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Sol ing 2D linea iso opic elas odynamics by means o
scala po en ials: a new challenge o ini e elemen s
Jo ge Albella Ma ínez, Sébas ien Impe iale, Pa ick Joly, Je ónimo Rod íguez
To ci e his e sion:
Jo ge Albella Ma ínez, Sébas ien Impe iale, Pa ick Joly, Je ónimo Rod íguez. Sol ing 2D linea
iso opic elas odynamics by means o scala po en ials: a new challenge o ini e elemen s. Jou nal
o Scien i ic Compu ing, 2018, �10.1007/s10915-018-0768-9�. �hal-01803536�
Sol ing 2D linea iso opic elas odynamics by
means o scala po en ials: a new challenge o
ini e elemen s
Jo ge Albella Ma ´ınez1, S´ebas ien Impe iale2,3, Pa ick Joly2,4,
and Je ´onimo Rod ´ıguez1
1Depa amen o de Ma em´a ica Aplicada, Uni e sidade de San iago
de Compos ela, 15706 San iago de Compos ela, Spain
2In ia, Uni e si ´e Pa is-Saclay, F ance
3LMS, Ecole Poly echnique, CNRS, Uni e si ´e Pa is-Saclay, F ance
4UMA, Ens a, CNRS, Uni e si ´e Pa is-Saclay, F ance
5IMAT, Uni e sidade de San iago de Compos ela, 15706 San iago
de Compos ela, Spain
6ITMATI, Campus Su , 15706 San iago de Compos ela, Spain
May 7, 2018
Abs ac
In his wo k we p esen a me hod o he compu a ion o nume i-
cal solu ions o 2D homogeneous iso opic elas odynamics equa ions by
sol ing scala wa e equa ions. These equa ions ac on he po en ials o
a Helmhol z decomposi ion o he displacemen ield and a e decoupled
inside he p opaga ion domain. We de ail how hese equa ions a e cou-
pled a he bounda y depending on he na u e o he bounda y condi ion
sa is ied by he displacemen ield. A e p esen ing he case o igid
bounda y condi ions, ha p esen s no speci ic di icul y, we ackle he
challenging case o ee su ace bounda y condi ions ha p esen s se e e
s abili y issues i a s aigh o wa d app oach is used. We in oduce an ad-
equa e unc ional amewo k as well as a ime domain mixed o mula ion
o ci cum en hese issues. Nume ical esul s con i m he s abili y o he
p oposed app oach.
1 In oduc ion
In his pape ou goal is o e isi a e y classical ques ion, namely, he nume -
ical solu ion o elas odynamics equa ions in iso opic media, which go e n he
1
p opaga ion o elas ic wa es in solids, in he ime domain. As a ma e o ac
he e exis al eady many nume ical me hods o sol ing hese equa ions. To
begin wi h, o ins ance, in he amewo k o ini e elemen s in space and ini e
di e ences in ime, s anda d con o ming ini e elemen s (possibly high o de )
me hods o he pu e displacemen o mula ion o he elas odynamics sys em (a
second o de hype bolic sys em) as well as mixed ini e elemen me hods o he
equi alen eloci y-s ess o mula ion o he same sys em ( i s o de di e en ial
sys em). This space disc e iza ion is hen coupled o explici ini e di e ence
ime s epping ha is subjec o a CFL s abili y condi ion.
On he o he hand, in many classical physics ex books au ho s used he
well-known Helmhol z decomposi ion o ec o ields (w i e a ec o ield as he
sum o a g adien and a cu l) o compu e analy ical solu ions in homogeneous
iso opic media. Such a decomposi ion ela es elas odynamic equa ions o wo
wa e equa ions and enligh ens he decomposi ion o he wa e ield as he sum
o p essu e wa es (P-wa es, ha a e g adien s o a p essu e po en ial ϕP) and
shea wa es (S-wa es, ha a e cu ls o a shea po en ial ϕS) ha p opaga e
independen ly wi h di e en eloci ies, he eloci y VPo he P-wa es being
la ge han VS he eloci y o he S-wa es. In he 2D case, o which we will
es ic ou sel es o simplici y, he simpli ica ion is ha bo h p essu e and shea
po en ials a e scala . Howe e he ex ension o 3D does no pose a p io i any
addi ional concep ual di icul y and will be he objec o u he de elopmen s.
In a piecewise homogeneous media, such a decomposi ion is alid locally
and he di e en ypes o wa es ecouple a bounda ies and in e aces. This
is he main sou ce o complexi y o he p opaga ion p ocess. Looking a he
li e a u e, i seems ha e y ew wo ks ha e been de o ed on he exploi a ion o
his idea o ini e elemen compu a ions (howe e , one can ind a ew e e ences
conce ning ini e di e ences compu a ions, see [1] in which a ini e di e ence
scheme is cons uc ed wi h app oxima ion p ope ies independen o he a io
VP/VS) al hough i has been used in o he domains o physics, in pa icula
in luid mechanics (cu en - o ici y o mula ions [2], chap e 2, [3] and [4]).
The i s mo i a ion o he p esen wo k is an in ellec ual cu iosi y: could we
use po en ials o sol e iso opic elas odynamics wi h ini e elemen s? The e is
also a mo e ele an mo i a ion conce ning applica ions. This would conce n
he p opaga ion o elas ic wa es in nea ly incomp essible media, such as so
issues, in which P-wa es p opaga e much as e han S-wa es. In such a case,
i is well known ha displacemen -based me hods, which do no dis inguish bo h
wa es along he calcula ion p ocess, a e g ea ly penalized by la ge alues o he
a io VP/VSdue o he CFL condi ion (assuming ha explici ime in eg a o s
a e used). Le us explain his by a simple compu a ion. Le us conside a d-
dimensional iso opic homogeneous medium o cha ac e is ic leng h Lon each
di ec ion, subjec o a sou ce e m in ol ing a minimal ime scale T?. This
sou ce gene a es wo di e en minimal wa elengh s, he S-wa eleng h λS=
VS ?, which is much smalle han he P-wa eleng h λP=VP ? i VP/VSis la ge.
Assuming ha we conside P1o Q1 ini e elemen s on a quasi- egula mesh o
s ep size h, o accu acy easons, hshould be chosen p opo ional o λS, ha
is, h∝λS. On he o he hand, conside ing a leap- og ime disc e iza ion o
2
ins ance, he ime s ep is cons ained by he s abili y condi ion ha in ol es he
as es eloci y VP, ha is, ∆ ∝h/VP. Conside ing a ime in e al in eg a ion
[0, T], he numbe o ime s eps is T/∆ and since, one uses an explici scheme,
he cos o each i e a ion is p opo ional o numbe o deg ees o eedom, namely
Ld/hd. As a consequence, we can oughly es ima e he o e all compu a ional
cos as
Cos ∝LdT
hdT?∝Ld
λd
S
T
T?
VP
VS
,(1)
whe e Ld/λd
S ep esen s he size o he p oblem in space and T/T? he size o
he simula ion in ime. Clea ly he penalizing ac o is VP/VSwhich would no
appea when sol ing a s anda d scala wa e equa ion.
Po en ial o mula ions will a p io i au ho ize he use o di e en meshes o
bo h he p essu e and shea po en ials in iew o adap ing he mesh size o
each wa e leng h which is smalle o S han o Pwa es which would esul in
an impo an sa ing o he compu a ion cos o la ge a io VP/VS. This will
be explained wi h mo e de ails in Sec ion 2.3. A by p oduc o his app oach
is ha one could bene i o well-known echniques o he nume ical ea men
o he s anda d scala wa e equa ion such as o example he use o pe ec ly
ma ch laye s (PMLs) o he ea men o unbounded domains. Indeed long
ime s able implemen a ions o PMLs o iso opic elas odynamic equa ions aise
some di icul ies especially i he a io VP/VSis la ge (e en hough he so-called
C-PML sol e he long ime s abili y issue as shown in [5]).
As he eade can expec , he main sou ce o di icul ies is he ea men o
bounda ies and in e aces because, con a y o he in e io equa ions, bounda y
and ansmission condi ions a e no easily exp essed in e ms o hese po en-
ials. The main goal and he main challenge we wish o add ess in his pape
is he ea men o he couplings induced by hese a ious su ace condi ions,
ha we need o handle in a gua an eed s able way and possibly wi h hope ully
no in luence on he CFL condi ion a e ime disc e iza ion. In [6, 7] we i s
add essed he case o a homogeneous medium wi h a clamped bounda y, ha
is o say he Di ichle bounda y condi ion, o which we succeeded in achie ing
his goal. The app oach and esul s o his pape will be ecalled in Sec ion
2. This second pape ollows he philosophy o he p e ious wo k and aims a
ea ing he ee su ace bounda y condi ion, o Neumann bounda y condi ion,
ha appea s as much mo e challenging. This wo k is also p epa a o y o he
ea men o in e ace condi ions: he e a e wo o hem, one is o Di ichle ype,
he second one o Neumann ype and we an icipa e ha hei ea men would
ely on bo h ea men s o Di ichle and Neumann bounda y condi ions.
The ou line o he es o he a icle is as ollows. In Sec ion 2, we i s
shall ecap how o educe he solu ion o 2D iso opic elas odynamics equa ions
o wo scala wa e equa ions and mo e impo an ly, explain how o ea he
Di ichle bounda y condi ion as i has been done in [6]. The main sec ion o his
pape is Sec ion 3 whe e we ea he ee su ace bounda y condi ion. In Sec .
