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Removing the Sti�ness of Curvature in Computing 3-DFilamentsThomas Y. Hou� Isaac Klapper y Helen Si zMarch 5, 1998AbstractIn this paper, we present a new formulation for computing the motion of a curvaturedriven 3-D �lament. This new numerical method has no high order time step stability con-straints that are usually associated with curvature regularization. This result generalizesthe previous work of Hou-Lowengrub-Shelley [7] for 2-D uid interfaces with surface ten-sion. Applications to 2-D vortex sheets, 3-D motion by curvature, the Kirchho� rod modeland nearly anti{parallel vortex �laments will be presented to demonstrate the robustnessof the method.1 IntroductionIn this paper, we present a new formulation for computing the motion of a curvature driven3-D �lament. This new numerical method has no high order time step stability constraintsthat are usually associated with curvature regularization. This result generalizes the previouswork of Hou-Lowengrub-Shelley [7] for 2-D uid interfaces with surface tension. Applicationsto 2-D vortex sheets, 3-D motion by curvature, the Kirchho� rod model and anti{parallelvortex �laments will be presented to demonstrate the robustness of the method.Accurate numerical computation of the evolution of a free interface is at the heart of alarge number of scienti�c and engineering problems. Examples include the evolution of aphase boundary in solidi�cation, the breakup of drops in sprays, multi- uid interfaces, andthe motion of cells in the blood. In many applications, local curvature (or surface tension)has an important e�ect on the dynamics of interfaces. On the other hand, curvature alsointroduces new di�culties at the numerical level since curvature contains high order spatialderivatives in the equations of motion for the interface. If an explicit method is used, theseterms induce strong stability constraints on the time step. These stability constraints aregenerally time dependent, and become more severe by the di�erential clustering of points alongthe interface. We refer to this time stability constraint as sti�ness. This sti�ness is especiallydi�cult to remove for uid interfaces with surface tension. Surface tension introduces sti�nessto the evolution equation in a nonlinear and nonlocal manner. A straightforward implicitdiscretization leads to a nonlinear and nonlocal system which is di�cult to invert in general.�Applied Mathematics, California Institute of Technology, Pasadena, CA 91125. Email :[email protected]. Research of this author is in part supported by a NSF Grant: DMS-9407030 and aDOE grant DE-FGO3-89ER25073.yDepartment of Mathematical Sciences, Montana State University, Bozeman, Montana 59717. Email :[email protected]. Research of this author is in part supported by NSF Grant: DMS-9704486.zApplied Mathematics, California Institute of Technology, Pasadena, CA 91125. Email : [email protected]. 1
Hou, Lowengrub, and Shelley [7] proposed to remove the sti�ness of surface tension for2-D uid interfaces by using the Small Scale Decomposition Technique and reformulating theproblem in the tangent angle and arclength metric variables. This is the so-called � � Lframe (see, e.g., [5]). This reformulation greatly improves the stability constraint. It allowsus to perform well-resolved calculations for large times. Many interfacial problems that werepreviously not amenable are now solvable using this method.However, this ��L frame cannot be generalized directly to 3-D �laments since the tangentangle is not well de�ned. In this paper, we propose to use curvature � as the new dynami-cal variable when computing 2-D and 3-D free interfaces with curvature regularization. Wefound that for 3-D �laments, the natural curvature and torsion variables are not suitable forcomputational purpose, see also [10]. The reason is that the torsion variable may be singu-lar whenever curvature is zero. This is purely an arti�cial parameterization singularity. Toovercome this di�culty, we propose to use generalized curvatures and the rate of rotation asthe dynamical variables, �1; �2; !. These variables can be related to the natural curvature �and torsion � variables but also have a physical interpretation in terms of the curvature andtwist of a thin non-isotropic rod. As in 2-D, the total arclength, L(t), is also used as anotherdynamical variable. Together we obtain a new set of evolution equations for the �lament.We show that by using this reformulation, we can easily remove the sti�ness associated withcurvature regularization.To demonstrate the robustness of the method, we apply our method to a number of in-teresting applications. Our numerical study for vortex sheets with surface tension indicatesthat our new formulation shares the same stability property as the � � L formulation intro-duced in [7]. Our numerical experiments also demonstrate convincingly that this idea worksequally well for 3-D �lament calculations, such as the motion by curvature, the Kirchho� rodmodel and anti{parallel vortex �laments. The Kirchho� rod model has received increasinginterest in recent years because it can be used as a model to study the dynamics of proteinsand super-coiled DNA [17]. Our numerical calculations reveal some interesting equilibriumstates for the Kirchho� rod model. With our new formulation, we can now a�ord to per-form well resolved and long time computations for problems in computational geometry andcomputational biology. The dynamics of vortex �laments has been studied analytically andnumerically over the past thirty years (see [13] for references). It has important applicationsinvolving secondary three-dimensional instability in mixing layers and boundary layers in highReynolds number ows. Here we apply our method to study the interaction of anti{parallelvortex �lament pair using a simpli�ed model proposed by Klein, Majda and Damodaran [13].Our results compare well with those obtained in [13]. Our approach can also be applied tothe case when the nonlocal term becomes important. Generalization of this idea to 3-D freesurfaces has also been carried out. This will be reported elsewhere.The organization of the rest of the paper is as follows. In Section 2, we derive the � � Lformulation for two dimensional interfaces, with an application to vortex sheets with surfacetension. Section 3 is devoted to 3-D �laments using the �1 � �2 � ! � L formulation. Asan example of illustration, we consider the motion of a 3-D curve by its local curvature. Wethen apply this idea to a couple of more interesting applications, the Kirchho� rod model andanti{parallel vortex �laments in Sections 4 and 5. We discuss some practical implementationissues in Section 6. Finally, in Section 7, we present some numerical results which includevortex sheets with surface tension, motion of a 3-D curve by curvature, the Kirchho� rodmodel and anti{parallel vortex �laments.2
2 The �� L Formulation for 2-D InterfacesIn this section, we derive the ��Lmethod for 2-D interfaces. We �rst motivate the formulationfor the simple model problem of motion by curvature. Then we derive the formulation for uid interfaces, and indicate how the � � L formulation can be used to remove the sti�nessof surface tension for uid interface problems.2.1 Motion by CurvatureWe motivate the �� L approach by considering the motion by curvature in 2-D. Let a curve� be given by X(�; t) = (x(�; t); y(�; t)); � 2 [0; 2�]; (1)where � parameterizes the curve. Then X evolves byXt = Un; U = �; (2)where n = (�ys; xs) is the right-handed normal and � = xsyss� xssys = (x�y�� � x��y�)=s3�is the signed curvature. Here s is arclength, and the s and � derivatives can be exchangedthrough the relation @=@s = (1=s�)(@=@�) , where s� = px2� + y2�. We assume X is 2�-periodic in �. If we discretize Eqn. (2) using an explicit method, this will give a time-stepstability constraint in the form of �t � C�h(t)2, where �h(t) is the minimum grid spacing attime t. An implicit integration method, like the backward Euler or Crank-Nicholson scheme,would give a more stable discretization. But since curvature is a nonlinear function of theinterface position, this would give rise to a nonlinear system for the implicit solution at thenext time-step.The ��L approach, on the other hand, makes the application of an implicit method mucheasier. It consists of two steps:(A) Formulate the evolution using the � and s� as the new dynamical variables.(B) Introduce a change of frame in the parameterization of � so that s� is independent of �and depends only on time. Thus, the equation for s� becomes an ODE for L, the lengthof the curve �. This reformulation of interface motion is motivated by the � � L framein [5] (also see [7]).We notice that the shape of the curve is determined solely by its normal velocity U . Atangential motion only results in a change in frame for the parameterization of the curve.Therefore, we can add a tangential motion to the dynamics without changing the interface'sshape, i.e., Xt = Un+ T s;where s = (xs; ys) is the unit tangent vector, T is the added tangential velocity which will bedetermined later. To derive the evolution equations for � and s�, we use the Fr�enet equations,@ss = �n and @sn = ��s. The evolution equations for s� and � are given bys�t = T� � U�s� (3)�t = 1s� �U�s� �� + T��s� + U�2: (4)3
