Lagrangian blob methods applied to biological fluid flow ... · leads to the Method of Regularized Stokeslets: u(x) = 1 8ˇ X. k. S (y. k. x)f. k. A. k. where S (x)f = f r H. 1 (r)

Lagrangian blob methods applied to biological fluid flow problems

Lagrangian blob methods applied to biological fluidflow problems

Ricardo CortezTulane University

[email protected]

Fluid dynamics, Analysis and Numerics 2010


Blob methods in general

Idea of blob methods

The idea is motivated by the vortex blob method.

Key: Develop a regularized version of the fundamentalsolutions by designing smooth approximations of Deltafunctions

Do not simply modify final expressions



Blobs

φδ(r)� unit mass� radially symmetric� concentrated near x = 0� decay properties� scalable: φδ(x) = 1δn φ(x/δ)



Singular vs. Regularized Forces

fδ(x)

fφδ(x)



Fundamental solution of Laplace’s equation (3D)

Singular Regularized

∆u = a δ(x) ∆u = a φδ(x)

⇒ u(x) = a G (x) = −a4πr

⇒ u(x) = a [G ∗ φδ] (x)

= a Gδ(x) =−a H0(r)

4πr

where r = |x|.



Singular vs. Regularized Green’s functions

G (x)

Gδ(x)



Comments

� The regularized expressions are exact solutions of the givenequations for regularized forces. No approximations.

� The spread of the force depends on the parameter δ

� limδ→0

φδ(x) = δ(x)

� For δ � r , Gδ(r) ≈ G (r)

� limδ→0

Gδ(r) = G (r)


Example 1: Impulse blob method

Background

Impulse blob method

Euler equations in two dimensions

ut + u · ∇u = −∇p + F, ∇ · u = 0

Define a new vector field as m = u +∇χ.We can rewrite Euler’s equations as

mt + u · ∇m = − (∇u)T m + F

where (∇u)ij = ∂ui/∂xj .



Background

Impulse blob method: compact support

m 6= 0

m = 0



Background

Impulse blob method: discretization

xj



Background

Impulse blob method: strengths

xjmj



Background

Impulse blob method

In a Lagrangian method, the particle xj carries impulse mj . Theparticles move according to

d

dtxj = u(xj)

where u is the incompressible component of m, which evolves with

d

dtmj = − (∇u)T mj + F(xj)

based on mt + u · ∇m = − (∇u)T m + F.



Background

Impulse blob method

A Lagrangian blob method:

1 approximate the impulse field m using the discretization

m(x) =

∫R2

m(y)δ(x− y)dy ≈j=N∑j=1

mj(t) φδ(x− xj)Aj

2 find u from m via a projection m = u +∇χ

u =(I −∇∆−1∇·

)m

3 advance the particle position using ẋj = u(xj)

4 update the impuse strengths using ṁj = − (∇u)T mj + F(xj)



Old blobs and new blobs, example

Impulse blob method

If m(x) occupies an area (in 2D), then Aj is an area element andan `-th order blob φδ(x) is chosen to satisfy the moment conditions∫

R2φδ(x)dx = 1, φδ(r) = δ

−2φ1(r/δ)∫R2

xα φδ(x)dx = 0, for 0 < |α| < `− 1∫R2|x|` φδ(x)dx




Old and New(er) blobs

For a radially symmetric blob φδ(x) consider

m(x) = moφδ(x− xo)

and define

∆G = δ and ∆Gδ = φδ ⇒ Gδ = G ∗ φδ

in R2. Then u =(I −∇∆−1∇·

)m is

dx

dt= uδ(x) = moφδ(x− xo)− (mo · ∇)∇Gδ(x− xo).




Old and New(er) blobs

(Loading movie...)

If m(x) is supported on a curve Γ in 2D or a surface in 3D, then

m(x) =

∫Γm(y)δ(x− y)dS(y) ≈

j=N∑j=1

mj(t) φδ(x− xj)Sj

is not a convolution and moment conditions do not help.

eelmovie.mpgMedia File (video/mpeg)




Beale’s result

In the context of single-layer potentials, Beale showed that in 3Dfor a homogeneous polynomial pn(x),∫∫

Γ[Gδ(x)− G (x)] pn(x) dx = Cn δn+1

∫ ∞0

[F (r)− 1] rn dr

where Γ is a plane and F (r) = 2π∫ r

0 φ1(s) s ds. The leading errorterm (n = 0) can be eliminated with the shape factor condition∫ ∞

0[F (r)− 1]dr = 0

yielding to a regularization error of O(δ3).




