[PPT] - Application of generalized Convolution Quadrature in Acoustics and PowerPoint Presentation

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Graz University of Technology Institute of Applied Mechanics

Application of generalized Convolution Quadrature in Acoustics and Thermoelasticity

Martin Schanz joint work with Relindis Rott and Stefan Sauter Space-Time Methods for PDEs Special Semester on Computational Methods in Science and Engineering RICAM, Linz, Austria, November 10, 2016 > www.mech.tugraz.at

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 3 / 39

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Convolution integral

Convolution integral with the Laplace transformed function ˆ f (s) y (t) = (f ∗ g)(t) =

ˆ

f (∂t)g

(t) =

t

f (t −τ)g (τ)dτ

=

1 2πi

C

ˆ

f (s)

t

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−∆tns gn =

n

∑

j=1

∆tjgj

n

∏

k=j

1 1−∆tks

Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39

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Convolution integral

Convolution integral with the Laplace transformed function ˆ f (s) y (t) = (f ∗ g)(t) =

ˆ

f (∂t)g

(t) =

t

f (t −τ)g (τ)dτ

=

1 2πi

C

ˆ

f (s)

t

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−∆tns gn =

n

∑

j=1

∆tjgj

n

∏

k=j

1 1−∆tks

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

ˆ

f (s)xn (s)ds

= 1

2πi

C

ˆ

f (s)∆tn 1−∆tns gn ds + 1 2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds

=ˆ

f

1

∆tn

gn + 1

2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds. Recursion formula for the implicit Euler y (tn) = 1 2πi

C

ˆ

f (s)

n

∑

j=1

∆tjgj

n

∏

k=j

1 1−∆tks ds

= ˆ

f

1

∆tn

gn +

n−1

∑

j=1

∆tjgj

1 2πi

C

ˆ

f (s)

n

∏

k=j

1 1−∆tks 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

ˆ

f (s)xn (s)ds

= 1

2πi

C

ˆ

f (s)∆tn 1−∆tns gn ds + 1 2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds

=ˆ

f

1

∆tn

gn + 1

2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds. Recursion formula for the implicit Euler y (tn) = 1 2πi

C

ˆ

f (s)

n

∑

j=1

∆tjgj

n

∏

k=j

1 1−∆tks ds

= ˆ

f

1

∆tn

gn +

n−1

∑

j=1

∆tjgj

1 2πi

C

ˆ

f (s)

n

∏

k=j

1 1−∆tks 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

ˆ

f (s)xn (s)ds

= 1

2πi

C

ˆ

f (s)∆tn 1−∆tns gn ds + 1 2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds

=ˆ

f

1

∆tn

gn + 1

2πi

C

ˆ

f (s) xn−1 (s) 1−∆tns ds. Recursion formula for the implicit Euler y (tn) = 1 2πi

C

ˆ

f (s)

n

∑

j=1

∆tjgj

n

∏

k=j

1 1−∆tks ds

= ˆ

f

1

∆tn

gn +

n−1

∑

j=1

∆tjgj

1 2πi

C

ˆ

f (s)

n

∏

k=j

1 1−∆tks 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) = ˆ f

1

∆t1

g1

with implicit assumption of zero initial condition For all steps n = 2,...,N the algorithm has two steps

1

Update the solution vector xn−1 at all integration points sℓ with an implicit Euler step xn−1 (sℓ) = xn−2 (sℓ) 1−∆tn−1sℓ

+ ∆tn−1

1−∆tn−1sℓ gn−1 for ℓ = 1,...,NQ with the number of integration points NQ.

2

Compute the solution of the integral at the actual time step tn y (tn) = ˆ f

1

∆tn

gn +

NQ

∑

ℓ=1

ωℓ ˆ

f (sℓ) 1−∆tnsℓ xn−1 (sℓ)

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

∆tmin

Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

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Algorithm

First Euler step y (t1) = ˆ f

1

∆t1

g1

with implicit assumption of zero initial condition For all steps n = 2,...,N the algorithm has two steps

1

Update the solution vector xn−1 at all integration points sℓ with an implicit Euler step xn−1 (sℓ) = xn−2 (sℓ) 1−∆tn−1sℓ

+ ∆tn−1

1−∆tn−1sℓ gn−1 for ℓ = 1,...,NQ with the number of integration points NQ.

2

Compute the solution of the integral at the actual time step tn y (tn) = ˆ f

1

∆tn

gn +

NQ

∑

ℓ=1

ωℓ ˆ

f (sℓ) 1−∆tnsℓ xn−1 (sℓ)

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

∆tmin

Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

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Numerical integration

Integration weights and points sℓ = γ(σℓ)

ωℓ =

4K

k2

2πi

γ′ (σℓ)

for N = 25,T = 5,tn =

n

N

α T,α = 1.5

Martin Schanz gCQM: Acoustics and Thermoelasticity 7 / 39

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 8 / 39

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Absorbing boundary conditions

Materials with absorbing surfaces Mechanical modell: Coupling of porous material layer at the boundary Simpler mechanical model: Impedance boundary condition Z = p v· n specific 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

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Absorbing boundary conditions

Materials with absorbing surfaces Mechanical modell: Coupling of porous material layer at the boundary Simpler mechanical model: Impedance boundary condition Z = p v· n specific 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

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Problem setting

Bounded Lipschitz domain Ω− ⊂ R3 with boundary Γ := ∂Ω

Ω+ := R3\Ω− is its unbounded complement.

