XX Cong eso de Ecuaciones Di e enciales y Aplicaciones
X Cong eso de Ma em´
a ica Aplicada
Se illa, 24-28 sep iemb e 2007
(pp. 1–8)
An op imal design p oblem in wa e p opaga ion
J.C. Bellido1, A. Donoso1
1Dp o. Ma em´a icas (ETSII), Uni e sidad de Cas illa la Mancha, Edi icio Poli ´ecnico, A da. Camilo
Jos´e Cela s/n, 13071, Ciudad Real. E-mails: [email p o ec ed], [email p o ec ed].
Palab as cla e: Op imal design, wa e p opaga ion, classical solu ions
Resumen
In his pape we conside an op imal design p oblem in wa e p opaga ion p oposed
in [1] in he one-dimensional si ua ion: Gi en wo ma e ials a ou disposal wi h di -
e en Young’s modulus and di e en densi y, he p oblem consis s o inding he bes
dis ibu ions o he wo ini ial ma e ials in a od in o de o minimize he ib a ion
ene gy in he s uc u e unde pe iodic loading o d i ing equency Ω. We commen
on elaxa ion and op imali y condi ions, and pe o m nume ical simula ions o he
op imal con igu a ions. We also p o e he exis ence o classical solu ions in ce ain
cases.
1. In oduc ion
In [1] a s uc u al model was p oposed o he design o ini e s uc u es made up o
wo gi en ma e ials in he con ex o wa e p opaga ion. The model is an op imal design
p oblem in which he dis ibu ion o wo gi en ma e ials is op imized so as o minimize
a cos unc ional ela ed o he ib a ion o p opaga ion o wa es along he medium. In
p ac ice, his model may be use ul o he sys ema ic design o wa e il e s, damping o
wa es, o wa e guides. The au ho s de elop a nume ical me hod o he op imiza ion o
hose s uc u es based on opology op imiza ion (see [2]), bu he e is no ma hema ical
analysis o he model. In he nume ical examples, hey also obse e he su p ising ac
ha no mic os uc u e appea s be ween he wo ma e ials when one ies o design a il e
o o minimize he ib a ion ene gy. Ou aim he e is o analyze ma hema ically he model
p oposed in [1] in he one-dimensional si ua ion o longi udinal p opaga ion.
We conside he one-dimensional wa e equa ion
ρw = (Ewx)x,
1
J.C. Bellido, A. Donoso
whe e is he ime a iable, x he space a iable, wis he displacemen , and Eand ρ
a e he Young’s modulus and ma e ial densi y espec i ely. No ice we do no conside
any damping e m in ou equa ion. We a e in e es ed in ime-ha monic solu ions unde
pe iodic loading o he p e ious equa ion, i.e. in solu ions o he o m
w(x, ) = u(x) exp(iΩ ),
whe e Ω is he d i ing equency and u he ampli ude unc ion. Then he ampli ude, u,
e i ies he equa ion
E(x)u00+ Ω2ρ(x)u= 0.
Now, le us assume ha we ha e wo ma e ials a ou disposal, and consequen ly he
elas ic coe icien , E(x), and he densi y, ρ(x), ake wo alues on, E1, E2(0 < E1, E2)
and ρ1, ρ2(0 < ρ1, ρ2) espec i ely. We make he echnical assump ion
E1≤E2, ρ1≤ρ2,
and ha a leas one o he inequali ies is s ic (o he wise we would jus ha e one ma e-
ial). I we ha e a od o leng h l, ha we ep esen by he in e al [0, l], and we dis ibu e
he wo ma e ials in he od, he elas ic coe icien unc ion and he densi y unc ion a e
espec i ely
E(x) = E1χ(x) + E2(1 −χ(x)), ρ(x) = ρ1χ(x) + ρ2(1 −χ(x)),
whe e χis he cha ac e is ic unc ion o he (measu able) subse o he od whe e we place
he i s ma e ial. Now i is na u al o conside he ollowing op imal design p oblem,
(P) MinAad J(χ),
whe e he admissible se o designs is
Aad ={χ: cha ac e is ic unc ion o a measu able subse o [0, l]},
he cos unc ional is he ib a ion ene gy
J(χ) = Zl
0Ω2(ρ1χ(x) + ρ2(1 −χ(x)))|u(x)|2+ (E1χ(x) + E2(1 −χ(x)))|u0(x)|2dx,
and uis compu ed om χ h ough he s a e equa ion
[(E1χ(x) + E2(1 −χ(x)))u0]0+ Ω2(ρ1χ(x) + ρ2(1 −χ(x)))u= 0,
(E1χ(0) + E2(1 −χ(0)))u0(0) = γ, (E1χ(l) + E2(1 −χ(l)))u0(l) = 0,
γis a ixed gi en nonze o numbe . He e we conside he case o bounda y condi ions
co esponding o one ex e nal load on he le ex eme o he od, and homogeneous
Neumann bounda y condi ions on he o he ex eme. Roughly speaking, he op imal design
p oblem consis s o de e mining he bes dis ibu ion (i.e., he bes χ) o he wo gi en
ma e ials in he od in o de o minimize he ib a ion ene gy in he s uc u e, so ha we
minimize he ib a ion ampli ude along he od. No ice ha ou op imal design p oblem
indeed is an op imal design p oblem o an ODE, as we ha e simpli ied he wa e equa ion
by conside ing ime ha monic solu ions.