3.2 we show ha he mos nai e app oach di ec ly inspi ed om he ea men
o he Di ichle condi ion gi es ise o se ious nume ical s abili y p oblems a -
3
e disc e iza ion. This is linked o he p oposed a ia ional o mula ion o he
con inuous model, he appa en ly na u al unc ional space being oo la ge and
au ho izing he de elopmen o uns able su ace modes a e space disc e iza-
ion. Mo e p ecisely i appea s ha he new mass bilinea o m which con ains
an addi ional bounda y e m ( his is he main di e ence be ween Di ichle and
Neumann p oblems) ails o be posi i e con a y o he s i ness ma ix (which
emains he same as o he Di ichle condi ion). So he key idea o he ci cum-
en ing he p oblem is o ind a smalle (bu s ill su icien ly la ge) a ia ional
space in which we eco e he posi i i y o he mass bilinea o m. This is
p ecisely he objec o Sec ion 3.3. Fo he cons uc ion o his space, we a e
guided by he compa ison o he ene gy na u ally associa ed o he new po en-
ials o mula ion wi h he classical elas ic ene gy associa ed o he displacemen
o mula ion. The de ini ion o his new space is qui e implici and in ol es
he solu ion o some elas os a ic p oblem. Fo una ely, such a space can be
cha ac e ized as he o hogonal (wi h espec o he new mass bilinea o m) o
ano he subspace which is i sel isomo phic o a space o scala unc ions de ined
on he bounda y. We can exploi his cha ac e iza ion by p oposing a mixed
a ia ional o mula ion in which he abo e-men ioned o hogonali y ela ion is
ea ed as a cons ain leading o he in oduc ion o Lag ange mul iplie s as
unc ions de ined along he bounda y. The esul ing o mula ion is p o en o
be s able: his is he majo achie emen o his pape . Finally we show some
nume ical expe imen s ha con i m he heo e ical esul s p e iously ob ained.
2 Decomposi ion in o po en ials: he case o a
Di ichle bounda y condi ion
This sec ion has been added o pedagogical pu pose, o making he pape sel
con ained and o p epa ing Sec ion 3. Sec ion 2.1 ecaps e y s anda d ma e ial
while Sec ions 2.2 and 2.3 a e a summa y o wha has been done in [6].
2.1 Decomposi ion in o po en ials in homogeneous media
P elimina y no a ion. Th oughou he pape we will wo k in 2D and x=
(x1, x2) will deno e he space a iable. We shall use bold le e s o ep esen ing
ec o ields such as u= (u1, u2) o he displacemen ield in a elas ic body
o = ( 1, 2) o he eloci y ield ( =∂ u). O dina y le e s will be used
o scala ields such as he componen s o he ec o ields o he o hcoming
po en ials o be in oduced. Finally, unde lined bold le e s will be used o
2×2 enso ields such as he de o ma ion o s ain enso ε(u) = εij(u)
whe e 1 ≤i, j ≤2 o he s ess enso σ=σij ha ep esen s he in e nal
e o s inside he body.
4
Ma hema ical model. Le us b ie ly ecap he 2D elas odynamics equa ions.
Fi s , he ime a ia ion o he displacemen ield uis go e ned by he unda-
men al law in mechanics
ρ ∂2
u−di σ= ,(2)
whe e di σis he ec o ield de ined by
di σi=∂jσij(u),
(wi h Eins ein’s con en ion o summa ion o e he epea ed indices) ρ=ρ(x)≥
ρ0>0 is he densi y o he body ha migh depend on he x a iable o he -
e ogeneous media and he sou ce e m ∈L1
loc(R+,(L2(Ω))2). Equa ion (2)
mus be comple ed by cons i u i e laws ha ela es he displacemen ield o
he s ess enso . In an iso opic medium his is gi en by Hooke’s law which
in ol es he (non nega i e) Lam´e pa ame e s λ(x) and µ(x)
σ=σ(u) := λdi u I + 2 µε(u),(3)
whe e Iis he 2 ×2 iden i y ma ix and (we use again Eins ein’s con en ion)
di u=∂juj, εij(u) = 1
2∂iuj+∂jui,1≤i, j ≤2.
One can elimina e he unknown σ(u) by subs i u ing (3) in (2) and ob ain a
second o de sys em in u. In he homogeneous case, i.e. when λ,µand ρa e
cons an , we easily compu e ha
di (σ(u)) = (λ+ 2µ)∇di u−µcu l cu l u,(4)
so ha he equa ions can be w i en as ollows (see [8, 9] o ins ance)
ρ ∂2
u−(λ+ 2µ)∇di u+µcu l cu l u= ,(5)
whe e we ha e in oduced he wo cu l ope a o s in 2D de ined by
cu l u:=∂1u2−∂2u1, o he scala cu l o a ec o ield u,
cu l ϕ:=∂2ϕ, −∂1ϕ, o he ec o cu l o a scala ield ϕ. (6)
Equa ion (5) is comple ed, in he p esence o bounda ies, wi h bounda y con-
di ions (see la e ) and, o he sake o simplici y, anishing ini ial condi ions
u( = 0) = 0, ∂ u( = 0) = 0.(7)
Decomposi ion in o po en ials. We a e now going o in oduce wo scala
po en ials ϕPand ϕS ha ealize a Helmhol z decomposi ion, when he e is no
sou ce e m, o he eloci y ield =∂ u,
ρ ∂ ϕP= (λ+ 2µ) di u, ρ ∂ ϕS=−µcu l u.(8)
5
Subs i u ing (8) in o (5) leads, o ∂ −∇ϕP−cu l ϕS−g= 0 whe e
g( ) = 1
ρZ
0
(s)ds. (9)
Imposing anishing ini ial condi ions o he po en ials
ϕP( = 0) = 0, ϕS( = 0) = 0,(10)
we ge (since ( = 0) = 0)
=∇ϕP+cu l ϕS+g,(11)
which p o ides a Helmhol z decomposi ion [10] o he ec o ield when g an-
ishes. To ob ain he equa ions sa is ied by he po en ials we simply subs i u e
(11) in o he wo equa ions in (8) di e en ia ed in ime o ge wo scala wa e
equa ions o ϕPand ϕS
1
V2
P
∂2
ϕP−∆ϕP= di g,1
V2
S
∂2
ϕS−∆ϕS=−cu l g,(12)
wi h VP( esp. VS) he eloci y o he P-wa es ( esp. S-wa es) de ined by
VP=sλ+ 2µ
ρ,P wa es eloci y, VS= µ
ρ,S wa es eloci y.(13)
F om (8) we ob ain he ini ial condi ions o he ime de i a i e o he po en ials
∂ ϕP( = 0) = 0, ∂ ϕS( = 0) = 0.(14)
No e ha in he ee space (in absence o any bounda y), he wo wa e equa ions
in (12) a e ully decoupled.
2.2 Decomposi ion in o po en ials o a clamped domain
We now conside a 2D homogeneous iso opic p opaga ion domain Ω R2, o
ins ance, Ω bounded, wi h bounda y Γ = ∂Ω ha we assume o be clamped
which means ha equa ions (5) a e comple ed wi h he bounda y condi ion
=0,in Γ.(15)
P oceeding as in he p e ious sec ion, we in oduce ϕPand ϕS ia equa ions
(8) so ha , inside Ω, he Helmhol z decomposi ion (11) holds and he po en ials
sa is y he scala wa e equa ions in (12). These equa ions mus be comple ed
by bounda y condi ions aducing (15). We assume in he ollowing ha Γ is a
ini e union o piecewise C1closed cu es and hus admi s almos e e ywhe e a
uni no mal ou wa d ec o nand uni angen ec o τin such a way ha he
ame (τ,n) is a di ec ame so ha , i n= (n1, n2), hen τ= (n2,−n1). Fo
6
any su icien ly smoo h scala ield ϕwe ha e he ollowing iden i ies o aces
in Γ
cu l ϕ·n=−∂τϕ, cu l ϕ·τ=∂nϕ, (16)
whe e as usual ∂nϕ=∇ϕ·nand ∂τϕ=∇ϕ·τ. Thus, w i ing ha =0is
equi alen o w i ing ·n= 0 and ·τ= 0, which leads, acco ding o (11), o
he ollowing bounda y condi ions o ϕPand ϕS
∂nϕP=∂τϕS−g·n, ∂nϕS=−∂τϕP−g·τ.(17)
No e ha he wo essen ial condi ions (15) o he displacemen o mula ion
become wo na u al condi ions ha couple he wo po en ials ϕPand ϕS.
