Level-set method: Difference between revisions

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{{Short description|Conceptual framework used in numerical analysis of surfaces and shapes}}

{{Tone|date=December 2023}}

[[File:Levelset-mean-curvature-spiral.ogv|thumb|Video of spiral being propagated by level sets ([[curvature flow]]) in 2D. Left image shows zero-level solution. Right image shows the level-set scalar field.]]

The '''Level-set methodsmethod''' ('''LSM''') areis a conceptual framework for using [[level set]]s as a tool for [[numerical analysis]] of [[Surface (topology)|surface]]s and [[shape]]s. The advantage of the level-set model is that oneLSM can perform [[Numerical computation|numerical computations]] involving [[curve]]s and surfaces on a fixed [[Cartesian grid]] without having to [[Parametric surface|parameterize]] these objects (this is called the ''Eulerian approach'').<ref>{{Citation |last1 = Osher |first1 = S. |last2 = Sethian |first2 = J. A.| title = Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations| journal = J. Comput. Phys.| volume = 79 |issue = 1 |year = 1988 |pages = 12&ndash;49 |url = http://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf |doi=10.1016/0021-9991(88)90002-2|bibcode = 1988JCoPh..79...12O |hdl = 10338.dmlcz/144762 |citeseerx = 10.1.1.46.1266|s2cid = 205007680 }}</ref> Also,LSM themakes level-setit methodeasier makesto itperform verycomputations easyon toshapes followwith sharp corners and [[Shape|shapes]] that change [[topology]], for(such example,as whenby a shape splitssplitting in two, develops holes, or thedeveloping reverse of these operationsholes). AllThese thesecharacteristics make theLSM level-seteffective methodfor a[[modeling]] greatobjects toolthat forvary modelingin time-varying objects, likesuch inflation ofas an [[airbag]], inflating or a drop of oil floating in water.

[[Image:level set method.png|thumb|right|400px|An illustration of the level-set method]]

== Overview ==

The figure on the right illustrates several important ideas about the level-set methodLSM. In the upper- left corner we see a shape; that is, a [[bounded region]] with a well-behaved boundary. Below it, the red surface is the graph of a level set function <math>\varphi</math> determining this shape, and the flat blue region represents the ''xyX-Y'' plane. The boundary of the shape is then the zero-level set of <math>\varphi</math>, while the shape itself is the set of points in the plane for which <math>\varphi</math> is positive (interior of the shape) or zero (at the boundary).

In the top row we see, the shape's changingtopology itschanges topologyas byit splittingis split in two. It would be quiteis hardchallenging to describe this transformation numerically by [[Parametrization (geometry)|parameterizing]] the boundary of the shape and following its evolution. OneAn wouldalgorithm needcan an algorithmbe ableused to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves. On the otherbottom handrow, ifhowever, wethe lookplane at the bottom row, we see thatwhich the level set function merelyis sampled is translated downward.upwards, Thison iswhich anthe exampleshape's ofchange whenin ittopology canis bedescribed. It is muchless easierchallenging to work with a shape through its level-set function rather than with the shapeitself directly, wherein usingwhich thea shape directlymethod would need to consider and handle all the possible deformations the shape might undergo.

Thus, in two dimensions, the level-set method amounts to representing a [[closed curve]] <math>\Gamma</math> (such as the shape boundary in our example) using an [[auxiliary function]] <math>\varphi</math>, called the level-set function. The curve <math>\Gamma</math> is represented as the zero-level set of <math>\varphi</math> by

:<math>\Gamma = \{(x, y) \mid \varphi(x, y) = 0 \},</math>

and the level-set method manipulates <math>\Gamma</math> ''implicitly'', through the function <math>\varphi</math>. This function <math>\varphi</math> is assumed to take positive values inside the region delimited by the curve <math>\Gamma</math> and negative values outside.<ref name="osher" /><ref name="sethian" />

==The level-set equation==

If the curve <math>\Gamma</math> moves in the normal direction with a speed <math>v</math>, then by chain rule and implicit differentiation, it can be determined that the level-set function <math>\varphi</math> satisfies the ''level-set equation''

:<math>\frac{\partial\varphi}{\partial t} = v|\nabla \varphi|.</math>

Here, <math>|\cdot|</math> is the [[Euclidean norm]] (denoted customarily by single bars in PDEspartial differential equations), and <math>t</math> is time. This is a [[partial differential equation]], in particular a [[Hamilton–Jacobi equation]], and can be solved numerically, for example, by using [[finite difference]]s on a Cartesian grid.<ref name=osher>{{cite book |last=Osher |first=Stanley J. |authorlink = Stanley Osher |author2=Fedkiw, Ronald P. |authorlink2=Ronald Fedkiw |title=Level Set Methods and Dynamic Implicit Surfaces|publisher=[[Springer-Verlag]] |year=2002 |isbn= 978-0-387-95482-0}}</ref><ref name=sethian>{{cite book |last=Sethian |first=James A. |authorlink = James Sethian |title= Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science|publisher=[[Cambridge University Press]] |year=1999 |isbn= 978-0-521-64557-7}}</ref>

