Institute of Applied Mechanics, TU Braunschweighttp://www.infam.tu-braunschweig.de

Numerical Aspects of a Poroelastic Time DomainBoundary Element Formulation

Martin Schanz, Dobromil Pryl, Lars Kielhorn

Adaptive Fast Boundary Element Methods in Industrial Applications

Sollerhaus, 29.9.-2.10.2004

amin Institut für Angewandte Mechanik

Technische Universität Braunschweig

amin

2/25Content

Governing equations

Biot’s theory

Differential equation

Poroelastic Boundary Element Method

Boundary integral equation

Spatial shape functions

Convolution Quadrature Method

Numerical results

Dimensionless variables

Mixed shape functions

Rock foundation in a soil half-space

amin

3/25Biot’s theory of poroelastic continua

constitutive equation σi j = σSi j +σFδi j

σi j = G(ui, j +u j,i)+((

K− 23

G

)uk,k−αp

)δi j

ζ = αuk,k +φ2

Rp

equilibrium ρ = ρs(1−φ)+φρ f

σi j , j +Fi = ρ∂2

∂t2ui +ρ f∂∂t

wi

Darcy’s law

qi =−κ(

p,i +ρ f∂2

∂t2ui +ρa +φρ f

φ∂∂t

wi

)

continuity equation

∂∂t

ζ+qi,i = a

pressure gradient

Flux

Nomenclature

σi j total stress qi specific flux ui solid displacement ζ ‘fluid strain’

α Biot’s stress coefficient p pore pressure wi seepage velocity Fi bulk body force

G,K shear, bulk modulus φ porosity ρa apparent mass density ρ bulk density

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4/25Governing equations

• representation in Laplace domain (L f (t)= f )

Gui, j j +(

K +13

G

)u j,i j − (α−β) p,i −s2(ρ−βρ f ) ui =−Fi

βρ f s

p,ii −φ2sR

p− (α−β)sui,i =−a

B∗

ui

p

=−

Fi

a

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4/25Governing equations

• representation in Laplace domain (L f (t)= f )

Gui, j j +(

K +13

G

)u j,i j − (α−β) p,i −s2(ρ−βρ f ) ui =−Fi

βρ f s

p,ii −φ2sR

p− (α−β)sui,i =−a

B∗

ui

p

=−

Fi

a

• weak singular fundamental solutions

USi j =

1+ν8πE (1−ν)

r,ir, j +δi j (3−4ν)

1r

+O(r0) PF =

ρ f s

4πβ1r

+O(r0)

TFi =

ρ f s2

8πβ1−2ν1−ν

(α−β) r,ir,n +ni

(α+β

11−2ν

)1r

+O(r0)

QSj =

1+ν8πE (1−ν)

α(1−2ν)(r,nr, j −n j)−2β(1−ν)(r,nr, j +n j)

1r

+O(r0)

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4/25Governing equations

• representation in Laplace domain (L f (t)= f )

Gui, j j +(

K +13

G

)u j,i j − (α−β) p,i −s2(ρ−βρ f ) ui =−Fi

βρ f s

p,ii −φ2sR

p− (α−β)sui,i =−a

B∗

ui

p

=−

Fi

a

• weak singular fundamental solutions

USi j =

1+ν8πE (1−ν)

r,ir, j +δi j (3−4ν)

1r

+O(r0) PF =

ρ f s

4πβ1r

+O(r0)

TFi =

ρ f s2

8πβ1−2ν1−ν

(α−β) r,ir,n +ni

(α+β

11−2ν

)1r

+O(r0)

QSj =

1+ν8πE (1−ν)

α(1−2ν)(r,nr, j −n j)−2β(1−ν)(r,nr, j +n j)

1r

+O(r0)

• strong singular fundamental solutions

TSi j =

− [(1−2ν)δi j +3r,ir, j ] r,n +(1−2ν)(r, jni − r,in j)8π(1−ν) r2 +O

(r0) QF =− 1

4πr,n

r2 +O(r0)

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5/25Content

Governing equations

Biot’s theory

Differential equation

Poroelastic Boundary Element Method

Boundary integral equation

Spatial shape functions

Convolution Quadrature Method

Numerical results

Dimensionless variables

Mixed shape functions

Rock foundation in a soil half-space

amin

6/25Boundary integral equation

x1x2

x3

FdΩ t

n

Γt

ΓuΓ = Γu +Γt

Ω

weighted residuals

∫Ω

GTB∗

ui (x,s)

p(x,s)

dΩ = 0

with G =

USi j (x,y,s) UF

i (x,y,s)

