Available online at ScienceDirect. Procedia Engineering 153 (2016 ) 16 23

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1 Availale olie at wwwsciecediectcom ScieceDiect Pocedia Egieeig 5 (06 6 XXV Polish Russia Slovak Semia heoetical Foudatio of Civil Egieeig Semiaalytical stuctual aalysis ased o comied applicatio of fiite elemet method ad discete-cotiual fiite elemet method Pat : hee-dimesioal theoy of elasticity Pavel A Akimov ac * Oleg A Negozov a a Moscow State Uivesity of Civil Egieeig (Natioal Reseach Uivesity 6 Yaoslavskoye Shosse Moscow 97 Russia Russia Academy of Achitectue ad Costuctio Scieces 4 Ulitsa olshaya Dmitovka Moscow 070 Russia c Reseach & Educatioal Cete StaDyO Office ya Ulitsa Yamskogo Polya Moscow 5040 Russia Astact his pape is devoted to semiaalytical stuctual aalysis ased o comied applicatio of fiite elemet method (FEM [] ad discete-cotiual fiite elemet method (DCFEM [-9] ouday polems of thee-dimesioal theoy of elasticity (static aalysis of thee-dimesioal stuctue [] ae ude cosideatio the give domai is emodeed y exteded oe he field of applicatio of DCFEM compises stuctues with egula (costat o piecewise costat physical ad geometical paametes i some dimesio ( asic dimesio DCFEM pesupposes fiite elemet mesh appoximatio fo o-asic dimesios of exteded domai while i the asic dimesio polem emais cotiual Coespodig discete ad discete-cotiual appoximatio models fo sudomais ad coupled multilevel appoximatio model fo exteded domai ae ude cosideatio ief ifomatio aout softwae ad veificatio sample ae peseted as well 06 he Authos Pulished y y Elsevie Elsevie Ltd Ltd his is a ope access aticle ude the CC Y-NC-ND licese ( Pee-eview ude esposiility of the ogaizig committee of the XXV Polish Russia Slovak Semia heoetical Pee-eview Foudatio ude of Civil esposiility Egieeig of the ogaizig committee of the XXV Polish Russia Slovak Semia heoetical Foudatio of Civil Egieeig Keywods: discete-cotiual fiite elemet method fiite elemet method semiaalytical stctual aalysis thee-dimesioal theoy of elasticity * Coespodig autho el: +7( fax: +7( addess: akimov@aasu he Authos Pulished y Elsevie Ltd his is a ope access aticle ude the CC Y-NC-ND licese ( Pee-eview ude esposiility of the ogaizig committee of the XXV Polish Russia Slovak Semia heoetical Foudatio of Civil Egieeig doi:006/poeg

2 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( Fomulatio of the polem ad otatio system Let s coside polem of static aalysis of thee-dimesioal stuctue loaded y cocetated foce with higed eds (coss-sectios alog asic dimesio (Fig Some elemets of otatio system is peseted at Fig Fig Cosideig thee-dimesioal stuctue Let s e domai occupied y stuctue {( x x :0 x l 0 x l 0 x l} whee ad {( x x x :0 x l 0 x l x x x } k x x x ae coodiates k k k ( x coespods to asic dimesio x 0 x l x l l l ae coodiates of coespodig ouday poits (coss-sectios alog asic dimesio ad ae sudomais of ad ae exteded sudomais emodeig sudomais ad dc N dc dc dc x x i N ae coodiates (alog x ad x of odes (odal lies of discete-cotiual fiite elemets which ae used fo appoximatio of domai ( N dc ad ( N dc ae the umes of discete-cotiual fiite elemets alog coodiates x ad x x x x i N N N ae coodiates (alog x x ad x of odes of fiite elemets which ae used fo appoximatio of domai ( N ( N ad ( N is the ume of fiite elemets alog coodiates x x ad x

3 8 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( 06 6 Fig Coss-sectios of cosideig thee-dimesioal stuctue Let s coside the case of ectagula mesh appoximatio of domai (Fig : x x x x x x i N N N i ( hee-idex otatio system is used fo umeig of discete-cotiual fiite elemets ypical ume of has the fom ( k whee k is the ume of sudoma i ad ae umes of elemets (alog x ad x Fou-idex system is used fo umeig of fiite elemets ypical ume of has the fom ( k whee k is the ume of dc dc sudoma i ad k ae umes of elemets (alog x x ad x Let s N N N ad N N N dc x x i N N N Let s coside the simples case: dc x x x i N N N ( x i x i N N x x i N N ( Discete-cotiual appoximatio model fo sudomai Discete-cotiual appoximatio model is used fo thee-dimesioal polems It pesupposes mesh appoximatio fo o-asic dimesios of exteded domai (alog x ad x while i the asic dimesio (alog x polem emais cotiual hus exteded sudomai is divided ito discete-cotiual fiite elemets N N ( x x x : x x x x x x x x x } (4 i { i i Lame costats fo discete-cotiual fiite elemet ae defied y fomulas: i (5

