analytical evaluation of 3d bem integral representations ... · analytical evaluation of 3d bem...

Analytical evaluation of 3D BEM integral representations using complex analysis

Sonia Mogilevskaya

Department of Civil, Environmental, and Geo-Engineering University of Minnesota

January 15, 2015

BIRS Workshop 15w5052 Modern Applications of Complex Variables: Modeling, Theory and Computation

Boundary Element Methods

The Boundary Element Methods represent a family of general numerical techniques for solving a large class of engineering problems that can be reduced to the mathematical boundary value problems

Engineering Problems as Boundary Value Problems

Prescribed data at the boundary

Region governed by a differential equation

Initial conditions for time-dependent problems

Boundary Value Problem

Boundary Element Methods

M V

S

Main Idea

Governing differential equation

Integral representations

describe the fields at the point M via the integrals over some data at the boundary S

Boundary Integral equations

M M0 ∈ S

M0

describe the fields at the point M0 on the boundary via the integrals over prescribed boundary data

BEM Structure

Fundamental solution

Indirect BEM Direct BEM

Potentials (integrals of fundamental

solutions)

Integral identities (e.g.

reciprocal theorem)

Boundary Integral Equation

Boundary Value Problem

exactly satisfies governing equation everywhere but one point

express the fields in the domain via boundary data and fundamental solutions

Integral Representations: Laplace Equation

Integral Representations

u(x) = 14π

ϕ(ξ )r

dSξS∫

G x,ξ( ) = 14π1r

∇2u = ∂2u∂x2

+∂2u∂y2

+∂2u∂z2

= 0x

ξ r

r = x k−ξk( )2k∑

source

∇2G x,ξ( ) = 0, r ≠ 0


u(x) = 14π

ψ(ξ ) ∂∂n ξ( )

1rdSξ

S∫

SRegion of interest

14π

∂∂n x( )

ϕ(ξ )1rdSξ

S∫ 1

4π∂

∂n x( )ψ(ξ ) ∂

∂n ξ( )1rdSξ

S∫

Single-layer potential Double-layer potential

S

ϕ (ξ)

Integral Representations: Helmholtz Equation


u(x) = 14π

ϕ(ξ )rexp ikr( )dSξ

S∫

G x,ξ( ) = 14π1rexp(ikr)

∇2u+ k2u = 0, u = u x,ω( ) = Re u x( )exp −iωr( )#$ %&, k =ω / c


u(x) = 14π

ψ(ξ ) ∂∂n ξ( )

1rexp ikr( )

"

#$%

&'dSξ

S∫

SRegion of interest

14π

∂∂n x( )

ϕ(ξ )1rexp ikr( )dSξ

S∫ 1

4π∂

∂n x( )ψ(ξ ) ∂

∂n ξ( )1rexp ikr( )

"

#$%

&'dSξ

S∫


wave number sound speed frequency

Integral Representations: Navier-Cauchy Equation


u(x) = 14π

ϕ(ξ )U x,ξ( )dSξS∫

Umj x,ξ( ) = 116πµ 1−ν( ) r

3− 4ν( )δmj + r,mr,j"#

$%

λuk,km +µ um,kk + uk,km( ) = 0


u(x) = 14π

ψ(ξ )TT ξ ,x( )dSξS∫

SRegion of interest

t(x) = 14π

ϕ(ξ )T x,ξ( )dSξS∫ t(x) = 1

4πψ(ξ )H x,ξ( )dSξ

S∫


u,k = ∂u / ∂xk

Discretization of the boundary ( elements)

Numerical Solution

Approximation of the functions

Evaluation of the integrals

Final system of equations

Solution of the system of equations

Computation of the fields at the boundary and inside the domain

limit after integration

limit before integration

Evaluation of Integrals

Integrals

ΦmG x,ξ( )dSxS ( )∫ , Φm

∂G x,ξ( )∂nx

dSxS ( )∫

shape functions

x ( ) ≈ Φq( )

q∑ xq( )

u ( ) ≈ Φm( )

m∑ um( )

Isoparametric elements

S = Skk=1

N

element

node node

node

Fundamental solutions (scalar or vector functions)

Most consuming part of the BEM procedure !

Why Analytical Integration?

•  Singular, hypersingular, and near singular integrals appear in the “limit before integration procedure.” Lack of reliable quadrature rules.

•  Analytical integration leads to higher accuracy of computation and to the reduction of its cost.