Linear acoustics for the pressure p

∂ttp − c2∆p = 0

in Ωσ ×R>0, p(x,0) = ∂tp(x,0) = 0 in Ωσ,

γσ

1 (p)−σα

c γσ

0 (∂tp)= f (x,t)

n Γ×R>0

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 10 / 39

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Integral equation

Single layer ansatz for the density

−

σϕ

2 −K′ ∗ϕ

−σα

c (V ∗ ˙

ϕ) = f

a.e. in Γ×R>0 Retarded potentials

(V ∗ϕ)(x,t) =

Γ

ϕ

y,t − x−y

c

4πx − y

dΓy

K′ ∗ϕ
(x,t) = 1

4π

Γ

n(x),y − x x − y2   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

4πx − y

dΓy

∀(x,t) ∈ Ωσ ×R>0

Martin Schanz gCQM: Acoustics and Thermoelasticity 11 / 39

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Integral equation

Single layer ansatz for the density

−

σϕ

2 −K′ ∗ϕ

−σα

c (V ∗ ˙

ϕ) = f

a.e. in Γ×R>0 Retarded potentials

(V ∗ϕ)(x,t) =

Γ

ϕ

y,t − x−y

c

4πx − y

dΓy

K′ ∗ϕ
(x,t) = 1

4π

Γ

n(x),y − x x − y2   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

4πx − y

dΓy

∀(x,t) ∈ Ωσ ×R>0

Martin Schanz gCQM: Acoustics and Thermoelasticity 11 / 39

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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 m

n = λ(Z) n

s

c

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) = 1

2 − 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

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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 m

n = λ(Z) n

s

c

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) = 1

2 − 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

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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 m

n = λ(Z) n

s

c

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) = 1

2 − 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

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Solution in time domain

Solution for σ = +1, i.e., outer space, and n = 0 Load function f (t) = (ct)υ e−ct Density function

ϕ+ (t) = −

2 1+α

⌊ct/2⌋

∑

ℓ=0

(ct − 2ℓ)υ e−(ct−2ℓ)

−(1+α)υ αυ+1 γ

υ+ 1,

α

1+α (ct − 2ℓ)

e− ct−2ℓ

1+α

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

z

0 ta−1e−tdt

Pressure solution p+ (r,t) = − (1+α)υ 2√παυ+1 γ

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

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Solution in time domain

Solution for σ = +1, i.e., outer space, and n = 0 Load function f (t) = (ct)υ e−ct Density function

ϕ+ (t) = −

2 1+α

⌊ct/2⌋

∑

ℓ=0

(ct − 2ℓ)υ e−(ct−2ℓ)

−(1+α)υ αυ+1 γ

υ+ 1,

α

1+α (ct − 2ℓ)

e− ct−2ℓ

1+α

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

z

0 ta−1e−tdt

Pressure solution p+ (r,t) = − (1+α)υ 2√παυ+1 γ

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

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Discretization

Spatial discretization: Linear continuous shape functions on linear triangles Temporal 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

2

Observation 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

0.2 0.4 0.6 0.8 1 1.2

·10−2 −0.2

0.2 0.4 0.6 time t [s] density ϕ+

α = 0 α = 0.25 α = 0.5 α = 1

analytic α = 0.25 analytic α = 0.5 analytic α = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 15 / 39

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Solution pressure

0.2 0.4 0.6 0.8 1 1.2

·10−2

0.05 0.1 0.15 time t [s] pressure u+ [N/m2]

α = 0 α = 0.25 α = 0.5 α = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 16 / 39

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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

errrel ∆tvar,β = 0.125 ∆tconst,β = 0.125 ∆tvar,β = 0.0625 ∆tconst,β = 0.0625 eoc = 0.5 eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 17 / 39

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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

errrel ∆tvar,β = 0.25 ∆tconst,β = 0.25 ∆tvar,β = 0.125 ∆tconst,β = 0.125 ∆tvar,β = 0.0625 ∆tconst,β = 0.0625 eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 18 / 39

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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

errrel

mesh 2, ∆tconst mesh 2, ∆tvar mesh 3, ∆tconst mesh 3, ∆tvar mesh 4, ∆tconst mesh 4, ∆tvar

eoc = 0.5 eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 19 / 39

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Relative pressure error: time step

10−5 10−4 10−2 10−1 time step size ∆t

errrel

mesh 2, ∆tconst mesh 2, ∆tvar mesh 3, ∆tconst mesh 3, ∆tvar mesh 4, ∆tconst mesh 4, ∆tvar

eoc = 2 eoc = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 20 / 39

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Problem description

Atrium at University of Zurich (Irchel campus) Mesh: 7100 elements Time interval [0,T = 0.15s] with grading tn =

n + (n − 1)2

N

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

Martin Schanz gCQM: Acoustics and Thermoelasticity 21 / 39

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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

