application of generalized convolution quadrature in acoustics … · 2016-11-10 · graz...
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Graz University of TechnologyInstitute of Applied Mechanics
Application of generalized Convolution Quadraturein Acoustics and Thermoelasticity
Martin Schanz
joint work with Relindis Rott and Stefan Sauter
Space-Time Methods for PDEsSpecial Semester on Computational Methods in Science and EngineeringRICAM, Linz, Austria, November 10, 2016
> www.mech.tugraz.at
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
Martin Schanz gCQM: Acoustics and Thermoelasticity 3 / 39
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Convolution integral
Convolution integral with the Laplace transformed function f (s)
y (t) = (f ∗g)(t) =(f (∂t )g
)(t) =
t∫0
f (t− τ)g (τ)dτ
=1
2πi
∫C
f (s)
t∫0
es(t−τ)g (τ)dτ
︸ ︷︷ ︸x (t,s)
ds
Integral is equivalent to solution of ODE
∂
∂tx (t,s) = sx (t,s) + g (t) with x (t = 0,s) = 0
Implicit Euler for ODE , [0,T ] = [0, t1, t2, . . . , tN ], variable time steps∆ti , i = 1,2, . . . ,N
xn (s) =xn−1 (s)
1−∆tns+
∆tn1−∆tns
gn =n
∑j=1
∆tjgj
n
∏k=j
11−∆tk s
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Convolution integral
Convolution integral with the Laplace transformed function f (s)
y (t) = (f ∗g)(t) =(f (∂t )g
)(t) =
t∫0
f (t− τ)g (τ)dτ
=1
2πi
∫C
f (s)
t∫0
es(t−τ)g (τ)dτ
︸ ︷︷ ︸x (t,s)
ds
Integral is equivalent to solution of ODE
∂
∂tx (t,s) = sx (t,s) + g (t) with x (t = 0,s) = 0
Implicit Euler for ODE , [0,T ] = [0, t1, t2, . . . , tN ], variable time steps∆ti , i = 1,2, . . . ,N
xn (s) =xn−1 (s)
1−∆tns+
∆tn1−∆tns
gn =n
∑j=1
∆tjgj
n
∏k=j
11−∆tk s
Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39
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Time stepping formula
Solution at the discrete time tn
y (tn) =1
2πi
∫C
f (s)xn (s)ds
=1
2πi
∫C
f (s)∆tn1−∆tns
gn ds +1
2πi
∫C
f (s)xn−1 (s)
1−∆tnsds
=f
(1
∆tn
)gn +
12πi
∫C
f (s)xn−1 (s)
1−∆tnsds .
Recursion formula for the implicit Euler
y (tn) =1
2πi
∫C
f (s)n
∑j=1
∆tjgj
n
∏k=j
11−∆tk s
ds
= f
(1
∆tn
)gn +
n−1
∑j=1
∆tjgj1
2πi
∫C
f (s)n
∏k=j
11−∆tk s
ds
Complex integral is solved with a quadrature formula
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Time stepping formula
Solution at the discrete time tn
y (tn) =1
2πi
∫C
f (s)xn (s)ds
=1
2πi
∫C
f (s)∆tn1−∆tns
gn ds +1
2πi
∫C
f (s)xn−1 (s)
1−∆tnsds
=f
(1
∆tn
)gn +
12πi
∫C
f (s)xn−1 (s)
1−∆tnsds .
Recursion formula for the implicit Euler
y (tn) =1
2πi
∫C
f (s)n
∑j=1
∆tjgj
n
∏k=j
11−∆tk s
ds
= f
(1
∆tn
)gn +
n−1
∑j=1
∆tjgj1
2πi
∫C
f (s)n
∏k=j
11−∆tk s
ds
Complex integral is solved with a quadrature formula
Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39
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Time stepping formula
Solution at the discrete time tn
y (tn) =1
2πi
∫C
f (s)xn (s)ds
=1
2πi
∫C
f (s)∆tn1−∆tns
gn ds +1
2πi
∫C
f (s)xn−1 (s)
1−∆tnsds
=f
(1
∆tn
)gn +
12πi
∫C
f (s)xn−1 (s)
1−∆tnsds .
