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

Fourier Spectral Moving Mesh Method for WillmoreEquation of Order SixGuoliang Fang ∗AbstractRecently a new adaptive Fourier-spectral semi-implicit method(AFSIM) has beendeveloped for solving the phase field equation by combining an adaptive moving meshmethod and the semi-implicit Fourier spectral algorithm. In this paper we presentAFSIM for solving the phase-field model for Willmore Problem with volume and areaconstrains. The primitive phase-field variables are defined on staggered mesh. Inthe mesh redistribution part, a semi-implicit Fourier Spectral Method is developed.Numerical experiments are carried out to demonstrate the effectiveness of the proposedmoving mesh method.Keywords: Willmore Problem,Semi-implicit Fourier Spectral Method,Moving Mesh Method1 IntroductionThe Willmore problem is the classical problem from differential geometry to find Γ in anadmissible class of surfaces embedded in R 3 , which minimizes the mean curvature energy∫ΓH 2 dS.where H = (κ 1 + κ 2 )/2 is the mean curvature; κ 1 and κ 2 are the principal curvatures of Γ.The elastic bending energy model for bilayer membranes, first developed by Canham,Evans and Helfrich, has been widely used to study the mechanical properties of vesiclemembranes. The elastic bending energy is formulated in the form of a surface integral of themembrane Γ:∫E =Γa 1 + a 2 (H − c 0 ) 2 + a 3 Gds,∗ School of Mathetical Sciences, Beijing Normal University, Beijing 100875, P.R. China;glfang2005@gmail.com1

where a 1 represents the surface tension, H = κ 1+κ 22is the mean curvature of the membranesurface, with κ 1 and κ 2 as the principle curvatures, and G = κ 1 κ 2 is the Gaussian curvature.a 2 is the bending rigidity and a 3 the stretching rigidity. c 0 is the spontaneous curvaturethat describes the asymmetry effect of the membrane or its environment. The equilibriummembrane configurations are the minimizers of the energy subject to given surface area andvolume constraints. The simplified case ofis like the Willmore Problem.∫E = (H − c 0 ) 2 ds,ΓPhase field method has been extensively applied to modeling microstructure evolutionfor various materials process such as the mesoscale pattern formation and interface motion.A set of spatially dependent field variables are described by the phase function. Particularly,the membrane surface Γ is defined as the zero level set of the phase field function φ . Thecorrespondent phase field model is given by [?]∫E(φ) =Γ12ξ (ξ∆φ + (1 ξ φ + c √0 2)(1 − φ 2 )) 2 dxThe surface area and volume constrains can be specified as∫B(φ) =Ω∫A(φ) =Ωφdx = α[ ξ 2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 ]dx = β.Semi-implicit Fourier-spectral methods have been employed as an efficient numericalalgorithms for phase field simulations. For the reason that this kind of method can bothincrease the time step and improve the stability, while achieving high accuracy in space.[?]However, as the width of interface ξ in the phase-field model goes to zero, in order tomaintain high resolution, more grid points should be used.On the other hand, uniformgrids are required in order to use FFT algorithm; which results in the high cost of numericalcomputation.The studies on adaptive meshing techniques have led to significant improvement of thecomputational efficiency of traditional fixed grid methods in many applications, includingphase field modeling. It is clear that adaptive mesh methods are useful for microstructureswith a very small interfacial width compared to the domain size. .2

Recently, a new adaptive Fourier-spectral semi-implicit method(AFSIM) is developedwhich takes advantages of both moving mesh method and the Semi-implicit Fourier Spectralscheme.[?].The work here uses the AFSIM algorithm.And to our best knowledge, for the firsttime demonstrate the effectiveness of AFSIM for solving six-order Willmore problem. Sincewe use the gradient flow algorithm for the phase-field equations, with two constrains, it isimportant to for any algorithm to guarantee the decay of energy and the maintenance ofthese constrains. Several numerical examples in this work demonstrate that the AFSIM issatisfactory qualified for these requirements.The paper is organized as follows: in section 2, we first review the phase-field equationscorresponding to Willmore problems. In section 3, the frame work for the MMPDEs, itsFourier-Spectral implementation will be presented.In section 4, the Willmore equationcoupled with MMPDEs and their numerical scheme has been presented. We then presentnumerical simulation results in section5. In section 6, my future plans and problems arepresented. In the the appendix some formulas which kept bothering me has been proofed.2 Phase field model for Willmore problemFor simplicity, our phase field model is the variational problem:min E(φ) = κφ∈L∫Ω 2ξ (ξ∆φ + 1 ξ φ(1 − φ2 )) 2 dx (1)with constrains∫B(φ) =∫A(φ) =ΩΩφdx = α, (2)ξ2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 dx = β (3)over the admissible setL = {φ ∈ H 2 | φ| ∂Ω = −1, ∇ · n| ∂Ω = 0} (4)where A(φ) approaches the volume difference between the inside volume and outside volume,and B(φ) approaches to 2 √ 2area(Γ)/3.3

