Qualitative and numerical aspects of a motion of a family of interacting curves in space

by   Michal Beneš, et al.
Comenius University in Bratislava

In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both local or nonlocal character where the entire curve may influence evolution of other curves. Such an evolution and interaction can be found in applications. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. Using the finite volume method, we construct an efficient numerical scheme solving the governing system of nonlinear parabolic equations. Additionally, a nontrivial tangential velocity is considered allowing for redistribution of discretization nodes. We also present several computational studies of the flow combining the normal and binormal velocity and considering nonlocal interactions.



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1. Introduction

In this article we investigate motion of a family of interacting curves evolving in three dimensional Euclidean space (3D) according to the geometric evolution law:


where the unit tangent , normal and binormal vectors form the Frenet frame. We explore the direct Lagrangian approach to treat the geometric motion law (1). The evolving curves are parametrized as where is a smooth mapping. Hereafter, denotes the periodic interval isomorphic to the unit circle with . We assume the scalar velocities to be smooth functions of the position vector , the curvature , the torsion , and of all parametrized curves , i.e.

Motion (1) of one-dimensional structures forming space curves can be identified in variety of problems arising in science and engineering. Among them, one of the oldest is the dynamics of vortex structures formed along a one-dimensional curve, frequently a closed one, forming a vortex ring. The investigation of these structures dates back to Helmholtz [26]. Since then, the importance of vortex structures for both understanding nature and improving aerospace technology is reflected in many publications, from which Thomson [64], Da Rios [15], Betchov [9], Arms and Hama [6] or Bewley [11] are a sample only. Vortex structures can be relatively stable in time and may contribute to weather behavior, e.g. tornados, or accompany volcanic activity (c.f. Fukumoto et al. [20, 21], Hoz and Vega [29], Vega [65]). Particular vortex linear structures can interact each with other and exhibit interesting dynamics, e.g. known as frog leaps (c.f. Mariani and Kontis [43]). A comprehensive review of research of vortex rings can be found in Meleshko et al. [44].

One-dimensional structures can also be formed within the crystalline lattice of solid materials. As described, e.g. by Mura [52], some defects of the crystalline lattice (voids or interstitial atoms) can be organized along planar curves in glide planes. These structures are called the dislocations and are responsible for macroscopic material properties explored in the everyday engineering practice (see Hirth and Lothe [27] or Kubin [40]). The dislocations can move along the glide planes, be influenced by the external stress field in the material as well as by the force field of other dislocations. Such interaction can lead to the change of the glide plane (cross-slip) where the motion becomes three-dimensional (see Devincre et al. [16] or Pauš et al. [53] or Kolář et al. [39]).

Certain class of nano-materials is produced by electrospinning - jetting polymer solutions in high electric fields into ultrafine nanofibers (see Reneker [56], Yarin et al. [69], He et al. [25]). These structures move freely in space according to (1) before they are collected to form the material with desired features. The motion of nano-fibers as open curves in 3D is a combination of curvature and elastic response to the external electric forces (see Xu et al. [68]). As the nano-fibers are produced from a solution, they are subject of a drying process during electrospinning and may be considered as 3D objects with internal mass transfer, in detailed models (see [66]).

Some linear molecular structures with specific properties exist inside cells and exhibit specific dynamics in terms of (1) in space, which is rather a result of chemical reactions. They can interact with other structures as described in Fierling et al. in [19] where the deformations and twist of fluid membranes by adhering stiff amphiphilic filaments have been studied, or in Shlomovitz et al. [61], Shlomovitz et al. [62], Roux et al. [58], Kang et al. [33] or in Glagolev et al. [24].

The motion of curves in space or along manifolds has also been explored, e.g. in optimization of the truss construction and architectonic design (see Remešíková et al. [55]), in the virtual colonoscopy [48], in the numerical modeling of the wildland-fire propagation (see Ambrož et al. [3]), or in the satellite-image segmentation (in Mikula et al. [47]).