•  Analytical integration may facilitate the use of the so-called fast methods (e.g. fast multipole method) for solving large systems of algebraic equations.

•  Analytical integration routines can be used as “black boxes” by the developers of the BEM-based software.

Wirtinger Calculus

f (x,y)

New independent variables

Wirtinger derivatives

∂∂z=12

∂∂x− i ∂∂y

#

$%

&

'(;∂∂z=12

∂∂x+ i ∂∂y

#

$%

&

'(

Wilhelm Wirtinger

1865 –1945

x = z+ z2, y = z− z

2i

Interrelations between variables

z = x + iy, z = x − iy

f (z,z)

Complex Integral Theorems

14

f∇2g−g∇2f( )S∫ dzdz = − f ∂g

∂zdz+g ∂g

∂zdz

%

&'

(

)*

∂S∫

Green’s representation formula

Cauchy- Pompeiu formulae

Dimitrie Pompeiu

1873 –1954

Gauss theorems

∂SS

∂f∂zS∫ dS = − i

2f dz

∂S∫ , ∂f

∂zS∫ dS = i

2fdz

∂S∫

12π i

f τ( )τ − z∂S∫ dτ − 1

π

∂f τ( )∂τ

dSτ − zS

∫ =f z( ) z ∈ S

0 z ∉ S∂S

&'(

)(

−12π i

f τ( )τ − z∂S∫ dτ − 1

π

∂f τ( )∂τ

dSτ − zS

∫ =f z( ) z ∈ S

0 z ∉ S∂S

&'(

)(

Complex Notations for Plane Elements

r2 = τ − z( ) τ − z( )+ h2, dSξ = i / 2( )dτdτn = n1 x( )+ in2 x( ), n3 x( )τ = ξ1 + iξ2 , z = x1 + ix2, h = ξ3 − x3

Complex notation geometry

Complex notations for the fields (elasticity)

ξ

ξ1

ξ2

ξ3

x

z

h r

n(ξ)

t = t1 x( )+ it2 x( ), t3 x( )u = u1 ξ( )+ iu2 ξ( ), u3 x( )

Generic Integrals

τ − z( )mτ − z( )

n

rS∫ dSξ , m,n = 0,1,…

Potential and Elasticity Problems

Acoustics

τ − z( )mτ − z( )

n

rS∫ exp(ikr)dSξ , m,n = 0,1,…

+ their derivatives of various orders over z, z, h, k

Representation of Generic Integrals

τ − z( )mτ − z( )

n

rS∫ dSξ , m,n = 0,1,…

can be re-written as

∂f z( )∂zS

∫dSξτ − z

where

f z( ) =−2 r / h2( ) τ − z( )

m+1τ − z( )

n+1

2F1 1,n +3 / 2,3 / 2,r2 / h2( ) h ≠ 0

−2r 2n +1( )−1τ − z( )

mτ − z( )

nh = 0

#

$%%

&%%

hypergeometric functions


Representation of Generic Integrals

τ − z( )mτ − z( )

n

rS∫ exp ikr( )dSξ , m,n = 0,1,…

can be re-written as

∂f z( )∂zS

∫dSξτ − z

where

∂f z( )∂z

=τ − z( )

m+1τ − z( )

n

rexp ikr( )

Acoustics

Reduction of Generic Integral to a Contour Integral


τ − z( )mτ − z( )

n

rS∫ dSξ =

12i

f τ( )dττ − z∂S

∫ − f z( )π z ∈ Sγ / 2 z ∈∂S0 z ∉ S

&

'(

)(

Constant, linear, and quadratic approximation

f τ( ) =2r τ − z( )

mn = 0

23 r r

2 −3h2( ) τ − z( )m−1

n =1

215 r 3r

4 −10h2r2 +15h4( ) τ − z( )m−2

n = 2

"

#

$$$

%

$$$

r2 = τ − z( ) τ − z( )+ h2

internal angle at the point z

Reduction of Generic Integral to a Contour Integral

Acoustics

τ − z( )mτ − z( )

n

rS∫ exp ikr( )dSξ =

12i

f τ( )dττ − z∂S

∫ − f z( )π z ∈ Sγ / 2 z ∈∂S0 z ∉ S

&

'(

)(

Constant, linear, and quadratic approximation

f τ( ) =2ik−1 τ − z( )

mexp(ikr) n = 0

2ik−3 τ − z( )m−1exp(ikr) −2+ 2ikr + k2 r2 − h2( )"

