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
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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