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Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case

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(1)Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case Houssem Haddar, Patrick Joly, Hoai Minh Nguyen. To cite this version: Houssem Haddar, Patrick Joly, Hoai Minh Nguyen. Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case. [Research Report] RR-5199, INRIA. 2004, pp.42. �inria00070793�. HAL Id: inria-00070793 https://hal.inria.fr/inria-00070793 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Asymptotic Models for Scattering from Strongly Absorbing Obstacles: the Scalar Case H. Haddar — P. Joly — H.M. Nguyen. N° 5199 Mai 2004. N 0249-6399. ISRN INRIA/RR--5199--FR+ENG. THÈME 4. apport de recherche.

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(212) $. / n0jk[]dRs‚st_'n‡st_q© jt[‡_ st_yzlon v lq} v _y}  j  GIBC 2 ` 4 dfs±n]luj}tlo¡]g‡sxjy© §[]dmhf_jk[]_ st_yz'lqn v lo} v _'} ±j  2 *  dfsy¥ / n¬lqjt[]_y}±§5lq} v s2 *  dfs‚i}klq¡]g‡sxj±œo_'}†sxdflqn1luŸ!2 ` ¥¦"lqn‡z'_'}kn]dfn]‘jt[‡_jt[‡dm} v lq} v _'} GIBC zjk'[]lq_ n v d vdžjk_'dmlonYn‡jtdms'jt©Ydf4 _yn]s_yR džjk[]_'}5jt[‡_  j  GIBC4 2 qŠ 4 n]lq}"jt[‡_ ±j  GIBC 2 4 *S* 4 dRsJ}klq¡]g sj¥ / n v _'_ v ©elqn]_[‡iqs Z. Z. Γ. Dε,3 ϕ · ϕ ds =. α ε3 2. ϕ · N ε,3 ϕ ds =. αε 2. Z. Z. Γ. |∇Γ ϕ|2 ds + ε¯ α. |∇Γ ϕ|2 ds +. α ¯ ε. Z. Z. [1+ Γ. [1+. εH ε2 −i (3H2 − G + ω 2 ) ] |ϕ|2 ds, α 2. ε2 εH +i (H2 − G + ω 2 ) ] |ϕ|2 ds. α ¯ 2. Ÿ£}klq^ §[]dfz†[1lqn‡_ _yiqstdfhmwz'lq^:p]grjk_ys$jk[‡ij O2 }t_y^-_y^,¡D_'} α = Γ. Γ. Γ. √ 2 2. +i. √ 2 2. 4R. √ 3Z √ Z  Z 2ε ε 2  2 ε,3  |∇Γ ϕ| ds − ρε |ϕ|2 ds, D ϕ · ϕ ds =   Im 4 2 Γ 1 Γ Γ √ Z √ Z Z   2ε 2  2 ε,3  Im |∇Γ ϕ| ds − ρε |ϕ|2 ds. ϕ · N ϕ ds = 4 2ε Γ 2 Γ Γ. §[‡_'}k_‚jk[]_#Ÿ£g]n‡zjtdflqn‡s zloneœq_'}k‘q_72£g]n‡džŸ£lo}t^:hfw lqn jtl §[]_'n jk_'n v jkl:M2 iun v iq}t_jt[eg‡s„p lYsº dmjtdfœq_Ÿ£lq} ε st^iuhfh_'n‡lqg]ρ‘o[ 4 ¥±Z$[‡_p]}klq¡]hf_'^jt[‡_'n¬dRsΓjk[‡4 ij‚jt1[]_ dmnYjt_y‘qε}†iuhRsdfn |∇ ϕ| zlq^:_§dmjt[Pjt[]_ §}klqn‡‘-stdm‘on4¥ F s-§"_1sx[‡iqhmh5n]l§_'{ep‡hfiqdmn4©Edmj:dfs-pDlosksxdf¡]hf_jtl¯zlon‡sxjt}kg‡z®j-}tlo¡]g‡sxj s lqŸ±lq} v _y} * ¥¢Z$[]_d v _yi dRsEjkl,g‡st_±stlq^:_±iqp]p]}klqp]}kdfiujt_|Ei v  iup]p‡}tl*{rdf^:iujtdflqn-luŸªjk[]_±df^iu‘qdfn‡iqGIBC }tw p‡iu}tjJluŸMjk[]_±¡Dlqg]n v iu}kw lop _y}xº iujtlo}ks5vjk[‡ijŸ£lo}t^iuhfhfw‘qdfœq_s5jt []_,skiu^:_ lo} v _y}$luŸAv iup]p‡* }tl*{r/ df^:v iujtdfv lqn¡]grj±}k_ysxjtlo}t_#iu¡‡stlq}kprjkdmlonp]}klqpD_'}tjwq¥ ¦"lon‡std _y}5Ÿ£lq}dfn‡sj†iun‡z'_#jt[]_ j  GIBC luŸ%lq} _'} ¥ n _'_ ε j. Γ. ImD. ε,3. 2. √   √ 2 ε2 2 2 = −ε 1 − ε 2H + (3H − G + ω + ∆Γ ) . 2 2. €‚n‡_#zyiunjt[]_y}t_'Ÿ£lq}k_Ÿ£lq}k^:iqhmhfw§}kdžjk_ ImD. ε,3. √    √ 2 ε2 ε2 1 − ε 2H + ∆Γ + O(ε4 ). = −ε 1 + (3H2 − G + ω 2 ) 2 2 2. lujk_±jk[‡ij©]iqs H − G = (c − c ) ©r§[]_'}k_ c iqn v c iq}t_±jt[]_j§5l p‡}tdfn‡zdfp‡iqhMz'g]}kœ*iujtg]}k_ys5iuhflqn‡‘ 2 sx_y_sx_z®jtdflqn1`‡¥f‹ 4 ©]§"_#[‡i*œo_ Γ . 2. 1 4. 1. 2. 2. 1. (1 +. 2. ε2 (3H2 − G + ω 2 )) > 0. 2. √ / j dRsjt[]_yn stg zdf_'nYj#jtl´st_'_y¨Pi pDlostdmjtdfœq_iup]p‡}tl*{rdf^:iujtdflqn¬lqŸ lo¡rjkiqdmn‡_ v ¡Ywzlqn sxd v _'}kdfn]‘,jk[]_#Ÿ£lq}k^iuhªdfneœq_'}†st_q©en‡iu^:_yhmw (1 − ε 2H + √ ε2 1 − ε 2H + ∆Γ = 2. ½