Recursion formula for the implicit Euler
y (tn) =1
2πi
∫C
f (s)n
∑j=1
∆tjgj
n
∏k=j
11−∆tk s
ds
= f
(1
∆tn
)gn +
n−1
∑j=1
∆tjgj1
2πi
∫C
f (s)n
∏k=j
11−∆tk s
ds
Complex integral is solved with a quadrature formula
Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39
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Algorithm
First Euler step
y (t1) = f
(1
∆t1
)g1
with implicit assumption of zero initial condition
For all steps n = 2, . . . ,N the algorithm has two steps1 Update the solution vector xn−1 at all integration points s` with an implicit Euler step
xn−1 (s`) =xn−2 (s`)
1−∆tn−1s`+
∆tn−1
1−∆tn−1s`gn−1
for ` = 1, . . . ,NQ with the number of integration points NQ .2 Compute the solution of the integral at the actual time step tn
y (tn) = f
(1
∆tn
)gn +
NQ
∑`=1
ω`f (s`)
1−∆tns`xn−1 (s`)
Essential parameter: NQ = N log(N), integration is dependent on q = ∆tmax∆tmin
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Algorithm
First Euler step
y (t1) = f
(1
∆t1
)g1
with implicit assumption of zero initial condition
For all steps n = 2, . . . ,N the algorithm has two steps1 Update the solution vector xn−1 at all integration points s` with an implicit Euler step
xn−1 (s`) =xn−2 (s`)
1−∆tn−1s`+
∆tn−1
1−∆tn−1s`gn−1
for ` = 1, . . . ,NQ with the number of integration points NQ .2 Compute the solution of the integral at the actual time step tn
y (tn) = f
(1
∆tn
)gn +
NQ
∑`=1
ω`f (s`)
1−∆tns`xn−1 (s`)
Essential parameter: NQ = N log(N), integration is dependent on q = ∆tmax∆tmin
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Numerical integration
Integration weights and points
s` = γ(σ`) ω` =4K(k2)
2πiγ′ (σ`)
for N = 25,T = 5, tn =(
nN
)αT ,α = 1.5
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
Martin Schanz gCQM: Acoustics and Thermoelasticity 8 / 39
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Absorbing boundary conditions
Materials with absorbing surfaces
Mechanical modell: Coupling of porousmaterial layer at the boundary
Simpler mechanical model: Impedance boundary condition
Z =p
v ·nspecific impedance
Z (x)
ρc= α(x) = cosθ
1−√
1−αS (x)
1 +√
1−αS (x)
with density ρ, wave velocity c, and absorption coefficient αS = f (ω)
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Absorbing boundary conditions
Materials with absorbing surfaces
Mechanical modell: Coupling of porousmaterial layer at the boundary
Simpler mechanical model: Impedance boundary condition
Z =p
v ·nspecific impedance
Z (x)
ρc= α(x) = cosθ
1−√
1−αS (x)
1 +√
1−αS (x)
with density ρ, wave velocity c, and absorption coefficient αS = f (ω)
Martin Schanz gCQM: Acoustics and Thermoelasticity 9 / 39
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Problem setting
Bounded Lipschitz domain Ω− ⊂ R3 with boundary Γ := ∂ΩΩ+ := R3\Ω− is its unbounded complement.
