Spectral Element Methods (SEM) May 12, D Wave Equations (Strong and Weak form)

Transcription

1 Spectral Element Methods (SEM) May 12, D Wave Equations (Strong and Weak form) Strong form (PDEs): ρ 2 t s = T + F in (1) Multiply a global test function W(x) to both sides and integrate over to obtain the weak form: ρw s i dv = W ˆn T i dσ WT i d + WF i d (2) 2 Meshing Σ Grid into spectral elements (iteration index ispec). Each element will have GLL grid points in 3 directions (index α,β,γ). Globally number all grid points (index I) and establish the proection (α,β,γ; ispec) (I) (3) Notice this proection is one-to-one for grid points inside any element (valence = 0), and multipleto-one for grid points on the boundaries that are shared by elements (valence 1). 2.1 Interpolation of Shapes For boundary and volume elements, we not only need to access location of grid points of the element, but also the location of any arbitrary point inside the element. We can interpolate the shape of the element by given anchors x a. 1. For boundary element: x(ξ,η) = N a=1 N a (ξ,η)x a (4) and the Jacobian matrix associated with this transformation J b (ξ,η) = ( x ξ x η ) (5) 1

2 and the scalar Jacobian J b = J b. The inverse transform is given by ( ξ x η x ) = J 1 b and the normal to the boundary is described by (6) n(ξ,η) = x x ξ η x x ξ η (7) 2. For volume element: x(ξ,η,ζ) = N a=1 N a (ξ,η,ζ)x a (8) J a (ξ,η,ζ) = ( x(ξ,η,ζ) (ξ,η,ζ) ) (9) J a = J a (10) (ξ,η,ζ) x(ξ,η,ζ) = J 1 a N a (ξ,η) and N a (ξ,η,ζ) are the 2-D and 3-D shape functions. They are double or triple products of degree 1 or 2 Lagrangian polynomials. For example, degree 1 Lagrangian polynomials are N 1 (ξ) = 1(1 + ξ) and N 2 2(ξ) = 1 (1 ξ) for given anchors at 1 and Interpolation of Function Field We interpolate function field on GLL points that satisfy: (11) (1 ξ 2 ) P N (ξ) = 0 (12) and f(x(ξ,η)) Σe = αβ f(x(ξ,η,ζ)) e = αβγ f αβ l α (ξ)l β (η) (13) f αβγ l α (ξ)l β (η)l γ (ζ) 3.1 Derivatives i f(x(ξ,η,ζ)) = αβγ [ f αβγ l α (ξ)l β (η)l γ (ζ) ξ + l α (ξ) l β (η)l γ (ζ) η + l α (ξ)l β (η) l γ (ζ) ζ ] x i x i x i 2

3 Notice ξ η ζ x i, x i, x i values can be calculated from the 2D and 3D shape functions. Specifically, to compute derivatives on GLL points: [ ] [ ] [ ] i f αβγ = f βγ l (ξ α ) αβγ i ξ+ f αγ l (η β ) i η αβγ + f αβ l (ζ γ ) i ζ αβγ (15) (14) 3.2 Integration Integration quadrature f(x)dx = Σ e αβ e f(x)dx = αβγ ω α ω β f αβ J αβ b (16) ω α ω β ω γ f αβγ J αβγ a (17) 4 Global Test Functions Define global test functions W I (x), such that { W I l α (ξ)l β (η)l γ (ζ) if I e, and I e = (α,β,γ) (x) e = (18) 0 if I e If I e = (α,β,γ) has zero valence, then W I (x) is simply a 3-D local Lagrangian function extended to the whole space; otherwise it consists several pieces (valence+1) of local lagrangian function (edge ones). Therefore, for integration, one needs to loop over spectral elements, then all the GLL points, and adds contributions to the corresponding global grid point. Notice W I e (α,β,ζ ) = δ αα δ ββ δ ζζ (19) this simplifies the integration results. 3

4 5 Application to the Wave Equation 5.1 LHS ρw I s i (t) dx = ρw I e s i (t) dx (20) e e = ω α ω β ω γ ρ α β γ J α β γ s α β γ i δ αα δ ββ δ ζζ (21) e α β γ = ω α ω β ω γ ρ αβγ αβγ J αβγ s i (t) (22) e 5.2 Spatial Derivatives For example, i s (x) αβγ e = [ = i [ s α β γ α β γ l α (ξ)l β (η)l γ (ζ) s βγ l (ξ α )] i ξ(ξ α,η β,ζ γ ) ] αβγ e + [ + [ s αγ l (η β )] i η(ξ α,η β,ζ γ ) s αβ l (ζ γ )] i ζ(ξ α,η β,ζ γ ) (23) in the SEM code, hprime(α,) = l (ξ α ). The stress tensor is given by: T kl = C kli i s R kl (24) For isotropic and no pre-stress case, T kl = (κ 2 3 µ)δ klǫ ii + µ(ǫ kl + ǫ lk ) (25) 5.3 Boundary terms Σ W Iˆn T i dσ = e = e Σ e W I t i dσ e ω α ω β t αβ i J αβ (26) Where Ith global grid point is on Σ e elements, with GLL index (α,β). Normal stress t i is zero for free surface, and continuous for any internal boundaries. (For fluid-solid coupling problems, 4