Computationally, it is more convenient to use penalty formulation for the two constrains.Denote M 1 , M 2 the two penalty constants, we obtain a modified Elastic bending energyE M (φ) = E(φ) + 1 2 M 1(A(φ) − α) 2 + 1 2 M 2(B(φ) − β) 2 (5)The nonlinear variational problem min E M (φ) , is solved via a standard gradient flow method,with H −1 being the variational space. The resulting equation isφ t =−∆(− δE M(φ))δφ= ξ∆ 3 φ − 1 ξ ∆2 W ′ (φ) − 1 ξ ∆(∆φW ′′ (φ)) + 1 ξ 3 ∆(W ′ (φ) · W ′′ (φ))∫ξ+M{Ω 2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 dx − β}[−ξ∆ 2 φ + 1 ξ ∆W ′ (φ)] (6)where W (φ) = 1 4 (1 − φ2 ) 2It is proved [?] that the minimum of E(φ) can be approximated by the minimum ofE M (φ) ormin E(φ) = lim min E M(φ)A(φ)=α,B(φ)=β M 1 ,M 2 →∞3 Variational formulation of moving mesh PDEsMoving-mesh PDEs can be formulated either on a computational domain or on a physicaldomain.The former has the advantage of being simple and efficient, though bearing aless rigorous derivation. The latter is derived on a more rigorous basis, but the resultingMMPDEs is slightly more complicated. In this paper, only the physical domain variationformulation, is implemented in the numerical simulation.Consider the problem of generating a moving mesh for a simply connected, domain Ω p .Itis useful to regard the physical area Ω p and the computational(logical) domain Ω c as imagesof one another under invertible mapand its inversex 1 = x 1 (ξ 1 , ξ 2 , ξ 3 , t), x 2 = x 2 (ξ 1 , ξ 2 , ξ 3 , t), x 3 = x 3 (ξ 1 , ξ 2 , ξ 3 , t) (7)ξ 1 = ξ 1 (x 1 , x 2 , x 3 , t), ξ 2 = ξ 2 (x 1 , x 2 , x 3 , t), ξ 3 = ξ 3 (x 1 , x 2 , x 3 , t) (8)4

where ⃗x = (x 1 , x 2 , x 3 ) T ∈ Ω p and ⃗ ξ = (ξ 1 , ξ 2 , ξ 3 ) T ∈ Ω c are the physical and computationalcoordinates, and t is time. A mesh covering Ω p is usually numerically obtained by uniformlypartitioning Ω c and then mapping it to the physical domain through mapping⃗x = ⃗x( ⃗ ξ, t).With the MMPDE approach developed in [?], the time-dependent coordinate transformationis determined by solving a parabolic PDE called a moving mesh PDE or MMPDE.The grid points of the physical area are expected to concentrate in regions where the physicalsolution is steep.Given an appropriate matrix monitor function G, the mapping can be obtained by solvingthe following variational problem∫I[ξ] =Ω p∑i(∇ξ i ) T G − ∇ξ i dx (9)where ∇ is the gradient operator with respect to ⃗x, and G the so-called monitor function,is a three-by-three symmetric positive define matrix which interconnects the mesh and thephysical solution.Since the fastest decent direction of I[ξ 1 , ξ 2 , ξ 3 ] in the function space is the oppositedirection of the first order function derivative δI , the gradient flow equationδξ∂ξ i∂t = − δIδξ i= ∇ · (G −1 ∇ξ i ), i = 1, 2, 3 (10)defines a flow which converges to the equilibrium state as t → ∞. In practice, it is oftenmore suitable to use different decent directions than the fastest one and to allow the userto adjust time scale of the mesh equation. Hence the MMPED is then modified as gradientflow equation of∂ξ i∂t = p τ ∇ · (G−1 ∇ξ i ), i = 1, 2, 3 (11)where τ > 0 is the user-defined parameter used for adjusting the time scale of mesh movementand p is a positive function to be chosen such that the MMPDE has some desired properties.The equations are defined in terms of inverse mapping ⃗ ξ(⃗x, t). It is well known that theso defined functional is less likely to result in a singular coordinate transformation than asimilar functional defined in terms of ⃗x = ⃗x( ⃗ ξ, t).In practice, however, it is more convenient to work directly with ⃗x = ⃗x( ⃗ ξ, t) since itexplicitly defines the locations of mesh points. To realize this purpose, let’s introduce the5