Given s� and �, the position (x(�; t); y(�; t)) can be reconstructed (see Section 5). For motionby curvature, we have U = �. The evolution in terms of � and s� iss�t = T� � �2s� (5)�t = 1s� ���s��� + T��s� + �3: (6)For an explicit integration method, the stability constraint from the di�usion term is of theform �t < C � (�s�h)2; (7)where �s� = min�s�, and h is the initial grid spacing in �. Therefore, the stability constraintis determined by the minimum grid spacing (i.e., hs� � �s), which is time dependent, andfor motion by curvature, is always decreasing.In the reformulated system consisting of Eqns. (5) and (6), an implicit discretizationbecomes much easier since the highest order terms are linear. The discretization can besimpli�ed further if s� does not depend on �. This can be easily accomplished by choosing aspecial tangential velocity T to force s� equal to its mean:s� = 12� Z 2�0 s�0(�0; t)d�0 = 12�L(t); (8)where L is the length of the curve �. It follows from Eqn. (5) that T satis�esT� � �2s� = 12� Z 2�0 (T�0 � �2s�0)d�0;which implies T (�; t) = T (0; t) + L2� Z �0 �2d�0 � �L(2�)2 Z 2�0 �2d�0: (9)Here T (0; t) is simply an arbitrary change of frame which can be taken to be 0. Now since s�just depends on time t, but not on �, the PDE for s� is reduced to an ODE for L, and L and� evolve by Lt = � L2� Z 2�0 �2d�0 (10)�t = �2�L �2 ��� + 2�L T�� + �3: (11)Notice that the highest order term has no spatially varying prefactor, an implicit methodcan be easily applied to the PDE for �. It is su�cient to treat the leading order termsimplicitly, and discretize the lower order terms explicitly. Also, the equation for L is free ofsti�ness, we can use an explicit method such as the Adams-Bashforth method. Then at everytime step, L can be updated explicitly, and the implicit solution � at the new time-step canbe obtained explicitly by using the Fourier Transform.4
2.2 The Formulation for 2-D Fluid InterfacesIn the next two subsections, we will show how to generalize the idea presented in the previoussubsection to uid interface problems. The uid interface problem is more di�cult than motionby curvature because it involves nonlocal singular integral operators. To derive an e�cientimplicit discretization, we also use the so-called \small scale decomposition" technique whichseparates the leading order contribution of a singular integral operator from the lower ordercontributions. Since sti�ness enters only at small scales, it is enough to treat the leading orderoperators implicitly. For uid interfaces, these leading order intergal operators are the Hilberttransform and its variants. They can be diagonalized using the Fourier transform. Thus weobtain an e�cient implicit discretization at the same cost as an explicit method.We consider the motion of an interface � given by X = (x(�); y(�)), separating twoinviscid, incompressible and irrotational uids. The density is assumed to be constant oneach side of �. The velocity on either side of � is evolved according to the incompressibleEuler equationujt + (uj � r)uj = �1�r(pj + �jgy); r � uj = 0: (12)Here j = 1 is for the uid below � and j = 2 for the uid above, pj is the pressure, �j is thedensity, and gy is the gravitational potential. The boundary conditions are:(i) [u]� � n = 0; the kinematic boundary condition; (13)(ii) [p]� = ��; the dynamic boundary condition; (14)(iii) uj(x; y)! (�V1; 0) as y! �1; the far �eld boundary condition; (15)where [�] denotes the jump taken from above to below the interface. The velocity has atangential discontinuity at �. The velocity away from � has the integral representation (seee.g. [2]), (u(x; y); v(x; y)) = 12� Z (�0) (�(y � y(�0)); x � x(�0))(x� x(�0))2 + (y � y(�0))2 d�0; (16)where is called the (unnormalized) vortex sheet strength. The true vortex sheet strength(i.e., the tangential velocity jump) is given by~ = (�)s� = [u]� � s: (17)While there is a discontinuity in the tangential component of the velocity at �, the normalcomponent, U(�), is continuous and given by (16) asU(�) = W � n (18)where W(�) = 12�P:V: Z (�0) (�(y(�) � y(�0)); x(�) � x(�0))(x(�) � x(�0))2 + (y(�) � y(�0))2 d�0: (19)The P.V. in front of the integral denotes the principal value integral. This integral is calledthe Birkho�-Rott integral. Using the representation (16) for the velocity, Euler's equation at5
the interface and the Laplace-Young condition, the equations of motion for the interface areXt = Un+ T s (20) t � @�((T �W � s) =s�)= �2A�(s�Wt � s+ 18@�( =s�)2 + gy�� (T �W � s)W� � s=s�) + S��: (21)Here A� = (�1��2)=(�1+�2) is the Atwood ratio and S is a rescaled surface tension parameter(see [2]). In the special case of A� = 0, i.e. �1 = �2, the evolution equation is greatly simpli�ed.It is reduced to a vortex sheet equation (see [16]).2.3 The Equations of Motion ReposedIn the previous subsection, boundary integral formulation is given for the motion of a vortexsheet in two-dimensional, inviscid uid. Numerical sti�ness arises through the presence of highorder terms (i.e., high spatial derivatives) in the evolution. In this subsection, we reformulatethe equations of motion using the Small Scale Decomposition for inertial ows. The SmallScale Decomposition (SSD), which identi�es and separates the dominant terms at small spatialscales, was �rst presented in [7]. The key idea is to identify the leading order contributionof certain singular operators at small spatial scales. Recall that the normal velocity U isgiven by Eqns. (18) and (19). Let the complex position of the interface be given by z(�; t) =x(�; t) + iy(�; t), then U can be expressed asU(�; t) = � 1s� Im� z�2�iP:V:Z +1�1 (�0; t)z(�; t)� z(�0; t)d�0� : (22)Note that the kernel in the Birkho�-Rott integral can be decomposed into two terms:1z(�; t) � z(�0; t) = 1z�(�� �0)+ � 1z(�; t) � z(�0; t) � 1z�(�� �0)� : (23)The most signi�cant contribution comes from the �rst term on the right hand side, since thebracketed term is analytic and corresponds to a smoothing operator. Therefore, we obtainthe leading order behavior of U at small scales as:U(�; t) � 12s�H[ ](�; t); (24)where H is the Hilbert transform de�ned as(Hf)(�) = 1� Z +1�1 f(�0)�� �0 d�0: (25)Its Fourier transform is given by(dHf)(k) = �i(sgn(k))f (k): (26)The notation f � g means that the di�erence between f and g is smoother than f and g.In terms of the new dynamic variables, s�, �, and , the equations of motion for the inertial6
