High order numerical schemes for linear elasticity

A high order method on unstructured staggered meshes ..... Numercal experiments. Diffuse interface. Numerical experiments time xi. 0. 0.2. 0.4. 0.6. 0.8. -0.4.
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Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

High order numerical schemes for linear elasticity Tavelli Maurizio and Michael Dumbser University of Trento

21-25 May 2018

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Governing equations

∂ ∂t ∂ ∂t ∂ ∂t

σyy − λ σzz − λ

∂ ∂x ∂ ∂x

u−λ

∂t ∂ ∂

∂ ∂t ∂ ∂t

(ρu) −

u−λ

(ρw ) −

∂y

∂ ∂y



 σxy − µ

∂x

σyz − µ

∂ ∂z

σxz − µ

∂x ∂ ∂x

σxx − σxy −

∂ ∂x

v −λ v −λ

v − (λ + 2µ)





∂t ∂

(ρv ) −



∂ ∂y



∂t



∂x

u − (λ + 2µ)



∂t



σxx −

σxz −

∂z ∂ ∂y ∂ ∂y ∂ ∂y

v+ v+ u+

σxy − σyy − σyz −

∂ ∂z ∂ ∂z ∂ ∂z

∂ ∂y ∂ ∂y ∂ ∂x ∂ ∂z ∂ ∂z ∂ ∂z

w = 0, w = 0, w = 0, 

u

= 0, 

w

= 0, 

w

= 0,

σxz = 0, σyz = 0, σzz = 0.



σxx σ =  σxy σxz

σxy σyy σyz

 σxz σyz  = σ T . σzz

v = (u, v , w ) is the velocity field ρ is the density λ, µ are the classical Lam´e constants

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Governing equations

∂σ − E · ∇v = 0, ∂t ∂ρv − ∇ · σ = 0, ∂t

E is the usual stiffness tensor according to the Hooke law σij = Eijkl kl

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Sharp Interface approach A high order method on unstructured staggered meshes Diffuse interface approach A diffuse interface method on Cartesian AMR grids

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Sharp Interface approach Unstructured staggered mesh

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Example Ωi = {n1 , n2 , n3 } Si = {j1 , j2 , j3 } Γj1 = {n2 , n3 }

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Example Ωi = {n1 , n2 , n3 } Si = {j1 , j2 , j3 } Γj1 = {n2 , n3 }

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Example Ωi = {n1 , n2 , n3 } Si = {j1 , j2 , j3 } Γj1 = {n2 , n3 } `(j1 ) = i and r (j1 ) = i1 ℘(i, j1 ) = i1

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Example Ωi = {n1 , n2 , n3 } Si = {j1 , j2 , j3 } Γj1 = {n2 , n3 } `(j1 ) = i and r (j1 ) = i1 ℘(i, j1 ) = i1 Ωj1 = {i, i1 , Γj1 }

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Example Ωi = {n1 , n2 , n3 } Si = {j1 , j2 , j3 } Γj1 = {n2 , n3 } `(j1 ) = i and r (j1 ) = i1 ℘(i, j1 ) = i1 Ωj1 = {i, i1 , Γj1 }

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Basis Functions

Fixed p ∈ N we want a polynomial basis able to reconstruct exactly a polynomial of degree p. It depends on the element where we want to reconstruct the polynomial. {(ξk , γk )}k=1...N a set of distinct points in the reference element Tstd or Rstd , the coefficients of the polynomials can be found by imposing  1 k =l φk (ξl , γl ) = 0 k 6= l and solving a linear system.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Basis Functions

Basis Function {φk }k=1...Nφ Nφ =

(p+1)(p+2) 2

Basis Function {ψk }k=1...Nψ Nψ = (p + 1)2

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Triangle Transformations N1 (ξ, γ) = 1 − ξ − γ

Ti−1

;

N2 (ξ, γ) = ξ



(ξ, γ) −→ x =

3 X p=1

;

Np Xp , y =

N3 (ξ, γ) = γ

3 X p=1

 Np Yp 

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Triangle Transformations

Ti−1



(ξ, γ) −→ x =

Nφ X p=1

Np Xp , y =

Nφ X p=1

 Np Yp 

Preliminary

Sharp Interface

Numerical Method

Space-Time extension

Ωi , {φk (x)}, Nφ = Nφ (p) Ωj , {ψk (x)}, Nψ = Nψ (p)