Linear acoustics for the pressure p
∂tt p− c2∆p = 0 in Ωσ×R>0,
p (x ,0) = ∂t p (x ,0) = 0 in Ωσ,
γσ1 (p)−σ
α
cγ
σ0 (∂t p)= f (x , t) on Γ×R>0
with σ ∈ +,−, wave velocity c, and α absorption coefficient
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Integral equation
Single layer ansatz for the density
−(
σϕ
2−K′ ∗ϕ
)−σ
α
c(V ∗ ϕ) = f a.e. in Γ×R>0
Retarded potentials
(V ∗ϕ)(x , t) =∫Γ
ϕ
(y , t− ‖x−y‖
c
)4π‖x− y‖ dΓy
(K′ ∗ϕ
)(x , t) =
14π
∫Γ
〈n (x) ,y− x〉‖x− y‖2
ϕ
(y , t− ‖x−y‖
c
)‖x− y‖ +
ϕ
(y , t− ‖x−y‖
c
)c
dΓy
Single layer potential for the pressure
p (x , t) = (S ∗ϕ)(x , t) :=∫Γ
ϕ
(y , t− ‖x−y‖
c
)4π‖x− y‖ dΓy ∀(x , t) ∈ Ωσ×R>0
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Integral equation
Single layer ansatz for the density
−(
σϕ
2−K′ ∗ϕ
)−σ
α
c(V ∗ ϕ) = f a.e. in Γ×R>0
Retarded potentials
(V ∗ϕ)(x , t) =∫Γ
ϕ
(y , t− ‖x−y‖
c
)4π‖x− y‖ dΓy
(K′ ∗ϕ
)(x , t) =
14π
∫Γ
〈n (x) ,y− x〉‖x− y‖2
ϕ
(y , t− ‖x−y‖
c
)‖x− y‖ +
ϕ
(y , t− ‖x−y‖
c
)c
dΓy
Single layer potential for the pressure
p (x , t) = (S ∗ϕ)(x , t) :=∫Γ
ϕ
(y , t− ‖x−y‖
c
)4π‖x− y‖ dΓy ∀(x , t) ∈ Ωσ×R>0
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Solution for the unit ball
Geometry is the unit ball
Right hand side of the impedance boundary condition is
f (x , t) := f (t)Yn,m
Spherical harmonics are eigenfunctions of the boundary integral operators
ZY mn = λ
(Z)n
(sc
)Y m
n for Z ∈V,K,K′,W
It holds
λ(V)n (s) =−sjn (is)h(1)
n (is) λ(K′)n (s) =
12− is2jn (is)∂h(1)
n (is)
with the spherical Bessel and Hankel functions jn, h(1)n and ∂jn, ∂h(1)
n denoting theirfirst derivatives
Analytical transformation yields time domain solution
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Solution for the unit ball
Geometry is the unit ball
Right hand side of the impedance boundary condition is
f (x , t) := f (t)Yn,m
Spherical harmonics are eigenfunctions of the boundary integral operators
ZY mn = λ
(Z)n
(sc
)Y m
n for Z ∈V,K,K′,W
It holds
λ(V)n (s) =−sjn (is)h(1)
n (is) λ(K′)n (s) =
12− is2jn (is)∂h(1)
n (is)
with the spherical Bessel and Hankel functions jn, h(1)n and ∂jn, ∂h(1)
n denoting theirfirst derivatives
Analytical transformation yields time domain solution
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Solution for the unit ball
Geometry is the unit ball
Right hand side of the impedance boundary condition is
f (x , t) := f (t)Yn,m
Spherical harmonics are eigenfunctions of the boundary integral operators
ZY mn = λ
(Z)n
(sc
)Y m
n for Z ∈V,K,K′,W
It holds
λ(V)n (s) =−sjn (is)h(1)
n (is) λ(K′)n (s) =
12− is2jn (is)∂h(1)
n (is)
with the spherical Bessel and Hankel functions jn, h(1)n and ∂jn, ∂h(1)
n denoting theirfirst derivatives
Analytical transformation yields time domain solution
Martin Schanz gCQM: Acoustics and Thermoelasticity 12 / 39
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Solution in time domain
Solution for σ = +1, i.e., outer space, and n = 0Load function f (t) = (ct)υ e−ct
Density function
ϕ+ (t) =− 2
1 + α
bct/2c∑`=0
((ct−2`)υ e−(ct−2`)
− (1 + α)υ
αυ+1 γ
(υ + 1,
α
1 + α(ct−2`)
)e−
ct−2`1+α
)with the incomplete Gamma function γ(a,z) :=
∫ z0 ta−1e−t dt
Pressure solution
p+ (r , t) =− (1 + α)υ
2√
παυ+1γ
(υ + 1,
α
1 + ατ+
)e−
τ1+α
r.