vortex sheets are given by Eqns. (3), (4) and (21). Observe that the dominant term in Eqn.(21) for t is S�� at small scales. Now, substituting Eqn. (24) into Eqn. (4) gives�t = 12s� � 1s� � 12s�H[ ]���� + P (27) t = S�� +Q; (28)where P and Q represent lower order terms at small spatial scales. This is the small scaledecomposition. If s� is given, the dominant small scale term is linear in � and , but non-local by the virtue of the Hilbert transform. An implicit discretization can be obtained bydiscretizing the leading order terms implicitly, but treating the lower order terms explicitly.However, if s� is independent of �, the implicit solution can be obtained easily by Fast FourierTransform, just as in the case of motion by curvature. By choosing a particular tangentialvelocity, s� can indeed be independent of �.2.4 The �� L FormulationAs we mentioned above, the tangential velocity T may be introduced into the dynamicswithout changing the shape of the interface. We can choose the particular expression for Tso that s� does not depend on � in its evolution. As in the case of motion by curvature, s�is set to be equal to its mean, which iss� = 12� Z 2�0 s�0(�0; t)d�0 = 12�L(t); (29)where L is the length of the interface. It follows from Eqn. (3) that T satis�es the followingequation: T� � U�s� = 12� Z 2�0 (T�0 � U�s�0)d�0) T (�; t) = T (0; t) + L2� Z �0 U�d�0 � �L(2�)2 Z 2�0 U�d�0: (30)T (0; t) just gives an arbitrary change of frame, and, for simplicity, can be taken to be 0. Thus,the expression for T is determined entirely by L; � and U . Assume that Eqn. (29) is satis�edinitially, then Eqn. (30) for T ensures that the constraint (29) is satis�ed for all time. Now,the evolution of the interface is reformulated in terms of L and � byLt = � L2� Z 2�0 �Ud�0 (31)�t = �2�L �2 U�� + 2�L T�� + �2U: (32)Given U , Eqns. (30), (31) and (32) give a complete formulation of the evolution problem.The small scale decomposition for the inertial vortex sheets in the � � L formulation isnow given as: �t = �2�L �3H[ ��] + P (33) t = S�� +Q; (34)7
where P and Q denote the lower order terms, which do not contribute to the sti�ness, andwill be treated explicitly. In Fourier Space, these equations are�t(k) = �ik22 �2�L �2 jkj (k) + P (k) (35) t(k) = �iSk�(k) + Q(k); (36)where P and Q are the Fourier transforms of P and Q in Eqns. (33) and (34), i is theimaginary unit. Now the implicit integration scheme can be easily applied together withan explicit discretization of Eqn. (31). Since the lower order terms, P and Q, are treatedexplicitly, the implicit solution for � and can be inverted explicitly. This gives an e�cientimplicit discretization of the uid interface problem at the same cost as an explicit method.The numerical method in our computation will be discussed in subsection 6.1.3 The �1 � �2 � ! � L Formulation for 3-D FilamentsIn this section, we generalize the � � L method to 3-D �laments. The formulation is moresubtle for 3-D �laments since there are two normal vectors (e.g. the normal and the bi-normalvectors). It turns out that the choice of orthonormal basis has a signi�cant impact on thecomputational method. In particular, the conventional Fr�enet frame for 3-D �laments is notsuitable for computational purpose. It can give rise to an arti�cial parametrization singularitywhen curvature vanishes. To overcome this di�culty, we use a more general orthonormal basiswhich corresponds to the �1 � �2 � ! � L formulation for 3-D �laments.Let us consider a space curve X(s; t) : [0; L] ! R3 where s is arc-length and L is thetotal length of the curve. Alternatively we may parameterize X by a material coordinate�, i.e., X(�; t) : [0; 2�] ! R3. The unit tangent vector along the curve X is given byT(s; t) = (d=ds)X(s; t). A local description of the curve is provided by an appropriate set ofcoordinate axes. One such set is the Fr�enet triad consisting of the unit vectors T;N and B,the tangent, normal and binormal vectors respectively. This orthonormal triad satis�es thewell known Fr�enet equations Ts = �N;Ns = ��T+ �B;Bs = ��N. � is the curvature and� is the torsion. We can now write the evolution equation for the curve in the Fr�enet frame:Xt(�; t) = �UN+ �VB+ �WT (37)where �U; �V ; �W are the normal, bi-normal and tangential velocity components respectively,each of which can depend on both � and t. In the 2-D case, we use s� and � as the newdynamical variables. Naturally, we would like to use s�, �, and � as the new dynamicalvariables for 3-D �laments. Using the Fr�enet equations, we can derive the evolution of thecurve � in terms of s�; �, and � as:s�t = �W� � �U�s� (38)�t = 1s� � �U�s� �� � 2� �V� + �� �V � �W��s� � �U(�2 � �2) (39)�t = 1s� 0@� �V�s� ���s� 1A� + 1s� �2 �U�� + �U�� � �V �2s��s� ��+� �V� + �W��s� + 2� �U�: (40)8
Consider the natural generalization of motion of a closed curve by curvature, namely,�U = �; �V = 0 in Eqn. (37). �W can be added to the motion of equations without altering theshape of the curve. Thus Eqns. (38), (39) and (40) become:s�t = �W� � �2s� (41)�t = 1s� ���s��� + �W��s� � �(�2 � �2) (42)�t = 1s� ���s��� + 2�����s��� + �W��s� + 2�2�: (43)Now, if s� is given, the highest order terms in the equations for � and � are linear in � and� respectively. Thus, an implicit integration method can be applied. Similar to the 2-D case,we can choose a special expression for �W to enforce s� to be independent of �. Then thehighest order terms in Eqns. (42) and (43) do not have spatially varying prefactors. So theimplicit discretization of � and � can be updated explicitly. The stability constraint has theform max� �2j��j�s� ; j �W js� ��t < h: (44)Note that the stability constraint depends on curvature. If � becomes very small, we willget very strong stability constraint in the numerical computation. In fact, in the Fr�enettriad, N, B and � are only de�ned when the curvature does not vanish. In general N variesdiscontinuously through points where � = 0 even for smooth curves. This would lead to theblowup of � since � depends on Ns. This discontinuity in N through points where � vanishesis arti�cial and is due to a poor choice of coordinate frame. For this reason, the Fr�enet frameis not a good choice for computational purpose.Instead we propose to use a more general orthogonal basis T;N1;N2 = T �N1 in ournumerical calculation of 3-D �laments. The Fr�enet system is replaced byddsT = �1N1 � �2N2ddsN1 = ��1T+ !N2ddsN2 = �2T� !N1; (45)where �1 = �N �N1 and �2 = ��N �N2.There are natural relations between �1; �2; ! and �; � :� = q�21 + �22 (46)� = ! + �2�1s � �1�2s� : (47)If we make the special choice of ! = � , then the new orthonormal basis is reduced to theFr�enet triad. In this case, we have �1 = �; �2 = 0.The unit tangent vector T(s) = (d=ds)X(s) is determined once the curve X(s) is known.Then we choose vectors N1(0);N2(0) such that (T(0);N1(0);N2(0)) are a set of orthonormalvectors. By choosing a smooth function for the rate of rotation, !, and using the relations�1 = dTds �N1; �2 = �dTds �N2, we integrate the last two equations in (45) along the arclength,9
s, to determine N1(s) and N2(s). Notice that the �rst equation in (45) is automaticallysatis�ed since we have used it to construct �1 and �2 . Also, the orthogonality of these threevectors T(s);N1(s) and N2(s) can be shown by using Eqs. (45) . Thus we obtain a smoothorthonormal basis set (T(s);N1(s);N2(s)) . Clearly, this orthonormal basis is smooth as longas the curvature is smooth, even though the curvature may vanish at some points.Now we rewrite the evolution equation for the curve in our newly chosen orthonormal basis(T;N1;N2): Xt(�; t) = UN1 + VN2 +WT:Since we have relationships between N;B and N1;N2, namely�N = �1N1 � �2N2�B = �2N1 + �1N2; (48)it is straightforward to determine the relationships between U; V;W and �U; �V ; �W .The fact that X has continuous second order derivatives in space and time implies thatthe cross derivatives of � and t commute. To carry out the computations associated with thisrelationship it is convenient to write the time derivatives of the basis T;N1;N2 asTt = �F �T; N1t = �F �N1; N2t = �F �N2; (49)where �F (�; t) = �1N1 + �2N2 + �3T is the rotation vector whose components �1; �2; �3 arerelated to U; V;W and hence �1; �2; !:�1 = �V�s� � U! +W�2�2 = U�s� � V ! +W�1; (50)and �3 will be determined later. It can be shown (also see [8]) that the equations of motionfor s�; �1; �2; ! in terms of U; V;W take the form:s�t = W� + (V �2 � U�1)s� (51)�1t = 1s� �U�s� �� � 2!V� + !�V �W�1�s� � U!2+ �1(U�1 � V �2) + !�2W � �3�2 (52)�2t = � 1s� �V�s��� � 2!U� + !�U �W�2�s� + V !2+ �2(U�1 � V �2)� !�1W � �3�1 (53)!t = �1V� + �2U� � !W�s� + 2!(U�1 � V �2) + �3�s� : (54)As a �nal remark, note that we now have four functions s,�1, �2 and ! to describe a curvein R3. ! measures the twist rate of the N1-N2 plane around T, and may (e.g. the Kirchho�rod model) or may not (e.g. motion by curvature) have physical signi�cance.As in the 2-D case, we can choose a tangential velocity W to force s� to be everywhereequal to its mean, s� = 12� Z 2�0 s�0(�0; t)d�0 = 12�L(t); (55)10