2.3 A nume ical app oach o he Di ichle p oblem:
A ecap
Va ia ional o mula ion. We i s ecall how o es ablish a weak o mula-
ion o he bounda y alue p oblem (12, 17). Assuming ha he solu ion is
su icien ly smoo h, we can mul iply he equa ions (12) by es unc ions ψP
and ψSin H1(Ω), in eg a e by pa s and use (17) o eplace he no mal de i a-
i es o he po en ials by angen ial de i a i es. A e summa ion o he wo
esul ing equa ions we can p opose a i s abs ac a ia ional o mula ion o
he p oblem. To do so we in oduce ϕ= (ϕP, ϕS) and ψ= (ψP, ψS) o ge
Find ϕ( ) : R+−→ H1(Ω)2such ha (ϕ, ∂ ϕ)( = 0) = (0,0) and
d2
d 2mΩ(ϕ( ),ψ) + a(ϕ( ),ψ) = l( , ψ),∀ψ∈H1(Ω)2,
(18)
whe e he linea o m l( , ·) is gi en by
l( , ψ) = −ZΩ
g·∇ψP+cu l ψSdx,(19)
and he mass bilinea o m mΩ(·,·) decouples ϕPand ϕS
mΩ(ϕ,ψ) = mP(ϕP, ψP) + mS(ϕS, ψS)
mQ(ϕQ, ψQ) = 1
V2
QZΩ
ϕQψQdx, Q ∈ {P, S}.(20)
The s i ness bilinea o m a(·,·) is gi en by
a(ϕ,ψ) = aΩ(ϕ,ψ) + aΓ(ϕ,ψ),(21)
whe e he olumic bilinea o m aΩ(·,·) decouples in he same way as mΩ(·,·)
aΩ(ϕ,ψ) = aP(ϕP, ψP) + aS(ϕS, ψS)
aQ(ϕQ, ψQ) = ZΩ∇ϕQ·∇ψQdx, Q ∈ {P, S},(22)
7
and he coupling su ace bilinea o m aΓ(·,·) is de ined by
aΓ(ϕ,ψ) = ZΓ∂τϕPψS−∂τϕSψPdγ,(23)
whe e he in eg als in he bounda y should be in e p e ed as duali y p oduc s
be ween elemen s in H1
2(Γ) and i s dual H−1
2(Γ). All he abo e bilinea o ms
a e symme ic ( o aΓ(·,·) use in eg a ion by pa s along he bounda y), howe e
in o de ha (18) i s he classical heo y o second o de pa ial di e en ial
equa ions [11], some adequa e posi i i y / coe ci i y o he o ms mΩ(·,·) and
a(·,·) need o be checked. The posi i i y o mΩ(·,·) is clea bu he posi i i y
o he a(·,·) is no ob ious om (21) bu elies on he ollowing lemma
Lemma 2.1. One has he iden i y: o any ϕ,ψ∈H1(Ω)2×H1(Ω)2
a(ϕ,ψ) = ZΩ∇ϕP+cu l ϕS·∇ψP+cu l ψSdx.(24)
P oo . Le us deno e ˜a(ϕ,ψ) he igh hand side o (24). We ob ain a e
expansion, using cu l ϕS·cu l ψS=∇ϕS·∇ψSand (22)
˜a(ϕ,ψ) = aΩ(ϕ,ψ) + ZΩ∇ϕP·cu l ψSdx+ZΩ∇ψP·cu l ϕSdx.
Nex we obse e ha (G een’s o mula and di cu l = 0)
ZΩ∇ψP·cu l ϕSdx=ZΓ
ψPcu l ϕS·ndγ=−ZΓ
ψP∂τϕSdγ.(25)
In he same way
ZΩ∇ϕP·cu l ψSdx=−ZΓ
ϕP∂τψSdγ=ZΓ
∂τϕPψSdγ,(26)
a e in eg a ion by pa s along he bounda y. To conclude we add (25) and
(26) o in e ha by de ini ion o aΓ(·,·) (see (23))
˜a(ϕ,ψ) = aΩ(ϕ,ψ) + aΓ(ϕ,ψ) = a(ϕ,ψ),
acco ding o he de ini ion (21).
This lemma p o es ha he bilinea o m a(·,·) is posi i e, bu also sugges s
ha H1(Ω)2is no he app op ia e a ia ional space o he weak o mula ion
because o he coe ci i y equi emen . Tha is why we in oduce he space
V:= ϕ= (ϕP, ϕS)∈L2(Ω)2such ha ∇ϕP+cu l ϕS∈L2(Ω)2.(27)
In e p e ing ϕas a ec o ield whose i s componen is ϕPand i s second
componen is ϕS, one no ices ha
∇ϕP+cu l ϕS= di ϕ
−cu l ϕ!(28)
8
3 The case o a ee su ace bounda y condi ion
In his sec ion, we conside he case o a ee su ace bounda y condi ion ( he
co esponding p oblem will be e e ed o as he Neumann p oblem in he se-
quel), ha is, w i en in e ms o he unknow
σ( )n≡λdi n + 2 µε( )n=0,on Γ.(51)
In Sec ion 3.1 we will ecap some p elimina y esul s on he Neumann p oblem.
As i will be shown in Sec ion 3.2, a nai e ex ension o he echnique explained in
he p e ious sec ion leads o an uns able a ia ional o mula ion. The unc ional
space in which i is se appea s o be oo la ge. To o e come his p oblem, in
Sec ion 3.3 his space is cons ained in such a way ha he new o mula ion is
s abilized and sui able o ini e elemen app oxima ions.
Fo he sake o simplici y ( he eade will easily con ince himsel ha his is no
es ic i e), we shall assume ha Ω is bounded and simply connec ed, hus ha
Γ is a closed cu e (see Figu e 1). We shall also assume ha Γ is pa ame e ized
by x(s)∈W1,∞(0, L) whe e sis he cu ilinea abscissa along Γ and Lis
he o al leng h o Γ. Finally, we shall use he ollowing no a ion o deno ing a
pa icula p imi i e o a unc ion de ined on Γ (whe e we a bi a ily pa icula ize
he poin associa ed o s= 0 bu his choice has no in luence)
∀η∈L2(Γ),Iη(s) := Zs
0
η(σ) dσ∈H1(Γ).(52)
I is clea ha Ican be ex ended as a linea con inuous ope a o
I ∈ L(H−1/2(Γ), H1/2(Γ)).(53)
3.1 P elimina y ecaps on he Neumann p oblem
In his sec ion we a e in e es ed in he p oblem
(ρ ∂2
u−di σ(u) = ,in Ω,
σ(u)n=0,on Γ, (54)
comple ed wi h he ini ial condi ions (7). In his p oblem, a pa icula ole is
played by he 3 dimensional space o he so-called igid displacemen s
R(Ω) = nwR∈L2(Ω)2/ε(wR) = 0o
=na(x2,−x1) + (b1, b2) ,(a, (b1, b2)) ∈R×R2o.
(55)
We in oduce he spaces
L2
R(Ω) = nw∈L2(Ω)2/ZΩ
w·wRdx= 0,∀wR∈R(Ω)o,
H1
R(Ω) = H1(Ω)2∩L2
R(Ω).
(56)
15
so ha we ha e he di ec sums
L2(Ω)2=L2
R(Ω) ⊕R(Ω), H1(Ω)2=H1
R(Ω) ⊕R(Ω),(57)
which a e o hogonal in L2(Ω)2. A classical bu impo an p ope y o p oblem
(54) is p o ided in he ollowing lemma.
Lemma 3.1. I (·, )∈L2
R(Ω),∀ ≥0, hen
∀ ≥0,u(·, )∈H1
R(Ω).(58)
P oo . Mul iply (54) by wRand in eg a e o e Ω. Using G een’s o mula,
d2
d 2ρZΩ
u(·, )·wRdx= 0 ∀wR∈R(Ω).(59)
One concludes using he ini ial condi ions.
In he sequel o his sec ion we es ic ou sel es o sou ce e ms sa is ying
(·, )∈L2
R(Ω),∀ ≥0.(60)
This is no es ic i e due o he ollowing ema k.
Rema k 3.2. Fo a gene al sou ce e m = R+ ⊥
R, wi h R(·, )∈L2
R(Ω)
and ⊥
R(·, )∈R(Ω), i is easy o see ha he solu ion uo (54) can be decom-
posed as u=uR+u⊥
R, whe e uRis he solu ion o (54) wi h sou ce e m gi en
by Rand u⊥
Ris gi en by
u⊥
R=1
ρZ
0
( −s) ⊥
R(·, s) ds.
Ano he impo an p ope y is Ko n’s inequali y in H1
R(Ω) (see [18]):
P oposi ion 3.3. The e exis s a cons an CΩ>0such ha
∀w∈H1
R(Ω),kwk2
H1(Ω) ≤ CΩZΩ|ε(w)|2dx.(61)
3.2 The nai e app oach. S abili y issues
3.2.1 The ee bounda y condi ion wi h po en ials
O cou se, he i s s ep consis s in ew i ing he ee bounda y condi ion (51)
in e ms o he po en ials de ined by (8) and (10) (as we did in sec ion 2.2 by
ans o ming he Di ichle condi ion (15) in o (17)). Howe e , om (11), we see
ha , a p io i, he condi ion (51) leads o an equa ion in ol ing he second o de
space de i a i es o he po en ials, hus no well adap ed o a ini e elemen
o mula ion. To o e come his, le us assume o a while ha we know he
alue Γo he eloci y ield on he bounda y, o , in o he wo ds, ha we wan
16
o ea a non homogeneous Di ichle bounda y condi ion wi h = Γon he
bounda y. Then, p oceeding as in sec ion 2.2 o ob aining (17), we would use
he non homogeneous bounda y condi ion o ob ain
∂nϕP=∂τϕS−g·n+ Γ·n, ∂nϕS=−∂τϕP−g·τ+ Γ·τ.(62)
Then, since we do no know Γ, we would like o compu e i as a unc ion o ϕ
using he ee bounda y condi ion (51). To do so, we i s ema k ha
ε( ) = 2 di I −cu l J +H( ),(63)
whe e J= 0 1
−1 0 !,H( ) = −∂2 2∂1 2
∂2 1−∂1 1!,(64)
so ha he ee bounda y condi ion (51) can be ew i en
λ+ 2µdi n −µcu l τ + 2 µH( )n= 0,on Γ.(65)
We deduce om (8), di e en ia ing in ime, ha
ρ ∂2
ϕP= (λ+ 2µ) di , ρ ∂2
ϕS=−µcu l ,in Ω,(66)
so ha , assuming su icien smoo hness, we ha e on he bounda y
λ+ 2µdi n −µcu l τ =ρ ∂2
ϕPn+ρ ∂2
ϕSτ,on Γ.
