CNF:Mesh Sizing for CNF data - Revision history

Ctsolakis: /* Sampling-based method */

2019-09-24T22:37:56Z

Sampling-based method

← Older revision		Revision as of 17:37, 24 September 2019
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	Is there a more generic way tha avoid all these complications ?, How about gradient limiting (of the sizing) <ref> http://persson.berkeley.edu/pub/persson05sizefunc.pdf </ref>? '''		Is there a more generic way tha avoid all these complications ?, How about gradient limiting (of the sizing) <ref> http://persson.berkeley.edu/pub/persson05sizefunc.pdf </ref>? '''
	See also Mesh Gradation control <ref> https://www.ljll.math.upmc.fr/frey/publications/ijnme4398.pdf </ref>'''		See also Mesh Gradation control <ref> https://www.ljll.math.upmc.fr/frey/publications/ijnme4398.pdf </ref>'''

			== Notes ==
			* For higher dimensions the stencil can be extended easily:
			** Evaluate BG at the "centroid" of each lower dimensional entity. For example edge midpoints in 2D, edge midpoints and face centroids in 3D, etc.
			** OR Evaluate always at edge midpoints
			** Or evaluate centroid and midpoints of (vertex,centoid) edge (maybe the best, because it creates may points inside the element)
			* For the randomized sampling, a [https://en.wikipedia.org/wiki/Dirichlet_distribution Dirichlet distribution] can be used with alpha = (1,1,1......)

	= References =		= References =
	https://userweb.jlab.org/~gavalian/articles/0203033.pdf		https://userweb.jlab.org/~gavalian/articles/0203033.pdf

Ctsolakis: /* Sampling-based */

2019-09-24T15:54:30Z

Sampling-based

← Older revision		Revision as of 10:54, 24 September 2019
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	== Sampling-based ==		== Sampling-based ==
	<gallery widths=450px heights=300px>		<gallery widths=450px heights=300px>
	Image:NT X min 1 limit 2e-3.png\|minimum size = 1, limit = 2e-3		Image:NT X min 1 limit 2e-3.png\|minimum size = 1, limit = 2e-3, # elements = 8617
	Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3		Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3, # elements = 14451
	Image:NT X min 1 limit 5e-3.png\|minimum size = 1, limit = 5e-3		Image:NT X min 1 limit 5e-3.png\|minimum size = 1, limit = 5e-3, # elements = 2159
	File:NT X min 1 limit 5e-3 qual off.png \| minimum size = 1, limit = 5e-3 qual off		File:NT X min 1 limit 5e-3 qual off.png \| minimum size = 1, limit = 5e-3 qual off , # elements = 2102
	</gallery>		</gallery>

	[[File:Gaussian2 min 10 limit 1e-1.png\|400px\|min size = 10, limit = 1e-1]]		[[File:Gaussian2 min 10 limit 1e-1.png\|400px\|min size = 10, limit = 1e-1]]

Ctsolakis: /* Sampling-based */

2019-09-24T15:46:11Z

Sampling-based

← Older revision		Revision as of 10:46, 24 September 2019
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	Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3		Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3
	Image:NT X min 1 limit 5e-3.png\|minimum size = 1, limit = 5e-3		Image:NT X min 1 limit 5e-3.png\|minimum size = 1, limit = 5e-3
			File:NT X min 1 limit 5e-3 qual off.png \| minimum size = 1, limit = 5e-3 qual off
	</gallery>		</gallery>

Ctsolakis: /* Method */

2019-09-24T15:39:52Z

Method

Ctsolakis: /* 3D */

2019-09-24T15:38:36Z

← Older revision		Revision as of 10:38, 24 September 2019
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	== 3D ==		== 3D ==
	~~CNF Data looks like it follows exponential distribution,~~		Is there a more generic way tha avoid all these complications ?, How about gradient limiting (of the sizing) <ref> http://persson.berkeley.edu/pub/persson05sizefunc.pdf </ref>? '''
			See also Mesh Gradation control <ref> https://www.ljll.math.upmc.fr/frey/publications/ijnme4398.pdf </ref>'''
	~~'''How do we map exponential distribution properly to element size ? '''~~

