Boundary Integral Methods and Applications to Fluid Mechanicsrkannan/lade2018/Part5_GPRS.pdfNumerical Methods in Mechanics Finite Element Boundary Element Finite Difference G P Raja

Outline Difficulties Numerical Finite part integration Meth. of Kolm and Rokhlin

Boundary Integral Methods and Applications toFluid Mechanics

G P Raja SekharProfessor, Department of Mathematics

Indian Institute of Technology KharagpurKharagpur 721302, India

24 June 2014, NIT Jamshedpur

June 23, 2014

G P Raja SekharProfessor, Department of MathematicsIndian Institute of Technology KharagpurKharagpur 721302, India 24 June 2014, NIT Jamshedpur

Boundary Integral Methods and Applications to Fluid Mechanics


1 Difficulties

2 Numerical Finite part integration

3 Meth. of Kolm and RokhlinNumerical tests




Computational Techniques

Numerical Methods in Mechanics

Finite Element Boundary Element Finite Difference




Discretisation of a Domain

based on Variational methods based on Boundary integral methods




Advantages of BEM

Reduction of problem dimension by 1.Less data preparation time.Easier to change the applied mesh.Useful for problems that require re-meshing.

High Accuracy.Stresses are accurate as there are no approximations imposedon the solution in interior domain points.Suitable for modeling problems of rapidly changing stresses.

Less computer time and storage.For the same level of accuracy as other methods BEM usesless number of nodes and elements.

Filter out unwanted information.Internal points of the domain are optional.Focus on particular internal region.Further reduces computer time.

BEM is an attractive option.G P Raja SekharProfessor, Department of MathematicsIndian Institute of Technology KharagpurKharagpur 721302, India 24 June 2014, NIT Jamshedpur



The Heart of Boundary Integral Formulation

Fundamental solution of Laplacian: Those functions φ thatsatisfy the Laplace equation ∇2φ(x) = δ(x) in a domain D subjectto homogeneous boundary conditions are known as Green’sfunctions for the Laplace equation.1 dim::

d2φ∗

dx2= δ(x − x0)

⇒ φ∗(x , x0) =

−1

2 (x − x0), x > x012 (x − x0), x < x0




2D Case:

G (x− x0) =1

2πlog r =

1

2πlog |x− x0|

3D Case:

G (x− x0) =1

4πr




Two-dimensional Potential Problem

∇2φ = 0 in Ω

Can guess easily that ”suitable boundary data” is required!Given a data, solution can be attempted!




Divergence Theorem (flux conservation)

∫SFinids =

∫Ω

∂Fi∂xi

dΩ (1)

∫S

∂φ

∂xinids =

∫S

∂φ

∂nds =

∫Ω∇2φdΩ (2)

Remark: Ω ⊂ Rn : bounded, simply connected; Γ: sufficientlysmooth (say, here C 2). Looking for classical solutions.




Green’s identities

Green’s first identity∫Sφ∂ψ

∂xinids =

∫Ω

∂

∂xi

(φ∂ψ

∂xi

)dΩ (3)

=

∫Ω

∂φ

∂xi

∂ψ

∂xidΩ +

∫Ωφ∇2ψdΩ (4)

Green’s second identity∫S

(φ∂ψ

∂n− ψ∂φ

∂n

)ds =

∫Ω

(φ∇2ψ − ψ∇2φ

)dΩ (5)




Consider the Green’s identity∫S

(φ∂ψ

∂n− ψ∂φ

∂n

)ds =

∫Ω

(φ∇2ψ − ψ∇2φ

)dΩ (6)

Choice: φ, the harmonic function that needs to be obtainedφ∗ the fundamental solutionx = (x1, x2); y = (y1, y2)∫

S

(φ∂φ∗

∂n− φ∗∂φ

∂n

)ds =

∫Ω

(φ(y)∇2φ∗(x , y)

−φ∗(x , y)∇2φ(y))dΩy (7)




Exclusion of internal source point

e

S

x

S




Two - Dimensional Potential Problem contd.

∫S

(φ∂φ∗

∂n− φ∗∂φ

∂n

)ds +

∫Sε

(φ∂φ∗

∂n− φ∗∂φ

∂n

)dsε =∫

Ω−Ωε

(φ(y)∇2φ∗(x , y)− φ∗(x , y)∇2φ(y)

)dΩy

• In (Ω− Ωε), ∇2φ = 0,∇2φ∗ = 0 ⇒ RHS = 0.