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

where g(⃗x) = 1 e − x2 1 +x2 22πσ 2 2σ 2is the Gauss kernel.5.2 The choice of some parametersAs to the parameter β, which is used to in the monitor function ω = √ 1 + β 2 u 2 x, β can beused to control the concentration of mesh grid points effectively. In our simulation work, thegrids points are more concentrated when β is bigger(β = 10), however, we do find that undersuch circumstance, the mesh grid points might concentrate in the areas where the solutionis not steep. And such concentration might be caused by the effect of mesh grid points inother areas with large gradient . And when we use a small value e.g. β 2 = 0.1, the meshgrid points have less deformation in irrelevant areas, while still concentrate effective on areawhere u is steep.The parameter τ which is used to control the evolution of the mesh, it is set to be thesame as the real time t.And we simply treat σ ∗ = 0 in our simulation.5.3 The constrainsOne primary requirement in our simulation is to guarantee that the final numerical solutionsare the minimizers of above energy, under the two constraints. During our simulation, whenthe energy doesn’t decay, we halve the time step δt and δτ, and the the energy will keepdecaying. And when the energy doesn’t change significantly, we stop our program.Since our variation space is taken as H −1 , the equation is of conservative form, hence wedon’t have to treat the volume constrains on purpose, and our numerical results shows thatthe volume is almost preserved as a constant.As to the area constrains, at first we neglect such constrains, and after a few time steps,we obtain a smooth solution, and we dynamically control the penalty constant M 2 and thedesired area β, while doing such adjustment, we neglect the decay of the energy. And wethe area is maintained, we then adjust the time step to guarantee the decay of the energy.9

5.4 Numerical exampleWe used a square as initial data to test the validity of our algorithm, and we put no constrainson the area by setting M 2 = 0.From the figure below we can see that the energykeeps decaying, and the plot for the energy is the ”elbow-shape”. Also we noticed that thedifference of areas inside and out side the interface have little change (0.006).10.90.80.70.60.50.40.30.20.100 0.2 0.4 0.6 0.8 110.90.80.70.60.50.40.30.20.100 0.2 0.4 0.6 0.8 110.90.80.70.60.50.40.30.20.100 0.2 0.4 0.6 0.8 1Figure 1: The parameters are ξ = 0.015, N = 128, β 2 = 0.1.10.90.80.70.60.50.40.30.20.110.90.80.70.60.50.40.30.20.110.90.80.70.60.50.40.30.20.10.2 0.4 0.6 0.8 10.2 0.4 0.6 0.8 10.2 0.4 0.6 0.8 1Figure 2: The interface {−0.1 < φ < 0.1} .3002902802702602502402302202102000 2 4 6 8 10x 10 5−0.9092−0.9093−0.9094−0.9095−0.9096−0.9097−0.9098−0.9099−0.910 1 2 3 4 5x 10 4Figure 3: The left is the energy line.interfaceRight is the difference of area inside and out side10

area is not unchanged. Since we have no requirement for the boundaries, we cannot assumethe physical area is an unit square. Hence, it’s hard to propose the boundary conditions forthe Navier-Stokes equations. I have some ideas as follows:In order to force an unit square we need to force the displacement dx(x, y)| ∂Ω = 0 aswell as dy(x, y)| ∂Ω = 0. But we also require both dx and dy are periodic. And dx, dy arethe solutions of the MMPDEs. Hence, we cannot guarantee the existence of such a solution.And I need your suggestion on such issue.Another problem is about the semi-implicit scheme. Let’s review the equation(1 + ξ(λ ∗ κ 2 ) 3 δt)(ˆ¯φ − ˆφ) = δt{ ˙⃗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))} ∧I remembered in Hangzhou you told me that the splittingu t = a max ∆u implicit + (a(x) − a max )∆u explicitwill enhance the the stability. Hence I’m not surprised to see such a splitting scheme, whichutilized ∆ x φ = 1 ∇ J ξ · (JA∇ ξ φ) and it is some what like the splitting scheme above. And myquestion is that, why would such an semi-implicit scheme more stable? I use such a schemeover and over again, but I don’t know why? I failed to think it out. Is there some papers torefer to? Or would you mind explaining it to me?7 AppendixThe formulas in this paper have been cited many times,and here we give the detailed proof.a i = 1 J a j × a k , a i = Ja j × a k , a i · a j = δ i,jJ = a 1 · (a 2 × a 3 )Proof:First of all, by the definition of Jacobian Matrix it is obvious that J = a i · (a j × a k ) as longas i, j, k are in right hand order.Secondly, using the fundamental of inverse function, we have a i · a l = δil14