A e p ojec ion on coo dina e axes, using τ= (n2,−n1), his can be ew i en
λ+ 2µdi n −µcu l n =ρ
∂2
ϕPn1+∂2
ϕSn2
∂2
ϕPτ1+∂2
ϕSτ2
≡ρ ∂2
ϕ·n
∂2
ϕ·τ!.
In he same way, one compu es ha
H( )n|Γ= ∂τ 2|Γ
−∂τ 1|Γ!≡
∂τ Γ,2
−∂τ Γ,1
on Γ.
Thus, ecalling ha µ=ρ V 2
S, he bounda y condi ion (65) can be ew i en as
∂τ Γ,1=1
2V2
S
∂2
ϕ·τ, ∂τ Γ,2=−1
2V2
S
∂2
ϕ·n,on Γ.(67)
This allows us, as desi ed, o compu e Γin e ms o (ϕP, ϕS), up o an addi i e
cons an , using he ope a o I(see (52)). Mo e p ecisely, P0(Γ) deno ing he
space o cons an unc ions on Γ, using he ac ha Γ is a closed cu e and he
ini ial condi ions (14), equa ions (67) a e easily seen o be equi alen o
Γ,2+1
2V2
SI∂2
ϕ·n∈P0(Γ),ZΓ
ϕ·n= 0,
Γ,1−1
2V2
SI∂2
ϕ·τ∈P0(Γ),ZΓ
ϕ·τ= 0.
(68)
17
In he sequel, he las wo columns o (68) will be e e ed o as gauge condi ions.
To summa ize his sec ion, we ha e shown ha
sa is ies (51) ⇔ he e exis s ( Γ,ϕ) such ha (62,68) a e sa is ied.(69)
The o m (62, 68) o he ee bounda y condi ion is he one ha is use ul o
es ablishing he a ia ional o mula ion o he p oblem (see sec ion 3.2.2).
Rema k 3.4. In mo e gene al si ua ions whe e he bounda y has Nc>1con-
nec ed componen s (each o hem being smoo h enough and closed) we would
in oduce one ope a o such as he one in (52) pe componen . In consequence,
he po en ials should sa is y 2Ncgauge condi ions simila o hose o he las
column o (68).
3.2.2 Nai e a ia ional o mula ion
Acco ding o sec ion 3.2.1 (and also sec ion 2), he p oblem we wan o sol e is
Find ϕ= (ϕP, ϕS):Ω×R+→R2, Γ: Γ ×R+→R2/
1
V2
P
∂2
ϕP−∆ϕP= di g,in Ω ×R+,(i)
1
V2
S
∂2
ϕS−∆ϕS=−cu l g,in Ω ×R+,(ii)
∂nϕP=∂τϕS−g·n+ Γ·n,on Γ ×R+,(iii)
∂nϕS=−∂τϕP−g·τ+ Γ·τ,on Γ ×R+,(i )
Γ,2+1
2V2
SI∂2
ϕ·n∈P0(Γ),on Γ ×R+,( )
Γ,1−1
2V2
SI∂2
ϕ·τ∈P0(Γ),on Γ ×R+,( i)
ZΓ
ϕ·n=ZΓ
ϕ·τ= 0,( ii)
(70)
comple ed wi h he ini ial condi ions
ϕ(·,0) = 0, ∂ ϕ(·,0) = 0.(71)
We a e going o p o ide a a ia ional o mula ion o (70, 71) which na u ally
elimina es Γand p o ides a p oblem in ϕonly. We i s ake in o accoun he
las equa ion (70)( ii) by seeking ϕ(·, ) in V0, whe e V0is de ined by
V0:=nϕ∈Vs. . ZΓ
ϕ·n=ZΓ
ϕ·τ= 0o.(72)
18
Nex , we conside es unc ions ψ= (ψP, ψS) in V0and p oceed as in Sec ion
2.3 o ob ain a a ia ional o mula ion o he p oblem. The main di e ence
comes om he addi ional bounda y e ms in he in eg a ion by pa s due o
he p esence o Γin he equa ions (70)(iii) and (i ). We ob ain
d2
d 2mΩ(ϕ( ),ψ)−ZΓ Γ·nψP+ Γ·τψSdσ+a(ϕ( ),ψ) = l( , ψ).(73)
Nex , we apply he ollowing iden i y ( icky bu s aigh o wa d, i s e i ica ion
is le o he eade )
Γ·nψP+ Γ·τψS= Γ,1ψ·n+ Γ,2ψ·τ,(74)
so ha , by using (70)( ) and ( i), we can elimina e Γ hanks o he ac ha
ψ∈V0. Mo e p ecisely
−ZΓ Γ·nψP+ Γ·τψSdσ=mΓ∂2
ϕ,ψ,(75)
whe e we de ine he bilinea o m
mΓ(ϕ,ψ) := 1
2V2
SZΓI(ϕ·n)ψ·τ−I(ϕ·τ)ψ·ndγ.(76)
Finally subs i u ing (75) in o (73), we see ha ϕis solu ion o he ollowing
a ia ional p oblem
Find ϕ( ) : R+−→ V0, sa is ying (71) and such ha
d2
d 2m(ϕ( ),ψ) + a(ϕ( ),ψ) = l( , ψ),∀ψ∈V0,
(77)
whe e he new mass bilinea o m m(·,·) is de ined by
m(ϕ,ψ) = mΩ(ϕ,ψ) + mΓ(ϕ,ψ).(78)
3.2.3 Well-posedness issues
A a i s glance, he a ia ional p oblem (77) looks like a nice hype bolic a i-
a ional p oblem in he sense o heo y o Lions-Magenes [11]. We al eady saw
ha he bilinea o m a(·,·) is con inuous and coe ci e in V. Ano he good
poin is ha he bilinea o m mΓ(·,·) is symme ic (so m(·,·) is oo) due o
he obse a ion ha , using in eg a ion by pa s along Γ, we can w i e, o all
(ϕ,ψ)∈V0×V0
mΓ(ϕ,ψ) = 1
2V2
SZΓI(ϕ·n)ψ·τ+I(ψ·n)ϕ·τdγ.(79)
In addi ion one obse es ha mΓ(·,·) is con inuous in V0because he ope a o
Imaps con inuously H−1/2(Γ) in o H1/2(Γ). Ano he nice p ope y o m(·,·)
is gi en by he ollowing lemma.
19
Lemma 3.5. We ha e he injec i i y esul
(i)m(ϕ,ψ)=0 ∀ψ∈V0=⇒(ii)ϕ= 0.(80)
P oo . Le ϕsa is ying (i), since D(Ω)2⊂V0, in pa icula
∀ψ∈ D(Ω)2, m(ϕ,ψ) = mΩ(ϕ,ψ) = 0
hus ϕ= 0 by densi y o D(Ω)2in L2(Ω)2.
Howe e , all hese p ope ies a e no su icien o i Lions-Magenes heo y which
also equi es he posi i i y o m(·,·). Un o una ely, his ails o be ue:
Theo em 3.6. Assume ha he e is a pa o he bounda y Γ ha is o class
C2. Then, he e e exis s ψ∈V0such ha
m(ψ,ψ)<0.(81)
P oo . Assume ha he unc ion x(s) ha pa ame izes Γ (see he begining
o sec ion 3) sa is ies
x(s)∈C2(a, b), o some [a, b]⊂[0, L].
Le n(s) be he uni no mal ec o o Γ a poin x(s), ou going wi h espec o
Ω, and c(s) be he cu a u e o Γ a his poin . Le us de ine ν+such ha
ν+=1
2sup
s∈(a,b)
1
|c(s)|.