∫S

(φ∂φ∗

∂n− φ∗∂φ

∂n

)ds +

∫Sε

(φ∂φ∗

∂n− φ∗∂φ

∂n

)dsε = 0 (BI )

• over Sε where dsε(y) = εdθ

∂φ∗(x , y)

∂ny=∂φ∗(x , y)

∂r

∂r

∂ny=

1

2πε




Two - Dimensional Potential Problem contd.

Consider the second term on the LHS of (BI)∫Sε

(φ∂φ∗

∂n− φ∗∂φ

∂n

)dsε(y) =

1

2π

∫0

θx(φ(y)

1

ε+ ln ε

∂φ(x , y)

∂ny

)εdθx

=1

2π

∫0

θx(φ(y) + ε ln ε

∂φ(x , y)

∂ny

)dθx

limε→0

∫Sε

(φ∂φ∗

∂n− φ∗∂φ

∂n

)dsε(y) =

θx2πφ(x) (8)




The general integral representation is

θx2πφ(x) =

∫Sφ∗(x , y)

∂φ(y)

∂nydsy −

∫Sφ(y)

∂φ∗(x , y)

∂nydsy (IR1)

where

θx =

2π, x∈ Ωπ, x∈ ∂Ω0, x /∈ Ω

analogy: general solution using separation of variables




H. Power, L. C. Wrobel, Boundary Integral Methods in FluidMechanics, WIT Press, Computational Mechanics Publications,Southampton, 1995.




Layer Potentials

Consider the Integral Representation for x ∈ Ω

φ(x) =

∫S

(φ∗(x , y)

∂φ(y)

∂ny− φ(y)

∂φ∗(x , y)

∂ny

)dsy (IR1)

Here σ = ∂φ∂n , τ = φ are the Cauchy data on Γ.

Vσ(x) =

∫Sφ∗(x , y)σ(y)dsy (Single − Layer)

W τ(x) =

∫S

∂φ∗(x , y)

∂nyτ(y)dsy (Double − Layer)




Boundary potentials

for x ∈ Γ

Vσ(x) := Vσ(x),

Kτ(x) := W τ(x) +1

2τ(x)

K ∗σ(x) := gradxVσ(x) · nx −1

2σ(x)

Dτ(x) := −gradxW τ(z) · nx ;




Consider the Integral Representation, when ξ ∈ ∂Ω

1

2φ(ξ) =

∫Sφ∗(ξ, y)

∂φ(y)

∂nydsy −

∫ pv

Sφ(y)

∂φ∗(ξ, y)

∂nydsy (IR2)

Dirichlet Problem: If the boundary value φ over S is given, themissing data is σ = ∂φ

∂n |Γ

Vσ(x) =1

2φ(ξ) + Kτ(x)∫ pv

Sφ∗(x , y)σ(y)dsy = f (x)(Fredholm I kind)




Neumann Problem:

If the boundary value ∂φ∂n over S is given, the missing data is φ |Γ

1

2σ(x)− K ∗(σ(x) = Dτ(x)

σ(x)− 2

∫ pv

S

∂φ∗(ξ, y)

∂nyσ(y)dsy = g(x)

(Fredholm II kind)




The challenges:

Discretizing the surface integrals (3D) or line integrals (2D)Handling ”Singular Integrals”




BEM Implementations with constant Elements - Kamal CDas

approximate the boundary Γ by line segments Γi withendpoints xi−1,xi , with a nodal point at the midpoint of Γi .

values at the mid-points of each element Γi are assumed equalto their values over the whole element.

i ∈ 1, ...,N where N is the no. of nodal points.




Discretization

i

x i

x i−1

x0, i

Figure: BEM with constant elements.G P Raja SekharProfessor, Department of MathematicsIndian Institute of Technology KharagpurKharagpur 721302, India 24 June 2014, NIT Jamshedpur



W T Ang, A beginner’s course in BEMs, Universals-Publishers,2007Prem K Kythe, An introduction to BEMs, CRC Press, 1995.T W Wu, Boundary Element Acoustics: Fundamentals andComputer Codes, WIT, 2001




1

2φ(x) =

∫Sφ∗(x , y)

∂φ(y)

∂nydsy −

∫Sφ(y)

∂φ∗(x , y)

