An optimal design problem in wave propagation

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. 2 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. 4 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. 8