By elemen a y di e en ial geome y i is well known ha he map
(s, ν)∈(a, b)×(0, ν+)→x(s)−νn(s)∈R2
is injec i e. Mo eo e , he e exis s 0 < ν∗≤ν+such ha
Ω∗
a,b := {x(s)−νn(s), s ∈(a, b), ν ∈(0, ν∗)} ⊂ Ω
and ha (s, ν)→x(s) + νn(s) de ines a change o a iable om (a, b)×(0, ν∗)
in o Ω∗
a,b wi h jacobian J(s, ν) := 1 −ν c(s) ha is uni o mly bounded by J∗on
(a, b)×(0, ν∗). Le θ∈ D(a, b) and θ6= 0 such ha
Zb
a
θ(s)ds = 0
and χ∈C∞(R+) such ha supp χ⊂[0,1] and χ(0) = 1. Le 0 < δ < ν∗be a
small pa ame e de o ed o end o 0, we de ine ψδ∈C1(Ω)2as
ψδ(x) = (θ(s)τ(s)−θ0(s)n(s)χ(ν/δ),i x=x(s)−νn(s)∈Ω∗
a,b,
0,else,
20
Thanks o he assump ion on θ,ψδbelongs o V0. On he one hand
mΩ(ψδ,ψδ)≤δ(J∗/ V 2
S)kχk2
L2(R+)kθk2
H1(a,b).(82)
On he o he hand, along Γ, ψδ·τ=θ(s) o s∈(a, b) and 0 o he wise. In he
same way ψδ·n=−θ0(s) o s∈(a, b) and 0 o he wise, so ha I(ψδ·n) =
−θ(s) o s∈(a, b) and 0 o he wise. The e o e
mΓ(ψδ,ψδ) = −kθk2
L2(a,b)/V 2
S.(83)
Compa ing (82) and (83) i is clea ha m(ψδ,ψδ) is s ic ly nega i e o δ
small enough.
Rema k 3.7. The echnical assump ion on Theo em 3.6 abou he local egu-
la i y o Γis mos likely unnecessa y (and a he same ime no e y es ic i e
in p ac ice) bu needed o he p oo abo e.
Figu e 1: Le : De ini ion o he pa ame iza ion o he bounda y. Righ : De -
ini ion o he cu ilinea coo dina es and no a ions o he cons uc ion o he
unc ion ψδin he p oo o Theo em 3.6.
As one can expec , he p ope y o Theo em 3.6 is he cause o se e e ins abili ies
o any ini e elemen app oxima ion o he a ia ional o mula ion (77). This
will be pu in e idence in he nex sec ion.
3.2.4 Nume ical ins abili ies o Gale kin disc e iza ions
In oduc ion. We in oduce a ini e dimensional app oxima ion o he space
V0 ha will be deno ed by V0h ha is cons uc ed as ollows
V0h=Vh∩V0(84)
whe e Vh=VP,h ×VS,h is a Gale kin app oxima ion o he space H1(Ω)2. The
p oblem o sol e is
Find ϕh( ) : R+−→ V0hsuch ha (ϕh, ∂ ϕh)( = 0) = (0,0) and
d2
d 2mh(ϕh( ),ψh) + a(ϕh( ),ψh) = l( , ψh),∀ψh∈V0h,
(85)
21
whe e mh(·,·) is an app oxima ion o he bilinea o m m(·,·) ( o ins ance i can
be compu ed using quad a u e o mulae). A he algeb aic le el he o mula ion
akes he ollowing o m
Mh
d2Φh
d 2+AhΦh=Fh,Mh:= MΩ
h+MΓ
h,Ah:= AΩ
h+AΓ
h(86)
whe e as in (43), Φ
h= (ΦP,h,ΦS,h) is he ec o s o deg ees o eedom o ϕP,h
and ϕS,h. I one hinks o con inuous Lag ange ini e elemen s o ins ance, he
ma ix MΓ
hhas he ollowing s uc u e
MΓ
h= MΓ,P
hMΓ,P S
h
(MΓ,P S
h) MΓ,S
h!,
whe e he only non-ze o alues en ies o he ma ices (MΓ,P
h,MΓ,S
h,MΓ,P S
h)
co espond o he deg ees o eedom loca ed on he bounda y Γ (no e ha due
o he double in eg al on he bounda y, each deg ee o eedom on he bounda y
is coupled wi h all he o he deg ees o eedom on he bounda y).
Despi e o he injec i i y p ope y (80), i is no clea ha he ma ix Mhis
in e ible (which is a necessa y p ope y i one wan s o use an explici scheme in
ime). Indeed he p oo was done a he con inuous le el using densi y p ope ies
o smoo h compac ly suppo ed unc ions. A he disc e e le el, i.e. in ini e
dimensional space, such an a gumen can no be used any mo e.
Mo eo e , e en in he case whe e Mhis in e ible, i is mos likely ha , because
o he esul o Theo em 3.6, he solu ion o he semi-disc e e e olu ion p oblem,
which is gi en by
Φh( ) = Z
0
(M−1
hAh)−1
2sin (M−1
hAh)1
2( −s)M−1
hFh(s)ds,
will blow up exponen ially in ime (i.e. he semi-disc e e scheme (86) is uns a-
ble). Such a phenomenen is linked o he exis ence o s ic ly nega i e eigen al-
ues o he ollowing symme ic eigen alue p oblem (no e ha , as soon as Mh
is in e ible, such eigen alues a e necessa ily eal)
Find Ψh6= 0 and λ∈Rsuch ha AhΨh=λMhΨh.(87)
O cou se, he a e o exponen ial blow up will be, a leas , gi en by he mos
nega i e eigen alue λo M−1
hAh. Mo e p ecisely, i σhdeno es he spec um o
M−1
hAhwi h minimum alue λ−(h)<0, hen
k(M−1
hAh)−1
2sin (M−1
hAh)1
2 k ≥ C e √|C(h)|,(88)
whe e C(h)∈Ris such ha 0 <C(h)<|λ−(h)|. In he ollowing, we come
back in mo e de ails on hese in e ibili y and s abili y issues. In pa icula , in
22
he nex pa ag aph we conside a simpli ied oy p oblem associa ed o a simple
geome y. In his case, he wo issues (in e ibili y and lack o posi i i y) can
be s udied analy ically. Then, in he las pa ag aph o his sec ion we conside
a mo e gene al ini e elemen disc e iza ion and illus a e he ins abili y o he
scheme h ough nume ical compu a ions.
Rema k 3.8. No e ha because o he gauge condi ion (70)( ii) he wo spaces
a e coupled a he bounda y. In p ac ice he gauge condi ion is imposed using a
Lag ange mul iplie as usually done when imposing ze o a e age condi ion [19].
S udy o a pa icula oy p oblem. We wan o sol e elas odynamic equa-
ions (2) on he cylinde
Ω = (0,+∞)×(−π, π) (89)
ob ained when iden i ying he uppe and lowe bounda ies, whe e x1= 0 is he
ee bounda y. In consequence, we impose pe iodic bounda y condi ions
u( , x1, π) = u( , x1,−π), ∂2u( , x1, π) = ∂2u( , x1,−π),∀x1∈R+,(90)
in such a way ha he bounda y o Ω can be iden i ied o a ci cle ( his is a
closed cu e). No e ha his pa icula example does no comple ely i he
assump ions made on he domain’s geome y since Ω is unbounded, howe e
he eade will easily con ince himsel ha his is no an essen ial issue. Nex
we disc e ize he space H1(Ω) as ollows. We deno e Vh⊂H1(R+) he uni o m
disc e iza ion o R+by P1 ini e elemen s o leng h h, i.e.
Vh:= ψh∈H1(R+,C) s. . ψh(x1) =
+∞
X
j=0
ψjwj(x1)(91)
whe e he se {wj}a e he piecewise a ine unc ions ha sa is y wj(kh) = δkj
(whe e δis he K onecke symbol) so ha ψj=ψh(jh). No e ha his space
is isomo phic o he space `2(N). In he di ec ion x2, we use a spec al me hod
consis ing in unca ing he na u al Foui ie se ies expansion o a unc ion in
L2(−π, π) a o de L > 0, whe e 2π/L is hus he minimal oscilla ion leng h
allowed in he app oxima e space. This co esponds o he ollowing Gale kin
app oxima ion space Vh=VL
h×VL
hwi h
VL
h:= nϕh∈H1(Ω,R) s. . ϕh=
L
X
`=−L
ϕ`
h(x1)e−i`x2wi h ϕ`
h∈Vho.(92)
No e ha he app oxima ion pa ame e is he couple (h, L) whe e his he space
s ep in x1, de o ed o end o 0, and L he unca ion pa ame e in equency,
de o ed o end o +∞(see also ema k 3.9). I is hen immedia e o see ha
he space V0h(see (84)) is no hing bu
V0h=nϕh= (ϕP,h, ϕS,h)∈Vhs. . ϕ0,0
P=ϕ0,0
S= 0o⊂V0.
23
whe e ϕ0,0
P:= ϕ0
P,h(0) and ϕ0,0
S:= ϕ0
S,h(0), acco ding o he no a ion in oduced
abo e.
Rema k 3.9. I one makes an analogy wi h Q1 ini e elemen s o ins ance, his
would co esponds o disc e ise Ωuni o mly wi h ec angula ini e elemen s o
leng h h1=hin he di ec ion x1and h2=π/2Lin he di ec ion x2.