where L is the length of the curve �. Speci�callyW� � (U�1 � V �2)s� = 12� Z 2�0 (W�0 � (U�1 � V �2))s�0d�0)W (�; t) = L2� Z �0 (U�1 � V �2)d� � �L(2�)2 Z 2�0 (U�1 � V �2)d�0: (56)Now since s� depends only on t and not �, the PDE for s� reduces to an ODE for L. Equationsfor L and �1; �2; ! then reduce toLt = � L2� Z 2�0 (U�1 � V �2)d�0 (57)�1t = �2�L �2 U�� � 2�L (2!V� + !�V �W�1�)� U!3+ �1(U�1 � V �2) + !�2W � �3�2 (58)�2t = ��2�L �2 V�� � 2�L (2!U� + !�U �W�2�) + V !2+ �2(U�1 � V �2)� !�1W � �3�1 (59)!t = 2�L (�1V� + �2U� � !W�) + 2!(U�1 � V �2) + 2�L �3�: (60)We now show that for motion by curvature this reformulation leads to e�cient implicitdiscretization. To obtain the velocity in this new basis, we project the original equationXt = �N into the new orthonormal basis. Using the relations between N and N1;N2, wehave U = �1 and V = ��2. Simply substitute U; V into Eqns. (57) - (60) and take �3 to be0 (see subsection 6.3), we derive the formulation in terms of �1; �2; ! and L asLt = � L2� Z 2�0 (�21 + �22)d�0 (61)�1t = �2�L �2 �1�� + 2�L (2!�2� + !��2 +W�1�)+ �1(�21 + �22 � !2) + !�2W (62)�2t = �2�L �2 �2�� � 2�L (2!�1� � !��1 +W�2�)+ �2(�21 + �22 � !2)� !�1W (63)!t = 2�L (�2�1� � �1�2�): (64)As in the 2-D case, L and ! can be updated using an explicit integration method. The highestorder terms in Eqns. (58) and (59) do not have spatially varying prefactors. We can invertthe implicit discretization for the di�usion terms in the �1 and �2 equations e�ciently.4 Application to the Kirchho� Rod ModelWe now apply our method to the physically interesting problem of the Kirchho� rod. Thestudy of elastic rods is the subject of continued scienti�c and mathematical interest. Applica-tions of the dynamics of rods and �laments include the dynamics of proteins and super-coiledDNA [17], writhing instability in �bers and cables [19], three-dimensional scroll waves [18],11
magnetic ux tubes and the formation of sunspots [14], etc. Under some simplifying as-sumptions, the motion of an elastic rod �lament can be well described by a one-dimensionalsystem of equations. One such set of equations, the Kirchho� rod equations [9], can be con-structed as follows [6]. The rod is represented by its center line X(s; t) : [0; L] ! R3 andtwist (de�ned below) !(s; t) : [0; L] ! R. Here s is arclength and L is the length of therod. For simplicity we assume that the cross section of the �lament is always circular withconstant radius in space. De�ne a reference ribbon by a pair of curves (X;X+ �N1) whereN1(s; t) : [0; L] ! R3 is a unit vector �eld such that N1 � T = 0 (T is the unit tangentvector along the curve X) and � is the width of the ribbon. The twist ! (with respect tothe reference ribbon (X;X+ �N1)) is de�ned to be the rotation rate of N1 around T movingalong X; i.e., !(s; t) = (N1(s; t)� (d=ds)N1(s; t)) �T. The Fr�enet triad is a particular choiceof ribbon which corresponds to choosing N1 = N. Recall that N has the same direction as(d=ds)T(s; t). More typically N1 might point in the direction of one of the principle axes ofthe cross-section of the rod. The equations of motion can be written asd2dt2X = ddsF� �1 ddtX+ g (65)ddsM = F�T+ ��jxT+W1 + �2( _�jx +W2) +H (66)M = ��1�B+ !T; (67)where g contains the other external forces such as gravity, contact force, etc. � and �i measurerespectively the relative energetic importance of twist and exure and the relative time scalesof viscosity and inertia. The rod shearing terms W1 and W2 are set to be 0 and � to be 1.We rewrite the velocity of X in terms of the ribbon basis asXt(�; t) = UN1 + VN2 +WT;whereN2 = T�N1. To compute the main force F, we decompose it into the local orthonormalribbon basis: F = F1N1 + F2N2 + FTT:The normal and bi-normal components of F can be determined immediately from Eqns. (66)and (67), that is F1 = ��1s and F2 = �2s. The determination of the tangential force FT = F �Tis more subtle. We will derive it later. Using Eqns. (45), (49), (65), (66) and (67) we obtainthe evolution equations for �1; �2; !, and s� as follows:s�t = W� � (U�1 � V �2)s� (68)�1t = 1s� �U�s� �� � 2!V� + !�V �W�1�s� � U!2+ �1(U�1 � V �2) + !�2W � �2�2 (69)Ut = � 1s� ��1�s� �� � !�2�s� + FT�1 �W�2 + V �3 � �1U + g1 (70)�2t = � 1s� �V�s��� � 2!U� + !�U �W�2�s� + V !2+ �2(U�1 � V �2)� !�1W � �2�1 (71)Vt = 1s� ��2�s� �� � !�1�s� � FT�2 +W�1 � U�3 � �1V + g2 (72)12
!t = �3�s� + �1V �+ �2U�s� + 2!(U�1 � V �2)� !W�s� (73)�3t = !�s� � �2�3s� ; (74)where g1 = g �N1 and g2 = g �N2 .Eqn. (74) is derived from Eqns. (66) and (67). To see this, we observe that Eqns. (66)and (67) give !� = ��jXs� + �2 _�jX; (75)where _�jX refers to _� at a �xed �lament position X (see [11]) (Here a dot denotes @=@t). Ifwe hold X steady and allow twisting, we have N1t = �3(�; t)T�N1. Over an element of the�lament from X(�) to X(�+��)ddt�� = �3(�+��; t)� �3(�; t);where � = R s(�) !ds = R � s�!d� is the angle of rotation of the reference ribbon at X(�; t).Thus when X is �xed, _�jX = �3, and ��jX = _�3. Substituting these relations to Eqn. (75)gives Eqn. (74).Now we are going to determine the tension FT = F �T. From @X@� = s�T, we get@X@� � @ _X@� = s�T � (s�tT+ s�Tt)= s�s�t = rs2�; (76)provided that the rod has prescribed extension rate r(�; t), i.e. s�t = r(�; t)s�, which is trueby our choice of W from Eqn. (56). Di�erentiating this equation with respect to time t, weget @X@� � @ �X@� = �@ _X@� � @ _X@� + _rs2� + 2rs�s�t= s2�(r2 + _r � jTtj2): (77)From Eqn. (49), we get jTtj =p�21 + �22. Furthermore, we have@X@� � @ �X@� = s�T � @@� �X= s�T � @@�(dFds � �dXdt + g)= s2�T � (Fss + gs): (78)Thus FT satis�esd2ds2FT � (�21 + �22)FT = 2F1s�1 � 2F2s�2 + F1�1s � F2�2s � !(F1�2 + F2�1)+ r2 + _r � (�21 + �22) + �1g1 � �2g2 � gts; (79)with F1 = ��1s, F2 = �2s and gt = g �T. The right-hand side of the equation for FT dependsonly on known qualities and hence the tension is determined with the appropriate periodicboundary conditions for closed �laments. 13
We now summarize the small scale decomposition in the �1; �2 and ! formulae as follows:�1t � �2�L �2 U�� + P1Ut � ��2�L �2 �1�� +Q1�2t � ��2�L �2 V�� + P2Vt � �2�L �2 �2�� +Q2;with !t � 2�L �3� + P3 and �3t � 2�L !� + Q3, where Pi and Qi; i = 1; 2; 3 are the lower orderterms. The highest order terms in the equations for �1; �2; U; V; ! and �3 now appear linearly.After updating L explicitly, it is a straightforward exercise to apply an implicit integrationmethod to these equations.5 Application to Nearly Parallel Vortex FilamentsAnother interesting problem we consider is a nearly parallel pair of vortex �laments. Vor-tex �laments with large strength and narrow cross section are prominent uid mechanicalstructures in mixed layers, boundary layers and trailing wakes. It is interesting to study theinteraction of nearly parallel and anti{parallel vortex �laments in high Reynolds number ows.An ensemble of vortex �laments interacts via the three-dimensional Biot-Savart integrals forthe induced velocities on the �lament centerlines. The induced motion of each �lament con-sists of self- and foreign- induced velocity contributions. It has been shown by Callegari andTing [3], Klein and Majda [12] that the geometrical evolution of the �laments in the regimeconsidered obeys the propagation law@Xi@t = (ln(1� ) + Ci) �i4� (�B)i +Qfi +Qouteri : (80)The �rst term points in the direction of the local binormal vector Bi, and via the expressionln(1� ) + Ci describes the in uence of the vortex core structure on the �lament motion. Here� � 1 relates to the small e�ective core sizes and Ci is a quadratically nonlinear functional ofthe detailed core vorticity distribution of the ith �lament(see [3] and [12]). Qfi is the �lamentmotion due to non-local self stretching [12] and the foreign-induced velocity Qouteri has beenanalyzed in [13]. Klein, Majda and Damodaran derived simpli�ed equations for a pair ofinteracting vortex �laments in [13]:@Xi@t = �i(�B)i + 2t0 � �j Xi �XjjXi �Xjj2 ; (81)where t0 = (0; 0; 1) and i; j = 1; 2. These simpli�ed equations retain the important physicale�ects of linearized local self-induction and nonlinear potential vortex interaction among �l-aments but neglect other non-local e�ects of self-stretching and mutual induction. Now weapply our method to nearly parallel vortex �lament pair using the equations above. Noticingthat �B = �2N1+�1N2. Using Eqns. (57) - (60), it is easy to derive the formulation in terms14