PSj (x,y,s) PF (x,y,s)

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6/25Boundary integral equation

x1x2

x3

FdΩ t

n

Γt

ΓuΓ = Γu +Γt

Ω

weighted residuals

∫Ω

GTB∗

ui (x,s)

p(x,s)

dΩ = 0

with G =

USi j (x,y,s) UF

i (x,y,s)

PSj (x,y,s) PF (x,y,s)

two partial integrations singular behavior transformation to time domain

t∫0

∫Γ

USi j (t− τ,y,x) −PS

j (t− τ,y,x)

UFi (t− τ,y,x) −PF (t− τ,y,x)

ti (τ,x)

q(τ,x)

dΓdτ =

t∫0

∫Γ

C

TSi j (t− τ,y,x) QS

j (t− τ,y,x)

TFi (t− τ,y,x) QF (t− τ,y,x)

ui (τ,x)

p(τ,x)

dΓdτ+

ci j (y) 0

0 c(y)

ui (t,y)

p(t,y)

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7/25Spatial discretization

ζ η

spatial discretization

ui (x, t) =E

∑e=1

F

∑f=1

N fe (x)ue f

i (t)

ti (x, t) =E

∑e=1

F

∑f=1

N fe (x) te f

i (t)

p(x, t) =E

∑e=1

F

∑f=1

N fe (x) pe f(t)

q(x, t) =E

∑e=1

F

∑f=1

N fe (x)qe f(t)

e.g. linear ansatz function

N1e (η,ζ) =

14

(1−η)(1−ζ)

N2e (η,ζ) =

14

(1−η)(1+ζ)

N3e (η,ζ) =

14

(1+η)(1+ζ)

N4e (η,ζ) =

14

(1+η)(1−ζ)

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7/25Spatial discretization

ζ η

spatial discretization

ui (x, t) =E

∑e=1

F

∑f=1

N fe (x)ue f

i (t)

ti (x, t) =E

∑e=1

F

∑f=1

N fe (x) te f

i (t)

p(x, t) =E

∑e=1

F

∑f=1

N fe (x) pe f(t)

q(x, t) =E

∑e=1

F

∑f=1

N fe (x)qe f(t)

e.g. linear ansatz function

N1e (η,ζ) =

14

(1−η)(1−ζ)

N2e (η,ζ) =

14

(1−η)(1+ζ)

N3e (η,ζ) =

14

(1+η)(1+ζ)

N4e (η,ζ) =

14

(1+η)(1−ζ)

discretized integral equationci j (y)ui (y, t)

c(y) p(y, t)

=E

∑e=1

F

∑f=1

t∫0

∫Γ

USi j (x,y, t− τ) −PS

j (x,y, t− τ)

UFi (x,y, t− τ) −PF (x,y, t− τ)

N fe (x)dΓ

te fi (τ)

qe f (τ)

dτ

−t∫

0

∫Γ

C

TSi j (x,y, t− τ) QS

j (x,y, t− τ)

TFi (x,y, t− τ) QF (x,y, t− τ)

N fe (x)dΓ

ue fi (τ)

pe f (τ)

dτ

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8/25Mixed shape functions

• Isoparametric elements - shape functions identical for all quantities and geometry

• Mixed elements – using different shape functions for different quantities

(common for finite elements), e.g. N fe (x) linear for u, t and constant for p,q

−1 0 1

1

ko-2D−1 0 1

1

li-2D−1 0 1

1

lk-2D

lk-dr

Shape functions in 2-d and 3-d

Element uN fe , tN f

epN f

e , qN fe

ko-2D, ko-dr constant constant

li-2D, li-dr linear linear

lk-2D, lk-dr linear constant

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9/25Convolution Quadrature Method

quadrature rule for n = 0,1, . . . ,N time steps:

y(t) = f (t)∗g(t) =t∫

0

f (t− τ)g(τ)dτ ⇒ y(n∆t) =n

∑k=0

ωn−k(

f ,∆t)

g(k∆t)

Lubich, C. (1988): Convolution Quadrature and Discretized Operational Calculus. I., Numerische Mathematik, Vol.