#$% n =1

2ik−5 τ − z( )m−2exp(ikr) 24+ k −8kh 2−24ir + k r2 − h2( )( ) k2 r2 − h2( )+ 4ikr −12( )"

#&$%' n = 2

(

)

***

+

***

r2 = τ − z( ) τ − z( )+ h2

Analytical Evaluation of Generic Integral


Straight boundary segment

( )jjj

jjj a

aaaa

a −−

−+=

+

+ ττ1

1

(a j,a j+1)

( ) 41,1

≤≤−−∫+

kdzrj

j

a

a

k ττ

f τ ,τ( )τ − z

dτ∂S∫ , f =

2r τ − z( )m

n = 0

23 r r

2 −3h2( ) τ − z( )m−1

n =1

215 r 3r

4 −10h2r2 +15h4( ) τ − z( )m−2

n = 2

$

%

&&&

'

&&&

Elementary integrals (evaluated in closed-form )

Circular arc of radius R with the center zc

τ = zc + R2

τ − zcj

r τ − zc( )kdτ

a j

a j+1

∫ , −1≤ k ≤ 4


Acoustics

Straight boundary segment

( )jjj

jjj a

aaaa

a −−

−+=

+

+ ττ1

1

(a j,a j+1)

rp τ − z( )kexp(ikr)dτ

a j

a j+1

∫

Elementary integrals

Circular arc of radius R with the center zc

τ = zc + R2

τ − zcj

rp τ − zc( )kexp(ikr)dτ

a j

a j+1

∫ , −1≤ k ≤ 4

In general cannot be evaluated in closed forms


Acoustics

Basic requirement for the BEM discretization: 5-10 elements per wavelength

Asymptotic expansion

rp τ − zc( )kdτ

a j

a j+1

∫exp(ikr) = exp(ikr0 ) 1+ ikδr +…+in

n!kδr( )

n+…

!

"#

$

%&

Resulting integrals

Rapidly converging series

Example Elements

Results: Potential & Elasticity Problems

1rS∫ dSξ , h =1

!!

Results: Potential & Elasticity Problems

1rS∫ dSξ , h =1

Table 3 Comparison of numerical and analytical integration for (S dS/r, h = 1 and a

circular-sector element. The numerical results use N × N Gauss points for three values of N

z Analytical N = 3 N = 9 N = 20

−0.2 + 0.4 i 0.64624210 0.64842122 0.64634795 0.64625259 −0.4 − 0.2 i 0.53858071 0.54060724 0.53867925 0.53859048 0.2 − 0.4 i 0.58590952 0.58854325 0.58603679 0.58592213 0.4 + 0.2 i 0.72395773 0.72701346 0.72410217 0.72397203 0.8 + 0.4 i 0.70127134 0.70456037 0.70142764 0.70128680

4 + 2 i 0.19479573 0.19577969 0.19484158 0.19480026

!!

Results: Acoustics

1rS∫ exp ikr( )dSξ , h = 0

!!

!!

Num: Gauss quadrature 50x50 points

Results: Acoustics

!!

1rS∫ exp ikr( )dSξ , h = 0

!!

Num: Gauss quadrature 50x50 points

Conclusions •  The approach provide closed form expressions for three-dimensional integrals involved in the integral representations of potential, elasticity, and acoustic scattering problems in case of planar elements bounded by straight segment and circular arcs

•  All the integrals involved can be reduced to a few generic integrals and their derivatives with respect to specific parameters

•  The proposed approach could be extended on the case of non-planar elements that could be mapped to planar ones using rational mapping functions

•  The proposed approach can be extended to elastodynamic problems

•  The approach could be employed to create integration subroutines or functions that can be used as “black boxes” by the developers of the BEM software •  The approach could potentially have applications in some FEM integral representations

Acknowledgements

Potential and elasticity problems: with Dmitry Nikolskiy “The use of complex integral representations for analytical evaluation of three-dimensional BEM integrals Potential and elasticity problems” 2014. Q. J. Mech. Appl. Math. 67, 505-523.

Acoustics: with Fatemeh Pourahmadian “Complex variables-based approach for analytical evaluation of three-dimensional integral representations of acoustic scattering” 2015. Eng. Anal. Bound. Elements. 53, 9-17.

This is a collaborative work with Dmitry Nikolskiy and Fatemeh Pourahmadian

Department of Civil, Environmental, and Geo Engineering, UMN

References

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