A Seconda y Field Based hp-Fini e Elemen Me hod o
he Simula ion o Magne o ellu ic Measu emen s
J. Al a ez-A ambe ia,b,c,∗, D. Pa dob,a,d, H. Ba ucqc,e
aBCAM (Basque Cen e o Applied Ma hema ics), Maza edo 14, 48009, Bilbao, Spain
bUni e si y o he Basque Coun y (UPV/EHU), Bilbao, Spain
cUni e si y o Pau (UPPA), Pau, F ance
dIke basque, Bilbao, Spain
eIn ia eam-p ojec Magique-3D, Pau, F ance
Abs ac
In some geophysical p oblems, i is some imes possible o di ide he sub-
su ace esis i i y dis ibu ion as a one dimensional (1D) con ibu ion plus
some wo dimensional (2D) inhomogenei ies. Assuming his scena io, we spli
he elec omagne ic ields in o hei p ima y and seconda y componen s, he
o me co esponding o he 1D con ibu ion, and he la e o he 2D inho-
mogenei ies. While he p ima y ield is sol ed ia an analy ical solu ion, o
he seconda y ield we employ a mul i-goal o ien ed sel -adap i e hp-Fini e
Elemen Me hod (FEM). To unca e he compu a ional domain, we design a
Pe ec ly Ma ched Laye (PML) ha au oma ically adap s o high-con as
ma e ials ha appea in he subsu ace and in he ai -g ound in e ace. Nu-
me ical esul s illus a e he obus ness o he p oposed PML and he gains
o he seconda y ield app oach, whe e we ob ain esul s wi h compa able
accu acy han wi h a ull ield based o mula ion bu wi h a much lowe
compu a ional cos .
Keywo ds: Fini e Elemen Me hod (FEM), hp-adap i i y, Magne o ellu ic
P oblem, Seconda y Field Fo mula ion, Pe ec ly Ma ched Laye s (PML).
∗Co esponding au ho
Email add ess: [email p o ec ed] (J. Al a ez-A ambe i)
Ma ch 13, 2015
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Jou nal o Compu a ional Science 11 :
137-144 (2015), which has been published in inal o m a h ps://doi.o g/10.1016/j.jocs.2015.02.005. © 2015 Else ie
unde CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
1. In oduc ion
The magne o ellu ic (MT) me hod is a passi e explo a ion echnique
based on elec omagne ic (EM) wa es [1, 2, 3]. I aims a es ima ing he
esis i i y dis ibu ion, and he e o e a p o iding an image o he Ea h’s
subsu ace. MT measu emen s a e go e ned by Maxwell’s equa ions wi h a
su ace sou ce loca ed a he ionosphe e. In pa icula , when he ma e i-
als and he sou ce depend only upon wo spa ial a iables, wo independen
and uncoupled modes a e de i ed, he so-called T ans e se Elec ic (TE)
and T ans e se Magne ic (TM) pola iza ions. The solu ion o he equa ions
a isen om hese wo modes can be nume ically sol ed wi h a hp-Fini e Ele-
men Me hod (FEM) [4, 5, 6, 7]. Wi h hose solu ions, i is hen possible o
compu e he impedance and/o he appa en esis i i y, wo sui able physical
quan i ies o pe o m he in e sion.
To co ec ly cap u e he complexi y o he Ea h’s subsu ace, we employ
adap i e g ids, which allow o app oxima e special ea u es o he solu ion
by e ining only in speci ic a eas. To build he e ined mesh, we employ a
goal-o ien ed adap i e s a egy [8], which minimizes he e o o a p esc ibed
quan i y o in e es ep esen ed by a linea unc ional (see [9, 8, 10, 11, 12]
o de ails). The abili y o he goal-o ien ed algo i hm o p o ide accu a e
solu ions in a egion o in e es in he con ex o hp-FEM has been desc ibed
in a ious wo ks [13, 14, 15]. The hp-FEM p o ides exponen ial con e gence
a es o ellip ic p oblems wi h a piecewise analy ic solu ion, whe eas ho
p e sions con e ge only algeb aically. This was p o ed in 1D by Gui and
Babuska [16] and in 2D by Babuska and Gui [17] and Schwab [18].Only he
hp-FEM is able o combine small elemen s (needed o cap u e geome ical
de ails such as hin edges) wi h high o de s o app oxima ion (necessa y o
dec ease he dispe sion e o o wa e p opaga ion p oblems [19, 20, 21]).