52, 129–145

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9/25Convolution Quadrature Method

quadrature rule for n = 0,1, . . . ,N time steps:

y(t) = f (t)∗g(t) =t∫

0

f (t− τ)g(τ)dτ ⇒ y(n∆t) =n

∑k=0

ωn−k(

f ,∆t)

g(k∆t)

integration weight:

ωn(

f ,∆t)

=1

2πi

∫|z|=R

f

(γ(z)∆t

)z−n−1dz≈ R−n

L

L−1

∑=0

f

γ(Rei` 2π

L

)∆t

e−in` 2πL

• γ(z) A-stable multi step method, e.g. BDF 2: γ(z) = 32 −2z+ 1

2z2

• ∆t time step size of equal duration

• L = N effective choice for determining ωn (FFT)

• RN =√

ε with ε ≈ 10−10

Lubich, C. (1988): Convolution Quadrature and Discretized Operational Calculus. I., Numerische Mathematik, Vol.

52, 129–145

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10/25Temporal discretization

temporal discretization with Convolution Quadrature Method yields for n = 0,1, . . . ,Nci j (y)ui (n∆t)

c(y) p(n∆t)

=E

∑e=1

F

∑f=1

n

∑k=0

ωe fn−k

(US

i j ,y,∆t)

−ωe fn−k

(PS

j ,y,∆t)

ωe fn−k

(UF

i ,y,∆t)

−ωe fn−k

(PF ,y,∆t

)te f

i (k∆t)

qe f (k∆t)

−

ωe fn−k

(TS

i j ,y,∆t)

ωe fn−k

(QS

j ,y,∆t)

ωe fn−k

(TF

i ,y,∆t)

ωe fn−k

(QF ,y,∆t

)ue f

i (k∆t)

pe f (k∆t)

with integration weights, e.g.

ωe fn−k

(Ui j ,y,∆t

)=

Rk−n

L

L−1

∑=0

∫Γ

Ui j

x,y,γ(Re−i` 2π

L

)∆t

N fe (x)dΓ e−i(n−k)` 2π

L

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10/25Temporal discretization

temporal discretization with Convolution Quadrature Method yields for n = 0,1, . . . ,Nci j (y)ui (n∆t)

c(y) p(n∆t)

=E

∑e=1

F

∑f=1

n

∑k=0

ωe fn−k

(US

i j ,y,∆t)

−ωe fn−k

(PS

j ,y,∆t)

ωe fn−k

(UF

i ,y,∆t)

−ωe fn−k

(PF ,y,∆t

)te f

i (k∆t)

qe f (k∆t)

−

ωe fn−k

(TS

i j ,y,∆t)

ωe fn−k

(QS

j ,y,∆t)

ωe fn−k

(TF

i ,y,∆t)

ωe fn−k

(QF ,y,∆t

)ue f

i (k∆t)

pe f (k∆t)

with integration weights, e.g.

ωe fn−k

(Ui j ,y,∆t

)=

Rk−n

L

L−1

∑=0

∫Γ

Ui j

x,y,γ(Re−i` 2π

L

)∆t

N fe (x)dΓ e−i(n−k)` 2π

L

quadrature formula

• regular integrals: Gauss formula

• weak singular integrals: Regularization with polar coordinate transformation

• strong singular integrals: Formula by GUIGGIANI and GIGANTE

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11/25Time stepping procedure

solution with point collocation, i.e. moving y in every node and solving the system in each time

step (U,T,u, t are generalized variables here)

ω0(T)u(∆t) =ω0(U) t (∆t)

ω1(T)u(∆t)+ω0(T)u(2∆t) =ω1(U) t (∆t)+ω0(U) t (2∆t)

ω2(T)u(∆t)+ω1(T)u(2∆t)+ω0(T)u(3∆t) =ω2(U) t (∆t)+ω1(U) t (2∆t)+ω0(U) t (∆t)

...