Theoretical analysis of the motion of space curves is contained, among first, in papers by Altschuler and Grayson in [1] and [2]. The motion of space curves became useful tool in studying the singularities of the two-dimensional curve dynamics. Nonlocal curvature driven flows, especially in case of planar curves, have been studied e.g. by Gage and Epstein [22], [18]. Nonlocal curvature flows were treated by the Cahn-Hilliard theory in [59] and in [12]. Conserved planar curvature flow has been further investigated by Beneš, Kolář, and Ševčovič in [36, 37, 38]. Recently, Beneš, Kolář, and Ševčovič analyzed the flow of planar curves with mutual interactions in [10].

Recent theoretical results in the analysis of vortex filaments are provided by Jerrard and Seis [32]. The dynamics of curves driven by curvature in the binormal direction is discussed by Jerrard and Smets in [31]. Particular issues were numerically studied by Ishiwata and Kumazaki in [30].

Curvature driven flow in a higher dimensional Euclidean space and comparison to the motion of hypersurfaces with the constrained normal velocity have been studied by Barrett et al. [7, 8], Elliott and Fritz [17], Minarčík, Kimura and Beneš in [50]. Gradient-flow approach is explored by Laux and Yip [41]. Long-term behavior of the length shortening flow of curves in has been analyzed by Minarčík and Beneš in [51].

More specifically, we focus on the analysis of the motion of a family of curves evolving in 3D and satisfying the law


where , and are bounded and smooth functions of their arguments, is the unit tangent vector to the curve and is the unit arc-length parametrization of the curve (see Section 2). The source forcing term is assumed to be a smooth and bounded function. It may depend on the position and tangent vectors of the -th curve and integrals over other interacting curves as follows:


and , are given smooth functions. Since and (see Section 2) the relationship between geometric equations (1) and (2) reads as follows:


The system of equations (2) is subject to initial conditions


representing parametrization of the family of initial curves .

As an example of nonlocal source terms we can consider a flow of interacting curves evolving in 3D according to the geometric equations:


where the nonlocal source term has the form:


It represents the Biot-Savart law measuring the integrated influence of points belonging to the second curve at a given point belonging to the first interacting curve . In this example and . Such a flow is analyzed in a more detail in Subsection 6.2. In the case of a special configuration of the initial curves the dynamics can be reduced to a solution to a system on nonlinear ODEs. On the other hand, if and there are no explicit or semi-explicit solutions, in general. Therefore a stable numerical discretization scheme has to be developed. The scheme involving a nontrivial tangential velocity is derived and presented in Subsection 6.1. For such a configuration of normal and binormal components of the velocity we establish local existence, uniqueness and continuation of classical Hölder smooth solutions in Section 4. Here, we generalize methodology and technique of proofs of local existence, uniqueness and continuation provided in [10] to the case of combined motion of closed space curves in normal and binormal direction with mutual nonlocal interactions. The novelty and main contribution of this part is the result on existence and uniqueness of classical solutions for a system on evolving curves in with mutual nonlocal interactions including, in particular, the vortex dynamics evolved in the normal and binormal directions and external force of the Biot-Savart type, or evolution of interacting dislocation loops.

To avoid singularities in (7) arising in intersections of and one can regularize the expression for as follows


where is a small regularization parameter.

In general, the flow of interacting curves involving the Biot-Savart law is governed by the system of evolutionary equations:


The paper is organized as follows. In the next Section, we recall principles of the direct Lagrangian approach for solving normal and binormal curvature driven flows of a family of interacting plane curves in 3D. In Section 2 we derive a system of nonlocal evolution partial differential equations for parametrizations of a family of evolving curves. Section 3 is focused on the role of a tangential velocity. We will show that a suitable choice of tangential velocity leads to construction of an efficient and stable numerical scheme for solving the governing system of nonlinear parabolic equations in Section 5. Secondly, it helps to simplify the proof of local existence of classical solutions (see Section 4). Local existence, uniqueness, and continuation of classical Hölder smooth solutions is shown in Section 4. The method of the proof is based on the abstract theory of analytic semi-flows in Banach spaces due to Angenent