of �1i; �2i; !i and Li and the small scale decomposition in the �1i; �2i formulae is as follows:�1it � �i�2�Li�2 �2i�� + Pi�2it � ��i�2�Li�2 �1i�� +Qi;where Pi and Qi are the lower order terms, i = 1; 2. As before, the implicit solutions areeasily obtained by the Fast Fourier Transform.6 Some Implementation IssuesThis section is devoted to addressing a few practical implementation issues. This includes thequestion of what implicit discretization scheme we will use, the reconstruction of the interfacefrom the curvature variable, and the choice of orthogonal basis in the Kirchho� rod model.6.1 Time-Stepping ConsiderationsThe time integration scheme we used in this paper is a fourth order multi-step implicit/explicitscheme studied in [1] by Ascher, Ruuth and Wetton. This is one of the better high orderimplicit/explicit schemes to use in the sense that it has a large stability region. Consider atime-dependent PDE in which the spatial derivatives have been discretized by either centraldi�erences or by pseudo-spectral methods. This gives rise to a large system of ODEs in timewhich typically has the form dudt = f(u) + �g(u); (82)where g is a linear operator containing high order derivatives and f(u) is a nonlinear functionwhich we do not want to integrate implicitly in time. To avoid using excessively small timesteps, we would like to treat the �g(u) implicitly while treating the nonlinear term f(u)explicitly. Typically f(u) involves only �rst order derivatives from the convective terms, sothe sti�ness induced from the nonlinear term is not as severe as that from the linear operatorg(u).The fourth order implicit/explicit scheme considered by Ascher, Ruuth and Wetton isgiven as follows: 1�t(2512un+1 � 4un + 3un�1 � 43un�2 + 14un�3) =4f(un)� 6f(un�1) + 4f(un�2)� f(un�3) + �g(un+1): (83)In this paper, we simply apply this fourth order implicit/explicit scheme to our problems. Forexample, we use this scheme in the inertial vortex sheet problem:�t = 12s� � 1s� � 12s�H[ ]���� + P (84) t = S�� +Q; (85)where P and Q represent the lower order terms. We obtain the following time discrete system:15
1�t(2512�n+1 � 4�n + 3�n�1 � 43�n�2 + 14�n�3) =12s� � 1s� � 12s�H[ n+1]���� + 4P n � 6P n�1 + 4P n�2 � P n�3and 1�t(2512 n+1 � 4 n + 3 n�1 � 43 n�2 + 14 n�3) =S�n+1� + 4Qn � 6Qn�1 + 4Qn�2 �Qn�3Then with our special choice of the tangential velocity, T , s� is independent of �, and we cansolve for �n+1 and n+1 explicitly using the Fast Fourier Transform.6.2 Reconstruction of the Interface from CurvatureIn our paper, the construction of the initial equal arclength parameterization is the same asin [7]. We will not repeat the details here. On the other hand, it is important to discussthe reconstruction of the 2-D interface (x; y) from (L; �), and the 3-D �lament (x; y; z) from(L; �1; �2; !).One natural way to reconstruct (x; y) from curvature is to integrate the Fr�enet equationsalong the interface. This will generate the tangent vector T. We can then integrate thetangent vector along the interface to obtain the interface position. This involves two numericalintegrations for each time step, and we need to keep track of two initial conditions at thebeginning point of the interface. An alternative is to use the evolution equation for theinterface. Recall that � evolves according to Xt = Un + T s. We can reconstruct (x; y)through integration of these original equations. In the inertial vortex sheets problem, weknow that U(�; t) � 12s�H[ ](�; t);so we get Xt = 12s�H[ ]n+ P; (86)where P includes the lower order terms. In the computation, we treat 12s�H[ ] implicitly andall the other terms explicitly. However, due to the numerical error, the points on the curveare no longer equally distributed (i.e., s� is not exactly L=(2�) everywhere). This makes Eqn.(86) incompatible with Eqns. (27) and (28). This di�culty is overcome by redistributing(x; y). For example, we can make use of the formulaX� = L2�T; (87)where T = X�=jX�j:We denote the solution of Eqn. (86) by �X. Then integrate the equationX� = L2�j �X�j �X� (88)16
with respect to � to get X for the new time step. Of course, in the absence of numerical errors,the coe�cient in front of �X� should be 1. We have considered other ways of redistribution,but we have found that this approach gave the best performance numerically. This methodof reconstruction using the original evolution equation for X also applies to 3-D surfaces.In the case of 3-D �laments, a space curve � evolves according to Xt = �UN+ �VB+ �WT,where X = (x(�; t); y(�; t); z(�; t)). If we simply reconstruct (x; y; z) by integrating thesethree equations, we will get a stability constraint of the form �t � Ch2; since N involves asecond derivative of X. So, we try to reconstruct X using the �rst approach we mentionedearlier. First, we integrate Eqs. (45) to get the tangential vector T, then integrate Eqn. (87)to get X. By doing this, we can still have a stability constraint of the form �t � Ch.6.3 Contact Force in the Kirchho� Rod ModelIn practice, a contact force g is added to Eqn. (65) to avoid self crossing of the �lament.The contact force becomes important when the rod deforms in such a manner that pointsseparated by large di�erences in arc-length become close to one another in space. The contactforce can be modeled by the following integral formulag(s) = �Z M(s; �)U 0(jr(s)� r(�)j)jr(s)� r(�)j [r(s)� r(�)]d�; (89)where U is a self-potential generating a central force between pairs of points along the rod, andM is a molli�er leading to total energy and corresponding, for example, to a nonzero radiusof the rod. In our example, we take the potential U to be proportional to jr(s)� r(�)j�9.Another point we should stress is that in the case of motion by curvature, we simplytake �3 to be 0 which makes the formulation much easier. But this cannot be done in thecase of the Kirchho� rod model. This is because in the case of motion by curvature, we areonly concerned with the shape of the curve � which is determined by the tangential vector.Therefore we can choose a particular N1 and N2 by taking �3 to be 0. In the Kirchho� rodmodel, we do not just study a space curve. Instead, we study a rod with some thickness. Herethe twist ! is important in the evolution of the rod and in fact depends on �3.7 Numerical ResultsIn this section, the results of numerical simulations are presented for several 2-D and 3-Dproblems. All of these simulations use the appropriate small scale decomposition, togetherwith the associated numerical methods discussed in the previous sections. In subsections7.1 and 7.2, we consider motion by curvature and motion by �� < � > in two dimensions.These tests demonstrate that our method has only a linear stability constraint. Subsection 7.3presents the motion of inertial vortex sheets which has been well studied by Hou, Lowengruband Shelley in [7]. We demonstrate that our numerical method shares the similar stabilityproperty as the equal arclength/tangent angle formulation. We can compute very close to thetime when a pinching singularity is formed. A comparison of the stability constraint betweenthese two methods will be given. In subsection 7.4, we compute the motion by curvaturein three dimensions. The result is consistent with our �ndings for 2-D interfaces. Again,our method has only a linear stability constraint. Comparison with straightforward explicitmethod in (x; y; z) coordinates shows that our method allows time step 3,200 times larger thanthat of the corresponding explicit discretization for N = 512. Motions for the Kirchho� RodModel and anti{parallel vortex �laments are presented in subsections 7.5 and 7.6 respectively.17
