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

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

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

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

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

Convolution integral

Convolution integral with the Laplace transformed function f̂ (s)

y (t) = (f ∗g)(t) =(f̂ (∂t )g

)(t) =

t∫0

f (t− τ)g (τ)dτ

=1

2πi

∫C

f̂ (s)

t∫0

es(t−τ)g (τ)dτ

︸ ︷︷ ︸x (t,s)

ds

Integral is equivalent to solution of ODE

∂∂t

x (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

+∆tn

1−∆tnsgn =

n

∑j=1

∆tjgjn

∏k=j

11−∆tk s

Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39

Convolution integral

Convolution integral with the Laplace transformed function f̂ (s)

y (t) = (f ∗g)(t) =(f̂ (∂t )g

)(t) =

t∫0

f (t− τ)g (τ)dτ

=1

2πi

∫C

f̂ (s)

t∫0

es(t−τ)g (τ)dτ

︸ ︷︷ ︸x (t,s)

ds

Integral is equivalent to solution of ODE

∂∂t

x (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

+∆tn

1−∆tnsgn =

n

∑j=1

∆tjgjn

∏k=j

11−∆tk s

Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39

Time stepping formula

Solution at the discrete time tn

y (tn) =1

2πi

∫C

f̂ (s)xn (s)ds

=1

2πi

∫C

f̂ (s)∆tn1−∆tns

gn ds +1

2πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds

=f̂

(1

∆tn

)gn +

12πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds .

Recursion formula for the implicit Euler

y (tn) =1

2πi

∫C

f̂ (s)n

∑j=1

∆tjgjn

∏k=j

11−∆tk s

ds

= f̂

(1

∆tn

)gn +

n−1∑j=1

∆tjgj1

2πi

∫C

f̂ (s)n

∏k=j

11−∆tk s

ds

Complex integral is solved with a quadrature formula

Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Time stepping formula

Solution at the discrete time tn

y (tn) =1

2πi

∫C

f̂ (s)xn (s)ds

=1

2πi

∫C

f̂ (s)∆tn1−∆tns

gn ds +1

2πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds

=f̂

(1

∆tn

)gn +

12πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds .

Recursion formula for the implicit Euler

y (tn) =1

2πi

∫C

f̂ (s)n

∑j=1

∆tjgjn

∏k=j

11−∆tk s

ds

= f̂

(1

∆tn

)gn +

n−1∑j=1

∆tjgj1

2πi

∫C

f̂ (s)n

∏k=j

11−∆tk s

ds

Complex integral is solved with a quadrature formula

Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Time stepping formula

Solution at the discrete time tn

y (tn) =1

2πi

∫C

f̂ (s)xn (s)ds

=1

2πi

∫C

f̂ (s)∆tn1−∆tns

gn ds +1

2πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds

=f̂

(1

∆tn

)gn +

12πi

∫C

f̂ (s)xn−1 (s)1−∆tns

ds .

Recursion formula for the implicit Euler

y (tn) =1

2πi

∫C

f̂ (s)n

∑j=1

∆tjgjn

∏k=j

11−∆tk s

ds

= f̂

(1

∆tn

)gn +

n−1∑j=1

∆tjgj1

2πi

∫C

f̂ (s)n

∏k=j

11−∆tk s

ds

Complex integral is solved with a quadrature formula

Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Algorithm

First Euler step

y (t1) = f̂

(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−11−∆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) = f̂

(1

∆tn

)gn +

NQ

∑̀=1

ω`f̂ (s`)

1−∆tns`xn−1 (s`)

Essential parameter: NQ = N log(N), integration is dependent on q =∆tmax∆tmin

Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

Algorithm

First Euler step

y (t1) = f̂

(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−11−∆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) = f̂

(1

∆tn

)gn +

NQ

∑̀=1

ω`f̂ (s`)

1−∆tns`xn−1 (s`)

Essential parameter: NQ = N log(N), integration is dependent on q =∆tmax∆tmin

Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 7 / 39

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

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) = cosθ1−√

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

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) = cosθ1−√

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

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)−σαc

γσ0 (∂t p)= f (x , t) on Γ×R>0

with σ ∈ {+,−}, wave velocity c, and α absorption coefficient

Martin Schanz gCQM: Acoustics and Thermoelasticity 10 / 39

Integral equation

Single layer ansatz for the density

−(

σϕ2−K′ ∗ϕ

)−σα

c(V ∗ ϕ̇) = 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‖ +ϕ̇(

y , t− ‖x−y‖c)

c

dΓySingle 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

Martin Schanz gCQM: Acoustics and Thermoelasticity 11 / 39

Integral equation

Single layer ansatz for the density

−(

σϕ2−K′ ∗ϕ

)−σα

c(V ∗ ϕ̇) = 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‖ +ϕ̇(

y , t− ‖x−y‖c)

c

dΓySingle 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

Martin Schanz gCQM: Acoustics and Thermoelasticity 11 / 39

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

ẐY mn = λ(Z)n

(sc

)Y mn 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 their

first derivatives

Analytical transformation yields time domain solution

Martin Schanz gCQM: Acoustics and Thermoelasticity 12 / 39

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

ẐY mn = λ(Z)n

(sc

)Y mn 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 their

first derivatives

Analytical transformation yields time domain solution

Martin Schanz gCQM: Acoustics and Thermoelasticity 12 / 39

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

ẐY mn = λ(Z)n

(sc

)Y mn 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 their

first derivatives

Analytical transformation yields time domain solution

Martin Schanz gCQM: Acoustics and Thermoelasticity 12 / 39

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) =− 21 + α

bct/2c∑̀=0

((ct−2`)υ e−(ct−2`)

