Solution of a diffusion problem in a non-homogeneous flow and diffusion field by the integral representation method (IRM)

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1 Appled and ompaonal Mahemacs 4; 3: 5-6 Pblshed onlne Febrary 4 hp:// do:.648/.acm.43.3 olon of a dffson problem n a non-homogeneos flow and dffson feld by he negral represenaon mehod IRM Hrosh Isshk hch Nagaa Yasaka Ima Inse of Ocean Energy aga nversy aga Japan Emal address: sshk@dab.h-ho.ne.p H. Isshk nagaa@oes.saga-.ac.p. Nagaa may@cc.saga-.ac.p Y. Ima To ce hs arcle: Hrosh Isshk hch Nagaa Yasaka Ima. olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM. Appled and ompaonal Mahemacs. Vol. 3 No. 4 pp do:.648/.acm.43.3 Absrac: Inegral represenaons are derved from a dfferenal-ype bondary vale problem sng a fndamenal solon. A se of negral represenaons s eqvalen o a se of dfferenal eqaons. If he bondary condons are sbsed no he negral represenaons he negral eqaons are obaned and he nknown varables are deermned by solvng he negral eqaons. In oher words an negral-ype bondary vale problem s derved from he negral represenaons. An effecve and fleble fne elemen algorhm s easly obaned from he negral-ype bondary vale problem. In he presen paper negral represenaons are obaned for he dffson of a maeral or hea n he sea where he convecve velocy and dffson consan change n space and me. A new nmercal solon of an advecon-dffson eqaon s proposed based negral represenaons sng he fndamenal solon of he prmary space-dfferenal operaor and he nmercal resls are shown. An nnovave generalzaon of he negral represenaon mehod: generalzed negral represenaon mehod s also proposed. The nmercal eamples are gven o verfy he heory. Keywords: Advecon-Dffson Problem Varable Dffson onsan Inegral Represenaon Mehod Prmary pace-dfferenal Operaor eneralzed Fndamenal olon eneralzed Inegral Represenaon Mehod omponen. Inrodcon enerally speakng physcal phenomena are descrbed as bondary vale problems n dfferenal eqaons. We refer o hs ype of problem as a dfferenal-ype bondary vale problem. If we se a fndamenal solon of he dfferenal eqaons we can derve negral represenaons from he dfferenal-ype bondary vale problem. If we sbse he bondary condons no he negral represenaons we oban he negral eqaons. We can deermne he nknown varables by solvng he negral eqaons. The negral represenaons are eqvalen o he dfferenal-ype bondary vale problem. Hence we refer o he bondary vale problem epressed by he negral represenaons as he negral-ype bondary vale problem. If he dffson coeffcen s consan a solon obaned by he bondary elemen mehod BEM s well known []. In he presen paper we dscss a solon obaned by he negral represenaon mehod IRM where he dffson consan s no acally consan. In hs case a solon can be obaned sng he fne elemen mehod FEM or he BEM wh eraon BEMI. For he ordnary FEM algorhm of a bondary vale problem we ms dvde he compaonal regon no elemens and nerpolae nknown fncons. In he FEM we se smple nerpolaon fncons n he elemens. Ths may redce he degrees of freedom of he nerpolaon fncons and we overcome hs dffcly by ncreasng he nmber of elemens. As sch we face a seros problem n he mesh dvson. Alhogh mesh-free mehods [7] can be sed o solve hs problem we enconer some dffcles n consrcng he eqlbrm eqaons of nodes and/or he sably of he calclaon resls. A collocaon mehod sch as ha descrbed n Reference [8] brngs abo dffcles n he consrcon of he nerpolaon fncon. On he oher hand he IRM can

