Numerical analysis of moving boundary problems using the boundary tracking method

Kimura, Mamoru

doi:10.1007/bf03167390

Cited by 47 publications

(41 citation statements)

References 24 publications

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“…The asymptotically uniform tangential redistribution keeps very good curve resolution even in cases of several subsequent curve selfintersections. On the other hand, if we consider for example the well-known redistribution preserving the relative local length [18,21,25], obtained by taking ω = 0 in (2.11) and (3.1) then this method is not capable to handle this situations properly. If we consider δ = 0.01 the backward diffusion effects are strongly dominating.…”

Section: Discussion On Numerical Experimentsmentioning

confidence: 99%

“…As a consequence, it can significantly stabilize numerical computations. The reader is referred to papers [18,21,24,25,26,27,28,8,2,3,5,35,34] for detailed discussion on how a suitable tangential stabilization can prevent a numerical solution from forming various undesired singularities. We will specify our choice of a tangential velocity α later.…”

Section: Governing Equationsmentioning

confidence: 99%

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A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

Mikula¹,

Ševčovič

Balažovjech³

2010

CICP

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Abstract.A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

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Section: Discussion On Numerical Experimentsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

Mikula¹,

Ševčovič

Balažovjech³

2010

CICP

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“…Let us consider a ow t ; t¿0, of plane curves with the normal velocity given by (17). Local in time existence of a smooth solution follows from Theorem 3.1 provided that the initial curve 0 ⊂ is smooth.…”

Section: Lyapunov Functionals and Qualitative Properties Of Solutionsmentioning

confidence: 98%

“…Notice that the presence of a tangential velocity functional has no impact on the shape of evolving curves and therefore a 'natural' setting = 0 has been chosen for analytical as well as numerical treatment in the literature [11][12][13][14][15]. An important role of a non-trivial tangential term has been discovered and utilized in References [16][17][18][19][20][21]. In Reference [21], Equation (1) has been solved with ÿ = ÿ(k; ) non-linearly depending on the curvature k. The tangential term appearing in (2) has been chosen in such a way that an initial distribution of grid points representing a curve is preserved during evolution.…”

Section: Introductionmentioning

confidence: 98%

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

Mikula

Ševčovič

2004

Math Methods in App Sciences

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SUMMARYA new method for solution of the evolution of plane curves satisfying the geometric equation v=ÿ(x; k; ), where v is the normal velocity, k and are the curvature and tangential angle of a plane curve ⊂ R 2 at the point x ∈ , is proposed. We derive a governing system of partial di erential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non-trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to ÿnd a numerical solution for 2D anisotropic interface motions and image segmentation problems.

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“…By solving the so-called intrinsic heat equation one can directly find a position vector of a curve (see, e.g., [17,18,19,33,39,40]). There are also other direct methods based on solution of a porous medium-like equation for curvature of a curve [31,32], a crystalline curvature approximation [22,23,44], special finite difference schemes [28,29], and a method based on erosion of polygons in the affine invariant scale case [34]. By contrast to the direct approach, level set methods are based on introducing an auxiliary function whose zero level sets represent an evolving family of planar curves undergoing the geometric equation (1.1) (see, e.g., [36,41,42,43,26]).…”

mentioning

confidence: 99%

Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy

ŠEVČOVIČ¹,

MIKULA²

2000

Equadiff 99

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Abstract. In this paper we study evolution of plane curves satisfying a geometric equation v = β (k, ν), where v is the normal velocity and k and ν are the curvature and tangential angle of a plane curve Γ. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation. The intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional α. We show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities. From an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when β(k, ν) = γ(ν)k m , 0 < m ≤ 2, and the governing system of equations includes a nontrivial tangential velocity functional.

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Numerical analysis of moving boundary problems using the boundary tracking method

Cited by 47 publications

References 24 publications

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy

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