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Space-Time extension

st ˜ Ωi , {φk (x)}, Nφ = Nφ (p) → Ωst i = Ωi × T , {φk (x, t)}, Nφ = Nφ (p) · (pγ + 1) st ˜ Ωj , {ψk (x)}, Nψ = Nψ (p) → Ωst j = Ωj × T , {ψk (x, t)}, Nψ = Nψ (p) · (pγ + 1)

Preliminary

Sharp Interface

Numerical Method

Space-Time extension

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Space-Time extension

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Sharp Interface approach Numerical method

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Definition of the quantities vi (x, t) = vh (x, t)|Ωsti and ρi (x) = ρh (x)|Ωsti are defined on the main grid; σj (x, t) = σh (x, t)|Ωstj and Ej (x) = Eh (x)|Ωstj are defined on the dual grid.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Definition of the quantities vi (x, t) = vh (x, t)|Ωsti and ρi (x) = ρh (x)|Ωsti are defined on the main grid; σj (x, t) = σh (x, t)|Ωstj and Ej (x) = Eh (x)|Ωstj are defined on the dual grid. N st

vi (x, t)

=

φ X

(i) n+1 ˜(i) (x, t)ˆ φ˜l (x, t)ˆ vl,i =: φ vin+1 ,

l=1 N st

ρi (x, t)

=

φ X

(i) n+1 ˜(i) (x, t)ˆ φ˜l (x, t)ˆ vl,i =: φ ρn+1 , i

l=1 N st

σj (x, t)

=

ψ X

(j) n+1 =: ψ˜(j) (x, t)σ ˆ jn+1 , ψ˜l (x, t)ˆ σl,j

l=1 N st

Ej (x)

=

ψ X

l=1

(j) ˆ l,j =: ψ˜(j) (x)E ˆj . ψ˜l (x)E

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

Z Ωst i

(i) ∂ρv φ˜k dx dt − ∂t

Z Ωst i

(i) φ˜k ∇ · σdx dt = 0.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation



 Z Ωst i

 (i) ∂ρv φ˜k dx dt −  ∂t

Z

∂Ωst i

(i) φ˜k σ · ni dS dt −

Z Ωst i

 (i) ∇φ˜k · σdx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

 Z Ωst i

 (i) ∂ρv φ˜k dx dt −  ∂t

 Z

∂Ωst i

(i) φ˜k σ · ni dS dt −

Z Ωst i

 (i) ∇φ˜k · σdx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation



 Z Ωst i

 (i) ∂ρv φ˜k dx dt −  ∂t

Z

∂Ωst i

(i) φ˜k σ · ni dS dt −

Z Ωst i

 (i) ∇φ˜k · σdx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

 Z Ωst i

 (i) ∂ρv φ˜k dx dt −  ∂t

 Z

∂Ωst i

(i) φ˜k σ · ni dS dt −

Z Ωst i

 (i) ∇φ˜k · σdx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

 Z Ωst i

 (i) ∂ρv φ˜k dx dt −  ∂t

 Z

∂Ωst i

(i) φ˜k σ · ni dS dt −

Z Ωst i

 (i) ∇φ˜k · σdx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation



 Z Ωst i

(i) ∂(ρv)i φ˜k dx dt − ∂t

X  j∈Si

Z

Γst j

(i) φ˜k σj · ni,j dS dt −

Z

Ωst i,j

 (i) ∇φ˜k · σj dx dt  = 0,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation Z Ωst i

˜(i) φ k

∂(ρv)i ∂t

Z dx dt

˜(i) (x, t n+1,− )ρvi (x, t n+1,− )dx − φ k

= Ωi

Z Ωi

Z − Ωst i

˜(i) ∂φ k ∂t

(ρv)i dx dt ,

˜(i) (x, t n,+ )ρvi (x, t n,− )dx φ k

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation Z Ωst i

˜(i) φ k

∂(ρv)i ∂t

Z dx dt

˜(i) (x, t n+1,− )ρvi (x, t n+1,− )dx − φ k

= Ωi

Z

˜(i) (x, t n,+ )ρvi (x, t n,− )dx φ k

Ωi

˜(i) ∂φ k

Z −

∂t

Ωst i

(ρv)i dx dt ,

⇓ 

 Z  

˜(i) (x, t n+1,− )φ ˜(i) (x, t n+1,− )dx φ m k

Z − Ωst i

Ωi

(i)