Extending Beale’s result

This velocity uδ(x) = moφδ(x− xo)− (mo · ∇)∇Gδ(x− xo) canalso be written as

uδ(x) = mo

(−1

2πr2

)s1(r/δ) +

(mo · x̂)x̂r2

(2

2πr2

)s2(r/δ)

where x̂ = x− xo and r = |x̂|. The smoothing functions s1 and s2are given by

s1(r) = F (r)− rF ′(r) and s2(r) = F (r)−1

2rF ′(r).

where F is the shape factor associated with the blob φδ:

F (r/δ) = 2π

∫ r0φδ(s) s ds




Extending Beale’s result

One can show that the condition∫ ∞0

[F (r)− 1] dr = 0

implies that∫ ∞0

[s1(r)− 1] dr =∫ ∞

0[s2(r)− 1] dr = 0

so that

uδ(x) = mo

(−1

2πr2

)s1(r/δ) +

(mo · x̂)x̂r2

(2

2πr2

)s2(r/δ)

may exhibit 3-rd order convergence.

Example: φ1(r) =1

π(3− 2r2)e−r2




Example

Initial curve (in 2D) is a perturbed circle

r(θ) =√

1− �2/2 + � cos(2θ)

Initial impulse: m(θ) = 0. The force at x(θ, t) is given by thecurve total arc length at t times the curvature at x(θ, t).Asymptotically, the solution is

r(θ, t) =

[1− �

2

4A2(t)

]+ �A(t) cos(2θ) + �2B(t) cos(4θ) + O(�4)

where

A(t) = cos

(√6π

(1− 223

480�2)t

)B(t) =

3

20+

1

3cos(2

√6π t)− 29

60cos(√

60π t).




Results from exponential blob

Parameter: δ = 0.075 and δ = 0.15

0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

A(t)

= 0.075, 0.15

EXPONENTIAL BLOB

0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

B(t)

TIME




Results from Beale blob

Parameter: δ = 0.075 and δ = 0.15

0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

A(t)

= 0.075, 0.15

BEALE BLOB

0 0.5 1 1.5 2 2.5 31

0.5

0

0.5

1

TIME

B(t)




Error comparison

Ratio of errors in A(t) between consecutive numerical solutionsusing δ = 0.075, 0.15 and 0.3.

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16RATIO OF ERRORS IN A(t) BEALE (o), EXPONENTIAL (+)


Example 2: Regularized Stokeslets

Background

Change gears

New example



Background

Example 1: Stokes flow past point obstacles in 2D



Background

Example 2: Stokes flow around a cilium in 3D

(Loading movie...)

ciliummovie.aviMedia File (video/avi)



Background

Fundamental solution of Stokes equations

∇p = µ∆u + f δ(x), ∇ · u = 0

Suppose that f is constant and r = |x|. Then formally,

∆p = ∇ · [f δ(x)]

⇒ p(x) = f · ∇G (x) = f · x4πr3

And finally,

8πµu(x) =f

r+

(f · x)xr3

∼ 1r



Background

Regularized solution of Stokes equations: Stokeslet

∇p = µ∆u + f φδ(x), ∇ · u = 0

Then, define Gδ by ∆Gδ(x) = φδ(x).

∆p = ∇ · [f φδ(x)] ⇒ p(x) = f · ∇Gδ(x)

Now the velocity satisfies µ∆u = (f · ∇)∇Gδ − f φδ so that

µu(x) = −f Gδ(x) + (f · ∇)∇Bδ(x)

where ∆Bδ(x) = Gδ(x).



Background

Summary of Stokes Flow due to a force at the origin

Singular Stokeslet:

8πµu(x) =f

r+

(f · x)xr3

Regularized Stokeslet [R. C., 2001]:

8πµu(x) =f

rH1(r) +

(f · x)xr3

H2(r)

where H1(r) = Bδ′(r)− rGδ(r) and H2(r) = rBδ ′′(r)− Bδ ′(r).

� Note: limδ→0

H1(r) = limδ→0

H2(r) = 1



Background

General formulation of regularized Stokeslets

[R. C., L. Fauci & A. Medovikov, 2005]

One can derive the expression∫R3u(y)φδ(y − x)dy =

1

8πµ

∫∂D

Sδ(y − x)f(y) ds(y)

leads to the Method of Regularized Stokeslets:

u(x) =1

8πµ

∑k

Sδ(yk − x)fk Ak

where

Sδ(x)f =f

rH1(r) +

(f · x)xr3

H2(r), r = |x|



Background

Biological applications

microorganism swimming, sperm motility

collective motion of bacteria and algae

flow in arteries or channels

dynamics of cells

Goal: compute simultaneously the motion of the fluid and theelastic structure

Questions: why are organisms designed as they are? How do ciliaorchestrate collective motion? What are the consequences ofdefects in organism design?



Background

Motion of headless helical organism

(Loading movie...)

TenUnitsAboveWall.aviMedia File (video/avi)



Background

Improvement for closed boundaries

Velocity computation on the boundary or far from the boundary isnot problematic. Very near the boundary, it is challenging(nearly-singular integrals) with fixed discretizations.