(213) ½ÆT"U9V@WW. . √ ε2 1 + ε 2H + (4H2 − ∆Γ ) 2. −1. ε2 2 ∆Γ ). + O(ε3 ).. §[]dRz†[ zyiun ¡ _.

(214) ‹9A.  S      S) S(. Z$[‡_'}k_Ÿ£lq}k_q© ImD. √ √ ε2 ε2 2 = −ε (1 + (3H2 − G + ω 2 )) (1 + ε 2H + (4H2 − ∆Γ ))−1 + O(ε4 ). 2 2 2. ε,3. luŸ%lq} v _'} * Rd s$lq¡rj†iudfn]_ v ¡ew }k_'p]hRiqz'dmn‡‘. }klq¡‡g‡sj  j ±º hfdm¨o_ F. ¡ew. GIBC Dε,3 √   2 ε2 2 2 1 − (3H − G + ω + ∆Γ ) := ε 2 2 √ √ ε2 ε2 2 −iε (1 + (3H2 − G + ω 2 )) (1 + ε 2H + (4H2 − ∆Γ ))−1 . 2 2 2. 2* Q 4. √ √   ε2 ε2 2 2 2 2 1 − (3H − G + ω + ∆Γ ) ϕ − iε (1 + (3H2 − G + ω 2 ))ψ := ε 2 2 2 2. 2* D 4. √ ε2 ∆Γ ψ + (1 + ε 2H + 2ε2 H2 )ψ = ϕ. 2. 2* A 4. Drε,3. Z$[‡dfs$_'{ep‡}t_ststdmlon §dfhmh v _ g‡st_ v dmn0p]}†iqzjtdRz_#dmn1jk[]_#Ÿ£lqhfhml§dfn]‘:sx_yn‡st_SR Drε,3 ϕ. §[‡_'}k_ ψ dRsstlqhfgrjtdflqnjtlER. €‚n‡_#zyiun1_yiosxdfhmw œq_'}kdmŸ£wjt[‡iuj Z. −. √ Z √ ε2 2 ε2 2 2 (1 + (3H − G + ω )) (1 + ε 2H + ε2 2H2 )|ψ|2 + |∇Γ ψ|2 ds. ds = −ε 2 2 2 Γ 2*. Œ Z$[‡_}kdf‘q[Yj[‡iun v std v _dRsn]lonpDlostdmjtdfœq_Ÿ£lq}„iqhmh ε §[]_'n‡z'_jt[]_iq¡‡sxlo}tp]jtdflqn p‡}tlop _y}xjwŸ£lo} D ¥ 4 €±ŸEzlog]}†sx_lqn]_zyiunŸ£lqhfhml§ isxdf^:dmhRiu}p‡}tlrz_ v g]}k_±jkl v _'}kdfœq_#}tlo¡]g‡sxj „j  jt[]df} v lo} v _'} GIBC ¥5Z$[]_ _'{rp]}t_ststdflqn lqŸjk[]dfs±zlqn v džjkdmlondRs¡‡iosx_ v lon jk[]_iup‡p]}tl*{rdf^ijtdflqn R √ 2O`o 4 ε ε 2 (1 + (H − G + ω )) {1 − ∆ } + O(ε ). ImN = 2ε 2 2 L±_'n‡z'_q©]}k_'p‡hfiozdfn]‘ N ¡Yw Γ. Drε,3 ϕ · ϕ. ε,3 r. 2. ε,3. 2. 2. 2. −1. Γ. 2. ε,3. Nrε,3. :=. √   √ 2 ε2 2 2 1 + ε 2H − (H − G + ω + ∆Γ ) 2ε 2 √ 2 ε2 ε2 +i (1 + (H2 − G + ω 2 )) {1 − ∆Γ }−1 , 2ε 2 2. 2O`‡‹ 4. dfn>2 ** 4 ‘odmœo_ys4iqn]lujk[]_'}4jt[]df} v lq} v _'} ±j  GIBC ¥%Z$[]dRszlon v dmjtdflqn dRs4}klq¡]g sjdfn œedm_y§¯luŸrjt[‡_EŸ£lqhfhml§dfn]‘ d v _ynojkdžjwo©]§[]_y}t_jt[‡_ }tdf‘q[Yj$[‡iun v std v _dRsn]lonpDlostdmjtdfœq_Ÿ£lq}„iqhmh ε © √ Z   Z 2£`e 4 ε ε 2 ϕ · N ϕ ds = − (1 + (H − G + ω )) |ψ| + |∇ ψ| ds Im 2. ε,3. Γ. 2ε. Γ. 2. 2. 2. 2. 2. 2. 2. Γ. »£¼%½4»¿¾.

(215) ‹Œ. 

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