TheHowever, the numerical solution of the level- set equation, however,may requiresrequire sophisticatedadvanced techniques. Simple finite- difference methods fail quickly. [[Upwind scheme|Upwinding]] methods, such as the [[Godunov's scheme|Godunov method]], fareare considered better; however, the level- set method does not guarantee the conservationpreservation of the volume and the shape of the set level set in an advection field that does conserve themaintains shape and size, for example, a uniform or [[rotational velocity]] field. Instead, the shape of the level set may get severelybecome distorted, and the level set may vanishdisappear over severala few time steps. For this reasonTherefore, high-order finite- difference schemes are generally required, such as high-order [[essentially non-oscillatory]] (ENO) schemes, are often required, and even then, the feasibility of long-timeterm simulations is questionable. FurtherMore sophisticatedadvanced methods tohave dealbeen withdeveloped thisto difficultyovercome havethis; beenfor developed, e.g.example, combinations of the level-setleveling method with tracingtracking marker particles advectedsuggested by the velocity field.<ref>{{Citation |last1 = Enright |first1 = D. |last2 = Fedkiw |first2 = R. P.| last3 = Ferziger |first3 = J. H. |authorlink3 = Joel H. Ferziger| last4 = Mitchell |first4 = I.| title = A hybrid particle level set method for improved interface capturing| journal = J. Comput. Phys.| volume = 183 |issue = 1 |year = 2002 |pages = 83&ndash;116| url = http://www.cs.ubc.ca/~mitchell/Papers/myJCP02.pdf |doi=10.1006/jcph.2002.7166|bibcode = 2002JCoPh.183...83E |citeseerx = 10.1.1.15.910}}</ref>

==Example==

Consider a unit circle in <math display="inline">\mathbb{R}^2</math>, shrinking in on itself at a constant rate, i.e. each point on the boundary of the circle moves along its inwards pointing normalnormally at some fixed speed. The circle will shrink and eventually collapse down to a point. If an initial distance field is constructed (i.e. a function whose value is the signed euclidean[[Euclidean distance]] to the boundary, positive interior, negative exterior) on the initial circle, the normalisednormalized gradient of this field will be the circle normal.

If the field has a constant value subtracted from it in time, the zero level (which was the initial boundary) of the new fields will also be circular and will similarly collapse to a point. This is due to this being effectively the temporal integration of the [[Eikonal equation]] with a fixed front [[velocity]].

== Applications ==

In [[combustion]], this method is used to describe the instantaneous flame surface, known as the [[G equation]].

*In mathematical modeling of [[combustion]], this methodLSM is used to describe the instantaneous flame surface, known as the [[G equation]].

A number of *[[level set (data structures)|levelLevel-set data structures]] have been developed to facilitate the use of the level-set method in computer applications.

* [[Computational fluid dynamics]]

* [[Trajectory|Trajectory planning]]

* [[Mathematical optimization|Optimization]]

* [[Image processing]]

* [[Computational biophysics]]

* Discrete [[complex dynamics]]: visulalisation(visualization of the [[b:Fractals/Iterations in the complex plane/Mandelbrot set|parameter plane]] and the [[b:Fractals/Iterations in the complex plane/Julia set|dynamic plane]])

==History==

The level-set method was developed in 1979 by Alain Dervieux,<ref>{{cite book |last1=Dervieux |first1=A. |last2=Thomasset |first2=F. |chapter=A finite element method for the simulation of a Rayleigh-Taylor instability |chapter-url= |title=Approximation Methods for Navier-Stokes Problems |publisher=Springer |series=Lecture Notes in Mathematics |volume=771 |date=1980 |isbn=978-3-540-38550-9 |pages=145–158 |doi=10.1007/BFb0086904}}</ref> and subsequently popularized by [[Stanley Osher]] and [[James Sethian]]. It has since become popular in many disciplines, such as [[image processing]], [[computer graphics]], [[computational geometry]], [[optimization (mathematics)|optimization]], [[computational fluid dynamics]], and [[computational biology]].

A number of [[level set (data structures)|level-set data structures]] have been developed to facilitate the use of the level-set method in computer applications.

==Applications==

* [[Computational fluid dynamics]]

* [[Combustion]]

* Trajectory planning

* [[Mathematical optimization|Optimization]]

* [[Image processing]]

* [[Computational biophysics]]

* Discrete [[complex dynamics]]: visulalisation of [[b:Fractals/Iterations in the complex plane/Mandelbrot set|parameter plane]] and [[b:Fractals/Iterations in the complex plane/Julia set|dynamic plane]]

==Computational fluid dynamics==

{{unreferenced section|date=August 2022}}

To run a Math Model in the interface of two different fluids we need to soften the interactions between the fluids. Therefore we need to apply a specific function: Compact Level Set Method.