4 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( (6 0 whee is the chaacteistic fuctio of elemet ( ( ( ( ( ( asic odal ukow fuctios ae displacemet compoets u u u ad thei deivatives v v v with espect to x (supescipt heeiafte coespods to the ume of cosideed sudomai ie hus fo ode ( ( ( ( ( ( ( i we have the followig ukow fuctios: u u u ad v v v iliea appoximatio is used fo ukow fuctios withi discete-cotiual fiite elemet DCFEM is educed at some stage to the solutio of systems of 6N N fist-ode odiay difetial euatios: U x A U ( x R ( (7 ( x whee U ( x is gloal vecto of odal ukow fuctios (suscipt coespods to the ume of sudomai U ] U( x [ ( u (8 u u ( x [ ( u (9 v v ( x [ ( ( ( u ( ( ( u (0 ( ( ( u u ( x [ u ( u ( u ] ( N ( N A is gloal matix of coefficiets of ode ( u ( u ( ( u ( ( ( N ( ( N ( u ( N ( u ( N ( N ( N v ] ( N ( u N ( N N ( ( ( ( ( v ( x [ v v v ( 6N N R ( x is the ight-side vecto of ode 6N N Coect aalytical solutio of (7 is defied y fomula ] ] U whee E ( x x E ( x C S ( ( x ( x x ( x x S x ( x R ( ( ( ( x ( x is the fudametal matix-fuctio of system (4 which is costucted i the special fom coveiet fo polems of stuctual mechaics [] is covolutio otatio C is the vecto of costats of ode 6N N Discete (fiite elemet appoximatio model fo sudomai Discete (fiite elemet appoximatio model fo the cosideig thee-dimesioal polems pesupposes fiite elemet appoximatio alog x x ad x hus exteded sudomai is divided ito fiite elemets N N N ( x x : x x x x x x x x x } (4 i { i i

5 0 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( 06 6 Lame costats fo fiite elemet ae defied y fomulas: i (5 0 (6 whee i is the chaacteistic fuctio of elemet i asic odal ukows ae displacemet compoets u u (supescipt heeiafte coespods to the ( ( u ume of cosideed sudomai ie hus fo ode ( i we have the followig ukows: ( ( ( u u i u iliea appoximatio of ukows is used withi fiite elemet (covetioal thee-dimesioal paallelepipedic 8-ode fiite elemet of thee-dimesioal polem of elasticity theoy As kow FEM is educed to the solutio of systems of N NN liea algeaic euatios: ( K (7 U R whee U is gloal vecto of odal ukows (suscipt coespods to the ume of sudomai ( { U } k i g u (8 i g is the gloal idex of elemet of vecto U k ae coespodig local idexes ig ig ( NN ig ( NN ( N k i (9 NN N i ( N N ( N i g (0 K is gloal stifss matix of ode NN vecto [a] is the itegal pat of a N R is gloal ight-side vecto of ode N NN (gloal load 4 Multilevel appoximatio model fo domai System (7 ca e ewitte fo all odes with idexes N (ie (esolvig system of N ( N liea algeaic euatios: x x x i the followig fom K U R ( whee K is educed gloal stifss matix of size [ NN ( N ] [NN N] R is educed ight-side vecto of ode N N ( N ouday coditios at sectio u (higed edge has the fom ( N N euatios: x x ( ( x N ( ( 0 0 u ( x 0 0 u ( x 0 0 i N (

6 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( 06 6 Euatios ( ca e ewitte i matix fom: U ( x 0 g ( whee is matix of ouday coditios of size N 6NN N with elemets defied y fomula { N N (4 } p p p NN 6 g is the zeo vecto of ode N N (ie g 0 p is Koecke delta p p 0 p (5 Afte sustitutio of ( ito ( it ca e otaied that E x 0 C g S ( x 0 o QC G (6 ( whee Q is the matix of size NN 6NN G is the vecto of ode N N (7 Q E x 0 G g S x 0 ( ( ouday coditios at sectio (pect cotact has the fom ( 6N N euatios: x x u ( ( ( ( x 0 u i N N (8 ( x 0 ( ( ( x 0 ( ( ( x ( 0 i N N ( ( whee ( ad ae odal fuctios (afte coespodig aveagig of stess compoets ( x ( x ( x ( x ad ( x fo discete-cotiual fiite elemet ( i ( compoets ad (afte coespodig aveagig fo fiite elemet ( i Euatios ( ad ( ca e ewitte i matix fom: (9 ( ( ad ae odal stess U x 0 ( U (0 whee is matix of ouday coditios of size 6NN 6NN which ca e costucted i accodace with so-called method of asis vaiatios [-9] is matix of ouday coditios of size 6NN NNN which ca e costucted i accodace with method of asis vaiatios [-9] Afte sustitutio of ( ito (0 it ca e otaied that E( x 0 C U S( x 0 o QC QU G ( whee Q is the matix of size 6NN 6NN Q is the matix of size 6NN NNN G is the vecto of ode 6N N