− (1 + α)υ

αυ+1γ(

υ + 1,α

1 + α(ct−2`)

)e−

ct−2`1+α

)with the incomplete Gamma function γ(a,z) :=

∫ z0 t

a−1e−t dt

Pressure solution

p+ (r , t) =− (1 + α)υ

2√

παυ+1γ(

υ + 1,α

1 + ατ+)

e−τ

1+α

r.

with r > 1, we define τ := ct− (r −1) and (τ)+ := max{0,τ}

Martin Schanz gCQM: Acoustics and Thermoelasticity 13 / 39

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) =− 21 + α

bct/2c∑̀=0

((ct−2`)υ e−(ct−2`)

− (1 + α)υ

αυ+1γ(

υ + 1,α

1 + α(ct−2`)

)e−

ct−2`1+α

)with the incomplete Gamma function γ(a,z) :=

∫ z0 t

a−1e−t dt

Pressure solution

p+ (r , t) =− (1 + α)υ

2√

παυ+1γ(

υ + 1,α

1 + ατ+)

e−τ

1+α

r.

with r > 1, we define τ := ct− (r −1) and (τ)+ := max{0,τ}

Martin Schanz gCQM: Acoustics and Thermoelasticity 13 / 39

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 = 12Observation time T = 0.002915905s and β = c∆t/hError 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

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 15 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 16 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 17 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 18 / 39

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, ∆tconstmesh 2, ∆tvarmesh 3, ∆tconstmesh 3, ∆tvarmesh 4, ∆tconstmesh 4, ∆tvareoc = 0.5eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 19 / 39

Relative pressure error: time step

10−5 10−4

10−2

10−1

time step size ∆t

err re

l

mesh 2, ∆tconstmesh 2, ∆tvarmesh 3, ∆tconstmesh 3, ∆tvarmesh 4, ∆tconstmesh 4, ∆tvareoc = 2eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 20 / 39

Problem description

Atrium at University of Zurich (Irchel campus)

Mesh: 7100 elementsTime interval [0,T = 0.15s] with grading

tn =

(n +

(n−1)2N

)∆tconst with ∆tconst = 0.00037s ⇒ N = 405, Ngraded = 248

Martin Schanz gCQM: Acoustics and Thermoelasticity 21 / 39

Sound pressure field

t ≈ 0.028 s

α = 0.1 α = 0.5 α = 1

t ≈ 0.064 s

Martin Schanz gCQM: Acoustics and Thermoelasticity 22 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 23 / 39

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 24 / 39

Uncoupled thermoelasticity

Governing equations for temperature θ(x, t) and displacement u(x, t)

κ θ,jj (x, t)− θ̇(x, t) = 0µ 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)}dΓwith

Θ(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

Martin Schanz gCQM: Acoustics and Thermoelasticity 25 / 39

Uncoupled thermoelasticity

Governing equations for temperature θ(x, t) and displacement u(x, t)

κ θ,jj (x, t)− θ̇(x, t) = 0µ 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)}dΓwith

Θ(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

Martin Schanz gCQM: Acoustics and Thermoelasticity 25 / 39

Boundary element formulation

Spatial discretisation on some mesh

θ(x, t) =ND

∑k=1

ψk (x) θk (t) q(x, t) =NN

∑k=1

χk (x) qk (t)

uj (x, t) =ND

∑k=1

ψk (x) uki (t) tj (x, t) =NN

∑k=1

χk (x) tkj (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)

Martin Schanz gCQM: Acoustics and Thermoelasticity 26 / 39

Boundary element formulation

Spatial discretisation on some mesh

θ(x, t) =ND

∑k=1

ψk (x) θk (t) q(x, t) =NN

∑k=1

χk (x) qk (t)

uj (x, t) =ND

∑k=1

ψk (x) uki (t) tj (x, t) =NN

∑k=1

χk (x) tkj (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)

Martin Schanz gCQM: Acoustics and Thermoelasticity 26 / 39

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)2N

)∆t

graded tn = N∆t( n

N

)2Spatial discretisations

Mesh 1 Mesh 2 Mesh 3

Martin Schanz gCQM: Acoustics and Thermoelasticity 27 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 28 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 29 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 30 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 31 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 32 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 33 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 34 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 35 / 39

Displacement error L2 mesh 3

10−1.8 10−1.6 10−1.4 10−1.2 10−110−3

10−2

time step size ∆t

err re

l

mesh 3, constmesh 3, gradedmesh 3, increasingeoc = 1.2

Martin Schanz gCQM: Acoustics and Thermoelasticity 36 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 37 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 38 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 38 / 39

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 38 / 39

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

Generalized convolution quadrature method (gCQM)Quadrature formulaAlgorithm

Acoustics: Absorbing boundary conditionsBoundary element formulationAnalytical solutionNumerical examples

Thermoelasticity: Uncoupled formulationBoundary element formulationNumerical example