˜ ∂φ k ∂t

˜(i) dx dt φ m

 ˆ n+1 −  (ρv) m,i 

Z

ˆ n ˜(i) (x, t n,+ )φ ˜(i) (x, t n,− )dx (ρv) φ m m,i k

Ωi





Z Z  X  n+1 ˜(i) ψ ˜ (j) ni,j dS dt − ˜(i) ψ ˜ (j) dx dt  − ∇φ ˆ m,j = 0  φ ·σ m m k k   j∈Si

ˆ M¯ (ρv)

n+1

i

i

M

Γst j

Ωst i,j



ˆ n+ = ¯ i− (ρv) i

X j∈Si

D i,j · σ ˆ jn+1 ,

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

Z Ωst j

(j) ∂σ ψ˜k dx dt − ∂t

Z Ωst j

(j) ψ˜k E · ∇vdx dt = 0.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

Z Ωst j

(j) ∂σ dx dt − ψ˜k ∂t

Z Ωst j

(j) ψ˜k E · ∇vdx dt = 0.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

Z Ωst j

˜ (j) ψ k

∂σj ∂t

Z dx dt −

˜ (j) Ej · ∇v ψ `(j) dx dt − k

Ωst `(j),j

Z

˜ (j) Ej · ∇v dx dt − ψ r (j) k

Ωst r (j),j

Z

Γst j

˜ (j) Ej · (v ψ r (j) − v`(j) ) ⊗ nj dS dt = 0. k

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Weak formulation

 Z Z   ψ ˜ (i) (x, t n+1,− )ψ ˜ (i) (x, t n+1,− )dx − m  k

Ωst j

Ωj

 ˜ (i) ∂ψ k ∂t



˜ (i) dx dt ψ m

Z   n+1 n ˜ (i) (x, t n,+ )ψ ˜ (i) (x, t n,− )dx σ ˆ − ψ ˆ m,j σ m k  m,j Ωi



 Z ˆ q,j ·  −E  

˜ (j) ∇φ ˜(`(j)) ψ ˜ (j) dx dt − ψ m q k

Ωst `(j),j

Z Γst j

 n+1 ˜ (j) φ ˜(`(j)) ψ ˜ (j) nj dSdt  ψ v ˆ m q k  m,`(j)





 Z ˆ q,j ·  −E  

˜ (j) ∇φ ˜(r (j)) ψ ˜ (j) dx dt + ψ m q k

Ωst r (j),j

M σˆ j

n+1 j

Z Γst j

=

M

 n+1 ˜ (j) φ ˜(r (j)) ψ ˜ (j) nj dSdt  ψ v = 0. ˆ m q k  m,r (j)

⇓ − n ˆj j σ

ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 . +E `(j) r (j)

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method

M  M σˆ

M

 ˆ n+1 = ¯ − (ρv) ˆ n + P D i,j · σ  ¯ i (ρv) ˆj , i i i j

n+1 j

=

M

− n ˆj j σ

j∈Si

ˆ j · Q`(j),j vˆ`(j) + E ˆ j · Qr (j),j vˆr (j) +E

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method

We get for every i and j ∈ Si  ˆ n+1 = ¯ − (ρv) ˆ n + P D i,j · σ  ¯ i (ρv) ˆ n+1 j , i i i

M  Mσ ˆ j

n+1 j

=

M

M

− n ˆj j σ

j∈Si

ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 +E `(j) r (j)

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method Formal substitution of σ ˆ j in the momentum equation we get

M¯ ρˆ vˆ i

n+1 i i



X j∈Si

D i,j ·

M

−1 j



 ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 = E `(j) r (j)



− ˆi vˆin i ρ

+

X j∈Si

D i,j ·

M M −1 j

− n ˆj j σ

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method Formal substitution of σ ˆ j in the momentum equation we get