2 6 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM realze an easer dvson no elemens and a hgher precson nerpolaon. In he IRM snce he conny of nknown varables beween he elemens s no reqred eplcly a mesh-free approach wold be possble n he case of a consan dsrbon of nknown varables n elemens. If we nrodce for eample he movng leas sqares M mehod no he IRM a mesh-free mehod wold be feasble. In he presen paper we propose a new nmercal mehod for solvng he dffson eqaon: The proposed mehod s a nqe nmercal mehod hned from negral represenaons of Naver-okes eqaon obaned by hlman [9]. We can derve negral represenaons ha do no nclde he dfferenaons of nknown varables wh respec o spaal varables. We can easly nrodce an rreglar elemen dvson of he regon. The presen mehod wold be favorable when he fld regon s geomercally comple and/or when he bondary changes n me and he elemen dvson nevably becomes rreglar. 3 nce we need o consder only he fld regon n whch he maeral or hea ess he reqred calclaon me and comper memory may be redced. We condc nmercal calclaons of D and D problems and demonsrae he effecveness of he IRM. able and precse resls are obaned n a shor me. Frhermore we developed a generalzed negral represenaon mehod IRM. The negral represenaon based on he prmary space-dfferenaon operaor dscssed above s one of he generalzed negral represenaons IR. The prmary space-dfferenaon operaor s closely relaed o he dfferenal operaor of he bondary vale-problem. On he oher hand n he generalzed heory he fndamenal fncon s chosen frs and a dfferenal eqaon s defned properly reflecng he fndamenal solon and he bondary vale problem. For eample we can se he assan fncon as he generalzed fndamenal solon.. Advecon-Dffson Eqaon n One-Dmenson D The spaal coordnae and me are denoed by and respecvely. e and σ be he maeral densy or emperare advecve velocy dffson coeffcen and maeral sorce respecvely. The D dffson eqaon n regon < < s hen wren as σ σ n < < eqaon yelds a IP. d a IP s hen obaned by d d. Hence f s gven by he nal condon we can solve he nal-bondary vale problem. Rewrng we have σ We defne he space fndamenal solon of as where s Drac s dela fncon : 3 for and d 4 We refer o he operaor as he prmary space-dfferenal operaor. Then we have. 5 The fndamenal solon sasfes From we have sgn. 6 σ d Hence we oban where ε d σ d 7 8 when < < ε when or. 9 oherwse Transformng he negral of he frs erm on he rgh-hand sde of 8 and echangng and we oban If a an nernal pon IP s known and or a a bondary pon BP s gven a me from he bondary condons B hen dfferenal

3 Appled and ompaonal Mahemacs 4; 3: ε d d d σ d If we remove he spaal dervaves of n he negrals n we oban ε d d d σ d If a IP s known and or a BP s known from he bondary condons hen he negral represenaons are negral eqaons wh he nknown varables a IP and or he nknown varables a BP. We oban d a IP from d d. If on IP s known from he nal condons hen we can solve he nal and bondary vale problem sng he negral represenaons. When he dffson coeffcen and he advecve velocy are gven smply by cons a cons b and 5 and 6 are sbsed no we oban ε d sgn d σ d 3 sgn sgn 3. Nmercal Applcaons o One-Dmensonal D Problems For smplcy we assme he condons n and σ. Frs we ransform he negral represenaon 3 no algebrac eqaons. We dvde he regon no N eqal elemens of lengh d and denoe he mdpon of each elemen as N as follows: d 4 N and Eqaon 3 s appromaed by.5 d. 5 N d ε d d We defne N d sgn d d sgn sgn α for he nernal pon IP for he bondary pon BP or d d 4 α d d d 6 and α BP as when 7a when d α BP d d. 7b d Then we oban he algebrac eqaons for IP N and BP BP respecvely as BP N α N d sgn d d sgn sgn N α BP N sgn BP d BP BP sgn BP sgn BP BP BP 8a 8b There are wo ypes of solon of hese algebrac eqaons 8: eplc and mplc solons. Eplc solon A In he negral represenaon 8 he man nknown varable. s sed as

