Analytical evaluation of 3D BEM integral representations using complex analysis

Transcription

1 BIR Wokshop 5w5052 Moden Applications of Complex Vaiables: Modeling, Theoy and Computation Analytical evaluation of 3D BEM integal epesentations using complex analysis onia Mogilevskaya Depatment of Civil, Envionmental, and Geo-Engineeing Univesity of Minnesota Januay 5, 205

2 Bounday Element Methods Engineeing Poblems as Bounday Value Poblems The Bounday Element Methods epesent a family of geneal numeical techniques fo solving a lage class of engineeing poblems that can be educed to the mathematical bounday value poblems Bounday Value Poblem Pescibed data at the bounday Region govened by a diffeential equation Initial conditions fo time-dependent poblems

3 Bounday Element Methods Main Idea V M Govening diffeential equation M0 Integal epesentations descibe the fields at the point M via the integals ove some data at the bounday Bounday Integal equations M M0 descibe the fields at the point M0 on the bounday via the integals ove pescibed bounday data

4 BEM tuctue Bounday Value Poblem Fundamental solution Indiect BEM Diect BEM exactly satisfies govening equation eveywhee but one point Potentials (integals of fundamental solutions) Integal identities (e.g. ecipocal theoem) expess the fields in the domain via bounday data and fundamental solutions Bounday Integal Equation

5 Integal Repesentations: Laplace Equation = k x ξ ( x k ξ k ) 2 souce 2 u = 2 u x u y u z 2 = 0 Fundamental solution G( x,ξ ) = 4π 2 G( x,ξ ) = 0, 0 Region of inteest ϕ (ξ) ingle-laye potential u(x) = 4π ϕ(ξ) Integal Repesentations d ξ Double-laye potential u(x) = 4π ψ(ξ) n ξ d ξ 4π n x ϕ(ξ) d ξ 4π n x ψ(ξ) n ξ d ξ

6 Integal Repesentations: Helmholtz Equation 2 u + k 2 u = 0, u = u x,ω = Re u x #$ exp( iω ) % &, k = ω / c wave numbe fequency Fundamental solution sound speed G( x,ξ ) = 4π exp(ik) Region of inteest u(x) = 4π ingle-laye potential ϕ(ξ) Integal Repesentations exp( ik)d ξ Double-laye potential u(x) = 4π ψ(ξ) n ξ " exp ik # $ % & ' d ξ 4π n x ϕ(ξ) exp( ik)d ξ 4π n x ψ(ξ) n ξ " exp ik # $ % & ' d ξ

7 Region of inteest Integal Repesentations: Navie-Cauchy Equation u(x) = 4π U mj λu k,km + µ ( u m,kk + u ) k,km = 0 ( x,ξ) = ingle-laye potential "( 3 4ν )δ 6πµ ( mj +,m $,j ν # % ) Integal Repesentations ϕ(ξ)u ( x,ξ)d ξ Fundamental solution t(x) = 4π ϕ(ξ)t ( x,ξ )d ξ t(x) = 4π u,k = u / x k Double-laye potential u(x) = ( 4π ψ(ξ)tt ξ,x)d ξ ψ(ξ)h ( x,ξ)d ξ

8 Numeical olution Discetization of the bounday ( elements) limit befoe integation Appoximation of the functions Evaluation of the integals Final system of equations limit afte integation olution of the system of equations Computation of the fields at the bounday and inside the domain

9 Evaluation of Integals ( x ) Φ q q Isopaametic elements ( ) x q node = N k= k shape functions Integals Φ m G( x,ξ )d, Φ G x,ξ x m n x ( ) u Φ m m u m ( ) node d x element Fundamental solutions (scala o vecto functions) node Most consuming pat of the BEM pocedue!

10 Why Analytical Integation? ingula, hypesingula, and nea singula integals appea in the limit befoe integation pocedue. Lack of eliable quadatue ules. Analytical integation leads to highe accuacy of computation and to the eduction of its cost. Analytical integation may facilitate the use of the so-called fast methods (e.g. fast multipole method) fo solving lage systems of algebaic equations. Analytical integation outines can be used as black boxes by the developes of the BEM-based softwae.

11 Witinge Calculus New independent vaiables z = x + iy, z = x iy Wilhelm Witinge f (x,y) f (z,z) Inteelations between vaiables x = z + z 2, y = z z 2i Witinge deivatives z = 2 # x i & % (; $ y ' z = # 2 x + i & % ( $ y '

12 Complex Integal Theoems Geen s epesentation fomula Dimitie Pompeiu 4 % ( f 2 g g 2 f ) dzdz = f g g ' dz + g & z z dz ( * ) Gauss theoems f z d = i 2 f dz, f z d = i 2 fdz Cauchy- Pompeiu fomulae πi 2πi f ( τ ) dτ τ z π f ( τ ) τ z f τ & d τ τ z = ( ' )( dτ f τ & d π τ τ z = ( ' )( f ( z) z 0 z f ( z) z 0 z

13 Complex Notations fo Plane Elements Complex notation geomety ξ 3 ξ ξ 2 τ z ( x) + in 2 x 2 = τ z n = n + h2, d ξ = i / 2, n 3 x dτdτ ξ n(ξ) x h z τ = ξ + iξ 2, z = x + ix 2, h = ξ 3 x 3 Complex notations fo the fields (elasticity) t = t ( x) + it 2 ( x), t 3 x u = u ( ξ) + iu 2 ( ξ), u 3 x