M¯ ρˆ vˆ i

n+1 i i



X j∈Si

D i,j ·

M

−1 j



 ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 = E `(j) r (j)



− ˆi vˆin i ρ

+

X

D i,j ·

M M −1 j

− n ˆj j σ

j∈Si

is a 5-points block system in 3D and a 4-points block system in 2D

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method Formal substitution of σ ˆ j in the momentum equation we get

M¯ ρˆ vˆ i

n+1 i i



X j∈Si

D i,j ·

M

−1 j



 ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 = E `(j) r (j)



− ˆi vˆin i ρ

+

X

D i,j ·

M M −1 j

− n ˆj j σ

j∈Si

is a 5-points block system in 3D and a 4-points block system in 2D for pγ = 0 and homogeneous material is symmetric and positive definite

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method Formal substitution of σ ˆ j in the momentum equation we get

M¯ ρˆ vˆ i

n+1 i i



X

D i,j ·

j∈Si

M

−1 j



 ˆ j · Q`(j),j vˆn+1 + E ˆ j · Qr (j),j vˆn+1 = E `(j) r (j)



− ˆi vˆin i ρ

+

X

D i,j ·

M M −1 j

− n ˆj j σ

j∈Si

is a 5-points block system in 3D and a 4-points block system in 2D for pγ = 0 and homogeneous material is symmetric and positive definite All the matrices

M , Q, D can be pre-computed ∀i, j ∈ S

i

It can be efficiently solved by the GMRES algorithm or better by the CG Method.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical Method

Further properties of the staggered space-time DG schemes for linear elasticity in the homogeneous case: For any p ≥ 0 pγ ≥ 0 the fully discrete method is energy stable For p ≥ 0, pγ = 0 and for the special case of a Crank-Nicolson time discretization the scheme is also exactly energy preserving

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Sharp Interface approach Numerical experiments

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments

p 1 1 1 1 p 2 2 2 2 p 3 3 3 3 p 4 4 4 4

Ni 1760 3960 7040 11000 Ni 1760 3960 7040 11000 Ni 1760 3960 7040 11000 Ni 1760 3960 7040 11000

u 1.253E-01 4.609E-02 2.479E-02 1.567E-02 u 1.512E-03 3.697E-04 1.416E-04 6.901E-05 u 5.522E-05 1.079E-05 3.414E-06 1.396E-06 u 2.480E-06 3.270E-07 7.724E-08 2.532E-08

2.5 2.2 2.1

3.5 3.3 3.2

4.0 4.0 4.0

5.0 5.0 5.0

v 2.675E-01 1.284E-01 7.356E-02 4.741E-02 v 3.249E-03 6.568E-04 2.118E-04 8.872E-05 v 3.323E-05 5.544E-06 1.677E-06 6.827E-07 v 1.216E-06 1.582E-07 3.733E-08 1.218E-08

1.8 1.9 2.0

3.9 3.9 3.9

4.4 4.2 4.0

5.0 5.0 5.0

σxx 5.111E-01 2.428E-01 1.387E-01 8.931E-02 σxx 6.081E-03 1.218E-03 3.882E-04 1.601E-04 σxx 4.781E-05 6.534E-06 1.824E-06 7.183E-07 σxx 1.400E-06 1.820E-07 4.292E-08 1.402E-08

1.8 1.9 2.0

4.0 4.0 4.0

4.9 4.4 4.2

5.0 5.0 5.0

σyy 3.003E-01 1.248E-01 6.938E-02 4.430E-02 σyy 3.156E-03 6.411E-04 2.086E-04 8.835E-05 σyy 3.835E-05 6.313E-06 1.906E-06 7.668E-07 σyy 1.434E-06 1.869E-07 4.418E-08 1.442E-08

2.2 2.0 2.0

3.9 3.9 3.9

4.4 4.2 4.1

5.0 5.0 5.0

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments

Plane wave scattering on a circular cavity. Comparison of the isocontours of the stress tensor component σxx between the reference solution given by an explicit ADER-DG scheme (left) and our new staggered space-time DG scheme (right).

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments Reference sxx Staggered DG syy Staggered DG sxy Staggered DG

0.5

Reference u Staggered DG v Staggered DG

0.2

0.4 0.3

0.1 0.2

xi

xi

0.1 0

0 -0.1

-0.1

-0.2 -0.3 -0.4

-0.2 0

0.2

0.4

0.6

0.8

0

0.2

0.4

time

Reference sxx Staggered DG syy Staggered DG sxy Staggered DG

0.5

0.6

0.8

time

Reference u Staggered DG v Staggered DG

0.3

0.4 0.2

0.3 0.2

0.1

xi

xi

0.1 0

0

-0.1 -0.1

-0.2 -0.3

-0.2

-0.4 -0.5

0

0.2

0.4

0.6

time

0.8

-0.3

0

0.2

0.4

0.6

time

0.8

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments

Reference Staggered DG

2E-06

Reference Staggered DG

2E-06

1.5E-06 1E-06 1E-06

0

0

v

u

5E-07

-1E-06

-5E-07 -1E-06

-2E-06 -1.5E-06 -2E-06

0

0.2

0.4

0.6

time

0.8

1

-3E-06

0

0.2

0.4

0.6

time

0.8

1

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments 1.5E-07

Reference Staggered DG

1E-07

v

5E-08

0

-5E-08

-1E-07

-1.5E-07

0

1

2

3

4

5

time

2E-07

Reference Staggered DG

1.5E-07 1E-07 5E-08

v

0 -5E-08 -1E-07 -1.5E-07 -2E-07 -2.5E-07

0

1

2

3

4

5

time

Reference Staggered DG

1.5E-07

1E-07

5E-08

v

0

-5E-08

-1E-07

-1.5E-07

-2E-07

0

1

2

3

time

4

5

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Numerical experiments

Z

Number of average iterations needed for the GMRES algorithm with different preconditioners on the uniform unstructured grid (mesh 1) and the one containing the sliver elements (mesh 2) with (p, pγ ) = (4, 2).

Z

X

X Y

Y

None Pre 1 Pre 2

PRE 1 Mesh 2 PRE 1 Mesh 1

10

2

10

1

50

100

150

n

200

250

300

10

3

10

2

10

1

PRE 2 Mesh 2 PRE 2 Mesh 1

Iter

3

Iter

Iter

NO PRE Mesh 2 NO PRE Mesh 1

10

50

100

150

n

200

250

300

10

3

10

2

10

1

50

100

150

n

200

250

300

Iter. Mesh 1 112.59 86.73 53.27

Iter. Mesh 2 611.95 191.77 53.38

Factor 5.43 2.21 1.00

Preliminary

Sharp Interface

Numerical Method

Numerical experiments

Numercal experiments

Diffuse interface

Preliminary

Sharp Interface

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Numerical experiments

u Stag DG Rec. 1 u SeisSol Rec. 1

0.0003

v Stag DG Rec. 1 v SeisSol Rec. 1

0.0003

0.00025

0.00025

0.0002

0.0002

0.00015

0.00015

0.0001

0.0001

0.0001

5E-05

5E-05

0

v

u

0

w

5E-05

-5E-05

-5E-05

0 -5E-05

-0.0001

-0.0001

-0.0001

-0.00015

-0.00015

-0.00015

-0.0002

-0.0002

-0.0002

-0.00025 -0.0003

-0.00025 0

0.5

1

1.5

-0.0003

-0.00025 0

0.5

time

800

800

600

600

400

400

200

200

0

0

-0.0003

-400

400 200 0 -200 -400

-800

-800

-800

-1000

-1000

-1200

-1400

-1200

-1400 0.5

1

-1600

1.5

-1400 0

0.5

time

1

-1600

1.5

σyz Stag DG Rec. 1 σyz SeisSol Rec. 1

1000 800

800

600

600

400

400

400

200

200

200

0

σyz

-400 -600

-400 -600

-200 -400 -600

-800

-800

-800

-1000

-1000

-1000

-1200

-1200

-1400

-1200

-1400 0.5

1

time

1.5

-1600

1.5

0

-200

σxz

0 -200

1

σxz Stag DG Rec. 1 σxz SeisSol Rec. 1

1000

800

0

0.5

time

600

-1600

0

time

σxy Stag DG Rec. 1 σxy SeisSol Rec. 1

1000

1.5

σzz Stag DG Rec. 1 σzz SeisSol Rec. 1

600

-200 -400

-600

-1000

1

800

-600

0

0.5

time

-600

-1600

0

1000

σzz

-200

σyy

σxx

1.5

σyy Stag DG Rec. 1 σyy SeisSol Rec. 1

1000

-1200

σxy

1

time

σxx Stag DG Rec. 1 σxx SeisSol Rec. 1

1000

w Stag DG Rec. 1 w SeisSol Rec. 1

0.0003

0.00025

0.0002 0.00015

-1400 0

0.5

1

time

1.5

-1600

0

0.5

1

time

1.5

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Numerical experiments

u Stag DG Rec. 2 u SeisSol Rec. 2

0.0004

v Stag DG Rec. 2 v SeisSol Rec. 2

0.0004 0.0003

0.0003

0.0002

0.0002

0.0002

0.0001

0.0001

0.0001 0

v

w

0

u

0 -0.0001

-0.0001

-0.0001

-0.0002

-0.0002

-0.0002

-0.0003

-0.0003

-0.0003

-0.0004 -0.0005

-0.0004

0

0.5

1

1.5

-0.0005

-0.0004

0

0.5

time

1

1.5

-0.0005

1500

1000

1000

500

σzz

0

0

0

-500

-500

-500

-1000

-1000

-1000

1

-1500

1.5

0

0.5

time

1

-1500

1.5

σyz Stag DG Rec. 2 σyz SeisSol Rec. 2

2000

1500

1000

1000

σyz

500

0

0

0

-500

-500

-500

-1000

-1000

-1000

0.5

1

time

1.5

1.5

σxz

500

σxy

500

1

σxz Stag DG Rec. 2 σxz SeisSol Rec. 2

2000

1500

1000

0

0.5

time

1500

-1500

0

time

σxy Stag DG Rec. 2 σxy SeisSol Rec. 2

2000

1.5

500

σyy

σxx

1500

500

0.5

1

σzz Stag DG Rec. 2 σzz SeisSol Rec. 2

2000

1000

0

0.5

time

σyy Stag DG Rec. 2 σyy SeisSol Rec. 2

2000

1500

-1500

0

time

σxx Stag DG Rec. 2 σxx SeisSol Rec. 2

2000

w Stag DG Rec. 2 w SeisSol Rec. 2

0.0004

0.0003

-1500

0

0.5

1

time

1.5

-1500

0

0.5

1

time

1.5

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Numerical experiments

u StagDG Rec.1 u PDESol Rec.1 v StagDG Rec.1 v PDESol Rec.1 w StagDG Rec.1 w PDESol Rec.1

0.0008

u StagDG Rec.3 u PDESol Rec.3 v StagDG Rec.3 v PDESol Rec.3 w StagDG Rec.3 w PDESol Rec.3

0.0004

0.0006 0.0003 0.0004 0.0002 0.0002

u, v, w

u, v, w

0.0001 0

-0.0002

0

-0.0001 -0.0004 -0.0002 -0.0006 -0.0003 -0.0008 -0.0004 -0.001

0

1

2

time

3

0

1

2

time

3

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Main results

Main results Arbitrary high order method for the two and three-dimensional linear elasticity; The grid can be eventually curved; High order is achieved with a very small stencil; for pγ = 0 the main system results symmetric and positive-definite; The method is energy stable ∀p, pγ , For the special case of a Crank-Nicolson time discretization, the method is proven to be exactly energy conserving.

References M. Tavelli and M. Dumbser, Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity, Journal of computational physics, 2018 M. Tavelli and M. Dumbser, A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers, Journal of computational physics, 2017

Preliminary

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Diffuse Interface approach

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Diffuse interface approach We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0 ≤ α ≤ 1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary.

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Diffuse interface approach We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0 ≤ α ≤ 1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary.

v

v ⊗ ∇α = 0,

∂σ 1 1 − E(λ, µ) · ∇(α ) + E(λ, µ) · ∂t α α ∂α α − ∇·σ− ∂t ρ ∂α ∂λ ∂µ = 0. = 0, = 0, ∂t ∂t ∂t

v

1 σ∇α = 0, ρ ∂ρ = 0. ∂t

Preliminary

Sharp Interface

Numerical Method

Numercal experiments

Diffuse interface

Diffuse interface approach

∂Q ∂Q ∂Q ∂Q + B1 (Q) + B2 (Q) + B3 (Q) = 0, ∂t ∂x ∂y ∂z >

Q = (σxx , σyy , σzz , σxy , σyz , σxz , αu, αv , αw , λ, µ, ρ, α) ,

Preliminary

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Diffuse interface approach



0  0   0    0   0   0  α B1 =   −ρ   0   0   0   0   0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 (λ + 2µ) −α 1 λ −α 1 λ −α 0 0 0 0

1 µ −α 0 0 0

0

0

0

0 0 0 0 0

−α ρ 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0

0

−α ρ

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0 1 µ −α 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0

0

0

0

0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 (λ + 2µ)u α 1 λu α 1 λu α 1 µv α

0

1 µw α 1σ −ρ xx 1σ −ρ xy 1 − ρ σxz

0 0 0 0

            ,           

Preliminary

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Numerical Method

Numercal experiments

Diffuse interface

Diffuse interface approach The matrix of right eigenvectors of the matrix B1 is given by ρcp2  ρ(c 2 − 2c 2 ) p s   2 2  ρ(cp − 2cs )   0   0   0 R=  cp   0   0   0   0  0 0 

0 0 0 ρcs2 0 0 0 cs 0 0 0 0 0

0 0 0 0 0 ρcs2 0 0 cs 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0

−σxx 0 0 −σxy 0 −σxz αu αv αw 0 0 0 α

0 0 0 0 0 ρcs2 0 0 −cs 0 0 0 0

0 0 0 ρcs2 0 0 0 −cs 0 0 0 0 0

ρcp2 ρ(cp2 − 2cs2 ) ρ(cp2 − 2cs2 ) 0 0 0 −cp 0 0 0 0 0 0

The eigenvalues associated with the matrix B1 are λ1 = −cp ,

λ2,3 = −cs ,

λ4,5,6,7,8,9,10 = 0,

λ11,12 = +cs ,

where s cp =

λ + 2µ ρ

are the p− and s− wave velocities, respectively.

s and

cs =

µ ρ

λ13 = +cp ,

            .         

Preliminary

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Diffuse interface

Diffuse interface approach

QL

=

L L L L L L (σxx , σyy , σzz , σxy , σyz , σxz , u L , v L , w L , λ, µ, ρ, 1),

QR

=

R R R R R R (σxx , σyy , σzz , σxy , σyz , σxz , 0, 0, 0, λ, µ, ρ, 0).



QGod

=

L 2 L 2 L 2 L 2 σxx cp + 2σxx cp + 2σxx cs + σyy cp2 σxx cs + σzz cp2 L , 0, , , 0, σyz cp2 cp2 ! L c ρv L − σ L L L cp ρu L − σxx s xy cs ρw − σxz , , , λ, µ, ρ, 1 , cp ρ cs ρ cs ρ

0,

Preliminary

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Numercal experiments

Diffuse interface

Diffuse interface approach ∂Q ∂Q ∂Q ∂Q + B1 (Q) + B2 (Q) + B3 (Q) = 0, ∂t ∂x ∂y ∂z Numerical scheme Arbitrary high order accurate (in space and time) explicit ADER-DG schemes on Cartesian meshes; Adaptive mesh refinement (AMR); a posteriori subcell finite volume limiter with a very robust second order TVD scheme.

∆t
(1 + η)ID , r < −(1 − η)ID ,

if

r ∈ [−(1 − η)ID , (1 + η)ID ].

p α(r ) = (1 − ξ(r )) d ,

α

−1

∼ =

α α2 + (α)

(α) = 0 (1 − α)

Preliminary

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Numerical experiments

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Numerical experiments

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Numerical experiments Reference AMR-DIM Ref=2 AMR-DIM Ref=3

1.5E-06

4E-07

5E-07

2E-07

0

u

u

1E-06

0

-5E-07

-2E-07

-1E-06

-4E-07

-1.5E-06

0

0.2

0.4

0.6

Reference AMR-DIM Ref=2 AMR-DIM Ref=3

6E-07

0.8

1

-6E-07

0

0.2

0.4

time

Reference AMR-DIM Ref=2 AMR-DIM Ref=3

1.5E-06

0.8

1

0.8

1

Reference AMR-DIM Ref=2 AMR-DIM Ref=3

1.5E-06

1E-06

1E-06

5E-07

5E-07

-5E-07

v

0

-5E-07

v

0

-1E-06

-1E-06

-1.5E-06

-1.5E-06

-2E-06

0.6

time

0

0.2

0.4

0.6

time

0.8

1

-2E-06

0

0.2

0.4

0.6

time

Preliminary

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Numerical experiments

6E-08

Reference DIM

4E-08

u

2E-08

0

-2E-08

-4E-08

-6E-08

0

0.5

1

1.5

2

time

Reference DIM

1.5E-07

1E-07

u

5E-08

0

-5E-08

-1E-07 0

0.5

1

1.5

2

time

1.5E-07

Reference DIM

1E-07

5E-08

u

0

-5E-08

-1E-07

-1.5E-07

-2E-07

0

0.5

1

time

1.5

2

Preliminary

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Numerical experiments

Reference DIM

0.0008

Reference DIM

0.00025

Reference DIM 0.0005

0.0006

0.0002

0.0004

0.00015

0.0002

0

w

v

u

0.0001 0

5E-05 -0.0002

-5E-05

-0.0006 -0.0008

-0.0005

0

-0.0004

0

1

2

3

4

-0.0001

0

1

time

Reference DIM

2E-05

2

3

4

-0.001

0

1

time

Reference DIM

8E-05

1.5E-05

2

3

4

time

Reference DIM

6E-05

6E-05

4E-05

1E-05 4E-05 5E-06

2E-05

v

u

w

2E-05

0 -5E-06

0

0

-1E-05

-2E-05 -2E-05

-1.5E-05

-2.5E-05

-4E-05

-4E-05

-2E-05

0

1

2

time

3

4

-6E-05

0

1

2

time

3

4

-6E-05

0

1

2

time

3

4

Preliminary

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Numercal experiments

Diffuse interface

Main results Main results A novel diffuse interface method (DIM) for the simulation of seismic wave propagation for arbitrary complex geometries; The free surface topology does not affect the CFL time restriction (no sliver elements) since α has no influence on the eigenvalues of the governing PDE system. It does not require any external mesh generation tools or any manual interaction with the user. References M. Tavelli, M. Dumbser, D. E. Charrier, L. Rannabauer, T. Weinzierl, M. Bader, A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography , Journal of computational physics, submitted to, 2018

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Acknowledgements

Thank you for the attention

This research was funded by the European Union’s Horizon 2020 Research and Innovation Programme under the project ExaHyPE, grant no. 671698 (call FETHPC-1-2014). The 3D simulations were performed on the HazelHen supercomputer at the HLRS in Stuttgart, Germany and on the SuperMUC supercomputer at the LRZ in Garching, Germany.

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Appendix



(αs ρs ) + ∇ · (αs ρs vs )

=

0,

(αs ρs vs ) + ∇ · (αs ρs vs ⊗ vs + αs σs ) − σI ∇αs

=

αs ρs Sv ,s ,

(αs ρs Es ) + ∇ · (αs ρs Es vs + αs σs vs ) − σI ∇αs · vI

=

αs ρs Sv ,s · vs ,

  αg ρ g + ∇ · αg ρ g v g

=

0,



αg ρg vg + ∇ · αg ρg vg ⊗ vg + αg σg − σg ∇αg

=

αg ρg Sv ,g ,

  αg ρg Eg + ∇ · αg ρg Eg vg + αg σg vg − σI ∇αg · vI

=

αg ρg Sv ,g · vg ,

=

0.

∂t ∂ ∂t ∂ ∂t

∂ ∂t ∂ ∂t ∂ ∂t



∂ ∂t

αs + vI ∇αs

(1)

Preliminary

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Diffuse interface

Appendix

Assumptions: The interface between solid and gas is not moving, i.e. vI = 0; all evolution equations related to the gas phase can be neglected; we assume the density ρs of the solid phase to be constant in time; the nonlinear convective term αs ρs vs ⊗ vs ; boundary condition at the interface leads to σs · ∇αs = 0.