The results match very well with the existing results ([10] and [13]) and no sti�ness wasobserved in our computation.7.1 Motion By Curvature in 2-DIn the next two subsections, we perform several numerical tests on motion by curvature in2-D to demonstrate the e�ectiveness of our method. These tests all demonstarte that ourreformulated implicit method has only a linear stability constraint, i.e. �t is of the sameorder of the spacial mesh size. This linear stability constraint is expected since we treat theconvection terms explicitly. From our stability analysis for the convection equation, we cansee a dependence of the CFL condition on the maximum curvature. This is also veri�ednumerically.We consider a plane curve � evolving according toXt = �n: (90)In our numerical calculations, we use the length of the curve and the curvature as our dynamicvariables. They evolve by Eqns. (10) and (11). The reconstruction of the position of the curveis done by directly integrating the equation Xt = �n + T s, where T is of the form given inEqn. (9).In our �rst example, we choose the initial curve as X = (�+ :2 cos(4��); :5 sin(2��)); 0 �� � 1: We graph the position of the curve at various times. In Fig. 1 (a)-(d), we show thecontinued evolution of the curve from t = 0:0 to t = 0:08. There are N = 128 mesh points inthe unit interval with time step �t = 0:00025.0 0.5 1
−0.5
0
0.5b
0 0.5 1−0.5
0
0.5a
0 0.5 1−0.5
0
0.5c
0 0.5 1−0.5
0
0.5d
Figure 1: Motion by curvature; initial sin curve; N = 128;�t = 0:00025; curve portrayedevery 0:005: a) 0 to 0:02; b) 0:02 to 0:04; c) 0:04 to 0:06; d) 0:06 to 0:08:18
In fact, �t can be increased as time progresses. We list the maximum time steps that canbe used at various times in Fig. 2.dt
0 0.01 0.02 0.03 0.050.04
0.00125
0.0004
0.0008
0.00025
0.002
tFigure 2: Maximum time steps at various timesThe reason �t is chosen to be so small initially is because the stability constraint is of theform max� j �T j�t < C � Lh; (91)from Eqns. (10) and (11). Here �T = R �0 (�2)d�0 � � R 10 (�2)d�0. Thus �t is still related to themagnitude of curvature through �T . We print out the curvature of this curve in Fig. 3. Themaximum curvature of this curve is around 130. Since the initially curvature is large along
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−100
−50
0
50
100
alpha
curv
atur
e
Figure 3: Curvature of the initial sin curvesome part of the curve, the time step has to be small to satisfy the stability constraint. Wesee that this periodic curve moves faster where it has bigger curvature and it relaxes to astraight line with increasing time. When we increase the number of points in the calculation,we do see the time step decreases linearly.We next consider the initial curve X = (�+ :1 sin(2��); :5 cos(2��)); 0 � � � 1, evolvingaccording to Eqn. (90). The maximum curvature of this initial curve is around 143. We19
graph the position of the curve at various times. In Fig. 4 (a){(d), we show the continuedevolution of the curve from t = 0:0 to t = 0:08. N = 128 mesh points were used and the timestep �t = 0:00005. This periodic curve relaxes quickly to a straight line as time increases.
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5a
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5b
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5c
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5d
Figure 4: Motion by curvature: N = 128;�t = 0:00005; curve portrayed every 0:004: a) 0 to0:02; b) 0:02 to 0:04; c) 0:04 to 0:06; d) 0:06 to 0:08:We also list the maximum time steps that can be used at various times in Fig. 5.We also consider the evolution of the initial closed curveX = (1 + 0:4 sin(10��))(cos(2��); sin(2��)); 0 � � � 1:according to Eqn. (90). With N = 256 mesh points, and time step �t = 0:001, we show inFig. 6 (a){(d) the continued evolution of the curve from t = 0:0 to t = 0:2. The plots showthat this star-shaped curve quickly relaxes to a circle.7.2 Motion by �� < � > in 2-DWe consider the initial curve X = (�2 sin(2��); cos(2��)) evolving according toXt = (�� < � >)n: (92)here < � > is the mean of �, i.e., R 10 �d�. With N = 256 mesh points and �t = 0:005, weshow the continued evolution from t = 0:0 to t = 2:0 in Fig. 7. We see that a circle is theequilibrium state for this ellipse under the motion by �� < � >.20
dt
0 0.01 0.02 0.03 0.050.04 t0.00005
0.00035
0.00075
0.0005
0.00125
Figure 5: Maximum time steps at various times
−2 −1 0 1 2−2
−1
0
1
2a
−2 −1 0 1 2−2
−1
0
1
2b
−2 −1 0 1 2−2
−1
0
1
2c
−2 −1 0 1 2−2
−1
0
1
2d
Figure 6: Motion by curvature; star-shaped curve: N = 256;�t = 0:001; curve portrayedevery 0:01: a) 0 to 0:05; b) 0:05 to 0:1; c) 0:1 to 0:15; d) 0:221
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
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2
Figure 7: N = 256;�t = 0:005; t = 0:0; 2:0(0:2)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 8: N = 256;�t = 0:0025; t = 0:0; 1:0(0:1)22
We also compute the same initial curve evolving according to Eqn. (90). We use N = 256mesh points and �t = 0:0025 and show the evolution from t = 0:0 to t = 1:0 in Fig. 8. Herewe see that the ellipse shrinks to a point under Eqn. (90).To test the dependence of �t on the magnitude of curvature and the spatial mesh size, h,we perform a series of resolution studies for three examples. These examples give the sameshapes of curves, but with increasing curvature by a constant factor, 2. In the �rst example,the initial curve is given by X1 = (�4 sin(2��); 2 cos(2��)). It evolves according to Eqn.(90) and Eqn. (92). In the following table the largest possible time steps that give stablediscretizations are shown. No. of points U = � U = �� < � >128 0.02 0.025256 0.01 0.0125512 0.005 0.00625Using Eqn. (90), we only calculate until t = 4:0 at which time the curve essentially becomesa point.In the second example, we scale the initial curve of the �rst example by a factor of 2, i.e.X2 = (�2 sin(2��); cos(2��)). We evolve it by the same equations, Eqn. (90) and Eqn. (92).The largest possible time steps that give stable discretizations are given below.No. of points U = � U = �� < � >128 0.005 0.0075256 0.0025 0.005512 0.00125 0.0025Using Eqn. (90), we only calculate until t = 1:0 before it is essentially a point.In the third example, we scale the initial curve of the �rst example by a factor of 4, i.e.X3 = (� sin(2��); 0:5 cos(2��)), and evolve it by the same equations. Again we list belowthe largest possible time steps that give stable discretizations.No. of points U = � U = �� < � >128 0.002 0.002256 0.001 0.001512 0.0005 0.0005Using Eqn. (90), it will essentially be a point after t = 0:25.Basically the curvature ofX3 is two times the curvature ofX2 and four times the curvatureof X1. From Eqns. (31) and (32), the stability constraint is of the formmax� j �T j�t < CLh; (93)under the motion by � or �� < � >. Here �T = R �0 (U�)d�0 � � R 10 (U�)d�0. Since �T isproportional to �2, the time step constraint for X3 is approximately four times smaller thanthat for X2. Similarly the time step constraint of X2 is approximately four times smallerthan that for X1. This is exactly what we observed from the numerical calculations.The above calculations all demonstrate that our numerical method is free of severe timestep constraint. The time step is proportional to the space grid size in all these calculations.In fact, the particle grid spacing is decreasing in almost all the cases since the curve shrinks toa point. Without using our implicit discretization, the method would have become unstablevery early on. 23