4 8 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM Assme known. Oban sng 8. 3 Oban d from d d 4 Repea he process.. Implc solon A In he negral represenaon 8 s sed as he man nknown varable. Assme d known. Appromae n 8 by [ d] d and solve for he nknown varable. 3 Repea he process. If we se mplc solon he sably of he nmercal calclaon ncreases mch and we can se mch larger d han he one sed n he eplc mehod. The non-homogeneos erm of he negral eqaon ms be evalaed a. 5d. In he case of eplc solon A we se he followng algebrac eqaons and progresson of me eqaon: sgn sgn 9a N N d sgn d d α BP BP BP sgn BP sgn BP BP BP N d N sgn d BP d α 9b d d. 9c The oal nmber of he nknowns n he algebrac eqaons s N : where for N and or and or. The oal nmber of eqaons s also N n oal: N eqaons for IP and eqaons for BP. In hs case all varables on IP and BP become nknowns. Namely we are facng a regon-bondary elemen problem. As an appromaon we se y [ ] d f s known. Oherwse we se [ ] d f s known. We se smlar appromaons for. In hs case he nmber of nknowns and he nmber of eqaons are boh N. In solon A we can oban he smlar algebrac eqaons as n solon A. 3.. Dffson of Maeral n Infne pace who Advecon We consder an nal vale problem who advecon n he regon < < < < n compaon wh he nal condon. As a resl of he symmery we consder he regon gven by < < < n compaon. We smmarze he condons as follows: Inal condon: N d where s he Kronecker dela: s f or oherwse. Bondary condon: Appromaon : In he case of procedre A we solve from 9 wh N α N 3a d d N. 3b The solon of hs problem s he well-known fndamenal solon of he lnear D dffson problem: 4 e. 4 πν Nmercal resls are shown n Fgs. and. We sed procedre A for he nmercal calclaon. The calclaon condons are N d. 5 and ν. 89. The precson of he calclaon s very hgh. Fgre. alclaed resls a each me sep.

5 Appled and ompaonal Mahemacs 4; 3: d d 3 Hence we have d cons. 3 The eac solon of he problem s gven by Fgre. omparson wh he eac solon Dffson of Maeral n Infne pace wh Advecon 4 e. 33 π In he calclaons n econ 3. we consdered n he regon gven by < < n compaon sng symmery. However snce hs problem s asymmerc we ms consder n he regon gven by < < < n compaon. When < s large enogh we can assme ± ± 5 If we consder a doble-ype nal dsrbon of maeral he nal condon s gven by d d d 6 Fgre 3. pace dsrbon of n case of doble ype nal dsrbon of maeral: a appromae solon; b eac solon.48 In hs case he sze and cener of elemens are gven by d 7a N.5 d N. 7b From 6 he nal condon n hs case s specfcally as follows: N N N 8 d d snce d n N / d < < N / d d 9a oherwse d n N / d < < N / d d. 9b oherwse If we adop he eplc solon A we have from 9 N N sgn d α N. 3 For he check of he calclaon we se Nmercal resls are shown n Fg. 3. We sed A procedre for he nmercal calclaon. The calclaon condons are 4 N 6 d and.. The precson of he calclaon s hgh. 4. Advecon-Dffson Eqaon n Two-Dmenson D The spaal coordnae and me are denoed as y and respecvely. e y y y v y y and σ y be he maeral densy or emperare he advecve velocy vecor he dffson coeffcen and he maeral sorce respecvely. The D dffson eqaon n regon s hen wren as Rewrng 34 we have σ σ n 34 σ. 35 The fndamenal solon of aplace operaor he prmary space-dfferenal operaor n hs case s defned as