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11/25Time stepping procedure

solution with point collocation, i.e. moving y in every node and solving the system in each time

step (U,T,u, t are generalized variables here)

ω0(T)u(∆t) =ω0(U) t (∆t)

ω1(T)u(∆t)+ω0(T)u(2∆t) =ω1(U) t (∆t)+ω0(U) t (2∆t)

ω2(T)u(∆t)+ω1(T)u(2∆t)+ω0(T)u(3∆t) =ω2(U) t (∆t)+ω1(U) t (2∆t)+ω0(U) t (∆t)

...

final recursion formula

ω0(C)dn = ω0(D) dn +n

∑m=1

(ωm(U) tn−m−ωm(T)un−m)

n = 1,2, . . . ,N

with the vector of unknown boundary data dn and the known boundary data dn in each time step

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12/25Content

Governing equations

Biot’s theory

Differential equation

Poroelastic Boundary Element Method

Boundary integral equation

Spatial shape functions

Convolution Quadrature Method

Numerical results

Dimensionless variables

Mixed shape functions

Rock foundation in a soil half-space

amin

13/25Column: Problem description

ty =−1 Nm2H(t),

permeable

free,impermeable

fixed, impermeable

3m

x

y

K [ Nm2 ] G [ N

m2 ] ρ [ kgm3 ] φ R [ N

m2 ] ρ f [ kgm3 ] α κ [m4

Ns]

rock 8·109 6·109 2458 0.19 4.7 ·108 1000 0.867 1.9 ·10−10

soil 2.1·108 9.8 ·107 1884 0.48 1.2 ·109 1000 0.981 3.55 ·10−9

324 elements on 188 nodes

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14/25Dimensionless variables

Dimensionless variables

x =xA

t =tB

K =KC

G =GC

ρ =A2

B2Cρ κ =

BCA2 κ

• Fall 1, 2, 3 ⇒ all material data O (λ)

A = κλ2√

ρC B=ρκλ2 C =

1λ

(K +

43

G+α2

φ2 R

)λ = 1,10−3,103

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14/25Dimensionless variables

Dimensionless variables

x =xA

t =tB

K =KC

G =GC

ρ =A2

B2Cρ κ =

BCA2 κ

• Fall 1, 2, 3 ⇒ all material data O (λ)

A = κλ2√

ρC B=ρκλ2 C =

1λ

(K +

43

G+α2

φ2 R

)λ = 1,10−3,103

• Fall 4, 5 ⇒ only normalization of modules

A = 1 B = 1 C = λE = λ9KG

6K +Gλ = 1,10

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14/25Dimensionless variables

Dimensionless variables

x =xA

t =tB

K =KC

G =GC

ρ =A2

B2Cρ κ =

BCA2 κ

• Fall 1, 2, 3 ⇒ all material data O (λ)

A = κλ2√

ρC B=ρκλ2 C =

1λ

(K +

43

G+α2

φ2 R

)λ = 1,10−3,103

• Fall 4, 5 ⇒ only normalization of modules

A = 1 B = 1 C = λE = λ9KG

6K +Gλ = 1,10

• Fall 6 ⇒ scaling of Young’s modules to the permeability

A = 1 B = 1 C =

√Eκ

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14/25Dimensionless variables

Dimensionless variables

x =xA

t =tB

K =KC

G =GC

ρ =A2

B2Cρ κ =

BCA2 κ

• Fall 1, 2, 3 ⇒ all material data O (λ)

A = κλ2√

ρC B=ρκλ2 C =

1λ

(K +

43

G+α2

φ2 R

)λ = 1,10−3,103

• Fall 4, 5 ⇒ only normalization of modules

A = 1 B = 1 C = λE = λ9KG

6K +Gλ = 1,10

• Fall 6 ⇒ scaling of Young’s modules to the permeability

A = 1 B = 1 C =

√Eκ

• Fall 7 ⇒ simple normalization

A = rmaxmaximum radius B = te maximum time C = E

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15/25Condition number in frequency domain

0 1000 2000 3000 4000 5000 6000

10E+6

10E+11

10E+16

10E+21

10E+26

10E+6

10E+11

10E+16

10E+21

10E+26

Fall 7Fall 6Fall 5Fall 4Fall 3Fall 2Fall 1

Frequency ω [1/s]

Con

ditio

nnu

mbe

r

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16/25Condition number in time domain

0 0.005 0.01 0.015 0.02 0.025

10E+8

10E+14

10E+20

10E+26

10E+32

10E+8

10E+14

10E+20

10E+26

10E+32

Fall 7Fall 6Fall 5Fall 4Fall 3Fall 2Fall 1

Time t [s]

