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* [[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 (CFD)==
{{unreferenced section|date=August 2022}}
To run a [[Math model|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 - abbreviated as CompactLSM.
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 generation|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.<ref>{{cite book |title=Computational Fluid Dynamics |author1=John David Anderson |edition=illustrated |publisher=McGraw-Hill Education |year=1995 |isbn=978-0-07-001685-9 |url=https://books.google.com/books?id=dJceAQAAIAAJ}}</ref>
==See also==
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