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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

10 20 30 time t [s] sound pressure level u [dB]

α = 0.1 α = 0.5 α = 1

Martin Schanz gCQM: Acoustics and Thermoelasticity 23 / 39

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Content

1 Generalized convolution quadrature method (gCQM) Quadrature formula Algorithm

2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples

3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example

Martin Schanz gCQM: Acoustics and Thermoelasticity 24 / 39

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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 constants

Boundary 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

with Θ(x,y,t) and Q(x,y,t) kernels of the heat equation

Uij(x,y) and Tij(x,y) kernels from elastostatics Gi(x,y,t) and Fi(x,y,t) kernels for the one sided coupling

Martin Schanz gCQM: Acoustics and Thermoelasticity 25 / 39

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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 constants

Boundary 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

with Θ(x,y,t) and Q(x,y,t) kernels of the heat equation

Uij(x,y) and Tij(x,y) kernels from elastostatics Gi(x,y,t) and Fi(x,y,t) kernels for the one sided coupling

Martin Schanz gCQM: Acoustics and Thermoelasticity 25 / 39

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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) uk

i (t)

tj(x,t) =

NN

∑

k=1

χk(x) tk

j (t)

Semi-discrete BEM Cθ

θ θ(t) = [Θ Θ Θ∗ q](t)−[Q∗θ θ θ](t)

Ceu(t) = Ut(t)− Tu(t)+[G∗ q](t)−[F∗θ

θ θ](t)

Temporal discretisation with gCQM

to solve the thermal equation to perform the convolution of known data for the coupling terms

[G∗ q](t) [F∗θ θ θ](t)

Martin Schanz gCQM: Acoustics and Thermoelasticity 26 / 39

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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) uk

i (t)

tj(x,t) =

NN

∑

k=1

χk(x) tk

j (t)

Semi-discrete BEM Cθ

θ θ(t) = [Θ Θ Θ∗ q](t)−[Q∗θ θ θ](t)

Ceu(t) = Ut(t)− Tu(t)+[G∗ q](t)−[F∗θ

θ θ](t)

Temporal discretisation with gCQM

to solve the thermal equation to perform the convolution of known data for the coupling terms

[G∗ q](t) [F∗θ θ θ](t)

Martin Schanz gCQM: Acoustics and Thermoelasticity 26 / 39

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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 = 1 λ = 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

Martin Schanz gCQM: Acoustics and Thermoelasticity 27 / 39

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Temperature solution: gCQM, graded

1 2 3 4 0.2 0.4 0.6 0.8 1 time t [s] temperature θ[K] mesh 1 mesh 2 mesh 3 analytic

Martin Schanz gCQM: Acoustics and Thermoelasticity 28 / 39

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Displacement solution: gCQM, graded

1 2 3 4 0.2 0.4 0.6 0.8 1 time t [s] displacement ux[m] mesh 1 mesh 2 mesh 3 analytic

Martin Schanz gCQM: Acoustics and Thermoelasticity 29 / 39

SLIDE 43

Temperature solution: error, mesh 2

1 2 3 4 0.5 1 1.5

·10−2

time t [s]

errabs

mesh 2, constant mesh 2, graded mesh 2, increasing

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SLIDE 44

Temperature solution: error, mesh 3

1 2 3 4 0.2 0.4 0.6 0.8 1

·10−2

time t [s]

errabs

mesh 3, constant mesh 3, graded mesh 3, increasing

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SLIDE 45

Displacement solution: error, mesh 2

1 2 3 4 0.5 1 1.5

·10−2

time t [s]

errabs

mesh 2, constant mesh 2, graded mesh 2, increasing

Martin Schanz gCQM: Acoustics and Thermoelasticity 32 / 39

SLIDE 46

Displacement solution: error, mesh 3

1 2 3 4 1 2 3 4 ·10−3 time t [s]

errabs

mesh 3, constant mesh 3, graded mesh 3, increasing

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SLIDE 47

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

errrel

mesh 3, const mesh 3, graded mesh 3, increasing

eoc = 2 eoc = 1

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SLIDE 48

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

errabs

mesh 3, const mesh 3, graded mesh 3, increasing

eoc = 0.7 eoc = 1

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SLIDE 49

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

errrel

mesh 3, const mesh 3, graded mesh 3, increasing

eoc = 1.2

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SLIDE 50

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

errabs

mesh 3, const mesh 3, graded mesh 3, increasing

eoc = 0.5 eoc = 1.3

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SLIDE 51

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

SLIDE 52

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

SLIDE 53

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

SLIDE 54

Graz University of Technology Institute of Applied Mechanics

Application of generalized Convolution Quadrature in Acoustics and Thermoelasticity

Martin Schanz joint work with Relindis Rott and Stefan Sauter Space-Time Methods for PDEs Special Semester on Computational Methods in Science and Engineering RICAM, Linz, Austria, November 10, 2016 > www.mech.tugraz.at