	~~'''~~ Is there a more generic way tha avoid all these complications ?, How about gradient limiting (of the sizing) <ref> http://persson.berkeley.edu/pub/persson05sizefunc.pdf </ref>? '''
	~~'''~~ See also Mesh Gradation control <ref> https://www.ljll.math.upmc.fr/frey/publications/ijnme4398.pdf </ref>'''

	= References =		= References =
	https://userweb.jlab.org/~gavalian/articles/0203033.pdf		https://userweb.jlab.org/~gavalian/articles/0203033.pdf

Ctsolakis: /* Sampling -based method */

2019-09-24T15:37:43Z

Sampling -based method

Ctsolakis: /* Sampling-based */

2019-09-24T15:34:10Z

Sampling-based

@@ Line 51: / Line 51: @@
 Image:Paraboloid_2d_mesh_min_10_max_1000_linear.png| min size =10, tranformation linear
 </gallery>
 == Sampling-based ==

Ctsolakis: /* Sampling-based */

2019-09-24T15:09:08Z

Sampling-based

← Older revision		Revision as of 10:09, 24 September 2019
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	</gallery>		</gallery>

	[[File:Gaussian2 min 10 limit 1e-1.png\|~~200px~~\|min size = 10, limit = 1e-1]]		[[File:Gaussian2 min 10 limit 1e-1.png\|400px\|min size = 10, limit = 1e-1]]

	== 3D ==		== 3D ==

Ctsolakis: /* Sampling-based */

2019-09-24T15:08:55Z

Sampling-based

← Older revision		Revision as of 10:08, 24 September 2019
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	</gallery>		</gallery>

	[[File:Gaussian2 min 10 limit 1e-1.png\|200px~~\|frame~~\|min size = 10, limit = 1e-1]]		[[File:Gaussian2 min 10 limit 1e-1.png\|200px\|min size = 10, limit = 1e-1]]

	== 3D ==		== 3D ==

Ctsolakis: /* Sampling-based */

2019-09-24T15:08:45Z

Sampling-based

← Older revision		Revision as of 10:08, 24 September 2019
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	== Sampling-based ==		== Sampling-based ==
	<gallery widths=~~500px~~ heights=300px>		<gallery widths=450px heights=300px>
	Image:NT X min 1 limit 2e-3.png\|minimum size = 1, limit = 2e-3		Image:NT X min 1 limit 2e-3.png\|minimum size = 1, limit = 2e-3
	Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3		Image:NT X min 0.5 limit 2e-3.png\|minimum size = 0.5, limit = 2e-3
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	</gallery>		</gallery>

	~~<gallery widths=500px heights=500px>~~		[[File:Gaussian2 min 10 limit 1e-1.png\|200px\|frame\|min size = 10, limit = 1e-1]]
	~~Image~~:~~NT X~~ min 1 limit 2e-3.png\|~~minimum size = 1, limit = 2e-3~~
	~~Image:NT X~~ min ~~0.5 limit 2e-3.png\|minimum~~ size = ~~0.5~~, limit = 2e-3
	~~Image:NT X min~~ 1 ~~limit 5e-3.png\|minimum size = 1, limit = 5e-3~~
	~~</gallery>~~

	== 3D ==		== 3D ==

← Older revision		Revision as of 10:39, 24 September 2019
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	After a set of values, V, have been sampled (with either approach), the quantity max(V) -min(V) is evaluated and if the difference		After a set of values, V, have been sampled (with either approach), the quantity max(V) -min(V) is evaluated and if the difference
	exceeds a user-specified limit, the element is split. <ref> Traditional Delaunay methods split elements at		exceeds a user-specified limit, the element is split. <ref> Traditional Delaunay methods split elements at
	~~circumenter~~ to guarantee certain element quality limits. However, since in this case quality is not a concern,		circumcenter to guarantee certain element quality limits. However, since in this case quality is not a concern,
	a more appropriate approach that "minimizes the error" can be used. A simple approach is to pick a point between		a more appropriate approach that "minimizes the error" can be used. A simple approach is to pick a point between
	the points that min and max value. (see also paper from Gagik) </ref>		the points that min and max value. (see also paper from Gagik) </ref>