6 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM and s gven by 36 lnr 37 π where y η and r yη. The fndamenal solon sasfes r. 38 π r π An eenson o 3D may be sragh forward. e r n 3D be r y η z ζ and he fndamenal solon be 39a 4π r r b 4π r 4π And we replace area and lne negrals by volme and area negrals respecvely. From 35 and 36 we have σ d Hence we oban where ε d σ d } 4 4 when ε when 4 oherwse and s he bondary of. We ransform he negral of he frs erm on he rgh-hand sde of 4. sng he vecor formla: we oban 43 ε dl n n d d d σ d 44 where n s he n oward normal on. Echangng and we have ε dl n n d d d σ d 45 Removng he spaal dervaves of n he area negrals n 45 we have ε dl n n ndl dl n d d d d σ 46 Ths eqaon does no nclde he spaal dervave of he nknown varable n he area negrals. If a IP and or n a BP are known from he bondary condons he negral represenaons are negral eqaons havng nknown varables a IP and n or a BP. We oban d a IP from d d. pecfcally f a IP s known from he nal condons we can solve he nal and bondary vale problems sng he negral represenaons. When he dffson coeffcen and he advecve velocy v are gven smply by cons 47a cons 47b and 37 and 38 are sbsed no 46 we oban ε ln d π ln d π ln n ln dl σ d π π ln ln dl dl π n π n 48 If we consder he seady sae 48 wh s he

7 Appled and ompaonal Mahemacs 4; 3: 5-6 negral represenaon of he Posson eqaon. bsng he bondary condon no 48 and consderng on he bondary we oban he negral eqaon sed n he BEM. 5. Nmercal Applcaons o Two-Dmensonal D Problems 5.. Dffson of Maeral n Infne pace who Advecon For smplcy we assme he condons gven n 47 and σ. We consder he dffson of a maeral placed nally a he cener of he coordnaes n an nfne regon. In oher words he nal condon y y s specfed n he regon < < and < y <. sng symmery we consder only he qarer regon gven by y B. Assmng σ and n hs case we have y ε y d π π where and B y η dη η y B η ln y η d dη πν 49 n are assmed o be zero on he bondares and y B. For smplcy we dvde he regon y B no M N eqal elemens wh sdes d and dy and denoe he cener of each elemen as y M N. In oher words we have B d dy M N 5 d d M 5a dy y dy N. 5b We smmarze he nal and bondary condons as follows: Inal condon: y 4ddy M N. 5 Bondary condons: y N 53a y M. 53b Eqaon 49 s hen appromaed by he followng algebrac eqaon: m M m d ε y y d n N m π m d m y n π yn dy n y η πν m M n N m n m d yn dy m d yn dy y yn dy dη m yn ln y η d dη 54 As dscssed n econ 3 here are wo ypes of solon: he eplc and mplc solons. We se he eplc solons below and consder he wo followng solons: olon A.a. Only y a IP s nknown We appromae he bondary vale m and yn as follows: m m y yn yn 55 and he algebrac eqaon s appromaed for M N as y y d πν m M m d m η π m d m y dη π n N yn dy yn y n dy n y η m M n N m n m d yn dy m d yn dy m yn ln y η ddη 56 The nknowns of hs algebrac eqaon are a IP and he nmber of he nknowns s M N. The nmber of eqaons s also M N. Hence we can solve hs eqaon. However he precson s no hgh becase of he appromaon n 55. olon A.b. Boh y a IP and and y a BP are nknown For he nernal pon y M N we have y y d m M m d m π m d m y y dη π πν n N yn dy n y n dy n y η m M n N m n m d yn dy m d yn dy m yn ln y η d dη 57a for he bondary pon on he -as M