[5, 4]. A numerical discretization scheme is derived in Section 5. We apply the flowing finite volume method for discretization of spatial derivatives and the method of lines for solving the resulting system of ODEs. Finally, examples of evolution of interacting curves are presented in Section 6. Interactions are modeled by means of the Biot-Savart nonlocal law. We show examples of interacting curves following the motion with binormal velocity only as well as evolution of arbitrary curves evolving in both normal and binormal directions.

2. Dynamic governing equations for geometric quantities

Assume the family of evolving curves is parametrized as follows: where is a smooth mapping. For brevity we drop the superscript and we let wherever it is not necessary. Then the unit arc-length parametrization is given by . The unit tangent vector is given by . In the case when the curvature is strictly positive, we can define the so-called Frenet frame. It means that the unit normal and binormal vectors and can be uniquely defined as follows: , . These unit vectors satisfy the following identities:

and the Frenet-Serret formulae:

where is the torsion of a curve. For the torsion is given by

Indeed, as , we obtain

Concerning the dynamical governing equations we have the following proposition. Some of these identities have been already discovered as a particular case by other authors (see e.g. [51], [50]). Our aim is to provide evolution equations general settings of normal , binormal , and tangent velocities . Although our approach is based on the analysis and numerical solution of the position vector equation (2), we provide the dynamic equations for the curvature and torsion in the following proposition.

Proposition 1.

Assume a family of curves is evolving in 3D according to the geometric law:

Then the unit vectors forming the Frenet frame satisfy the following system of evolution partial differential equations:

The local length element and the commutator satisfy

The curvature and torsion (for ) satisfy the evolution equations:


Denote . Then . Using Frenet-Serret formulae we have

Since we have

and, as a consequence, , and because . Next

Since we have

and, as a consequence,

Finally, as and and we have

In the case when the curvature is strictly positive, the evolution equation for the torsion can be deduced from the fact , i.e.

As a consequence of the previous proposition we obtain the following results concerning temporal evolution of global quantities integrated over the evolving curves:

Proposition 2.

Assume a family of curves evolving in 3D according to the geometric law:

Then, the length and the generalized area enclosed by satisfy the following identities:

In particular, if the family of curves evolves in parallel planes then is the area enclosed by , and .


The first statement follows from the identity . Indeed,

because is a closed curve. Therefore .

As for the second statement, we have , and so

In particular, if the family of 3D curves evolves in parallel planes with the normal vector then the binormal vector is a constant vector perpendicular to this plane. As a consequence, , and the proof of the last statement of the proposition follows from the fact that the enclosed area of a curve belonging to the plane is given by , and for any rotation matrix transforming the vector to the vector . ∎

3. The role of tangential redistribution

The tangential velocity appearing in the geometric evolution (2) has no impact on the shape of evolving family of curves . It means that the curves evolving according to the system of geometric equations:


do not depend on a particular choice of the total tangential velocity given by

However, the tangential velocity has a significant impact on the analysis of evolution of curves from both the analytical as well as numerical points of view. It was shown by Hou et al. [28], Kimura [35], Mikula and Ševčovič [45, 46, 49], Yazaki and Ševčovič [60]. Barrett et al. [7, 8], Elliott and Fritz [17], investigated the gradient and elastic flows for closed and open curves in . They constructed a numerical approximation scheme using a suitable tangential redistribution. Kessler et al. [34] and Strain [63] illustrated the role of suitably chosen tangential velocity in numerical simulation of the two-dimensional snowflake growth and unstable solidification models. Later, Garcke et al. [23] applied the uniform tangential redistribution in the theoretical proof of nonlinear stability of stationary solutions for curvature driven flow with triple junction in the plane.