Con

ditio

nnu

mbe

r

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17/25Condition number: Different parameters

0 0.005 0.01 0.015 0.02 0.025

10E+9

10E+15

10E+21

10E+27

10E+33

10E+9

10E+15

10E+21

10E+27

10E+33

Fall 4Fall 3Fall 2Fall 1

Time t [s]

Con

ditio

nnu

mbe

r

Fall 1∧= A = rmax B = 1 C = 1 Fall 2

∧= A = 1 B = tmax C = 1

Fall 3∧= A = rmax B = tmax C = 1 Fall 4

∧= A = 1 B = 1 C = E

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18/25Mixed Elements in 2-d

0.0 0.005 0.01 0.015 0.02

time t [s]

-1.6*10-9

-1.4*10-9

-1.2*10-9

-1.0*10-9

-8.0*10-10

-6.0*10-10

-4.0*10-10

-2.0*10-10

0.0*100

disp

lace

men

t uy [

m]

li-2D, t=0.00012s li-2D 128, t=0.00005slk-2D, t=0.00012s lk-2D 128, t=0.00005sko-2D, t=0.0004s ko-2D 128, t=0.00005sanalytic 1-d

0.0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

time t [s]

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

pore

pre

ssur

e p

[N/m

2 ]

li-2D, t=0.00012s li-2D 128, t=0.00005slk-2D, t=0.00012s lk-2D 128, t=0.00005sko-2D, t=0.0004s ko-2D 128, t=0.00005sanalytic 1-d

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19/25Mixed Elements in 3-d: Fixed surfaces

0.0 0.005 0.01 0.015 0.02

time t [s]

-1.5*10-9

-1.0*10-9

-5.0*10-10

0.0*100

disp

lace

men

t uy [

m]

li-dr, t=0.0002slk-dr, t=0.0002sko-dr, t=0.0002sanalytic 1-d

00.25

0.50.75

1

0

1

2

3

0

0.25

0.5

0.75

1

00.25

0.50.75

1

0

1

2

3

700 elementson 392 nodes

0.0 0.005 0.01 0.015 0.02

time t [s]

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

pore

pre

ssur

e p

[N/m

2 ]

li-dr, t=0.0002slk-dr, t=0.0002sko-dr, t=0.0002sanalytic 1-d

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20/25Mixed Elements in 3-d: Free surfaces

0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

time t [s]

0.0*100

5.0*10-9

1.0*10-8

1.5*10-8

disp

lace

men

t uy [

m]

li-dr, t=0.0005slk-dr, t=0.0005sko-dr, t=0.0005s

0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

time t [s]

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

pore

pre

ssur

e p

[N/m

2 ]

li-dr, t=0.0005slk-dr, t=0.0005sko-dr, t=0.0005s

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21/25Half space: Problem description

2m

3m

2m

1m

2m

5m

p(t)

p(t)

p(t)

p(t)

0,02s

p(t)

t

1,25MN

Rock foundation embedded in a soil half-space

Foundation: 124 elements on 68 nodes

Half space: 334 elements on 192 nodes

load

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22/25Half space: Condition number

0 0.02 0.04 0.06 0.08 0.1 0.12

10E+18

10E+20

10E+22

10E+24

10E+26

10E+18

10E+20

10E+22

10E+24

10E+26

Fall 2

Fall 1

Time t [s]

Con

ditio

nnu

mbe

r

Fall 1∧= old dimensionless variables Fall 2

∧= new more simpler suggestion

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23/25Half space: Numerical results

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24/25Summary

Poroelastic BEM

Biot’s theory

Based on Convolution Quadrature Method

Only Laplace transformed fundamental solutions are required

Dimensionless variables

Normalization w.r.t. time, space, and Young’s modulus

Largest influence due to the normalization to Young’s modulus

Mixed shape functions

Only sometimes improvement of stability and accuracy

Very CPU-time consuming

No justification for numerical effort

Institute of Applied Mechanics, TU Braunschweighttp://www.infam.tu-braunschweig.de

Numerical Aspects of a Poroelastic Time DomainBoundary Element Formulation

Martin Schanz, Dobromil Pryl, Lars Kielhorn

Adaptive Fast Boundary Element Methods in Industrial Applications

Sollerhaus, 29.9.-2.10.2004

amin Institut für Angewandte Mechanik

Technische Universität Braunschweig