Besides, i is obus o singula ly pe u bed p oblems, ha is, i s ill pe -
o ms app op ia ely when a pa ame e in ol ed in a gi en ellip ic p oblem
app oaches a c i ical alue [18].
In some geophysical applica ions, as in MTs, he da a is acqui ed a se -
e al ecei e s loca ed a he Ea h’s su ace. I becomes hen necessa y o
ob ain accu a e esul s a mul iple posi ions, being his he eason o ex end
he goal-o ien ed s a egy o a mul igoal-o ien ed one. The e exis wo possi-
ble app oaches owa ds mul igoal-o ien ed adap i i y. The i s one consis s
o using one g id o each goal, as in [22], whe e he implemen a ion needs
o handle mul iple g ids, which in gene al may be complica ed. The second
2
one consis s o de ining a new quan i y o in e es ha akes in o accoun
all goals (see [23, 24]). Based on his second app oach, we implemen he
algo i hm p oposed by Pa do in [25].
In geophysics in gene al and in MT in pa icula , when he subsu ace
dis ibu ion o he esis i i y depends upon mul iple spa ial a iables, i is
some imes possible o in e p e i as a 1D o ma ion plus some 2D (o 3D)
he e ogenei ies. In his wo k, we conside a ho izon ally laye ed Ea h model
wi h 2D he e ogenei ies. Then, in o de o sol e he TE and TM modes,
we spli he elec ic and magne ic ields in o hei p ima y and seconda y
componen s. The i s co esponds o he ields a isen om some e e ence
conduc i i y model (1D), while he second a ises om he di e ence be ween
he ac ual conduc i i y dis ibu ion wi h espec o he e e ence model (2D).
Since he 1D solu ion is known analy ically, he main ad an age o his
app oach is ha we only need o accu a ely sol e he seconda y ield a i-
a ions ( he e m “seconda y ield” is also known as “sca e ed ield” in he
elec ical enginee ing communi y), which in gene al a e easie o sol e, since
hey exhibi less a ia ions (smalle g adien s) han he p ima y ield. Hence,
i is gene ally possible o employ coa se g ids, and hence educe he com-
pu a ional cos .
Addi ionally, in MTs he compu a ional domains a e usually e y la ge i
one models he inciden plane wa e sou ce. In he seconda y ield o mula-
ion, he sou ce e m is no a he ionosphe e, bu whe e he inhomogenei ies
a e conside ed. Since i is no necessa y anymo e o model he ionosphe e
sou ce, his allows us o conside ably educe he compu a ional domain. Fi-
nally, since we sepa a e he p ima y om he seconda y ield, we may ob ain
addi ional physical ele an in o ma ion by analyzing each ield (p ima y and
seconda y) sepa a ely.
The main con ibu ion o his wo k is hen o sol e, ia he hp-FEM,
he MT di ec p oblem using he seconda y ield o mula ion o simula e
MT measu emen s. The men ioned bene i s o his app oach will hen be
no o ious in he in e sion. On he one hand, he e exis s he possibili y o
analyzing 1D and 2D e ec s sepa a ely. On he o he hand, since he solu ion
o he in e se p oblem is based on ei e a ed solu ions o he di ec p oblem,
educing he compu a ional cos o sol ing he di ec p oblem p oduces la ge
sa ings in he compu a ional cos s.
Addi ionally, we p o ide an au oma ic echnique o unca e he com-
pu a ional domain, a u he p oblem ha appea s when applying a FEM
o unbounded egion p oblems such as MT. Di e en app oaches can be em-
3
ployed o his pu pose. We employ a Pe ec ly Ma ched Laye (PML), which
is an exac me hod a he con inuous le el, and hus, i ma ches he high-
accu acy deli e ed by he hp-adap i e FEM. The wo k o Gomez-Re uel o
e al. [26] shows he sui abili y o he u iliza ion o PMLs in his con ex .