			In summary,
	~~Since a "dense" solution is available each time~~

	The constrains build in the SF of the example is below are:		The constrains build in the SF of the example is below are:
	* min,max size (area) of the element (user defined).		* min,max size (area) of the element (user-defined).
	** Elements below ''min'' are skipped		** Elements below ''min'' are skipped
	** Elememts above ''max'' are split		** Elememts above ''max'' are split
	* if max(V)-min(V)> user_limit (where V is the sample of values from the BG image), split element		* if max(V)-min(V)> user_limit (where V is the sample of values from the BG image), split element

	== Sampling-based ==		== Sampling-based ==

← Older revision		Revision as of 10:37, 24 September 2019
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	</gallery>		</gallery>

	= Sampling -based method =		= Sampling-based method =
	== Introduction ==		== Introduction ==
	As an simpler alternative one can also use directly the input data/image values to guide the refinement of the mesh.		As a simpler alternative one can also use directly the input data/image values to guide the refinement of the mesh.
	This approach is also more close to the specific domain since, in contrast to the CFD cases the starting point is		This approach is also more close to the specific domain since, in contrast to the CFD cases the starting point is
	a finer "solution" instead of a coarser.		a finer "solution" instead of a coarser.

	== Method ==		== Method ==
	For this approach the starting mesh is a triangulation of the bounding box of the image.		For this approach, the starting mesh is a triangulation of the bounding box of the image.
	The input image is then used as a background mesh while refining the mesh.		The input image is then used as a background mesh while refining the mesh.
	In particular, ~~the~~ the refinement algorithm will query a sizing function (SF) in order to		In particular, the refinement algorithm will query a sizing function (SF) in order to
	verify whether each element satisfies the user constrains.		verify whether each element satisfies the user constrains.

	Each time SF is called upon an element it will check if its size meets the '''global''' user constrains		Each time SF is called upon an element it will check if its size meets the '''global''' user constrains
	and then it will create a sample of values from the background mesh by picking points that fall ~~with in~~ the		and then it will create a sample of values from the background mesh by picking points that fall within the
	element.		element.
	The ~~avaliable~~ picking strategies include a deterministic stencil and a random sample.		The available picking strategies include a deterministic stencil and a random sample.
	For the stencil approach SF is evaluated on element vertices, barycenter and edge midpoints.		For the stencil approach, SF is evaluated on element vertices, barycenter, and edge midpoints.
	In the randomized strategy, SF is evaluated on a uniform random sample of points inside the triangle.		In the randomized strategy, SF is evaluated on a uniform random sample of points inside the triangle.
	The size of the random sample is user-defined.		The size of the random sample is user-defined.

	After a set of values, V, have been sampled (with either approach), the ~~quanity~~ max(V) -min(V) is evaluated		After a set of values, V, have been sampled (with either approach), the quantity max(V) -min(V) is evaluated and if the difference
	and if the difference exceeds a user specified limit, the element is split. <ref> Traditional Delaunay methods		exceeds a user-specified limit, the element is split. <ref> Traditional Delaunay methods split elements at
	split elements at circumenter to ~~quarantedd~~ certain element quality limits. However, since in this case quality is not a concern,		circumenter to guarantee certain element quality limits. However, since in this case quality is not a concern,
	a more ~~sofisticated~~ approach that "minimizes the error" can be used. A simple approach is to pick a point between		a more appropriate approach that "minimizes the error" can be used. A simple approach is to pick a point between
	the min and max value </ref>		the points that min and max value. (see also paper from Gagik) </ref>