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Op imal design in wa e p opaga ion
We do no conside any olume cons ain in his p oblem, and his is in acco dance
wi h he physical na u e o he p oblem since wa es p opaga e be e h ough homogeneous
ma e ials han h ough mix u es o wo ma e ials. Howe e he p oblems makes pe ec
sense i we impose a es ic ion on he amoun o he ma e ials ha we may use, as o
ins ance he cons ain
Zl
0
χ(x)dx ≤δ, o Zl
0
χ(x)dx =δ,
wi h δ∈(0, l) a ixed cons an .
The solu ions o he s a e equa ions a e unde s ood as weak solu ions. Tha equa ion
is ob iously nonellip ic; howe e by he F edholm’s al e na i e, we can claim he exis ence
o a unique solu ion o any bounda y da a i and only i Ω2is no an eigen alue o he
p oblem
−(E1χ(x) + E2(1 −χ(x)))u00=λ(ρ1χ(x) + ρ2(1 −χ(x)))u, u ∈H1(0, l),(1a)
(E1χ(0) + E2(1 −χ(0)))u0(0) = 0,(1b)
(E1χ(l) + E2(1 −χ(l)))u0(l) = 0.(1c)
Recall ha , as he ope a o in ol ed is compac and sel -adjoin , we know ha he e exis s
an inc easing (maybe no s ic ly) sequence o eigen alues. Fo he sake o simplici y, and
in o de o o mula e he op imal design p oblem, we will assume ha Ω2is a away o
any eigen alue o he p oblem (1a-1c) o any cha ac e is ic unc ion χ. This hypo hesis
makes physical sense (see [1]).
I is well-known ha in p inciple we canno hope he p e ious op imal design p oblem
o admi op imal solu ions, and in gene al we need o s udy elaxa ion o i (see [4] and
he e e ences he ein).
The plan o he pape is he ollowing. Sec ion 2 is de o ed o s a e a elaxa ion esul
o ou op imal design p oblem and o gi e necessa y condi ions o op imali y o he
elaxed o mula ion o he p oblem. In Sec ion 3 we ob ain quali a i e p ope ies o he
op imal solu ions o he elaxed p oblem, being able o p o e, in ce ain cases, ha he e
a e op imal solu ions in he o m o cha ac e is ic unc ions, wha implies he exis ence o
minimize s o he o iginal op imal design p oblem (P). Finally, in Sec ion 4 we pe o m
nume ical simula ions by using a g adien me hod based on he sensi i i ies compu ed in
Sec ion 2 and analyze nume ically examples o p ac ical in e es .
2. Relaxa ion
Relaxa ion o op imal con ol o op imal design p oblems o ODE’s is a classical
subjec . In his sense ou op imal design p oblem i s in o his amewo k and elaxa ion
o i is s aigh o wa d, and so o p o e he elaxa ion esul s a ed below is s anda d
(basic e e ences on he subjec a e [4, 5, 6]).
