keynote presentation 3A-LS-O1.3 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.
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Example: superconduc8ng wire
• You can give it a try now: • Download the code from hfp://onelab.info • Uncompress the archive and launch Gmsh ( icon) • Open the file “models/superconductors/helix.pro” with the “File>Open” menu • Change some parameters and click on “Run” • The code combines the mesh generator Gmsh (hfp://gmsh.info) and the finite element solver GetDP (hfp://getdp.info). Both are free soiware, released under GNU GPL. Demo!
22
IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.
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Conclusion & thanks
• Many exciJng developments in computaJonal electromagneJcs • All examples from this talk solved with the open source codes Gmsh and GetDP — more examples on hfp://onelab.info • SuperconducJng wire example developed in collaboraJon with • Ank Stenvall (Tampere University) • Abelin Kameni (Supelec & Université Paris XI) Thank
you
for
your
attention
Available on hfp://onelab.info/wiki/SuperconducJng_wire • For more informaJon (references, preprints, codes, etc.), see
hfp://www.montefiore.ulg.ac.be/~geuzaine
Papers/preprints:
[email protected]
[email protected]
23