∂nydsy , x ∈ ∂Ω

1

2φi = −

N∑j=1

∫Γj

φ∗(xi , y)∂φ(y)

∂nydsy +

N∑j=1

∫Γj

φ(y)∂φ∗(xi , y)

∂nydsy

φ = φj ; ∂φ∂n = ∂φj on the j th element




−1

2φi +

N∑j=1

(∫Γj

∂φ∗

∂ndS

)φj =

N∑j=1

(∫Γj

φ∗dS

)∂φj

Hij =

∫Γj

∂φ∗(xi , y)

∂nydS , G ∗ij =

∫Γj

φ∗(xi , y)dS




−1

2φ∗i +

N∑j=1

Hijφ∗j =

N∑j=1

Gij∂φ∗j

Let, Hij = Hij − 12δij , we have

N∑j=1

Hijφ∗j =

N∑j=1

Gij∂φ∗j

[H]N×Nφ∗N×1 = [G ]N×N∂φ

∗N×1




Example:

∇2φ = 0, 0 < x < 1, 0 < y < 1

φ(0, y) = 0;φ(1, y) = cosπy

∂φ

∂n(x , 0) = 0;

∂φ

∂n(x , 1) = 0

φ(x , y) =sinhπx cosπx

sinhπ




Laplacian

Figure: Boundary nodes (o) and midpoints of boundary elements (x).




Laplacian

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Surface Plot for solution

x

y

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 1

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

y

u

Exact (x=0.5)Exact (x=0.8)BEM (x=0.5)BEM (x=0.8)

Figure: (a) Surface plot of solution φ, (b) BEM solution Vs exactsolution of along x=0.5 and x=0.8.




Helmholtz equation-application to Acoustics

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x axis (m)

y ax

is (

m)

Figure: The geometry of the circular cylinderG P Raja SekharProfessor, Department of MathematicsIndian Institute of Technology KharagpurKharagpur 721302, India 24 June 2014, NIT Jamshedpur



Helmholtz

20 40 60 80 100 120

0.5

0.51

C(p

)

HIEAnalytic

20 40 60 80 100 120

379

379.5

380

real

ps

(Pa)

HIEAnalytic

20 40 60 80 100 120

95.8

95.9

96

Number of Node (n)

imag

iner

ps

(Pa)

HIEAnalytic

Figure: Distribution of solid angles C (P) and surface pressure (real andimaginary part) along the boundary nodes




Helmholtz

2 4 6 8 10 12 14 16 18 20

-200

-100

0

100

200

300

distance from the surface (m)

real

and

imag

iner

pre

ssur

e (P

a)

re(pf), analyticre(pf), HIEim(pf), analyticim(pf), HIE

Figure: Distribution of field pressure pf away from the surface of thecylinder




Stokes flow-Example

−∇p +∇2v = 0 in Ω, (9)

∇ · v = 0 in Ω, (10)

vθ(a, θ) = f (θ) = 4a3 cos 2θ − 3a2(1− sin θ) cos θ

p(a, θ) = g(θ) = −2a(1− 12 sin θ) cos θ

A. Zeb, L. Elliott, D.B. Ingham, D. Lesnic, The boundary elementmethod for the solution of Stokes equations in two-dimensionaldomains, Engineering Analysis with Boundary Elements, 22 (1998),317326




Stokes flow

‐25

‐20

‐15

‐10

‐5

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0v r(R,θ)

θ/2π

Analytical Solution

BE Solution, Elements=12

Figure: The normal components of velocity along the boundary with 12constant elements




Stokes flow

‐40

‐30

‐20

‐10

0

10

20

30

40

50

0.0 0.2 0.4 0.6 0.8 1.0

v θ(R,θ)

θ/2π

Analytical

BE Solution, Elements=12

Figure: tangential components of velocity along the boundary with 12constant elements




Difficulties as per M.Sc. project by Aftab Yusuf Patel

off diagonal coefficients are regular. Standard integrationmethods used.

diagonal coefficients involve singular, hypersingular integrals,for which singular terms cancel(see[Kohr, Raja Sekhar, 2008]). Standard methods do not workdue to complexity of kernels(esp. corresponding to Brinkmaneq.).

standard method of taking distorted paths of integrationaround singularities is too slow to be of practical utility.Program written in MATHEMATICA proved to be tooinefficient to run on available hardware.