Theo em 2.1 The op imal design p oblem
(˜
P) MinA?
ad
˜
J(E(θ), ρ(θ)),
3
J.C. Bellido, A. Donoso
whe e he admissibili y se is
A?
ad ={θ∈L∞(0, l) : 0 ≤θ≤1},
E(θ) and ρ(θ) a e gi en by
E=E(θ) = E1E2/(E1+θ(E2−E1)), ρ =ρ(θ) = ρ1θ+ρ2(1 −θ),
he cos unc ional is
˜
J(E, ρ) = Zl
0Ω2ρ(x)|u(x)|2+E(x)|u0(x)|2dx,
and uis compu ed om he pai (E, ρ) h ough he s a e equa ion
Eu00+ Ω2ρu = 0,(2a)
E(0)u0(0) = γ, E(l)u0(l) = 0 (2b)
is a elaxa ion o he op imal design p oblem (P).
We will need o he nume ical simula ions he g adien o he cos unc ional ˜
Jin he
elaxed p oblem (˜
P). I s compu a ion is di ec by using he classical adjoin me hod.
Theo em 2.2 The unc ional ˜
Jin p oblem (˜
P) is Gˆa eaux di e en iable on he se A?
ad,
and he Gˆa eaux de i a i e a ¯
θin he di ec ion o he admissible a ia ion θ(admissible
a ia ions means ha θ∈L∞(0, l) such ha he e exis s h0>0 such ha o e e y
h≤h0,¯
θ+hθ ∈ A?
ad) is gi en by
˜
J0(¯
θ;θ) = Zl
0
{[E0(¯
θ(x))(¯u0(x)¯p0(x) + |¯u0(x)|2)
+ Ω2(ρ2−ρ1)(¯u(x)¯p(x)− |¯u(x)|2)]θ(x)
−(E0(¯
θ(x))¯u(x)¯p(x)θ(x))0}dx,
whe e ¯uis he s a e associa ed o ¯
θ(i.e. he solu ion o (2a-2b) o he pai (E(¯
θ)ρ(¯
θ))),
and ¯pis he adjoin s a e, being in his case he unique solu ion o
[E(¯
θ)¯p0]0+ Ω2ρ(¯
θ)¯p= 2(Ω2ρ(¯
θ)¯u−[E(¯
θ)¯u0]0),(3a)
E(¯
θ(0))¯p0(0) = −2γ, E(¯
θ(l))¯p0(l) = 0.(3b)
Using he Taylo expansion o he cos unc ional we ob ain, as a di ec consequence
o he p e ious esul , he ollowing maximum p inciple.
Co olla y 2.1 I ¯
θ(wi h ¯
θan elemen o he in e io o A?
ad) is an op imal solu ion o
p oblem (˜
P) hen ˜
J0(¯
θ;θ−¯
θ)≥0
o any θsuch ha θ−¯
θis an admissible a ia ion.
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Op imal design in wa e p opaga ion
3. Exis ence o Solu ions o P oblem (P)
In his sec ion we explo e he consequences o he op imali y condi ions gi en in he
o m o a maximum p inciple in Co olla y 2.1 on he op imal solu ions. Ou objec i e is
o check whe he i is possible o p o e exis ence esul s o he o iginal op imal design
p oblem (P). In he case ha E1=E2(and we assume hen ha E1=E2= 1), he
design a iable does no ac on he p inciple pa o he s a e equa ion and he p oblem
is ma hema ically much simple , we a e able o p o e ha indeed any op imal solu ion
o p oblem (˜
P) is a cha ac e is ic unc ion, so ha he o iginal op imal design p oblem
admi s minimize s. We use he op imali y condi ions in his pa icula case o show he
esul , which is gi en in Theo em 3.1. The gene al si ua ion is mo e di icul and does
no seem possible o use he same a gumen , howe e we gi e a condi ion implying he
exis ence o a leas one op imal solu ion in he o m o a cha ac e is ic unc ion. We di ide
his sec ion in o wo subsec ions co esponding o he wo cases.
3.1. Case E1=E2
We assume ha E1=E2= 1 and ou p oblem now, ha we deno e as (˜
P1), is he
ollowing
MinA?
ad
˜
J(ρ(θ)),
whe e he admissibili y se is
A?
ad ={θ∈L∞(0, l) : 0 ≤θ≤1},
ρ(θ) is gi en by
ρ(θ) = ρ1θ+ρ2(1 −θ),
he cos unc ional is
˜
J(θ) = Zl
0Ω2ρ(θ(x))|u(x)|2+|u0(x)|2dx,
and uis compu ed om θ h ough he s a e equa ion
u00+ Ω2ρ(θ(x))u= 0,
u0(0) = γ, u0(l) = 0
Following Theo em 2.1, his p oblem is in he elaxed o m and he e o e i admi s a
leas one op imal solu ion. The ollowing esul es ablishes ha any op imal solu ion is
ac ually a classical one (classical solu ion means a solu ion in he o m o a cha ac e is ic
unc ion).
