Level-set method: Difference between revisions

Revision as of 01:02, 15 December 2023

Revision as of 18:40, 15 December 2023

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

The figure on the right illustrates several important ideas about the level-set method. 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 ''xy'' 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, the shape can be seen changing its topology by splitting in two. It would be difficult to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. One would need an algorithm able to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves. On the bottom row, however, the level set function accomplishes this change by translating downward. This is an example of when it can be easier to work with a shape through its level-set function than with the shape directly, where the method would need to consider and handle all the possible deformations the shape might undergo.

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Here, <math>|\cdot|</math> is the [[Euclidean norm]] (denoted customarily by single bars in PDEs), 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, numerical solution of the level- set equation, however,may requiresrequire sophisticatedcomplex techniques. Simple finite- difference methods fail quickly. [[Upwind scheme|Upwinding]] methods, such as the [[Godunov's scheme|Godunovmethod method]],are fareconsidered 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 sophisticatedcomplex 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 normal 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 distance to the boundary, positive interior, negative exterior) on the initial circle, the normalized 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]].

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

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{{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.<ref>{{Cite web |title= |url=https://www.airloads.net/Downloads/Textbooks/Computational-Fluid-Dynamics-the-Basics-With-Applications-Anderson-J-D.pdf}}</ref>

==See also==