5 we may solve both side independently, and the interchange of two fields will occur through this boundary condition.) For regional problems, other boundaries are absorbing, where t i = ˆn T i = ρα(ˆn ṡ )ˆn i + ρβ(δ i ˆn iˆn )ṡ (27) 5.4 Volume integral terms W I T i d = W I e T i d e e e = [ l α ξl β l γ + l α lβ ηl γ + l α l β lγ ζ]t i dx e e = {[ ] ω lα (ξ ) ξ(ξ,η β,ζ γ )T βγ i J βγ ω β ω γ e [ ] + ω lβ (η ) η(ξ α,η,ζ γ )T αγ i J αγ ω α ω γ + [ ω lγ (ζ ) ζ(ξ α,η β,ζ )T αβ i J αβ ] ω α ω β } (28) in the SEM code, sums over are named as temp[x y z][1 2 3]. 5.5 Forcing terms 1. For point force F i (x,t) = f i (t)δ(x x 0 ), W I F i d = W I (x 0 )f i (t) f i (t) if x 0 I = 0 if x 0 other grid point (29) l α (ξ(x 0 ))l β (ξ(x 0 ))l γ (ξ(x 0 ))f i (t) else x 0 e if x 0 has local parameter {ispec;ξ,η,ζ}, then compute the Lagrange polynomial at the source location. 2. For Moment tensor point force F i (x,t) = M i δ(x x 0 )g(t), W I F i d = M i W I (x 0 )g(t) (30) 5

6 Since W = W x and define G ik (ξ,η,ζ) = M i x, M i W I (x 0 ) = G ik W (ξ 0 ) = r,t,v = r,t,v l r (ξ 0 )l t (η 0 )l v (ζ 0 )G ik (ξ r,η t,ζ v ) ξk (l α (ξ r )l β (η t )l γ (ζ v )) l r (ξ 0 )l t (η 0 )l v (ζ 0 ) [G i1 (ξ r,η t,ζ v )l α(ξ r )l β (η t )l γ (ζ v )) + ] (31) 3. For body force field F i (x,t), W I F i d = e = e e W I e F i d e ω α ω β ω γ F αβγ i (t)j αβγ (32) 4. For surface force field F i (x,t) = Σ 0 τ i (x,t)δ(x x 0 )dσ 0, (monopoles on Σ 0 ) [ ] W I F i d = W I (x) τ i (x,t)δ(x x 0 )dσ 0 d (33) Σ 0 = τ i (x 0,t)W I (x 0 )dσ 0 (34) Σ 0 to compute this in practice, loop over surface elements, and then for each surface GLL point I = (α,β), Contr. to I th test function + = ω α ω β τ αβ J αβ (35) 5. For surface double couple force field F p (x,t) = Σ 0 u i (x,t)n C ipq q δ(x x 0 )dσ 0, assume that Σ 0 do not overlap with the surface Σ that encompasses, then [ ] W I F p d = W I (x) u i (x,t)n C ipq q δ dσ 0 d Σ [ 0 ] = W I (x)u i (x,t)n C ipq q δ(x x 0 )d W I (x) Σ 0 ( = q ui (t)n C ipq W I) (x 0 )dσ 0 (36) Σ 0 We define the corresponding moment density tensor m pq (t) = u i (t)n C ipq on Σ 0. In the case of a fault in an isotropic medium where u is perpendicular to n, m pq = µ(u p n q + u q n p ) is the standard moment density tensor that describes a seismic source. Now rewrite the above 6

7 expression: W I F i d = [ q m pq (t)w I + m pq(t) q W I ]dσ 0 Σ 0 (37) In practice, we store f p (t) = q m pq and m pq (t) in advance for each grid point of the surface Σ 0, then the above two terms can be calculated with ease in runtime. 5.6 Assemble Both Sides M s(t) = B + T + F(t) (38) We can update the acceleration at time t by s(t) = (M) 1 (B + T + F(t)) (39) Notice that M is diagonal (I I), B,T,F, s(t) are all vectors (I 1). Try loop over ispec and then (α,β,γ) to assemble the contribution of each grid point (in an element) into global matrices. This is true for all matrices, except the forcing term in which the contribution is added by looping over only elements that contain the forces and related GLL points. 6 Time Marching Schemes Newmark scheme Predictor: d n+1 = d n + v n t an ( t) 2 v n+1 = v n an t a n+1 = 0 (40) Corrector: a n+1 = (M) 1 (B + T + F n+1 ) v n+1 = v n an+1 t d n+1 = d n+1 (41) 7