8 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM n N yn dy yn dη π y n dy πν m M n N m n m d yn dy m d yn dy n m yn η ln y η d dη 57b and for he bondary pon on he y-as y N m M m d y y m d π m d y πν m M n N m n m d yn dy m d yn dy m y y ln η d dη m n 57c The nmber of nknowns and he nmber of eqaons are boh M N M N. Ths we can solve he eqaon. The eac solon of hs problem s gven by y 4ν y e 58 4πν The nmercal resls are shown n Fg. 4. The condons of he calclaons are B M N d. 5.5 and. 89. The resls ndcae ha olon A.b s more precse han olon A.a. B d dy M N 59.5 d y B.5 dy M N. 6 The nal and bondary condons are as follows. Inal condon: M N ddy and M N. 6 Bondary condon: on 6 n are assmed o be zero on he bondares ± and y ± B. We have from 48 B ε y η ddη π B y η 63 B η ln y η d dη π B The eac solon of hs problem s gven by y 4 y e 64 4π We se he eplc olon A. The nmercal resls are shown n Fgs. 5 and 6. The condons of he calclaons are B M N 4 d and.. Accordng o he resls he accracy of he nmercal resls s hgh. The nmercal resl a. 5 s affeced by he bondary. Fgre 4. Nmercal resls a y ; a olon A.a; b olon A.b. 5.. Dffson of Maeral n Infne pace wh Advecon For smplcy we assme he condons gven n 47 and σ. We assme ha he advecon velocy s no zero. In he calclaons n econ 5. we consdered n he regon gven by < y < < y < B n compaon sng symmery. However snce hs problem s asymmerc we ms consder n he regon gven by < < y < < B y < B n compaon. Namely he sze and cener of elemens are gven by Fgre 5. pace dsrbon of n case of sorce ype nal vale y : a appromae solon; b eac solon.

9 Appled and ompaonal Mahemacs 4; 3: Fgre 6. pace dsrbon of obaned by appromae and eac solons: a appr. 75. ; b appr. 5. ; c eac 75. ; d eac eneralzaon o eneralzed Inegral Represenaon Mehod IRM 6.. eneralzed Inegral Represenaon Mehod IRM n D We nrodce a generalzed space fndamenal solon defned as 65 where s derved from when s specfed. If we se he assan fncon: ep π 66 hen and sasfy 3 ep π 67a 3 ep π 5 ep π 67b and. 68 Frhermore when s defned as 66 and ends o zero hen and sasfy 69 7 where s he Drac dela fncon: and for d 7 Assmng 66 s no snglar a and decreases rapdly as ncreases. These properes are sefl n he nmercal calclaons. We can ncrease he accracy of he nmercal negraon and redce he compaon me. From and 65 we have σ d. 7 Rewrng 7 we have d d d d σ. 73 Transformng he negral of he frs and hrd erms on he rgh-hand sde of 73 and echangng and we derve he generalzed negral represenaon: d [ ] d σ d. 74 Ths eqaon does no nclde he spaal dervave of he nknown varable n he negrals. When ends o zero hen 69 and 7 make he generalzed negral represenaon 74 ends o he dfferenal eqaon a IP.

10 4 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM If a IP and or a BP are known from he bondary condons hen he generalzed negral represenaons are negral eqaons wh he nknown varables a IP and or he nknown varables a BP. We oban d a IP from d d. If on IP s known from he nal condons hen we can solve he nal and bondary vale problem sng he generalzed negral represenaons. The generalzed negral represenaons are eqvalen o he dfferenal eqaons. When he dffson coeffcen and he advecve velocy are gven smply by cons 75a cons 75b f we assme and 74 are gven by ep π ep 5 π 3 ep 3 π d 76 d σ d. 77 [ ] [ ] 6.. Dffson of Maeral n D Infne pace wh Advecon We dscssed he same problem n econ 3.. For smplcy we assme 75 and σ. We assme he ranglar nal dsrbon of a maeral and he same compaon regon gven by < < s sed. If we assme >> and ± and ± 78 he generalzed negral represenaon s from 77 gven by d d 79 The nal vale s specfed as f < f < oherwse. 8 Nmercal resls are shown n Fg. 7. We sed procedre A for he nmercal calclaon. The calclaon condons are 4 N d and In he calclaons N or d was chosen assmng a proper vale of he local Pecle nmber d and hen d and were deermned keepng he local oran nmber d d and he seepness of he assan fncon d consan:. and. 65 respecvely. If N was ncreased hen he compaonal nose was elmnaed and he precse resls were obaned. Fgre 7. pace dsrbon of n case of ranglar nal dsrbon of a maeral: a appromae solon N 8 ; b appromae solon N 6 ; c appromae solon N 3 ; d eac solon eneralzed Inegral Represenaon Mehod IRM n D In he case of D he dffson eqaon s gven by 34 and we replace 65 by. 8 The eenson of he heory o 3D s sraghforward. We can se he assan fncon as a generalzed space fndamenal solon : r ep 8 π where r y η. Then we have r ep 4 83a π

