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)
M¯
− 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)
M¯
− 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)
M¯
− 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)
M¯
− 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
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Numerical experiments
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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
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Diffuse interface
Preliminary
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Numerical Method
Numerical experiments
Numercal experiments
Diffuse interface
Preliminary
Sharp Interface
Numerical Method
Numerical experiments
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Diffuse interface
Preliminary
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Numerical experiments
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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
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Preliminary
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Numerical Method
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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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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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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
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Diffuse Interface approach
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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
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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
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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) ,
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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
,
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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 ,
.
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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,
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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)
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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
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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
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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
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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)
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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.