PMLs we e p oposed by Be enge [27] (1994) in an elec omagne ic con-
ex as an a i icial laye in ended o educe e lec ions om he bounda y
o a unca ed compu a ional domain. In his me hod, one has o selec he
decay p o ile o he wa e in o he PML egion. This p o ile needs o ensu e
ha e lec ions om he bounda y a e a bi a ily small, which implies ha
he solu ion decays a bi a ily as , c ea ing hen a “bounda y laye ” wi h
s ong g adien s wi hin he PML egion. Thus, while a low decay p oduces
incoming wa es e lec ed om he bounda y, an excessi e decay equi es a
e y ine g id o app oxima e he bounda y laye . To ind an equilib ium
be ween a as and a slow decay, i is necessa y o p ope ly adjus he PML
pa ame e s, which is usually icky since hey depend on he p oblem i -
sel . Mo eo e , when we ha e a laye ed ma e ial wi h high con as s on he
ma e ial p ope ies, his selec ion o he pa ame e s is e en mo e challenging.
Thus, in his wo k we also p o ide a me hod o au oma ically adjus
he PML pa ame e s, e en in he mos complex scena ios whe e he ma e-
ial con as p ope ies among neighbo ing ma e ials a e as high as six een
o de s o magni ude. These ype o scena ios o en appea in geophysical
elec omagne ic (EM) applica ions ha in ol e bo h, ai and g ound. We
show ha he p oposed PML p oduces an app op ia e decay o he solu ion
in he ai and in he subsu ace wi hou in oducing spu ious e lec ions, and
hus, p o iding accu a e solu ions.
The p esen wo k is o ganized as ollows. In Sec ion 2 we de ine he
o mula ion o he p oblem. Sec ion 3 desc ibes he o mula ion o he PML
and how he pa ame e s a e adjus in he Au oma ically Adap ed PML. We
de i e he seconda y ield o mula ion in Sec ion 4 and nume ical esul s
based on he MT p oblem a e illus a ed in Sec ion 5. Sec ion 6 is de o ed
o he conclusions.
2. Fo mula ion o he Me hod
MT measu emen s a e go e ned by he elec omagne ic phenomena, which
is desc ibed by Maxwell’s equa ions. Assuming a ime-ha monic dependence
4
o he o m ejω , hese equa ions can be exp essed in equency domain as:
∇×E=−jωµH −Mimp (Fa aday),
∇×H= (σ+jωε)E+Jimp (Amp`e e), (1)
whe e Eand Ha e he elec ic and magne ic ields, espec i ely. These
ields a e d i en by an imp essed p esc ibed elec ic and magne ic densi y
cu en sou ces, Jimp = (0, Jy,0) and Mimp = (0, My,0), espec i ely. We
emphasize ha , as explained in [28], magne ic imp essed cu en s a e only
ma hema ical symbols u ilized o ep esen sou ces. jis he imagina y uni ,
ωis he angula equency, and σs ands o he conduc i i y o he media.
We assume ha
σ=
σ0 0
0σ0
0 0 σ
,ε=
ε0 0
0ε0
0 0 ε
,µ=
µ0 0
0µ0
0 0 µ
,(2)
whe e he elec ical pe mi i i y εand he magne ic pe meabili y µa e as-
sumed o be ha o he acuum (ε0and µ0 espec i ely) and σ(x, y, z) o be
piecewise cons an , non-nega i e, and bounded abo e.
2.1. T ans e se Elec ic (TE) and T ans e se Magne ic (TM) Modes
P e-mul iplying bo h sides o Fa aday’s Law by µ−1, applying he cu l,
and using Amp`e e’s Law, we ob ain he educed wa e equa ion,
∇×(µ−1∇×E)−k2E=−jωJimp −∇×µ−1Mimp,(3)
whe e k2=ω2ε−jωσ. A simila equa ion is ob ained in e ms o he
magne ic ield by mul iplying bo h sides o Amp`e e’s Law by ˆ
σ−1= (σ+
jωε)−1and applying he cu l o Fa aday’s Law
∇×(ˆ
σ−1∇×H) + jωµH =−Mimp +∇׈
σ−1Jimp.(4)
When he ma e ials and he sou ce depend only upon wo spa ial a iables
(x, z), hen ∂/∂y = 0 and wo independen and uncoupled modes a e de i ed
om Maxwell’s equa ions. The uncoupled TE mode in ol es (Ey, Hx, Hz)
ield componen s, while TM only conside s (Hy, Ex, Ez). Ou aim is o ind
he ycomponen o he elec ic and magne ic ields Ey(x, z), Hy(x, z)∈
H1(Ω) ha sa is y he BCs and equa ions (3), and (4), espec i ely.
5
To de i e he a ia ional o mula ion, we i s de ine he L2-inne p oduc
o wo possible complex and ec o alued unc ions g1and g2as:
hg1,g2iL2(Ω) =ZΩ
g1
∗g2dΩ,(5)
whe e g∗deno es he adjoin ( anspose o he complex conjuga e) o g.
2.1.1. TE Va ia ional Fo mula ion
To ob ain he co esponding a ia ional o mula ion, we p e-mul iply (3)
by he complex conjuga e o a es unc ion F∈V(Ω), whe e V(Ω) =
H1
ΓD(Ω) = {F∈L2(Ω) : F|ΓD= 0,∇F∈L2(Ω)}is he space o admis-
sible es unc ions. Then, we in eg a e by pa s and we inco po a e he
homogeneous Di ichle BC ( he ones conside ed in he p esen wo k) o e
ΓD=∂Ω. Thus, we ob ain:
Find Ey∈V(Ω),such ha :
h∇F, µ−1∇EyiL2(Ω) − hF, k2EyiL2(Ω) =−jωhF, Jimp
yiL2(Ω) ∀F∈V(Ω),
(6)
2.1.2. TM Va ia ional Fo mula ion
In a simila way, om (4) we ob ain he co esponding a ia ional o mu-
la ion o he magne ic ield
Find Hy∈V(Ω),such ha :
h∇F, ˆσ−1∇HyiL2(Ω) +jωhF, µHyiL2(Ω) =−hF, Mimp
yiL2(Ω) ∀F∈V(Ω),
(7)
We employ an hp-Fini e Elemen Me hod [4] o sol e bo h p oblems (6)
and (7). The objec i e o he adi ional goal-o ien ed me hod is o cons uc
an op imal hp-g id in he sense ha i minimizes he p oblem size needed o
achie e a gi en ole ance e o o a gi en quan i y o in e es (solu ion a
he ecei e ) Li(u), being uei he Eyo Hy. This quan i y is a linea and
con inuous unc ional [14, 15] in uassocia ed o he i- h ecei e and de ined
as:
Li(u) = 1
|ΩRi|ZΩRi
u dΩ,(8)
whe e ΩRiis he domain occupied by he i- h ecei e .
6
Since we ha e mo e han one ecei e , we need o p ope ly compu e se -
e al quan i ies o in e es . The e o e, we employ a mul igoal-o ien ed s a -
egy, p oposed in [25], whe e a new linea quan i y o in e es ha akes in o
accoun all ecei e s is employed.
F om he solu ion o he a ia ional p oblems, we compu e he impedance
and/o he appa en esis i i y, which a e wo pos -p ocessed ans e unc-
ions ha a e ypically used du ing in e sion in MT p oblems. The impedance
Zis de ined as
Zi
T E =Zi
yx =Li(Ey)
Li(Hx),Zi
T M =Zi
xy =Li(Ex)
Li(Hy),(9)
whe e Hxand Eya e ob ained om Maxwell’s equa ions as
Hx=1
jωµ
∂Ey
∂z , Ex=−1
σ+jωε
∂Hy
∂z .(10)
The appa en esis i i y ρapp is de ined as
ρapp
mn =|Zmn|2
ωµ .(11)
Fo he sake o simplici y in he no a ion, we omi he ysubsc ip om Ey,
Hy,Jimp
y, and Mimp
y om now on.
3. T unca ion o he Domain
When applying a FEM o unbounded egion p oblems such as MT, he
compu a ional domain mus be unca ed. We employ PMLs o his pu -
pose, and we ollow he in e p e a ion in oduced by Teixe a and Chew
in [29, 30], whe e hey conside a PML as an analy ic con inua ion o he
go e ning equa ions in o he complex plane (see also [31]).
PMLs ans o m p opaga ing and e anescen wa es in o exponen ially
as decaying e anescen wa es. Since wa es a e s ongly a enua ed inside
he PML egion, he bounded compu a ional domain can be limi ed by a
su ace on which one can se pe ec ly e lec ing BCs (in ou case, we se
homogeneous Di ichle BCs). Indeed, any e lec ed wa e is so much abso bed
inside he laye ha i does no pollu e he solu ion inside he domain o
s udy. Then, he selec ed BCs o bo h p oblems imply ha he angen ial
componen o he ields a e ze o on he ou e pa o he bounda y. Fo a
ecen e iew o he s a e o he a o his unca ion echnique, see [32]
and [33].
7
3.1. PML De ini ion
Le he Ca esian coo dina e sys em x= (x, z) be he e e ence sys em
o coo dina es in a 2D scena io, whe e o simplici y, we selec he e ical
coo dina e zas he di ec ion pe pendicula o he g ound-ai in e ace. Gi en
an a bi a y complex sys em o coo dina es ζ= (ζ1, ζ2), we de ine ou change
o coo dina es by x=ψ(ζ)and we deno e he Jacobian ma ix and i s
de e minan by Jand de (J). The change o coo dina es is assumed o
be ep esen ed by an injec i e di e en iable unc ion wi h con inuous pa ial
de i a i es and nonze o de e minan a any poin .
We de ine a one dimensional change o a iables in he posi i e di ec ion
o he i- h coo dina e as
ζxi(xi) = Zxi
0h(η)dη, o i= 1,2, x1=x, x2=z, (12)
whe e h(·) is a possibly complex alued unc ion o be de e mined in sec-
ion 3.3. The case co esponding o he nega i e di ec ion can be de ined
analogously. The Jacobian is gi en by [J]i,j ="∂ζi
∂xj#i,j
, o i, j = 1,2. Thus,
i is exp essed as
J=
h(x) 0
0h(z)
,whe e de (J) = h(x)h(z),(13)
deno es he de e minan o he Jacobian.
Wi h his pa icula change o coo dina es, he Jacobian is diagonal.
Howe e , we de i e he ein he a ia ional o mula ion o a gene al, non
o hogonal change o a iables. This is use ul o o he pu poses, e.g. de el-
opmen o non-o hogonal Fou ie FEMs in ce ain geome ies (see [34, 35]).
3.2. Va ia ional Fo mula ion in an A bi a y Sys em o Coo dina es
We de ine he change o coo dina es e
E:= E◦ψ=e
E(ζ), e
F:= F◦
ψ=e
F(ζ), and e
Jimp := Jimp ◦ψ=Jimp(ζ). Using Eins ein’s summa ion
con en ion, acco ding o he chain ule, deno ing wi h he uppe ba he
complex conjuga e, and aking in o accoun ha i ∈C1(Ω), hen o all i
∂
∂ζi
=∂
∂ζi
,∂
∂ζi
=∂
∂ζi
,(14)
8
we ob ain ha
∇ζe
E=∂e
E
∂xi
∂xi
∂ζn
exn= (J−1)T∇E,
∇ζe
E=∂e
E
∂xi
∂xi
∂ζn
exn= (J−1)∗∇E.
(15)
The e o e, mul iplying (3) by he complex conjuga e o a es unc ion e
F,
in eg a ing by pa s, and inco po a ing he homogeneous Di ichle BC o e
e
ΓD, we ob ain:
h∇ζe
F, e
µ−1∇ζe
EiL2(e
Ω) =h(J−1)∗∇F, µ−1(J−1)T∇E de (J)iL2(Ω) =
=h∇F, J−1µ−1(J−1)T∇E de (J)iL2(Ω) ,
he
F, e
k2e
EiL2(e
Ω) =hF, k2E de (J)iL2(Ω),
he
F, e
JimpiL2(e
Ω) =hF, Jimpde (J)iL2(Ω),
(16)
whe e e
µ:= µ◦ψ,e
k:= k◦ψ,e
Ω := Ω ◦ψ, and e
ΓD:= ΓD◦ψ.