Then we look o he solu ion o (85) wi h he ollowing exp ession o he bilinea
o m a(·,·) and m(·,·) ha ake in o accoun he ac ha we deal wi h complex
alued unc ions,
mh(ϕh,ψh) = 1
V2
PZπ
−πIR+
ϕP,hψP,h dx1dx2+1
V2
SZπ
−πIR+
ϕS,hψS,h dx1dx2
−1
2V2
SZπ
−πZx2
−π
ϕP,h dsψS,h +Zx2
−π
ψP,h dsϕS,h dx2,
a(ϕh,ψh) = ZΩ∇ϕP,h +cu l ϕS,h·∇ψP,h +cu l ψS,h dx,
whe e he symbol His used o accoun o he use o he ollowing quad a u e
o mula
IR+
ϕhψhdx1:= h
2ϕ0ψ0+h
+∞
X
j=1
ϕjψj,kϕhk2
h:= IR+|ϕh|2dx1,(93)
ha allows us o ge mass lumping. Since we ha e used a spec al app oxima ion
in x2, he p e ious p oblem decouples as a amily o 2L+1 p oblems in 1Dwi h
espec i e unknowns {ϕ`
h:= (ϕ`
P,h, ϕ`
S,h)}. Mo e p ecisely, choosing
ψh=e−i`x2ψ`
h(x1) wi h ψ`
h:= (ψ`
P,h, ψ`
S,h)
as es unc ions in (85) and using he o hogonali y p ope ies o igonome ic
unc ions, we ge ha , o each `∈ {−L, . . . , L},ϕ`
h( ) : R+7→ Vh×Vhsa is ies
he ollowing 1D a ia ional p oblem: o all ψ`
h∈Vh×Vh,
d2
d 2m`ϕ`
h,ψ`
h+a`ϕ`
h,ψ`
h=l`( , ψ`
h),(94)
24
The i em (ii) will guide he cons uc ion o VNand is expec ed o gua an ee he
well-posedness o (108) and he s abili y o i s Gale kin app oxima ion.
3.3.1 Cons uc ion o he new a ia ional space
The ene gy na u ally associa ed wi h (77), which is conse ed as soon as he
igh hand side anishes, is
EN( ):=ED( ) + 1
2mΓ(∂ ϕ( ), ∂ ϕ( ))
=1
2[m(∂ ϕ( ), ∂ ϕ( )) + a(ϕ( ),ϕ( )) ].
(110)
The posi i i y o his ene gy is ob iously ela ed o he posi i i y o he bilinea
o m m(·,·) on a space o which ϕbelongs. Fo he Di ichle p oblem, he
(ob ious) posi i i y o he ene gy ED( ) was con i med by he iden i ies
(i)ρ
2mΩ(∂ ϕ, ∂ ϕ) = Ep( ) and (ii)ρ
2a(ϕ,ϕ) = Ec( ),(111)
whe e he po en ial ene gy Ep(·) ( esp. kine ic ene gy Ec(·)) a e de ined by
(37) ( esp. (36)). We expec simila iden i ies o he solu ion o he ee
bounda y p oblem. In ac , i is clea ha he iden i y (111)(ii) s ill holds
o he solu ion o he Neumann p oblem because i s p oo does no e e o
he Di ichle condi ion: i only uses (11) which is alid independen ly o he
bounda y condi ion. We jus ha e o ob ain an equi alen o (111)(i) wi h
m(·,·) ins ead o mΩ(·,·), i.e.,
ρ
2m(∂ ϕ, ∂ ϕ) = Ep( ).(112)
This is ela ed o he iden i y (38) in Lemma 2.3 which yields
Ep( ) = ρ
2mΩ(∂ ϕ, ∂ ϕ)−2µZΓ
u2∂τu1dγ,(113)
so ha we ha e o check ha , since 2µ= 2V2
Sρ,
mΓ(∂ ϕ, ∂ ϕ) = −4V2
SZΓ
u2∂τu1dγ.(114)
He e is whe e we a e going o use he ac ha usa is ies he ee bounda y
condi ion. Mo e p ecisely, in eg a ing in ime (67), we ha e
∂τu1=1
2V2
S
∂ ϕ·τ, u2+1
2V2
SI∂ ϕ·n∈P0(Γ).(115)
Then, since ϕ( ) belongs o V0, (114) ollows. The eade should no ice ha
which was impo an o ob aining (112) is ha we could w i e (see (8))
∂ ϕP=V2
Pdi u, ∂ ϕS=−V2
Scu l u,
31
wi h ua smoo h enough unc ion sa is ying he ee bounda y condi ion (51).
The abo e obse a ion gi es he idea o he cons uc ion o he space VN. Le
us i s in oduce he space
D:={w∈H1(Ω)2such ha di σ(w)∈L2(Ω)2}
≡ {w∈H1(Ω)2/−V2
P∇(di w) + V2
Scu l (cu l w)∈L2(Ω)2},
(116)
whe e he second line comes om using (4). This space is a Hilbe space o
he no m:
kwk2
D=kwk2
H1(Ω) +kdi σ(w)k2
L2(Ω).(117)
No e ha , by cons uc ion o Dand de ini ion o V,
∀u∈D,Fu:= V2
Pdi u,−V2
Scu l u∈V.(118)
Nex , we conside he closed subspace o Do ec o ields ha a e o hogonal
o he igid displacemen s and sa is y he ee bounda y condi ion
DN:={w∈D∩H1
R(Ω) such ha σ(w)n= 0 on Γ},(119)
and inally he space
VN:= F(DN)≡nV2
Pdi w,−V2
Scu l w,w∈DNo.(120)
A i s ema kable p ope y o his space is gi en by he lemma
Lemma 3.12. The space VNis a subspace o V0.
P oo . By (118) we al eady know ha VN⊂V. So we simply ha e o check
he gauge condi ions in (72). Le ψ∈VN, i.e.,
ψ= (V2
Pdi w,−V2
Scu l w),w∈DN.(121)
The p oo is essen ially a ma e o ep oducing he compu a ion in sec ion 3.2
o p o ing (67), wi h ψins ead o ∂2
ϕand wins ead o . Simply no e ha
σ(w)n=0on Γ ollows om w∈DNwhile he equi alen o (66) is no hing
bu (121). Then we ob ain he equi alen o (67), namely,
∂τw1=1
2V2
S
ψ·τ, ∂τw2=−1
2V2
S
ψ·n,on Γ.(122)
Finally, he gauge condi ions a e simply ob ained by in eg a ing he abo e equal-
i ies on Γ.
We shall use la e ano he nice p ope y o he space VN:
Lemma 3.13. Fo any ψ= (ψP, ψS)∈VN,∇ψP+cu l ψS∈L2
R(Ω).
32
P oo . Le ψ∈VN. Then, ψ= (V2
Pdi w,−V2
Scu l w) wi h w∈DN. By
(4), di σ(w) = ρ∇ψP+cu l ψS, ha we mul iply by any wR∈R(Ω) and
in eg a e o e Ω o ob ain, using G een’s o mula
ZΩ
σ(w) : ε(wR) dx+ZΓ
σ(w)n·wRdγ=−ρZΩ∇ψP+cu l ψS·wRdx= 0,
since ε(wR) = 0 and σ(w)n= 0 o w∈DN.
Now, we ema k ha , hanks o he p ope y (58) (see Lemma 3.1) and Lemma
3.12, he ec o o po en ials ϕ(·, ) belongs o VN o all ≥0. This implies
ha he space VNsa is ies he equi emen (108). Nex we p o e ha he space
VNsa is ies he equi emen (109).
Theo em 3.14. The bilinea o m m(·,·)is posi i e de ini e in he space VN.
Fu he mo e, he e exis s C>0, only depending on Ω,λ,µand ρsuch ha
m(ψ,ψ)≥CZΩ|ψ|2dx,∀ψ∈VN.
P oo . Le ψ= (ψP, ψS)∈VN, i.e., ψ= (V2
Pdi w,−V2
Scu l w),w∈DN.
Applying he iden i y (38)
ZΩ|ε(w)|2dx=ZΩ|di w|2dx+1
2ZΩ|cu l w|2dx−2ZΓ
w2∂τw1dγ
so ha we deduce om (3) ha
ZΩ
σ(w) : ε(w) dx= (λ+ 2µ)ZΩ|di w|2dx+µZΩ|cu l w|2dx
−4µZΓ
w2∂τw1dγ.
Since λ+ 2µ=ρ V 2
Pand µ=ρ V 2
S, we compu e om he de ini ion o ψ ha
(λ+ 2µ)ZΩ|di w|2dx+µZΩ|cu l w|2dx=ρ mΩ(ψ,ψ).
On he o he hand, using (122) (see he p oo o Lemma 3.12) we ob ain
−4µZΓ
w2∂τw1dγ=ρ mΓ(ψ,ψ).
which, combined o he wo p e ious equali ies, gi es
m(ψ,ψ) = 1
ρZΩ
ε(w) : σ(w) dx.
The e o e, since, by (3) again,
σ(w) : ε(w) = λdi w2+ 2 µ|ε(w)|2≥2µ|ε(w)|2,
33
we ge
m(ψ,ψ)≥2V2
SZΩ|ε(w)|2dx.
In he space DN⊂H1
R(Ω), we can use he Ko n’s inequali y (61) o ob ain
ZΩ|ε(w)|2dx≥2CΩV2
SZΩ|∇w|2dx.
This allows us o conclude since, ob iously, om he de ini ion o ψ,
ZΩ|∇w|2dx≥e
CZΩ|ψ|2dx,
wi h e
Conly depending on VPand VS.
A his le el, we ha e iden i ied a good space VNsa is ying he equi emen s
a he beginning o his sec ion. Howe e , he de ini ion o his space (120) is
qui e heo e ical and implici and hus a he ha d o use i nume ically: in
pa icula , i e e s o displacemen ields, which we wan p ecisely o a oid.
The goal o he nex sec ion is o gi e a mo e sui able cha ac e iza ion o VN.