Theo em 3.1 Any op imal solu ion, ¯
θ, o p oblem (˜
P1) akes on only he alues {0,1},
i.e. any op imal solu ion o (˜
P1) is a cha ac e is ic unc ion.
5

J.C. Bellido, A. Donoso
3.2. Gene al case
The exis ence o classical solu ions o he gene al si ua ion o (˜
P) is no clea in gene al
as a as he au ho s a e able o claim, al hough some hing may be said abou his. We
do no s a e a o mal esul in his sec ion, bu show using new ideas ha he p oblem
admi s op imal solu ions in ce ain cases. All he echnical de ails a e included in [3], bu
we gi e nume ical e idence o his in he nex sec ion.
4. Nume ical App oach and Examples
A e he analysis ca ied ou in he p e ious sec ions, his sec ion is de o ed o show
se e al nume ical simula ions o ou p oblem. The app oach ha we ollow in he p e-
sen wo k is o app oxima e di ec ly he op imal solu ions o he elaxed op imal design
p oblem (˜
P) by a g adien me hod. O cou se his implies app oxima ion o he s a es
and adjoin s a es, i.e. solu ions o he homogenized s a e equa ion (see Theo em 2.1) and
he co esponding adjoin equa ion (see Theo em 2.2). We belie e ha his app oach will
ce ainly gi e good app oxima ions o he op imal designs o p oblem (˜
P).
The nume ical algo i hm used in his wo k is a a iable s ep-size descen me hod gi en
by
θk+1 =θk−˜
J0(θk)sk,
whe e skis a small posi i e s ep such ha ˜
J(θk+1)<˜
J(θk), ˜
J0(θk) is he g adien o he
cos unc ional espec o he con ol, compu ed in Sec ion 2, and he s ep skis looked
o as he s ep ha minimizes ˜
J(θk+1(sk)) (we use a descen me hod o compu ing sk).
On he o he hand, since he con ol mus e i y ha 0 ≤θk+1 ≤1, hen he upda ing
o mula can be easily compac ed by
θk+1 = m´ax(0,m´ın(1, θk−˜
J0(θk)sk)),
being he same ha we ob ained by using a p ojec ed g adien me hod. Conce ning he
s opping c i e ia, i has been checked ha he cos unc ional sligh ly changes i s alue
and he e o e, he L∞−no m o he di e ence ˜
J(θk+1)−˜
J(θk) educes o ze o, as we
app oxima e o he minimum. All nume ical examples belonging o his wo k ha e been
ob ained unde he s opping c i e ia, ol = 10−5, equi ing o con e gence abou 15
i e a ions o he algo i hm, on a e age. I has also been es ed ha e y simila esul s
a e ob ained when we un he same nume ical examples wi h di e en mesh sizes.
Fo he sake o b ie ness, we illus a e ou app oach h ough wo nume ical examples.
In bo h o hem, he design domain is an elas ic od o uni leng h subjec ed o ime
ha monic longi udinal exci a ion o ampli ude γin he le end. Ma e ial p ope ies a e
E2= 200 GPa, ρ2= 7,800 kg/m3, and E1= 70 GPa, ρ1= 2700 kg/m3co esponding o
s eel and aluminum, espec i ely. In p e ious sec ions i has been possible o p o e ha
op imized solu ions a e classical when he Young’s modulus Ekeeps cons an along he
od and o show ha he e a e cases wi h E a ying i s alue in which s ill he e a e
classical solu ions. In gene al, nume ical simula ions indica e ha op imized layou s a e
classical, and he e o e, he e is no mic os uc u e among he wo phases we a e mixing.
In he i s example we ha e used he equency alue Ω = 50 kHz. E en hough we le
he con ol θ a y con inuously be ween 0 and 1 du ing he op imiza ion p ocess, we can
6
Op imal design in wa e p opaga ion
see in Figu e 1(a) ha he op imized solu ion ob ained is classical a he end o he p ocess.
No ice ha his con igu a ion make he wa e damp i sel in an op imized way, as i a els
along he od (see Figu e 1(b)). Simila conclusions as be o e can be ob ained in he second
example o Ω = 200 kHz (see Figu e 1(c) and (d)). We can epea he same nume ical
examples bu keeping cons an he Young’s modulus, o ins ance, E2=E1= 70 GPa.