As a “spin off”, the CompactLSM is a complement of the LSM, that helps solving LSM equations. It can be used in numerical simulation of flow, for example, if we are working with discretization of the interface water-air, compacts at sixth order, ensures the accurate and fast calculation of the interface equations (Monteiro 2018).

The LSM uses a distance function to locate different fluids. A distance function is that whose value represents the smallest distance from the point where it is being analyzed to the interface. This distance function is identified by isolines (2D) or isosurfaces (3D), showing that  the negative values refer to one of the fluids, positive values refer to the other and the zero value corresponds to the position of the interface.

But, how is the Heaviside function inserted in the ''Compact Level Set Method?''

Since the specific mass and viscosity are discontinuous at the interface, both excess diffusion problem (interface widening) and numerical oscillations are expected if there is no adequate treatment of the fluid near the interface. To minimize these problems, the Level Set method uses a smooth, cell-related Heaviside function that explicitly defines the interface position (<math>\varphi=0</math>).

The transition in the interface is kept smooth, but with a thickness of the order of magnitude of the cell size, to avoid the introduction of disturbances with a length scale equal to that of the mesh, since the interface infers an abrupt jump property from one cell to the next (Unverdi and Tryggvason, 1992). To reconstruct the material properties of the flow, such as specific mass and viscosity, another marker function, <math>I(\varphi)</math>, of the Heaviside type is used:

{{NumBlk|:| <math>I (\varphi) = \begin{cases}

0, & \text{if }\varphi<-\delta\Delta \\[8pt]

\dfrac 1 2 \left[ 1 + \dfrac \varphi {\delta\Delta} + \dfrac 1 \pi \sin\left( \dfrac{\pi\varphi}{\delta\Delta} \right) \right], & \text{if } |\varphi| \leq \delta\Delta \\[8pt]

1, & \text{if }\varphi>\delta\Delta

\end{cases}</math>|{{EquationRef|1}}}}

where <math>\delta</math> is an empirical coefficient, usually equal to 1; 5 and <math>\Delta</math> is the characteristic discretization of the problem, which varies according to the phenomenon to be simulated. The value of <math>\delta</math> represents an interface with a thickness of three cells, and thus <math>\delta\Delta</math> represents half the thickness of the interface. Note that in this method, the interface has a virtual thickness, as it is represented by a smooth function. Physical properties, such as specific mass and kinematic viscosity, are calculated as:

{{NumBlk|:| <math>\rho = (1-I)\rho_1+I\rho_2 \qquad e\qquad v=(1-I)v_1+Iv_2 </math>|{{EquationRef|2}}}}

where <math>\rho_1</math>, <math>\rho_2</math>, <math>v_1</math> and <math>v_2</math> are the specific mass and kinematic viscosity of fluids 1 and 2. Equation {{EquationNote|2}} can be applied analogously to the other properties of the fluids.

==See also==

{{Div col|colwidth=20em}}

* [[Contour boxplot]]

* [[Zebra striping (computer graphics)analysis]]

* [[G equation]]

* [[Advanced Simulation Library]]

* [[Volume of fluid method]]

* [[Image segmentation#Level-set methods]]

* [[Immersed boundary method]]s

* [[Stochastic Eulerian Lagrangian method]]s

* [[Level set (data structures)]]

* [[Posterization]]

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==External links==

* See [[Ronald Fedkiw]]'s [http://graphics.stanford.edu/~fedkiw/ academic web page] for many stunning pictures and animations showing how the level-set method can be used to model real-life phenomena, like fire, water, cloth, fracturing materials, etc.

* [http://vivienmallet.net/fronts/ Multivac] is a C++ library for front tracking in 2D with level-set methods.

* [[James Sethian]]'s [http://math.berkeley.edu/~sethian/ web page] on level-set method.

* [[Stanley Osher]]'s [https://www.math.ucla.edu/~sjo/ homepage].

* [https://math.mit.edu/classes/18.086/2007/levelsetpres.pdf The Level Set Method. MIT 16.920J / 2.097J / 6.339J. Numerical Methods for Partial Differential Equations by Per-Olof Persson. March 8, 2005]

* [https://ocw.mit.edu/courses/18-086-mathematical-methods-for-engineers-ii-spring-2006/resources/lecture-11-level-set-method/ Lecture 11: The Level Set Method: MIT 18.086. Mathematical Methods for Engineers II by Gilbert Strang]

{{Numerical PDE}}

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[[Category:Computational fluid dynamics]]

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[[Category:Implicit surface modeling]]