11 Appled and ompaonal Mahemacs 4; 3: r r r ep ep b π π and. 84 When s defned as 8 and ends o zero hen sasfes 85 where s he Drac dela fncon n D: calclaon condons are B 4 M N 4 d.5.. and. 6. The accracy of he nmercal resls s hgh. y η. 86 From 34 and 8 we have { σ [ d. 87 ]} From 87 we derve he generalzed negral represenaon: d dl n n ndl d [ ] σ d. 88 Ths eqaon does no nclde he spaal dervave of he nknown varable n he negrals on. When ends o zero hen he generalzed negral represenaon 88 ends o he dfferenal eqaon gven by 34 a IP Dffson of Maeral n D Infne pace wh Advecon We dscssed a smlar problem n econ 5.. The compaon regon s gven by < < y < < B y < B n compaon. For smplcy we assme 47 and σ. The generalzed negral represenaon s from 88 gven by d d. 89 The nal vale s specfed as 8 8y y ep. 9 B Nmercal resls are shown n Fg. 8. We sed he eplc procedre A for he nmercal calclaon. The Fgre 8. pace dsrbon of n case of assan ype nal vale y : a appromae solon; b eac solon. 7. onclsons An negral represenaon s obaned sng a fndamenal solon of a dfferenal-ype bondary vale problem. If he bondary condons are sbsed no he negral represenaon an negral eqaon s obaned. In he D-problems he dffson of a maeral nally concenraed a he cener of he space wh and who he advecon was dscssed. The D-eamples revealed he basc aspecs of he negral represenaon mehod. mlar problems were also dscssed n D. As shown n Fgs. hrogh 6 he agreemen of he nmercal resls wh he analycal ones was sasfacory. The dffson and he advecon are properly calclaed n he eamples. Frhermore we proposed a generalzed negral represenaon mehod. The negral represenaon based on he prmary space-dfferenaon operaor s one of he generalzed negral represenaons. In he generalzed heory he fndamenal fncon s chosen frs and he dfferenal eqaon s defned properly reflecng he fndamenal solon and he bondary vale problem. In he nmercal eamples we sed he assan fncon as he generalzed fndamenal solon. The sably and he precson of he compaon were sasfacory as shown n Fgs. 7 and 8. Assmng he assan fncon as he generalzed fndamenal solon we can no only elmnae he snglary b also localze he effec of a pon on oher pons n s neghborhood. These properes are sefl n he nmercal calclaons. References [].A. Brebba J..F. Telles.. Wrobel 4.3 opled bondary elemen - Fne dfference mehods Bondary Elemen Technqes Theory and Applcaons n Engneerng prnger-verlag 984. [] B. Nayroles. Tozo P. Vllon eneralzng he fne elemen mehod: Dffse appromaon and dffse elemens omp. Mechancs

12 6 Hrosh Isshk e al.: olon of a Dffson Problem n a Non-Homogeneos Flow and Dffson Feld by he Inegral Represenaon Mehod IRM [3] T. Belyschko Y.Y.. Elemen-free alarkn mehods Inernaonal Jornal for Nmercal Mehods n Engneerng [4]. Yagawa Y. Yamda Free mesh mehod: A new meshless fne elemen mehod ompaonal Mechancs [5].B. cy A nmercal approach o he esng of he fsson hypohes The Asronomcal Jonal [6].R. M.B. moohed Parcle Hydrodyndmcs a meshfree parcle mehod. World cenfc; IBN [7] H. Isshk Dscree dfferenal operaors on rreglar nodes DDIN Inernaonal Jornal for Nmercal Mehods n Engneerng [8] H. Isshk Random ollocaon Mehod RM IMEE-3954 Vancover anada. [9] J.. hlman An negral eqaon formlaon of he eqaons of moon of an ncompressble fld NW-NPT Techncal Repor 86 5 Jly 99.