with r > 1, we define τ := ct− (r −1) and (τ)+ := max0,τ
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Solution in time domain
Solution for σ = +1, i.e., outer space, and n = 0Load function f (t) = (ct)υ e−ct
Density function
ϕ+ (t) =− 2
1 + α
bct/2c∑`=0
((ct−2`)υ e−(ct−2`)
− (1 + α)υ
αυ+1 γ
(υ + 1,
α
1 + α(ct−2`)
)e−
ct−2`1+α
)with the incomplete Gamma function γ(a,z) :=
∫ z0 ta−1e−t dt
Pressure solution
p+ (r , t) =− (1 + α)υ
2√
παυ+1γ
(υ + 1,
α
1 + ατ+
)e−
τ1+α
r.
with r > 1, we define τ := ct− (r −1) and (τ)+ := max0,τ
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Discretization
Spatial discretization: Linear continuous shape functions on linear trianglesTemporal discretization: gCQM with time grading
tn = T( n
N
)χ
, n = 0, . . . ,N with grading exponent χ = 1/υ
Meshes of the unit sphere
h1 = 0.393m h2 = 0.196m h3 = 0.098m h3 = 0.049m
Material data: Air (c = 343.41 m/s)Load function: f (t) = (ct)υ e−ct with υ = 1
2Observation time T = 0.002915905s and β = c∆t/h
Error in time
errrel =
√N
∑n=0
∆tn (u (tn)−uh (tn))2/
√N
∑n=0
∆tn (u (tn))2 eoc = log2
(errh
errh+1
)Martin Schanz gCQM: Acoustics and Thermoelasticity 14 / 39
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Solution density
0 0.2 0.4 0.6 0.8 1 1.2·10−2
−0.2
0
0.2
0.4
0.6
time t [s]
dens
ityϕ
+α = 0α = 0.25α = 0.5α = 1analytic α = 0.25analytic α = 0.5analytic α = 1
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Solution pressure
0 0.2 0.4 0.6 0.8 1 1.2·10−2
0
0.05
0.1
0.15
time t [s]
pres
sure
u+[N/m
2 ]
α = 0α = 0.25α = 0.5α = 1
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Relative density error: mesh size
10−1.2 10−1 10−0.8 10−0.6 10−0.4
10−2.5
10−2
10−1.5
mesh size h
err re
l
∆tvar ,β = 0.125∆tconst ,β = 0.125∆tvar ,β = 0.0625∆tconst ,β = 0.0625eoc = 0.5eoc = 1
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Relative pressure error: mesh size
10−1.2 10−1 10−0.8 10−0.6 10−0.4
10−2
10−1
mesh size h
err re
l
∆tvar ,β = 0.25∆tconst ,β = 0.25∆tvar ,β = 0.125∆tconst ,β = 0.125∆tvar ,β = 0.0625∆tconst ,β = 0.0625eoc = 1
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Relative density error: time step
10−5 10−4
10−2.5
10−2
10−1.5
time step size ∆t
err re
l
mesh 2, ∆tconst
mesh 2, ∆tvar
mesh 3, ∆tconst
mesh 3, ∆tvar
mesh 4, ∆tconst
mesh 4, ∆tvareoc = 0.5eoc = 1
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Relative pressure error: time step
10−5 10−4
10−2
10−1
time step size ∆t
err re
l
mesh 2, ∆tconst
mesh 2, ∆tvar
mesh 3, ∆tconst
mesh 3, ∆tvar
mesh 4, ∆tconst
mesh 4, ∆tvareoc = 2eoc = 1
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Problem description
Atrium at University of Zurich (Irchel campus)
Mesh: 7100 elementsTime interval [0,T = 0.15s] with grading
tn =
(n +
(n−1)2
N
)∆tconst with ∆tconst = 0.00037s ⇒ N = 405, Ngraded = 248
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Sound pressure field
t ≈ 0.028 s
α = 0.1 α = 0.5 α = 1
t ≈ 0.064 s
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Sound pressure level
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16−30
−20
−10
0
10
20
30
time t [s]
soun
dpr
essu
rele
velu
[dB
]α = 0.1α = 0.5α = 1
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Content
1 Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm
2 Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples
3 Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example
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Uncoupled thermoelasticity
Governing equations for temperature θ(x, t) and displacement u(x, t)
κ θ,jj (x, t)− θ(x, t) = 0
µ ui,jj (x, t) + (λ + µ)uj,ij (x, t)− (3λ + 2µ)α θ,i (x, t) = 0
κ thermal diffusivity, α thermal expansion coefficient, λ,µ Lamé constantsBoundary integral formulation
c (y)θ(y, t) =∫Γ
[Θ∗q](x,y, t)− [Q ∗θ](x,y, t)dΓ
cij (y)uj (y, t) =∫Γ
Uij (x,y)tj (x, t)−Tij (x,y) uj (x, t)
+ [Gi ∗q](x,y, t)− [Fi ∗θ](x,y, t)dΓ
withΘ(x,y, t) and Q(x,y, t) kernels of the heat equationUij (x,y) and Tij (x,y) kernels from elastostaticsGi (x,y, t) and Fi (x,y, t) kernels for the one sided coupling
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Uncoupled thermoelasticity
Governing equations for temperature θ(x, t) and displacement u(x, t)
κ θ,jj (x, t)− θ(x, t) = 0
µ ui,jj (x, t) + (λ + µ)uj,ij (x, t)− (3λ + 2µ)α θ,i (x, t) = 0
κ thermal diffusivity, α thermal expansion coefficient, λ,µ Lamé constantsBoundary integral formulation
c (y)θ(y, t) =∫Γ
[Θ∗q](x,y, t)− [Q ∗θ](x,y, t)dΓ
cij (y)uj (y, t) =∫Γ
Uij (x,y)tj (x, t)−Tij (x,y) uj (x, t)
+ [Gi ∗q](x,y, t)− [Fi ∗θ](x,y, t)dΓ
withΘ(x,y, t) and Q(x,y, t) kernels of the heat equationUij (x,y) and Tij (x,y) kernels from elastostaticsGi (x,y, t) and Fi (x,y, t) kernels for the one sided coupling
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Boundary element formulation
Spatial discretisation on some mesh
θ(x, t) =ND
∑k=1
ψk (x) θ
k (t) q(x, t) =NN
∑k=1
χk (x) qk (t)
uj (x, t) =ND
∑k=1
ψk (x) uk
i (t) tj (x, t) =NN
∑k=1
χk (x) tk
j (t)
Semi-discrete BEM
Cθθθ(t) = [ΘΘΘ∗q](t)− [Q∗θθθ](t)
Ceu(t) = Ut(t)−Tu(t) + [G∗q](t)− [F∗θθθ](t)
Temporal discretisation with gCQMto solve the thermal equationto perform the convolution of known data for the coupling terms
[G∗q](t) [F∗θθθ](t)
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Boundary element formulation
Spatial discretisation on some mesh
θ(x, t) =ND
∑k=1
ψk (x) θ
k (t) q(x, t) =NN
∑k=1
χk (x) qk (t)
uj (x, t) =ND
∑k=1
ψk (x) uk
i (t) tj (x, t) =NN
∑k=1
χk (x) tk
j (t)
Semi-discrete BEM
Cθθθ(t) = [ΘΘΘ∗q](t)− [Q∗θθθ](t)
Ceu(t) = Ut(t)−Tu(t) + [G∗q](t)− [F∗θθθ](t)
Temporal discretisation with gCQMto solve the thermal equationto perform the convolution of known data for the coupling terms
[G∗q](t) [F∗θθθ](t)
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Problem setting
Cube under restrictive boundary conditions to enforce a 1-d solution
q = 0
q = 0
q = 0
θ(t > 0) = 1
x
y or z
• •
Material data:α = 1 κ = 1λ = 0 µ = 0.5
Time discretisation:constant tn = n∆t
increasing tn =
(n +
(n−1)2
N
)∆t
graded tn = N∆t( n
N
)2
Spatial discretisations
Mesh 1 Mesh 2 Mesh 3
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Temperature solution: gCQM, graded
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
time t [s]
tem
pera
ture
θ[K
]
mesh 1mesh 2mesh 3analytic
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Displacement solution: gCQM, graded
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
time t [s]
disp
lace
men
tux[m
]
mesh 1mesh 2mesh 3analytic
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Temperature solution: error, mesh 2
0 1 2 3 4
0
0.5
1
1.5
·10−2
time t [s]
err a
bs
mesh 2, constantmesh 2, gradedmesh 2, increasing