Consider a closed boundary parametrized by x(s) for 0 ≤ s ≤ L.Strategy: write the Stokeslet formula in terms of

I1(x, f ) =

∫ L0

f (s) x′(s)×∇G (x− x(s))ds

I2(x, g) =

∫ L0

g(s) x′(s) · ∇G (x− x(s))ds



Background

Single and double layer potentials

Beale and Lai developed correction terms for the trapezoid rule onthe regularized Green’s functions that guarantee second-orderaccuracy. Use the identities

I1(x, 1) =

∫ L0

x′(s)×∇G (x− x(s))ds ={

1, for x inside0, for x outside

I2(x, 1) =

∫ L0

x′(s) · ∇G (x− x(s))ds = 0



Background


For example

I1(x, f ) =

∫ L0

f (s) x′(s)×∇G (x− x(s))ds

= f (s∗) χ(x) +

∫ L0

[f (s)− f (s∗)] x′(s)×∇G (x− x(s))ds



Background


For example

I1(x, f ) =

∫ L0

f (s) x′(s)×∇G (x− x(s))ds

= f (s∗) χ(x) +

∫ L0

[f (s)− f (s∗)] x′(s)×∇G (x− x(s))ds

S1(x, f ) = f (s∗) χ(x) +

∑k

[fk − f (s∗)] x′(sk)×∇Gδ(x− xk))∆s



Background


For example

I1(x, f ) =

∫ L0

f (s) x′(s)×∇G (x− x(s))ds

= f (s∗) χ(x) +

∫ L0

[f (s)− f (s∗)] x′(s)×∇G (x− x(s))ds

S1(x, f ) = f (s∗) χ(x) +

∑k

[fk − f (s∗)] x′(sk)×∇Gδ(x− xk))∆s

I1(x, f ) = S1(x, f ) + T(n)1 (x) + T

(n)2 (x) + O(∆s

2) + O(δ3)



Background


Similarly

I2(x, g) =

∫ L0

g(s) x′(s) · ∇G (x− x(s))ds

=

∫ L0

[g(s)− g(s∗)] x′(s)×∇G (x− x(s))ds

S2(x, g) =∑k

[gk − g(s∗)] x′(sk) · ∇Gδ(x− xk))∆s

I2(x, g) = S2(x, g) + T(t)1 (x) + T

(t)2 (x) + O(∆s

2) + O(δ3)



Background

Comments

T(n)1 and T

(t)1 are corrections due to the regularization of G

with a Gaussian blob

T(n)2 and T

(t)2 are corrections to the quadrature

In 2D

4πµu(x) =

∫Γf(s) log |x− x(s)|+ [f · (x− x(s))](x− x(s))

|x− x(s)|2ds

must be written in terms of single and double-layer potentials.



Background

Formulation

p(x) =

∫Γf(s) · ∇G (x− x(s))ds = I2(x, f)

4πµu(x) =

∫Γf(s) log |x− x(s)|+ [f · (x− x(s))](x− x(s))

|x− x(s)|2ds

= I2(x,F) + I1(x,Qn) + I2(x,Q

t)

where

F(s) = −2π∫ s

0f(α)dα,

Qn(s) = −2π(x− x(s)) f(s) · n̂(s),Qt(s) = −2π(x− x(s)) f(s) · x′(s)



Background

Example: Circle with f(θ) = 2 sin(3θ)x(θ)

Pressure

Corrected Uncorrected/Exact



Background

Example: Circle with f(θ) = 2 sin(3θ)x(θ)

Errors as a function of discretization



Background

Example: Circle with f(θ) = 2 sin(3θ)x′(θ)

Pressure



Background

Example: Circle with f(θ) = 2 sin(3θ)x′(θ)

Velocities



Background

Final Remarks

The method of Regularized Stokeslet provides smoothing forforce distributions at points or filaments. The integrability ofStokeslets over surfaces allows modifications (jumpconditions) for increased accuracy. Thanks Tom.

The regularization approach can be applied to single- anddouble-layer potentials. This allows derivatives of thepotentials to be computed similarly [S. Tlupova’s]. ThanksTom.

Current applications for Stokes flows: cilia, spirochetes, otherorganisms, individual and collective motion; extensions toDarcy’s law and Brinkman flows.



Background

Final Remarks

Does the methodology extend to other fluids (viscoelastic,complex)?

Regularized slender body theory

Properties of the matrix relating the velocity to the force isgeometry-dependent. Sometimes there is a non-trivial nullspace. Invertibility questions remain.

Modifications for bounded flows



Background

Thank you

Collaborators: L. Fauci, M. Bees, J. Kessler, S. Tlupova, B.Cummins, Hoa Nguyen, R. Ortiz, M. Nicholas, D. Varela, K.Leiderman, M. Hernandez-Viera, Shilpa Boindala.

Work supported by NSF & Louisiana Board of Regents.

Blob methods in generalExample 1: Impulse blob methodBackgroundOld blobs and new blobs, example

Example 2: Regularized StokesletsBackground

Documents

Lagrangian blob methods applied to biological fluid flow ... · leads to the Method of Regularized Stokeslets: u(x) = 1 8ˇ X. k. S (y. k. x)f. k. A. k. where S (x)f = f r H. 1 (r)