14 Geneic Integals Potential and Elasticity Poblems n ( τ z) m τ z d ξ, m,n = 0,, Acoustics ( τ z) m ( τ z) n exp(ik)d ξ, m,n = 0,, + thei deivatives of vaious odes ove z, z, h, k

15 Repesentation of Geneic Integals Potential and Elasticity Poblems n ( τ z) m τ z can be e-witten as f ( z) z d ξ, m,n = 0,, d ξ τ z f ( z) = # % $ % &% whee 2( / h 2 )( τ z) m+ ( τ z) n+ 2( 2n +) ( τ z) m τ z hypegeometic functions F,n + 3 / 2,3 / 2 ( 2,2 / h 2 ) h 0 n h = 0

16 Repesentation of Geneic Integals ( τ z) m ( τ z) n can be e-witten as whee Acoustics f ( z) z exp( ik)d ξ, m,n = 0,, d ξ τ z f ( z) z = ( τ z) m+ ( τ z) n exp( ik)

17 Reduction of Geneic Integal to a Contou Integal Potential and Elasticity Poblems n ( τ z) m τ z d ξ = 2i f ( τ )dτ τ z f z & ( ' ( ) π γ / 2 z z 0 z Constant, linea, and quadatic appoximation intenal angle at the point z f ( τ ) = " $ $ # $ $ % $ 2( τ z) m n = ( 2 3h 2 ) τ z τ z h h 4 m n = m 2 n = 2 2 = ( τ z) ( τ z) + h2

18 Reduction of Geneic Integal to a Contou Integal Acoustics ( τ z) m ( τ z) n exp( ik)d ξ = 2i f ( τ )dτ τ z f z & ( ' ( ) π z γ / 2 z 0 z Constant, linea, and quadatic appoximation f ( τ ) = ( * * ) * * + * 2ik ( τ z) exp(ik) n = 0 2ik 3 ( τ z) exp(ik) " 2 + 2ik + k 2 ( 2 h 2 $ # )% n = 2ik 5 ( τ z) exp(ik) " 24 + k( 8kh 2 24i + k( 2 h 2 ))( ( k2 2 h 2 ) + 4ik 2) $ #& %' n = 2 2 = ( τ z) ( τ z) + h2

19 Analytical Evaluation of Geneic Integal taight bounday segment τ = a j + Potential and Elasticity Poblems a a j+ j+ a a j j ( τ a ) (a j,a j+ ) j Cicula ac of adius R with the cente z c τ = zc + R 2 τ z cj $ & 2( τ z) m n = 0 f ( τ,τ ) τ z dτ, f = & 2 % 3 ( 2 3h 2 )( τ z) m n = & & 2 5 ( 34 0h h 4 )( τ z) m 2 n = 2 '& Elementay integals (evaluated in closed-fom ) a j+ a j k ( τ z) dτ, k 4 a j+ k dτ τ z c, k 4 a j

20 Analytical Evaluation of Geneic Integal Acoustics taight bounday segment τ = a j + a a j+ j+ a a j j ( τ a ) (a j,a j+ ) j Cicula ac of adius R with the cente z c τ = zc + R 2 τ z cj Elementay integals a j+ a j p ( τ z) k exp(ik)dτ a j+ k exp(ik)dτ p τ z c, k 4 a j In geneal cannot be evaluated in closed foms

21 Analytical Evaluation of Geneic Integal Acoustics Basic equiement fo the BEM discetization: 5-0 elements pe wavelength Asymptotic expansion! exp(ik) = exp(ik 0 )# + ikδ + + in " n! kδ Rapidly conveging seies n + $ & % Resulting integals a j+ a j p ( τ z c ) k dτ

22 Example Elements

23 Results: Potential & Elasticity Poblems d, h = ξ

24 Results: Potential & Elasticity Poblems d, h = ξ Table 3 Compaison of numeical and analytical integation fo ( d/, h = and a cicula-secto element. The numeical esults use N N Gauss points fo thee values of N!! z Analytical N = 3 N = 9 N = i i i i i i

25 Results: Acoustics exp ( ik )d ξ, h = 0! Num: Gauss quadatue 50x50 points!

26 Results: Acoustics exp ( ik )d ξ, h = 0! Num: Gauss quadatue 50x50 points!

27 Conclusions The appoach povide closed fom expessions fo thee-dimensional integals involved in the integal epesentations of potential, elasticity, and acoustic scatteing poblems in case of plana elements bounded by staight segment and cicula acs All the integals involved can be educed to a few geneic integals and thei deivatives with espect to specific paametes The poposed appoach could be extended on the case of non-plana elements that could be mapped to plana ones using ational mapping functions The poposed appoach can be extended to elastodynamic poblems The appoach could be employed to ceate integation suboutines o functions that can be used as black boxes by the developes of the BEM softwae The appoach could potentially have applications in some FEM integal epesentations

28 Acknowledgements This is a collaboative wok with Dmity Nikolskiy and Fatemeh Pouahmadian Depatment of Civil, Envionmental, and Geo Engineeing, UMN Refeences Potential and elasticity poblems: with Dmity Nikolskiy The use of complex integal epesentations fo analytical evaluation of thee-dimensional BEM integals Potential and elasticity poblems 204. Q. J. Mech. Appl. Math. 67, Acoustics: with Fatemeh Pouahmadian Complex vaiables-based appoach fo analytical evaluation of thee-dimensional integal epesentations of acoustic scatteing 205. Eng. Anal. Bound. Elements. 53, 9-7.