Following he ideas o [36] conce ning he inclusion o me ic-dependen a i-
ables wi hin ma e ial coe icien s, we de ine he ollowing unc ions:
µT E
NEW =JTµJ1
de (J)=
µh(x)
h(z)0
0µh(z)
h(x)
,
k2
NEW =k2de (J) = k2h(x)h(z),
Jimp
NEW =Jimp de (J) = Jimph(x)h(z).
(17)
The new sou ce and new ma e ial enso s inco po a e he in o ma ion abou
he change o coo dina es. Thus, he a ia ional o mula ion can be exp essed
in e ms o an a bi a y sys em o coo dina es by simply conside ing he new
sou ce and ma e ials. The new a ia ional o mula ion o he elec ic ield
9
σ1σ2σ3
Model 1 1 1 1
Model 2 1 1/10 1/3
Model 3 1 1/10 1/10
Model 4 1 1/100 1/3
Table 1: Di e en models o he o ma ion o he subsu ace. Conduc i i ies a e
gi en in S/m.
equencies be ween he nume ical hp-FEM solu ion and he exac solu ion.
We ob ain ela i e e o s below 1.5%, a supe b accu acy o hese ype o
simula ions.
10−5 10−4 10−3 10−2 10−1 100
0
0.5
1
1.5
F equency (Hz.)
Rela i e e o in pe cen
Model 1
Model 2
Model 3
Model 4
Figu e 5: Rela i e e o be ween he exac and nume ical solu ions o di e en
subsu ace o ma ions agains equency.
To s udy he beha io o he solu ion in o he PML egion, we conside
Model 4 wi h equency equal o 10−4Hz and we display he loga i hm o he
module o he impedance along all sides o he compu a ional domain. Thus,
we ep esen log(|ZT E |) in Figu e 6. We app ecia e ha he PML beha es
p ope ly e e ywhe e, wi h a smoo h decay o he solu ion and wi hou in-
oducing nume ical e lec ions e en in he a eas wi h high con as be ween
ma e ial p ope ies. Panels (a) and (b) co espond o he in e sec ion be-
ween ai and g ound. The e, he con as be ween esis i i ies is abou
six een o de s o magni ude and e en in his scena io, he decay seems o be
supe b.
16
(a) (b)
(c) (d)
Figu e 6: log(|ZT E |) co esponding o Model 4, wi h a 5 km hick PML and α=
10−5. Panel (a) co esponds o he le side o he domain. (b), (c), and (d)
co espond o he igh , op and bo om pa s o he domain, espec i ely. The
black line indica es he egion whe e he PML s a s.
5.3. Seconda y Field Fo mula ion
We conside now a 2D scena io wi h he ollowing conduc i i y dis ibu-
ion: σ1= 1/3, σ2= 1/2, σ3= 1/4, σ4= 1/200 S/m. The mos sensi i e
equency o he a ge a ea, ha is, he equency a which he p esence
o he a ge a ec s mos he measu emen s a he ecei e s, co esponds o
0.05 Hz. Figu e 7 shows he inal g ids a e execu ing he mul i-goal o i-
en ed adap i i y o he ull o mula ion (le ) and o he seconda y ield
based p oblem ( igh ) a his equency. The le panel shows a zoom o he
inal g id wi h he o igin o coo dina es a he cen e . The size o he ep e-
sen ed domain is o 40 ×70 km2. The g id in he igh panel is he comple e
g id o he seconda y ield p oblem (50 ×70) km2.
Figu e 8 displays he ela i e e o s in he appa en esis i i y be ween
he ull ield and seconda y ield solu ions. The e, posi ions 1 o 4 co espond
o measu emen s ob ained a 0, 4, 8 and 20 km om he cen e o he do-
main, espec i ely. Due o he low e o s obse ed in Figu e 8, we conclude
ha bo h app oaches p o ide analogous esul s. Howe e , he numbe o
unknowns needed o achie e hese small e o s a e no he same.