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Temperature solution: error, mesh 3
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
·10−2
time t [s]
err a
bs
mesh 3, constantmesh 3, gradedmesh 3, increasing
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Displacement solution: error, mesh 2
0 1 2 3 4
0
0.5
1
1.5
·10−2
time t [s]
err a
bs
mesh 2, constantmesh 2, gradedmesh 2, increasing
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Displacement solution: error, mesh 3
0 1 2 3 4
1
2
3
4·10−3
time t [s]
err a
bs
mesh 3, constantmesh 3, gradedmesh 3, increasing
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Temperature error L2 mesh 3
10−1.8 10−1.6 10−1.4 10−1.2 10−1
10−5
10−4
time step size ∆t
err re
lmesh 3, constmesh 3, gradedmesh 3, increasingeoc = 2eoc = 1
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Temperature error Lmax mesh 3
10−1.8 10−1.6 10−1.4 10−1.2 10−1
10−2.5
10−2
10−1.5
time step size ∆t
err a
bsmesh 3, constmesh 3, gradedmesh 3, increasingeoc = 0.7eoc = 1
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Displacement error L2 mesh 3
10−1.8 10−1.6 10−1.4 10−1.2 10−1
10−3
10−2
time step size ∆t
err re
l
mesh 3, constmesh 3, gradedmesh 3, increasingeoc = 1.2
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Displacement error Lmax mesh 3
10−1.8 10−1.6 10−1.4 10−1.2 10−1
10−3
10−2
time step size ∆t
err a
bs
mesh 3, constmesh 3, gradedmesh 3, increasingeoc = 0.5eoc = 1.3
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Conclusions
Indirect BE formulation in time domain for absorbing BC in acoustics
Direct BE formulation for uncoupled thermoelasticity
Time discretisation with generalized Convolution Quadrature Method
Expected rate of convergence in time
Application to real world problems possible
Fast methods to compress matrices is to be done
Fast method only for matrix-vector products
Possible extension to variable space-time formulation
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Conclusions
Indirect BE formulation in time domain for absorbing BC in acoustics
Direct BE formulation for uncoupled thermoelasticity
Time discretisation with generalized Convolution Quadrature Method
Expected rate of convergence in time
Application to real world problems possible
Fast methods to compress matrices is to be done
Fast method only for matrix-vector products
Possible extension to variable space-time formulation
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Conclusions
Indirect BE formulation in time domain for absorbing BC in acoustics
Direct BE formulation for uncoupled thermoelasticity
Time discretisation with generalized Convolution Quadrature Method
Expected rate of convergence in time
Application to real world problems possible
Fast methods to compress matrices is to be done
Fast method only for matrix-vector products
Possible extension to variable space-time formulation
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Graz University of TechnologyInstitute of Applied Mechanics
Application of generalized Convolution Quadraturein Acoustics and Thermoelasticity
Martin Schanz
joint work with Relindis Rott and Stefan Sauter
Space-Time Methods for PDEsSpecial Semester on Computational Methods in Science and EngineeringRICAM, Linz, Austria, November 10, 2016
> www.mech.tugraz.at