3.3.2 A cha ac e iza ion o he new space
Le ΠRbe he L2(Ω)2o hogonal p ojec ion on o he space L2
R(Ω) ≡R(Ω)⊥.
Gi en ψ∈V, le us deno e w?:= SNψ he solu ion o he ollowing elas os a ic
p oblem (whose well-posedness ollows om F edholm al e na i e)
−di σ(w?) = −ρΠR∇ψP+cu l ψS,in Ω,
w?∈H1
R(Ω),σ(w?)n=0,on Γ.
(123)
The eade will easily e i y ha , by cons uc ion,
SN∈ L(V;D) and Im(SN)⊂DN.
E en mo e we ha e he ollowing esul :
Lemma 3.15. The ope a o SNis a le in e se o F es ic ed o he space
DN. Mo e p ecisely,
∀w∈DN,w=SNFw.
As a consequence, he image o SNcoincides wi h he space DN.
P oo . Le w∈DN. As −di σ(w) = −ρ∇ψP+cu l ψSwi h ψ=Fw
(c . he p oo o Lemma 3.13), we deduce om Lemma 3.13 ha
di σ(w)∈L2
R(Ω).
Then di σ(w) coincides wi h i s own p ojec ion on o L2
R(Ω) and we can w i e
di σ(w) = ρΠR∇ψP+cu l ψS.(124)
34
Mo eo e , since w∈DNwe ha e
w∈H1
R(Ω),σ(w)n=0on Γ.(125)
Finally, (125) and (124) p o e ha w=SNψ, i.e., w=SNFw. O cou se,
w i ing w=SNFw o any w∈DN, p o es ha DN⊂ ImSN.
Since F ∈ L(D,V) we can de ine
T:= F ◦SN∈ L(V),and ImT=VN,(126)
whe e he equali y de i es om he de ini ion o VNand Lemma 3.15. We
summa ize in Figu e 6 he images and p eimages o he ope a o in oduced so
a , i.e. F,SNand T.
Figu e 6: Rep esen a ion o he images and p e-images o he ope a o F,SN
and T.
No e ha , by de ini ion o SN(see (123)) and F(118)
Tψ= ( V2
Pdi w?,−V2
Scu l w?) whe e w?is he solu ion o (123).(127)
A s aigh o wa d, bu impo an , consequence o Lemma 3.15 is ha Tis a
p ojec o in o VN. Indeed,
∀ψ∈V,T2ψ=F SNF SNψ=FSNFSNψ.(128)
Since SNψ∈DN, by Lemma 3.15, SNFSNψ=SNψand hus
∀ψ∈V,T2ψ=FSNψ=Tψ⇐⇒ T =T2.(129)
35
As o any p ojec o , we can w i e he di ec sum
V= ke T ⊕ImT= ke T ⊕VN.(130)
Con a y o Vo VN, i is possible o gi e an explici desc ip ion o he space
ke Twhich is comple ely independen o he spaces Do DN. Mo e p ecisely,
Lemma 3.16. The ke nel o he ope a o Tis cha ac e ized by
ke T={ψo= (ψo
P, ψo
S)∈V/∇ψo
P+cu l ψo
S∈R(Ω)}.(131)
P oo . Le ϕ∈ke T, and w?=SNϕ, solu ion o (123). Then Tϕ= 0
means ha di w?= cu l w?= 0, hus di σ(w?) = 0by (4). In consequence
o (123), ΠR∇ϕP+ cu l ϕS=0. The e e se implica ion is i ial.
The decomposi ion in (130) is no o hogonal in he classical sense bu o hog-
onal wi h espec o m(·,·) in he sense o he ollowing heo em.
Theo em 3.17. We ha e VN=ke T⊥,m whe e
ke T⊥,m := {ψ∈V/∀ψo∈ke T, m(ψ,ψo)=0}.
P oo . S ep 1:VN⊂ke T⊥,m.
Le ϕ= (ϕP, ϕS)∈VNand ψo= (ψo
P, ψo
S)∈ke T. We know ha , by
de ini ion o VN,ϕ=V2
Pdi w,−V2
Scu l wwi h w∈DN. Le us compu e
m(ϕ,ψ) = mΩ(ϕ,ψo) + mΓ(ϕ,ψo). We ha e, by G een’s o mulas,
1
V2
PZΩ
ϕPψo
Pdx=ZΩ
di wψo
Pdx=−ZΩ
w·∇ψo
Pdx+ZΓ
(w·n)ψo
Pdγ,
1
V2
SZΩ
ϕSψo
Sdx=−ZΩ
cu l wψo
Sdx=−ZΩ
w·cu l ψo
Sdx+ZΓ
(w·τ)ψo
Sdγ.
Le us add he wo equali ies. Since by Lemma 3.16, ∇ψo
P+cu l ψo
S∈R(Ω),
while w∈DN⊂L2
R(Ω), he olume in eg al anishes and
mΩ(ϕ,ψo) = ZΓ(w·n)ψo
P+ (w·τ)ψo
Sdγ. (132)
Acco ding o he p oo o Lemma 3.12, we can use (122), wi h ϕins ead o ψ,
which gi es, o come cons an s C1and C2,
1
2V2
SI(ϕ·τ) = w1+C1,−1
2V2
SI(ϕ·n) = w2+C2,on Γ.(133)
Subs i u ing (133) in o he exp ession (76) o mΓ(·,·), we ge , using he gauge
condi ions o ψo,
mΓ(ϕ,ψo) = −ZΓw1(ψo·τ) + w2(ψo·n)dγ. (134)
36
Finally, adding (132) and (134) gi es m(ϕ,ψo) = 0 hanks o he iden i y (74).
S ep 2:(ke T)⊥,m ⊂VN.
Le ϕ∈(ke T)⊥,m, i.e. such ha m(ϕ,ψo) = 0 o all ψo∈ke T. Le ψ∈V.
Using (130), we decompose ϕ,ψas:
ϕ=ϕo+ϕN,ψ=ψo+ψN,(ϕo,ψo)∈(ke T)2,(ϕN,ψN)∈V2
N.
Ou goal is o p o e ha ϕo= 0. No e ha by s ep 1,
m(ϕo,ψN) = m(ϕN,ψo)=0.(135)
The e o e, we compu e
m(ϕo,ψ) = m(ϕo,ψo) + m(ϕo,ψN) = m(ϕo,ψo).
Nex , since ϕo=ϕ−ϕNand ϕ∈(ke T)⊥,m,
m(ϕo,ψ) = m(ϕ,ψo)−m(ϕN,ψo) =
(135) m(ϕ,ψo) = 0.
Thus m(ϕo,ψ)=0,∀ψ∈V. In pa icula , o ψ∈(D(Ω))2⊂V,
mΩ(ϕo,ψ) = m(ϕo,ψ) = 0.
We conclude by densi y o D(Ω)2in L2(Ω)2 ha ϕo=0, hus ϕ∈VN.
3.3.3 A i s s abilized mixed o mula ion
In his sec ion, we a e going o exploi Theo em 3.17, namely
ϕ∈VN⇐⇒ ϕ∈Vand m(ϕ,ψ)=0,∀ψ∈ke T,
by ein e p e ing he las condi ion as an equali y cons ain on ϕ. As i is usual
o ea ing equali y cons ain s (see [19, 20, 21]), we a e going o in oduce a
Lag ange mul iplie in he space ke T. This leads us o w i e he ollowing
mixed p oblem
Find (ϕ( ),ϕo( )) : R+−→ V×ke Tsa is ying (71) and
d2
d 2m(ϕ( ),ψ) + a(ϕ( ),ψ) + m(ϕo( ),ψ) = l( , ψ),∀ψ∈V,
m(ϕ( ),ψo) = 0,∀ψo∈ke T.
(136)
Then we ha e he ollowing equi alence heo em be ween he abo e mixed p ob-
lem and he a ia ional p oblem (108) posed in he space VN.
Theo em 3.18. The p oblem (136) admi s a unique solu ion gi en by (ϕ( ),0)
whe e ϕ( )is he solu ion o he p oblem (108).
37
P oo . Le ϕ( ) be he solu ion o (108). Fi s , i is clea ha i sa is ies
he second equa ion in (136) since ϕ( )∈VNand VN= (ke T)⊥,m (Theo em
3.17). This also implies ha
d2
d 2m(ϕ( ),ψo) = 0,∀ψo∈ke T.
On he o he hand, by de ini ion (31) o a(·,·), we ha e
a(ϕ( ),ψo) = ZΩ∇ϕP( ) + cu l ϕS( )·∇ψo
P+cu l ψo
Sdx.
By Lemma 3.16, we know ha ∇ψo
P+cu l ψo
S∈R(Ω). Then, since ϕ( )∈VN
we deduce a(ϕ( ),ψo) = 0 om Lemma 3.13. Finally, hanks o he assump ion
(60) on he sou ce e m, de ined by (9, 19), we ha e l( , ψo) = 0. Thus, we
conclude ha
d2
d 2m(ϕ( ),ψo) + a(ϕ( ),ψo) = l( , ψo),∀ψo∈ke T.
Since ϕis he solu ion (108), we also ha e
d2
d 2m(ϕ( ),ψ) + a(ϕ( ),ψ) = l( , ψ),∀ψ∈VN,
and hen, by linea i y and using he decomposi ion (130) o V, we ob ain he
abo e equali y also o all ψ∈Vwhich is no hing bu he i s equa ion in
(136) wi h ϕo=0. We hus ha e p o en ha (ϕ( ),0) is solu ion o (136).