7.3 Inertial Vortex SheetsNext, we apply our reformulated implicit scheme to the inertial vortex sheet problem withsurface tension. This problem has been well studied by Hou, Lowengrub and Shelley in [7]using the ��L formulation. Signi�cant improvement on stability constraint was observed overconventional explicit discretizations, e.g. the fourth order Runge-Kutta method. It is naturalfor us to compare the performance of these two reformulated methods. Our numerical exper-iments indicate that these two formulations give the same stability constraint. This is alsoexplained analytically in this subsection. This is an important and encouraging comparison,because our reformulation can be applied to 3-D problems.In order to compare our methods with the � � L frame presented by Hou, Lowengruband Shelley in [7] , we examine the long-time evolution of inertial vortex sheets with surfacetension. We use the same initial condition as in [7]:x(�; 0) = �+ 0:01 sin 2��; y(�; 0) = �0:01 sin 2��; (�; 0) = 1:0 ; (94)and choose S = 0:005 as in their calculation. In Fig. 9, a time sequence of interface positionsis given, starting from the initial condition. Also we plot the vortex sheet strength and thecurvature � at various times in Figs. 10 and 11 respectively. The calculation uses N = 1024and �t = 1:25�10�4. We also compare directly our numerical solutions with those obtainedby the � � L frame presented in [7]. We �nd that the � � L frame and the �� L frame giveus essentially the same numerical result. Also we have checked the stability constraint usingthese two formulations. We �nd that using the same number of points, the largest possibletime steps that give stable discretizations are of the same order for the two methods. This canalso be explained analytically. Using the ��L frame (assume that � 2 [0; 2�]), the equationsof motion are given by Lt = �Z 2�0 ��0Ud�0 (95)�t = �2�L � (U� + ��T ) (96) t = 2�L S��� + @�((T �W � s) =s�); (97)where T is given by T (�; t) = Z �0 ��0Ud�0 � �2� Z 2�0 ��0Ud�0: (98)By using an implicit discretization like the one we discussed before, we will get a stabilityconstraint of the form max� jT j�t < C � L2�h: (99)Using the ��L frame, the equations of motion are given by Eqns. (31), (32) and (21), so thestability constraint is of the formmax� jT1j�t < C � L2�h; (100)24
0 0.5 1
−0.2
0
0.2
T=0
(a)
0 0.5 1
−0.2
0
0.2
T=0.60
(b)
0 0.5 1
−0.2
0
0.2
T=0.80
(c)
0 0.5 1
−0.2
0
0.2
T=1.20
(d)
0 0.5 1
−0.2
0
0.2
T=1.40
(e)
0.4 0.45 0.5 0.550
0.05
0.1
T=1.40
(f)
Figure 9: Inertial vortex sheets; sequence of interface positions: S = 0:005; N = 1024;�t =1:25 � 10�4 : (a)t = 0; (b) : t = 0:60; (c) : t = 0:80; (d) : t = 1:20; (e) : t = 1:40; (f) close-up oftop pinching region, t = 1:40:
25
0 0.5 1−4
−2
0
2
T=0.0
(a)
0 0.5 1−4
−2
0
2
T=0.60
(b)
0 0.5 1−4
−2
0
2
T=0.80
(c)
0 0.5 1−4
−2
0
2
T=1.20
(e)
0 0.5 1−4
−2
0
2
T=1.00
(d)
0 0.5 1−4
−2
0
2
T=1.40
(f)
Figure 10: Inertial vortex sheets; sequence of : S = 0:005; N = 1024;�t = 1:25 � 10�4 :(a)t = 0; (b) : t = 0:60; (c) : t = 0:80; (d) : t = 1:00; (e) : t = 1:20; (f) : t = 1:40:
0 0.5 1−200
0
200
T=1.40
(f)
0 0.5 1−200
0
200
T=1.20
(e)
0 0.5 1−200
0
200
T=0.0
(a)
0 0.5 1−200
0
200
T=0.60
(b)
0 0.5 1−200
0
200
T=0.80
(c)
0 0.5 1−200
0
200
T=1.00
(e)
Figure 11: Inertial vortex sheets; sequence of �: S = 0:005; N = 1024;�t = 1:25 � 10�4 :(a)t = 0; (b) : t = 0:60; (c) : t = 0:80; (d) : t = 1:00; (e) : t = 1:20; (f) : t = 1:40:26
where T1 is given by T1(�; t) = L2� Z �0 �Ud�0 � �L(2�)2 Z 2�0 �Ud�0: (101)Using the relation between � and �, � = ��=s�, for here s� = L=(2�), it is easy to see tothat T = T1. This shows that the ��L frame and ��L frame have the same order stabilityconstraints.The reason that we use this � � L frame is that we could use the similar idea to thecomputation of 3-D curves and surfaces. The comparison of the results by using the � � Lframe and � � L frame shows that our �� L frame shares the same stability property as the� � L frame, and yet has the advantage of being applicable to 3D �laments and surfaces.7.4 Motion by Curvature in 3-DWe now turn our attention to 3-D �laments. First we test our method for the simple mo-tion by curvature in 3-D. We basically con�rm the similar performance we observed for thecorresponding 2-D problem. We perform a careful comparison with an explicit fourth orderRunge-Kutta discretization. For N = 512, the maximum allowable time step for our methodis 3200 times larger than that for the Runge-Kutta method. We also test the reformulationusing the Fr�enet frame. We found that the computation breaks down at a relative early timedue to the formation of a vanishing curvation point. This corresponds to a blowup in thetorsion variable. This is an arti�cial parametrization singularity. The �lament is very smoothat this time. Using the generalized curvature �1 and �2, we can compute well beyond thistime without any di�culty.Consider the 3-D curveX = (sin(2�); cos(�); sin(�) + 2 cos(2�)); � 2 (0; 2�);evolving according to motion by curvature, Xt = �N. Using our �1��2�!�L formulation,with N = 256 mesh points, and time step �t = 0:0005, we show in Fig. 12 the continuedevolution of the curve from t = 0:0 to t = 1:4. We observe that this space curve relaxes to acircle and eventually shrinks to a point.
−10
12
34
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−1.5
−1
−0.5
0
0.5−5
−4
−3
−2
−1
0
1
xy
z
(a)
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12
34
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−1
0
1−2
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−1
−0.5
0
0.5
1
xy
z
(b)
We compare our method with a straightforward explicit discretization of Xt = �N in(x; y; z) coordinates. This involved using a spectral method for the spatial derivatives andfourth order Runge-Kutta method in time. We list below the maximum time step that can betaken to get a stable solution using these two methods. Due to the particle clustering, we can27
−10
12
34
−2
−1
0
1−2
−1.5
−1
−0.5
0
0.5
1
xy
z
(c)
Figure 12: N = 256;�t = 0:0005; a)t = 0:0; 0:6(0:1); b)t = 0:7; 1:0(0:1); c)t = 1:4only compute up to t = 1:0 using the explicit method. Clearly we can see the huge advantageof using our implicit discretization.No. of points Explicit Method �1-�2-!-L Method128 0.000125 0.0750256 0.00003125 0.0375512 0.00000625 0.0200The motion of a 3-D �lament by curvature is somewhat di�erent from that of the 2-Dcounterpart. In the 2-D case, it is possible to interpret the geometrical signi�cance of positiveor negative curvature. However, for 3-D curves, the curvature is de�ned by� =pXss �Xss = jXssj: (102)The positive square root is taken in Eqn. (102) and thus the curvature is always nonnegative,� � 0. When it passes through points where � = 0, the normal vectorN varies discontinuously.Moreover, at points where � = 0, the torsion is not well de�ned. Recall that the torsion isde�ned by � = ��2(Xs �Xss �Xsss): (103)It is obvious that the torsion is only de�ned when the curvature does not vanish.We have tried the same example using ����L formulation (41), (42) and (43). Numericaldi�culties developed around T = 1:015 when the curvature became close to zero at some pointon the curve. In fact, we were only able to calculate up to T = 1:015 using 256 points, nomatter how small a time step we took due to the stability constraints we derived from Eqn.(44). On the other hand, we had no di�culty computing past T = 1:015 when using the�1 � �2 � ! � L formulation . In Fig. 13, we compare the plots of curvature at T = 1:015using the �� � �L formulation and �1��2�!�L formulation by taking the time step to bedt = 0:00125 and dt = 0:01 respectively. Here we have used the relationship � =p�21 + �22 .We note the similarity in the two plots of the curvature. We also note the jump in thederivative of the curvature as the curvature approaches zero. This means that �� is notcontinuous and the Eqns. (42) and (43) break down.We also plot the variables �1 and �2 at T = 1:02 in Fig. 14. Note that these variablesremain smooth along the entire curve. Thus we see the advantage and indeed, necessity ofusing the �1 � �2 � ! � L formulation instead of the �� � � L formulation.28
0 2 4 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N=256,dt=0.00125 at T=1.015
Curva
ture
By using curvature and torsion
0 2 4 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N=256,dt=0.01 at T=1.015
By using k1,k2 and w
Figure 13: Comparison of curvature using �-� -L and �1-�2-!-L formulation
0 2 4 6−2.5
−2
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−1
−0.5
0
0.5
1
K2
0 2 4 6−2
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1
K1
Figure 14: �1 and �2 at T = 1.02 with N = 256 and dt = 0.0129