Thirdly, as long as the Jacobian Matrix ≠ 0,a i , a j , a k , can be viewed as a group of basis inthe R 3 space.a i · a j = 0, a i · a k = 0, ⇒ a i = λ i a j × a k , ⇒a i · a i = λ i a i · (a j × a k ) ⇒ 1 = λ i J ⇒ a i = 1 J a j × a k .By the same way we can prove thata i = Ja j × a k .Next we proof the following transformation relations∇ · −→ f = ∑ ia i ·∂ −→ 1 ∑ ∂f =∂ξ i Ji∂ξ i Jai · −→ f ,Proof∇ · −→ f =3∑i=1∂f i∂x i=3∑3∑i=1 j=1∂f i∂ξ j∂ξ j∂x i=3∑a j ·j=1∂∂ξ j−→ fAnd3∑a i ∂ −→ 13∑ ∂ −→ · f = a j × a k · fi=1∂ξ i Ji=1∂ξ i∂(a 2 × a 3 · −→ f ) = a 21 × a 3 · −→ f + a 2 × a 31 · −→ ∂ −→f + a 2 × a 3 · f∂ξ 1 ∂ξ 1∂(a 3 × a 1 · −→ f ) = a 32 × a 1 · −→ f + a 3 × a 12 · −→ ∂ −→f + a 3 × a 1 · f∂ξ 2 ∂ξ 2∂(a 1 × a 2 · −→ f ) = a 13 × a 2 · −→ f + a 1 × a 23 · −→ ∂ −→f + a 1 × a 2 · f∂ξ 3 ∂ξ 3where a ij =have∂2 x∂ξ i ∂ξ j= a ji , and a × b = −b × a, adding the last three equations together we3∑a j × a k ·i=1∂∂ξ i−→ f =3∑i=1∂∂ξ i(a j × a k · −→ f ) =3∑i=1∂∂ξ iJa i · −→ f∂J∂ξ l = J ∑ ia i · ∂a i∂ξ l15

Proof:∂J∂ξ l = ∂ ξ l a 1 · (a 2 × a 3 ) =3∑i=1∂a i∂xi l · (a j × a k ) = J3∑i=1a i · ∂a i∂ξ l∂a i∂ξ l = − ∑ s(a i · ∂a s∂ξ l )asProof:and we also havea i · a j = δ j i ⇒ ∂∂ξ l (ai · a j ) = 0 ⇒ ( ∂ai∂ξ l ) · a j = −( ∂a j∂ξ l ) · ai3∑− (a i · ∂a ss=1∂ξ l )as · a j = −( ∂a j∂ξ ) · l ai ,Hence, as long as the Jacobian Matrix is fully ranked, a 1 , a 2 , a 3 are a group of basis in R 3and we can attain the desired equation.∂x∂t = − ∑ ia i∂ξ i∂tProof:∀, −→ x 0 = (x 1 , x 2 , x 3 ), −→ x (ξ( −→ x 0 , t), t) = −→ x 0 ⇒3∑i=1∂x∂ξ i ∂ξ i∂t + ∂x∂t = 0 ⇒ ∂x∂t = − ∑ ia i∂ξ i∂tIt is very easy to verify thatHence∇ x f =3∑i=1a i ∂f∂ξ i∆ 2 xf = ∇ x · ∇f = ∇ x · (= 1 J ∇ ξ · (JA∇ ξ f)3∑j=1a i ∂f∂ξ i ) = 1 J3∑ ∂3∑Ja j · (j=1∂ξ j j=1a i ∂f∂ξ i )16

And∫Ω∫f(x, y)dxdy = f(ξ, η)J(x, y)dxdyΩ cReferences[1] Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, A phase field formulationof the Willmore problem, Nonlinearity, pp.1249-1267,(2005)[2] Qiang Du, Chun Liu, and Xiaoqiang Wang, A phase field approach in the numericalstudy of elastic bending energy for vesicle membranes, Journal of ComputationalPhysics, 198(2004), pp.450-468. (2004)[3] Chun Liu, and Jie Shen, A phase field model for the mixture of two incompressiblefluids and its approximation by a Fourier-spectral method,Physica D,(2003)[4] L.Q. Chen, and Jie Shen, Applications of semi-implicit Fourier-spectral method tophase field equation,Computer Physics Communications,108, pp.147-158(1998)[5] Weizhang Huang, Practical Aspects of Formulation and Solution of MovingMesh Partial Differential Equations,,Journal of Computational Physics,171(2001),pp.753-775(2001)[6] W.M.Feng, P. Yu, S.Y. Hu, Z.K. Liu, Q.Du, L.Q. Chen, Spectral implementationof an adaptive moving mesh method for phase-field equations, Journal of ComputationalPhysics, pp.498-510,(2006)[7] W.M.Feng, P. Yu, S.Y. Hu, Z.K. Liu, Q.Du, L.Q. Chen, A Fourier Spectral MovingMesh Method for the Cahn-Hillard Equation with Elasticity, To be publish,(2007)[8] W.Cao, W.Huang and R.D. RussellA study of Monitor Functions for Two DimensionalAdaptive Mesh Generation,SIAM Journal of Scientific Computing, 20 (1999),pp. 1978-1994(1999)17

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