The nume ical esul s o Ω = 50 kHz and 200 kHz (Figu e 2) co obo a e he exis ence
o classical solu ions such as i was analy ically p o ed.
0 0.25 0.5 0.75 11
0
0.5
11
Axial posi ion
Design a iable,
θ
(a) Op imized design
0 0.25 0.5 0.75 11
−5
0
5
10x 10
−6
Axial posi ion
Ampli ude, u
(b) Wa e ampli ude
0 0.25 0.5 0.75 11
0
0.5
11
Axial posi ion
Design a iable,
θ
(c) Op imized design
0 0.25 0.5 0.75 11
−1
0
1
2x 10
−6
Axial posi ion
Ampli ude, u
(d) Wa e ampli ude
Figu a 1:
I is impo an o no ice ha ( o his pa icula case E2=E1) we ini ially ied wi h
a nume ical app oach based on op imali y condi ions. Al hough in some si ua ions such
algo i hm ga e us he same esul s as he ones ob ained by he g adien me hod, he ac
is ha i did no enjoy good con e gence p ope ies in gene al, bu when i did we go
simila esul s.
In iew o he nume ical esul s we can s a e ha , e en hough he design p oblem
ea ed is his pape is a he di e en o he band-gap one, as we commen ed in he
In oduc ion, he op imized ini e s uc u es (almos pe iodic in some cases) ob ained ( o
minimum ib a ion ene gy) o a pa icula alue o Ω, no a oid (like in he band-gap
si ua ion) bu make a he di icul he wa e can p opaga e i sel along he od only a
ha co esponding alue o Ω.
To conclude, i is wo h emphasizing ha i s , all his analysis makes sense o high
equency alues ( ypically g ea e han 10 kHz), o he wise he op imal designs a e “bo-
ing” because hey would co espond o ake only he ma e ial wi h highe a io (Ei/ρi),
o i= 1,2, and second, he same layou s we p esen he e a e ob ained whene e he
p oduc Ω lkeeps cons an .
7
J.C. Bellido, A. Donoso
0 0.25 0.5 0.75 11
0
0.5
11
Axial posi ion
Design a iable,
θ
(a) Op imized design
0 0.25 0.5 0.75 11
−1
0
1x 10
−5
Axial posi ion
Ampli ude, u
(b) Wa e ampli ude
0 0.25 0.5 0.75 11
0
0.5
11
Axial posi ion
Design a iable,
θ
(c) Op imized design
0 0.25 0.5 0.75 11
−2
0
2
4x 10
−6
Axial posi ion
Ampli ude, u
(d) Wa e ampli ude
Figu a 2:
Acknowledgemen
The au ho s acknowledge many s imula ing discussions on he subjec o his pape
wi h Ma in Bendsøe, Jacob Jensen and Ole Sigmund. Financial suppo was p o ided
by MEC (Spain) G an MTM2004-07114, and by Jun a de Comunidades de Cas illa-La
Mancha (Spain) G an PAI-05-027.
Re e encias
[1] SIGMUND O. and JENSEN J.S. Sys ema ic Design o Phononic Band-Gap Ma e ials and S uc u es
by Topology Op imiza ion, The Royal Socie y o London, Philosophical T ansac ions, Se ies A, Vol.
361, pp. 1001–1019, 2003.
[2] BENDSØE M.P. and SIGMUND O. Topology Op imiza ion: Theo y, Me hods and Applica ions,
Sp inge -Ve lag, Be lin, Ge many, 2003.
[3] BELLIDO J.C. and DONOSO A. An op imal design p oblem in wa e p opaga ion, o appea in
Jou nal o Op imiza ion Theo y and Applica ions.
[4] LURIE K.A. Applied Op imal Con ol Theo y o Dis ibu ed Sys ems, Plenum P ess, New Yo k,
New Yo k, 1993.
[5] WARGA J. Op imal Con ol o Di e en ial and Func ional Equa ions, Academic P ess, New Yo k,
New Yo k, 1972.
[6] YOUNG L.C. Lec u es on he Calculus o Va ia ions and Op imal Con ol Theo y, W. B. Saunde s
Coo po a ion, Philadelphia, Pennsyl ania, 1969.
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