I emains o p o e he uniqueness o solu ions o (136). Le (ϕ,ϕo) be a
solu ion o (136) wi h l(·,·) = 0 . Then, by es ic ing he es unc ion ψin
he second line o (136) o ψ∈VN, using again he m−o hogonali y o VNand
ke T, we deduce ha ϕis he solu ion o (108) wi h l(·,·) = 0. Thus ϕ=0.
We hen deduce om he second line in (136) again ha
m(ϕo( ),ψ)=0,∀ψ∈V,
which leads o ϕo=0due o he injec i i y p ope y (80) (see Lemma 3.5).
The eade which is no amilia wi h mixed a ia ional o mula ion could be
su p ised ha we in oduce an addi ionnal unknown ha we know is 0. All he
in e es o his new o mula ion is when Gale kin disc e iza ion is conce ned:
a e disc e iza ion we ge a s able p oblem and he disc e e app oxima ion o
he unknown ϕois no longe 0(see Rema k 3.23).
3.3.4 Cha ac e iza ion o he mul iplie s space ke T
E en hough he mixed a ia ional p oblem (136) is nice han (108) in he sense
ha any e e ence o displacemen ields is emo ed, i is s ill no comple ely
38
sa is ac o y o ini e elemen app oxima ion since we need a p io i o cons uc
a Gale kin app oxima ion space o ke T. In o de o wo k a ound his p oblem
we a e going o cha ac e ize he space ke Tup o an explici h ee dimensional
subspace (see Lemma 3.19) as he image by an explici mapping Eo a space
Mo unc ions along he bounda y Γ (see (143)). This is hen sa is ac o y
om he nume ical poin o iew because i is easy o app oxima e he space M
wi h ini e elemen s (de ined on he bounda y Γ) while he ope a o Eis easy
o app oxima e nume ically. This will esul in an al e na i e e o mula ion o
he mixed p oblem which will be gi en in sec ion 3.3.5. We i s begin wi h a
lemma.
Lemma 3.19. The space ke Tcan be decomposed as he di ec sum
ke T=KR⊕K0,(137)
whe e K0is he closed space o Vo he so-called ha monic ields de ined by
K0=nψ= (ψP, ψS)∈V/∇ψP+cu l ψS=0o
≡nψ∈V/di ψ= cu l ψ= 0o,
(138)
and KRis he 3dimensional space KR= spanϕ1, ϕ2, ϕ3whe e
ϕ1=1
2(x1, x2) ,ϕ2=1
2(x2,−x1) ,ϕ3=1
2(0, x2
1+x2
2) .(139)
P oo . Fi s , we easily check ha
di ϕ1
−cu l ϕ1=1
0,di ϕ2
−cu l ϕ2=0
1,di ϕ3
−cu l ϕ3=x2
−x1.(140)
By (55), i ψ∈ke T, he e exis (a1, a2, b)∈Rsuch ha
di ψ
−cu l ψ=a11
0+a20
1+bx2
−x1.
Then ψ−a1ϕ1+a2ϕ2+bϕ3∈K0. To conclude is su ices o ema k, using
he o mulae (140), ha KR∩K0=∅.
Nex , we show ha he space K0can be iden i ied wi h a space o unc ions
de ined on he bounda y Γ. To do so, we in oduce he classical no mal ace
map
γ∈ LH(di ,Ω), H−1
2(Γ), γ :ϕ−→ ϕ·n|Γ.(141)
Theo em 3.20. The map γis an isomo phism om K0on o
M:= nν∈H−1
2(Γ) /ZΓ
νdγ= 0o.
39
P oo . The ac ha γϕ∈M o ϕ∈K0 ollows om G een’s o mula
ZΓ
ϕ·ndγ=ZΩ
di ϕdγ= 0,since di ϕ= 0 (second line o (138)).
To conclude, i su ices o show ha o any ν∈M, he e exis s a unique
ϕ∈K0such ha γϕ=ν. Fo he exis ence, le pbe he unique solu ion o he
Neumann p oblem (no e ha ν∈Myields he compa ibili y condi ion equi ed
o he unique sol abili y o his p oblem)
Find p∈H1(Ω)/Rsuch ha
−∆p= 0,in Ω
∂np=ν, in Γ,
(142)
Then se ing ϕ=∇pwe ha e di ϕ= cu l ϕ= 0 and ϕ·n=∂np=νon Γ,
hence ϕ∈K0.
Fo he uniqueness, we ha e simply o ema k ha cu l ψ= 0 in Ω implies ha
ψ=∇q, wi h q∈H1(Ω)/R(see Theo em 2.9 in [2]). Then i ψ·n= 0 on Γ
and di ψ= 0 we ha e ∆q= 0 as well as ∂nq= 0 he e o e q= 0 and ψ=0.
The abo e p oo shows ha he in e se o he map γis he li ing ope a o
E ∈ L(M, K0),
whe e Eµ:= ∇p, wi h p he unique solu ion o (142). In o he wo ds, Theo em
3.20 can be eph ased as
K0=Eν / ν ∈M.(143)
This cha ac e iza ion b ings a new ligh on he absence o posi i i y o he
bilinea o m m(·,·).
Co olla y 3.21. Fo all ψ∈K0,m(ψ,ψ)≤0.
P oo . F om (143) we know ha he e exis s νsuch ha ψ=Eν=∇pwi h
p he unique solu ion o (142), hen
m(∇p, ∇p) = 1
V2
PZΩ|∂1p|2dx+1
V2
SZΩ|∂2p|2dx−1
V2
SZΓI(∇p·τ)∇p·ndγ.
Since I(∇p·τ) = p+c o a cons an cand since ∇p·n=νhas ze o a e age
along Γ ZΓI(∇p·τ)∇p·n=ZΩ|∇p|2dx+ZΩ
p∆pdx=ZΩ|∇p|2dx.
The e o e combining he wo p e ious equa ion we ob ain
m(∇p, ∇p) = 1
V2
P−1
V2
SZΩ|∂1p|2dx≤0.
40
quasi- egula mesh o app oxima ely 16000 iangles, he ime s ep is ∆ = 0.01
and he ime scheme used is he explici scheme (148). We plo Figu e 8 and 9
snapsho s o he ob ained solu ion. Fo compa ison we also plo a snapsho o
he eloci y ield e
hob ained by he s anda d P1 ini e elemen disc e iza ion o
he elas odynamics equa ions (2). The esul s ob ained a e s able in ime, e en
o long ime o simula ions, mo eo e he econs uc ed eloci y ield de ined by
h=∇ϕP,h +cu l ϕS,h −gshow good ag eemen s wi h he di ec compu a ions
o he eloci y ield.
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
P,h(1,·)
S,h(1,·)
S,h(1.5,·)
P,h(1.5,·)
Figu e 8: Snapsho o (ϕP,h, ϕS,h), solu ion o p oblem (148) o di e en ime
o simula ion.
4 Conclusions and pe spec i es
We ha e p esen ed a me hod o he compu a ion o solu ions o an iso opic
elas odynamics p oblem by sol ing scala decoupled wa e equa ions ac ing on
he po en ials o a Helmhol z decomposi ion o he displacemen ield. We de-
ailed how hese equa ions a e coupled a he bounda y and how his coupling
47
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
|P,h |(1.5,·)
|cu l S,h|(1.5,·)
|
h|(1.5,·)
| h|(1.5,·)
Figu e 9: Snapsho o he econs uc ed eloci y ield om he solu ion o p ob-
lem (148) as well as he solu ion e
hcompu ed by sol ing (2).
akes in o accoun he na u e o he bounda y condi ions sa is ied by he dis-
placemen ield. Al hough he case o igid bounda y condi ions p esen ed no
speci ic di icul y, he challenge appea ed o deal wi h he ee su ace bounda y
condi ions since se e e s abili y issues was e ealed. We sol e hese issues by
in oducing an adequa e unc ional amewo k in which one has o look o he
solu ions o a oid hese ins abili ies. A mixed o mula ion was cons uc ed o
ensu e, using a Lag ange mul iplie , ha he sough solu ions indeed belongs
o he adequa e unc ional space. Nume ical esul s con i m he s abili y o he
p oposed app oach. Among he pe spec i es o his wo k we can men ion, i s ,
he cons uc ion and analysis o an e icien nume ical me hod, including he
analysis o he disc e e li ing ope a o in oduced in he las sec ion. Then,
he analysis o he ansmission p oblem be ween wo iso opic media could be
add essed, howe e we do no expec heo e ical di icul ies since one can see
ansmission condi ions as bo h he e ogeneous Di ichle and Neumann bound-
48
a y condi ions. Finally, he 3D case should be add essed, he di icul y being
ha he po en ial co esponding o he shea wa es in he Helmhol z decompo-
si ion is no longe scala and is associa ed wi h a gauge condi ion ha should
be aken in o accoun .
Acknowledgmen s The esea ch o he i s and ou h au ho s was pa ially
unded by FEDER and he Spanish Minis y o Science and Inno a ion h ough
g an s MTM2013-43745-R and MTM2017-86459-R and by Xun a de Galicia
h ough g an ED431C 2017/60.
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