7 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( 06 6 Q E x 0 ( Q G S x 0 ( ouday coditios at sectio x x ( (higed edge has the fom ( N N euatios: u ( ( ( u i 0 u 0 i N N 0 N ( Euatios ( ca e ewitte i matix fom: (4 U g whee is matix of ouday coditios of size N NN N N with elemets defied y fomula { N N N (5 } p p p NN g is the zeo vecto of ode N N (ie g 0 hus the total ume of euatio is eual to NN N 6NN Coespodig coupled system of NN N 6NN liea algeaic euatios with NN N 6NN ukows has the fom: Q Q G Q G (6 K U R g It should e oted that ouday coditios (4 ca e take ito accout automatically withi costuctio of gloal stifss matix ad gloal ight-side vecto coespodig to sudomai he we get (istead of (6: Q Q 0 0 G Q G (7 U K R whee K is coespodig educed gloal stifss matix of size [ N ( ] [ ] N N NN N R is coespodig educed gloal ight-side vecto of ode N N ( N Stai ad stess compoets ae computed accodig to well-kow fomulas afte solvig of system (7 5 Softwae ad veificatio samples We should stess that all methods ad algoithms cosideed i this pape have ee ealized i softwae he mai pupose of Aalysis system CSASAD (DCFEM + FEM is semiaalytical stuctual aalysis (static aalysis of thee-dimesioal stuctues withi thee-dimesioal theoy of elasticity ased o comied applicatio of FEM ad DCFEM Pogammig eviomet is Micosoft Visual Studio 0 Commuity ad Itel Paallel Studio 05XE [0] with Itel MKL Liay Softwae is desiged fo Micosoft Widows 8/0 Coespodig veificatio samples (ANSYS Mechaical 50 [67] was used fo veificatio puposes poved that DCFEM is moe efctive i the most citical vital potetially dageous aeas of stuctue i tems of factue

8 Pavel A Akimov ad Oleg A Negozov / Pocedia Egieeig 5 ( 06 6 (aeas of the so-called edge efcts whee some compoets of solutio ae apidly chagig fuctios ad thei ate of chage i may cases ca t e adeuately take ito accout y the stadad FEM [] Ackowledgemets he Repoted study was Fuded y Govemet Pogam of the Russia Fedeatio Developmet of sciece ad techology (0-00 withi Pogam of Fudametal Reseaches of Miisty of Costuctio Housig ad Utilities of the Russia Fedeatio ad Russia Academy of Achitectue ad Costuctio Scieces the Reseach Poect 7 Reeces [] OC Ziekiewicz RL aylo DD Fox he Fiite Elemet Method fo Solid ad Stuctual Mechaics uttewoth-heiema 0 [] PA Akimov Coect discete-cotiual fiite elemet method of stuctual aalysis ased o pecise aalytical solutios of esultig multipoit ouday polems fo systems of odiay difetial euatios Applied Mechaics ad Mateials Vols ( [] PA Akimov AM elostosky ML Mozgaleva M Aslam OA Negozov Coect multilevel discete-cotiual fiite elemet method of stuctual aalysis Advaced Mateials Reseach Vol 040 ( [4] PA Akimov ML Mozgaleva Method of exteded domai ad geeal piciples of mesh appoximatio fo ouday polems of stuctual aalysis Applied Mechaics ad Mateials Vols ( [5] PA Akimov ML Mozgaleva VN Sidoov Aout veificatio of discete-cotiual fiite elemet method of stuctual aalysis Pat : hee-dimesioal polems Pocedia Egieeig Vol 9 ( [6] PA Akimov OA Negozov O the use of discete-cotiual fiite elemet method with ustuctued meshes Pocedia Egieeig Vol (05 8- [7] PA Akimov OA Negozov O the use of discete-cotiual fiite elemets with tiagula coss-sectio fo semiaalytical stuctual aalysis Pocedia Egieeig Vol ( [8] PA Akimov ML Mozgaleva OA Negozov Aout veificatio of discete-cotiual fiite elemet method fo two-dimesioal polems of stuctual aalysis Pat : Deep eam with costat physical ad geometical paametes alog asic diectio Advaced Mateials Reseach Vols ( [9] PA Akimov ML Mozgaleva OA Negozov Aout veificatio of discete-cotiual fiite elemet method fo two-dimesioal polems of dtuctual aalysis Pat : Deep eam with piecewise costat physical ad geometical paametes alog asic diectio Advaced Mateials Reseach Vols ( [0] RJ Haso Hopkis Numeical computig with mode fota (Applied mathematics SIAM-Society fo Idustial ad Applied Mathematics 0 [] EM Alawadh Fiite elemet simulatios usig ANSYS CRC Pess 009

Mathematics; 26, Yaroslavskoe Shosse, Moscow, , Russia

Mathematics; 26, Yaroslavskoe Shosse, Moscow, , Russia MAEC We of Coreces 7, 000 (07 DOI: 0.05/ mateccof/077000 XXVI R-S-P Semiar 07, heoretical Foudatio of Civil Egieerig Aout solutio of multipoit oudary prolem of static aalysis of deep eam with the use of