Fourier Spectral Moving Mesh Method for Willmore Equation of ...

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covariant and contravariant base vectorsa i = ∂⃗x∂ξ i , ai = ∇ξ i , i = 1, 2, 3, (12)which are related bya i = 1 J a j × a k , a i = Ja j × a k , a i · a j = δ i,j (13)With the help of the following transformation relationsJ = a 1 · (a 2 × a 3 ) (14)∇ = ∑ ia i∂∂ξ = 1 i J∑i∂∂ξ i Jai ,∂J∂ξ l = J ∑ ia i · ∂a i∂ξ l (15)∂a i∂ξ l = − ∑ s(a i · ∂a s∂ξ l )as ,∂x∂t = − ∑ i(See the appendix for detail proof) The original MMPEDa i∂ξ i∂t(16)can be transformed intoor in a fully non-conservative formτ ∂⃗x∂t = p[∑ i,j∂ξ i∂t = p τ ∇ · (G−1 ∇ξ i ), i = 1, 2, 3 (17)τ ∂⃗x∂t = − p J∑ ∂a ii,j∂ξ j (Jaj · G −1 a i ) (18)(a i · G −1 a j ∂⃗x)∂ξ i ∂ξ − ∑ j i,j(a i · ∂G−1∂ξ j a j )∂⃗x∂ξ i ] (19)And if we take Winslow’s monitor function G = ωI, where ω = √ 1 + β 2 u 2 x, the final MMPDEisτ ∂x∂t = p ∑(a i · a j ) ∂ ∂x(ωω 2 ∂ξi ∂ξ ) (20)ji,jOur motivation for choosing p is to make the MMPDE behave similarly to a simplediffusion equation, hence here we setp = µω 2 /λ, (21)here λ is the largest eigenvalue of the positive-define matrix A = (A i,j ) = (a i · a j )Since period boundary conditions are the most commonly used in phase-field simulations,we discuss an implementation of the periodic boundary conditions for the MMPDEs. We6
don’t require that the mapping ⃗x( ⃗ ξ, t) maps a unit squre onto a unit square, in stead weassume that−→ x ( ⃗ ξ + (j, k, l), t) =−→ Id( ⃗ ξ, t) + (j, k, l) (22)where −→ Id is the identical mapping. Hence −→ X ( ⃗ ξ) = −→ x ( ⃗ ξ)− −→ Id( ⃗ ξ) is periodic. Interestingly, thiscondition does not require that ⃗x( ⃗ ξ, t) maps a unit square onto a unit square. The Fourierspectral implementation considered here allows the mesh moving equations be solved in thesimilar fashion as the phase field equations. With periodic boundary conditions, the physicaldomain does not necessarily maintain a square shape, while the computational domain does.Although the physical domain boundaries are curvy, the condition (??) guarantees that theperiodic copies of Ω p (a non-square shape) cover the whole 2-dimensional space as effectivelyas periodic copies of the unit square. In particular, it can be seen that (??) implies thatthe area of the physical domain Ω p is the same as that of Ω c . and the MMPDEs can betransformed into∂ ⃗ X∂t = µ ∑ i,ja i · a jλ∂∂ξ [ω(∂ X ⃗i ∂ξ + j ej )] (23)where e j is the canonical unit vector. The semi-implicit Fourier spectral scheme can be usedto solve it numerically. A particular form as used in [?], is give by(1 + µW δτκ 2 )( ˆ¯⃗X∑ a − ˆ⃗X) i · a j ∂= µδτ{λ ∂ξ [ω(∂ X ⃗i ∂ξ + j ei )]} ∧ (24)where ¯⃗ X is the value of ⃗ X at the next time step. W is the maximum of ω on Ωc , κ denotesthe wave vector, and ∧ represents the Fourier transform.4 Moving mesh formulation of the Willmore equationTreating φ as a function of ξ and t, we havewhere ⃗x = ∂i,j∂φ( ξ, ⃗ t)=∂t˙⃗x · ∇ x φ + ξ∆ 3 φ − 1 ξ ∆2 W ′ (φ) − 1 ξ ∆(∆φW ′′ (φ)) +∫1ξ ∆(W ξ′(φ)W ′′ (φ)) + M{3 Ω 2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 dx − β}(−ξ∆ 2 φ + 1 ξ ∆W ′ (φ))(25)x( ξ,t) ⃗∂tis the mesh velocity determined from the MMPDE. The convection term˙⃗x · ∇ x φ represents the change of inverse image of the field variable φ on the computational7
Page 1 and 2: Fourier Spectral Moving Mesh Method
Page 3 and 4: Recently, a new adaptive Fourier-sp
Page 5: where ⃗x = (x 1 , x 2 , x 3 ) T
Page 10: 5.4 Numerical exampleWe used a squa
Page 15 and 16: Thirdly, as long as the Jacobian Ma
Page 17 and 18: And∫Ω∫f(x, y)dxdy = f(ξ, η)J

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