We now conside he same model wi h he same equency o 0.05 Hz. We
compu e an o e kill solu ion wi h a much ine g id ob ained a e pe o ming
17
Figu e 7: Final mul i-goal o ien ed hp-g ids. Di e en colo s indica e di e en
alues o p. Le : ull o mula ion based p oblem (zoom). Righ : seconda y ield
based p oblem.
adap i i y. We use i o es ima e he ela i e e o s co esponding o he sec-
onda y ield and ull o mula ions a e se e al hand/o pglobal e inemen s.
Figu e 9 displays he esul s o hese compu a ions o he ecei e loca ed
a he cen e o he domain. We app ecia e ha , o ins ance, o achi e a
(small) ela i e e o o 0.1%, one only equi es a ound 7000 unknowns wi h
he seconda y ield o mula ion, while o sol e he ull o mula ion p oblem
wi h he same accu acy, we need a ound 17000 unknowns. The e o e, wi h
he i s app oach we only need app oxima ely 40% o he unknowns.
6. Conclusions
The mul i-goal o ien ed hp-FEM p o ides accu a e solu ions o he MT
p oblem a di e en ecei e s simul aneously. We show ha by employing
he seconda y ield app oach, we ob ain signi ican bene i s in compa ison
wi h di ec ly using he ull ield o mula ion: we can ob ain addi ional phys-
ical ele an in o ma ion by analyzing each ield (p ima y and seconda y)
sepa a ely, and u he mo e, his is ob ained employing a signi ican ly lowe
numbe o unknowns. Since he solu ion o he in e se p oblem is based on
i e a ed solu ions o he di ec p oblem, educing he compu a ional cos o
sol ing he di ec p oblem induces high sa ings in he in e sion p ocess.
We also p o ide a me hod o au oma ically unca e he compu a ional
domain employing PMLs. To ind an equilib ium be ween a as and a slow
decay on he PML egion is usually icky. I depends on he p oblem i sel
and i is e en mo e complica ed when he e exis s high con as s on adjacen
18
10−5 10−4 10−3 10−2 10−1 100
0
0.5
1
1.5
F equency (Hz.)
Rela i e e o in pe cen
Posi ion 1
Posi ion 2
Posi ion 3
Posi ion 4
Figu e 8: Compa ison be ween he esul s when using he ull o mula ion and he
seconda y ield o mula ion.
ma e ial p ope ies. We ha e shown ha in hese complica ed scena ios,
he Au oma ically Adap ed PML p o ides an adequa e decay, no so as o
equi e a oo ine g id and no so slow o in oduce a i icial e lec ions. Since
he choice o PML pa ame e s is au oma ic, he p oposed app oach is also
sui able o in e se p oblems.
E en i he educ ion o he compu a ional cos is i sel bene icial, he
main ad an age o sol ing he in e se p oblem wi h his app oach consis s
on he ac ha i allows o sepa a e analysis o 1D and 2D e ec s. This
will be analyzed in u u e esea ch.
Acknowledgmen s
Julen Al a ez-A ambe i and Da id Pa do we e pa ially unded by he
P ojec o he Spanish Minis y o Economy and Compe i i eness wi h e e -
ence MTM2013-40824-P, he BCAM “Se e o Ochoa” acc edi a ion o excel-
lence SEV-2013-0323, he CYTED 2011 p ojec 712RT0449, and he Basque
Go e nmen h ough he BERC 2014-2017 p og am and he Consolida ed Re-
sea ch G oup G an IT649-13 on “Ma hema ical Modeling, Simula ion, and
Indus ial Applica ions (M2SI)”. Da id Pa do has ecei ed unding om he
Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og amme unde
he Ma ie Sklodowska-Cu ie g an ag eemen No 644602, by he RISE Ho i-
zon 2020 Eu opean P ojec GEAGAM (644602). Julen Al a ez-A ambe i
was also pa ially unded by he Uni e si y o he Basque Coun y UPV/EHU
unde he g an PIFG05/2011.
19
0 0.5 1 1.5 2
x 104
0,01
0,1
1
10
Numbe o deg ees o eedom
Rela i e e o in pe cen
seconda y
ull
Figu e 9: Rela i e e o in loga i hmic scale o he appa en esis i i y compu ed
wi h he ull and seconda y ield o mula ions.
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