Principal value and finite part integrals

Attacking the subproblems

∫ b

a

φ(x)

x − ydx (11)

where y ∈ (a, b) do not exist in the classical Riemann or Lebesguesense.

Cauchy Principal Value:

limε→0

(∫ y−ε

a

φ(x)

x − ydx +

∫ b

y+ε

φ(x)

x − ydx

)(12)




Strongly singular or hypersingular

∫ b

a

φ(x)

(x − y)2dx (13)

with y ∈ (a, b) are divergent in the classical sense.

Hadamard Finite Part interpretation:

limε→0

(∫ y−ε

a

φ(x)

(x − y)2dx +

∫ b

y+ε

φ(x)

(x − y)2dx − 2φ(y)

ε

)(14)




Method Of Kolm and Rokhlin, 2001

In [Kolm, Rokhlin, 2001] a method was developed to numericallyintegrate functions of the form,

f (x) = A(x) + B(x)log |x |+ C (x)

x+

D(x)

x2(15)

• 0 belongs to the interior of the interval of integration• the functions A,B,C ,D are not available separately• Method was implemented in C for maximum efficiency and is thefirst such publicly available implementation to the best of ourknowledge




Numerical tests

Numerical tests

The implementation of the method was tested on the test cases:∫ 1

−1

(sin(x + π3 ) + cos(x)

xdx

has a singularity of the 1/x type.∫ 1

−1K2(|x |)dx

has singularities of log |x | and 1/x2 type. The method was foundto be accurate to 10 decimal places. The degree M was taken tobe M = [N/4] + 1, [·] being the greatest integer function.




Numerical tests

Test case 1

10 20 30 40 50 60 70 8010-15

10-12

10- 9

10-6

0.001

1

N

Ε

Figure: Error in computing integral∫ 1

−1((sin(x + π

3 ) + cos(x))/x)dx from−1 to 1 with M = [N/4] + 1, vs. number of quadrature points N.




Numerical tests

Test case 2

10 20 30 40 50 60 70 8010-15

10-12

10- 9

10-6

0.001

1

N

Ε

Figure: Error in computing integral∫ 1

−1K2(|x |)dx from −1 to 1 with

M = [N/4] + 1, vs. number of quadrature points N.




Numerical tests

Summary

try to avoid handling the entire domain via boundary integralsGreen’s function is requiredfriendly to linear PDEsBoundary and Domain integral methods for Non-linear PDEspossibleRich theory of existence and uniqueness for the Integral Operators:Lipschitz domains, Sobolev spacesNumerical methods: Boundary Elementshandling singular integrals is the bottle neckwhen we succeed, the expenses are going to be very less comparedto domain methods




Numerical tests

H. Power, L. C. Wrobel, Boundary Integral Methods in FluidMechanics, WIT Press, Computational MechanicsPublications, Southampton, 1995.

M. Kohr, G. P. Raja Sekhar. Existence and uniqueness resultfor two-dimensional porous media flows with porous inclusionsbased on Brinkman equation, Engineering Analysis BoundaryElements pp 604-613, No. 7, Vol. 31, July 2007.

M. Kohr, G. P. Raja Sekhar. Boundary integral equations for2D Brinkman flow problem past a void, Proceedings of 53rdconference of ISTAM, September 2008.

R.Cortez. The Method of Regularized Stokeslets, SIAM Journalof Scientific Computing, Vol. 23, No. 4, pp. 1204-1225, 2001.




Numerical tests

P. Kolm, V. Rokhlin. Numerical Quadratures for Singular andHypersingular Integrals, Computers and Mathematics withApplications, Vol. 41, pp. 327-352, 2001.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration,Academic Press, New York, 1975.

Mirela Kohr, Wolfgang L Wendland, G P Raja Sekhar,Boundary integral equations for two-dimensional low Reynoldsnumber flow past a porous body, Math. Meth. Appl. Sci. 2009;32:922962.

Mirela Kohr, G P Raja Sekhar, Wolfgang L Wendland,Boundary integral equations for a three-dimensionalStokes-Brinkman cell model, Mathematical Methods andModels in Applied Sciences, Vol. 18, No. 12 (2008) 20552085




Numerical tests

ThAnKs FoR yOuR aTtEnTiOn !



Documents

Boundary Integral Methods and Applications to Fluid Mechanicsrkannan/lade2018/Part5_GPRS.pdfNumerical Methods in Mechanics Finite Element Boundary Element Finite Difference G P Raja