7.5 Motion of the Kirchho� Rod ModelNext we test our numerical methods on the motion of the elastic rods. Two interestingequilibrium states are reached using two di�erent initial perturbation of a circular initial�lament. As before, no sti�ness is observed using our reformulated implicit schemes. Asequence of snapshots of the dynamics approaching to equilibrium for two examples (radiusr = 1) are shown in Fig. 15 and Fig. 16. In both examples, we choose as initial conditionsa circular conformation with total twist T! = 5. Here T!(X) = 12� H !(X(s))ds. In the�rst example, we choose the initial twist to be distributed uniformly with a small localizedperturbation. In particular, we choose !(s; 0) = (5 + !1)=(5 + 12� H !1ds), where!1 = 8>><>>:0 jx� �j � �2 ;14 cosh2( x��2x�� ) �2 < x � �;14 cosh2( ��x3��2x ) � < x < 3�2 : (104)In the second example, we use the same parameters and a similar initial condition as the�rst one, except that the initial twist includes an order one non-localized perturbation fromuniformity. More precisely, we choose !(s; 0) = 2�L�1T! � (1 + 0:8 � sin(2�s=L)). In both ofthese examples, we use 256 grid points in our calculations, and a time step dt = 0:00125. Forthe �rst example, the solutions are plotted at T = 0; 1:6; 2:1; 2:6; 4; 12 respectively. For thesecond example, the solutions are plotted at T = 0; 1:2; 2:4; 2:8; 4; 6; 12 respectively.In these two examples, the rods start twisting around T = 1:6 and T = 1:2 respectively.Because of the contact force, the rods cannot self-cross, thus it would keep twisting until itapproaches to the equilibrium con�gurations. We have also investigated using di�erent valuesfor the parameters �1; �2 in Eqns. (65) and (66). We �nd that there is little change in theequilibrium states in both examples, but the rate at which the rods evolve to these states isa�ected.We should mention the construction of the initial condition for these two examples. Inour methods, it is necessary to specify initial values of �1; �2 and !. The twist of the circle !is already given, so we need to determine �1 and �2 from the curvature � and the torsion � .Since �21 + �22 = �2, we parameterize �1; �2 by � and � as follows:�1 = � cos ��2 = � sin�: (105)Note that the torsion � is zero everywhere for an unit circle. Substituting the above equationsinto Eqn. (47), we get �s = !:Thus we are able to calculate �. The curvature of the unit circle is 1. Thus �1 and �2 arecompletely determined by Eqn. (105).Finally, it is necessary to include some sort of contact force g(s) to prevent the elastic rodfrom self-crossing. In a similar way to [10] we have setg(s) = �Z M(r(s); r(�)) r(s)� r(�)(jr(s) � r(�)j)10 d�: (106)The purpose of the molli�erM is three-fold. First, some distinction must be made betweennearest neighbor points along the curve and other points that are far away in arclength but30
Figure 15: Approach to equilibrium \clover" con�guration. T = 0; 1:6; 2:1; 2:6; 4; 12.31
Figure 16: Approach to equilibrium \plectonemic" con�guration. T = 0; 1:2; 2:4; 2:8; 4; 6; 1232
are close in space. Clearly, for those points which are nearest neighbors along the curve, nocontact force is necessary and therefore M is set to be zero. However, if two points whichare separated by a large distance in arc-length become close to another in space, M must benon-zero to activate the contact force. Therefore, M helps prevent self-crossing while ensuringthat points along the curve are not forced apart.Secondly, the magnitude of the contact force needs to be controlled to prevent overly largeforces from destabilizing our solution. The contact force has the form of a sti� inverse powerlaw ( / r�10) so some care must be taken in choosing a constant of proportionality. This isthe other role that M plays when the contact force is in e�ect. We assume the radius of therod is approximately 3 times the grid spacing, i.e. hs�, thus M needs to be chosen so thatthe distance between any two points which are not close in arc-length cannot be smaller than6hs�. We do not have an explicit expression for M here. In our �rst example, we simply takeM to be 0.005 if the distance is less than 12hs� and 0.1 if the distance is less than 8hs�. Inour calculation s� = L=2� is very close to 1. In our second example, we take M to be 0.004if the distance is less than 14hs� and 0.04 if the distance is less than 8hs�.Third, by setting M = 0 when r(s) and r(�) are distant we reduce the computational costin evaluating (106) from what would be O(n2) to O(n). This step is absolutely necessary inorder to prevent the evalution of g(s) from dominating the entire computation.By way of comparison, reference [10] used a similar model to calculate the evolutionof an elastic rod. The method there was to directly discretize Eqns. (65) { (67) usingsecond-order centered di�erences. Here we have the considerable advantage that no high ordertime step stability constraints are imposed. This advantage is crucial if accurate, long-timecomputations (such as DNA modelling) are to be attempted.It is interesting that both of these examples start from unit circles with the same totaltwist. The only di�erence is the distribution of the initial twist. But they approach to totallydi�erent equilibrium states. The clover-like structures are also observed in Langevin dynamicssimulations [15] and the plectonemic conformation is similar to DNA studies.7.6 Motion of Anti{parallel Pair of Vortex FilamentsFinally, we are going to test our method on the motion of anti{parallel vortex �laments. Weconsider large amplitude antisymmetric helical initial perturbations of anti{parallel pair ([4]& [13]): X1 = (�0:5 + 0:3 cos�; 0:3 sin�; �) (107)X2 = ( 0:5 + 0:3 cos�; 0:3 sin�; �) � 2 (0; 2�): (108)The circulation strengths �1;�2 in Eqn. (81) are taken to be 1 and �1 respectively. Weapply the fourth order implicit-explicit scheme in our numerical experiments and �nd thatthe time step is indeed linearly dependent on the spacial mesh size as we expected. However,the fourth order scheme for this particular problem requires a small time step for stabilityconstraint. Instead, we use the second order implicit-explicit scheme in our computation. Thesecond order implicit-explicit scheme (see subsection 6.1) simply uses the leap frog scheme forthe lower order term and the implicit Crank-Nicolson scheme for the leading order term:12�t(un+1 � un�1) = f(un) + �2 [g(un+1) + g(un�1)]: (109)Snapshots of the evolving �laments at times t = 0; 0:73 and 0:79 are given in Figs. 18 - 20where 1024 mesh points and time step �t = 0:00125 are used. The initial separation distance33
between the two �laments is constant, and as time evolves, the minimum separation distancedecreases until the pair collapses around t = 0:79. In Fig. 17, we also show the curvature �and the twist ! of the second �lament X2 at time t = 0:79. Using our method, we are alsoable to include the other non-local e�ects that are neglected in the simpli�ed equations (81).
0 2 4 60
5
10
15
T = 0.79
curvature
1.8 2 2.2 2.40
2
4
6
8
10
12
14
closer look at T = 0.79
0 2 4 60.6
0.8
1
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1.4
1.6
T = 0.79
twist
1.5 2 2.5 30.8
1
1.2
1.4
1.6
closer look at T = 0.79Figure 17: Curvature and twist of the second �lament at t = 0:798 SummaryA new formulation and new methods are presented for computing the motion of a curvaturedriven 3-D �lament. These numerical methods have no high order time step stability con-straints. Our methods are applied to compute the motion of 2-D vortex sheets with surfacetension, motion of 3-D �lament by curvature, the Kirchho� rod model and anti{parallel vor-tex �laments. Our numerical results demonstrate convincingly that our method removes thesevere time step stability constraint associated with explicit discretizations for both 2-D and3-D curves. It shares a similar stability property and computational e�ciency as the � � Lformulation derived by Hou-Lowengrub-Shelley in [7] for 2-D interfaces. There are many in-teresting physical and biological applications of motion of 3-D curvature driven �laments. Ourmethod provides an e�ective numerical technique for studying these problems. This techniquecan also be extended to compute 3-D free surfaces. This will be the topic of a future paper.34
y-axis
z-axis
x-axis
x−axis
y−axis
y−axis
z−axis
z−axis
x−axis
Figure 18: Snapshot of �laments for antisymmetric perturbation at time t = 035
y-axis
z-axis
x-axis
x−axis
y−axis
y−axis
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x−axis
Figure 19: Snapshot of �laments for antisymmetric perturbation at time t = 0:7336
y-axis
z-axis
x-axis
x−axis
y−axis
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x−axis
Figure 20: Snapshot of �laments for antisymmetric perturbation at time t = 0:7937
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