A suitable choice of can be very useful in order to prove local existence of solution. Furthermore, it can significantly help to construct a stable an efficient numerical scheme preventing from undesirable accumulation of grid points during curve evolution. Calculating the derivative ratio with respect to time we obtain


where . As a consequence, the relative local length is constant with respect to the time , i.e.

provided that the total tangential velocity satisfies:


(c.f. Hou and Lowengrub [28], Kimura [35], Mikula and Ševčovič [45]). Since the additional tangential velocity given by


, ensures that the relative local length is constant with respect to time, and

where . The tangential velocity is subject to the normalization constraint .

Another suitable choice of the total tangential velocity is the so-called asymptotically uniform tangential velocity proposed and analyzed by Mikula and Ševčovič in [46, 49]. If


then, using (11) we obtain

uniformly with respect to provided . It means that the redistribution becomes asymptotically uniform. In the context of evolution of 3D curves or the curves evolving on a given surface, the concept uniform and asymptotically uniform redistribution has been analyzed and successfully implemented for various applications by Mikula and Ševčovič in [46, 55], Mikula et al. [47], Beneš et al. [54], Ambrož et al. [3], and others.

Remark 1.

Suppose that the initial curve is uniformly parametrized, i.e. . If is a tangential velocity preserving the relative local length then

4. Existence and uniqueness of classical solutions

In this section we provide existence and uniqueness results for the system of nonlinear nonlocal equations (10) governing the motion of interacting closed curves in 3D. The method of the proof of existence and uniqueness is based on the abstract theory of analytic semi-flows in Banach spaces due to DaPrato and Grisvard [13], Angenent [5, 4], Lunardi [42]. Local existence and uniqueness of a classical Hölder smooth solution is based on analysis of the position vector equation (10) in which we choose the uniform tangential velocity . It leads to a uniformly parabolic equation (10) provided the diffusion coefficients are uniformly bounded from below by a positive constant. As a consequence, assumptions on strict positivity of the curvature and the existence of the Frenet frame are not required, in our method of the proof. The main idea is to rewrite the system (10) in the form of an initial value problem for the abstract parabolic equation:


in a suitable Banach space. Furthermore, we have to show that, for any , the linearization generates an analytic semigroup and it belongs to the so-called maximal regularity class of linear operators mapping the Banach space into Banach space .

Note that the principal part of the velocity vector can be expressed in the matrix form as follows:

where is a matrix,

Clearly, the symmetric part is a positive definite matrix for . If then is an indefinite and antisymmetric matrix, i.e., . For given values and a unit vector

, the eigenvalues of the matrix

are: . It means that the governing equation:


is of the parabolic type provided whereas it is of the hyperbolic type if and . In the case of interacting curves the system of governing equations reads as follows:


where for .

4.1. Maximal regularity for parabolic equations with complex valued diffusion functions

Assume and is a nonnegative integer. Let us denote by the so-called little Hölder space, i.e. the Banach space which is the closure of smooth functions in the norm Banach space of smooth functions defined on the periodic domain , and such that the -th derivative is -Hölder smooth. The norm is being given as a sum of the norm and the Hölder semi-norm of the -th derivative.

Among many important properties of Hölder spaces

there is an interpolation inequality. Let

be such that . Then, for any there exists such that


for any , where .

In what follows, we shall assume that the functions , and is strictly positive. Let us define the following linear second order differential operators :


The spectra , consist of discrete real eigenvalues. Furthermore, the linear operators generate the group of linear operators . It means that the function is a solution to the Schrödinger equation

Recall that the spectrum consists of real eigenvalues. Hence the linear operator is bounded in the space uniformly with respect to . Since is an interpolation space between and there exists a constant depending on the function only and such that


Moreover, in the respective norms of linear operators, .

Next, we shall prove the maximal regularity of solutions to the linear evolutionary equation:


That is to show the existence of a unique solution for the given right-hand side and initial condition and Here we have denoted by the following Banach spaces:


Consider the transformed function . Then is a solution to (21) if and only if is a solution to the equation: