Meshless Local Strong Form Method (MLSM) - Revision history

Jureslak at 16:39, 8 September 2019

2019-09-08T16:39:56Z

← Older revision		Revision as of 16:39, 8 September 2019
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	which enable us to approximate operators as $$(\mathcal{L}u)(p) = \sum_{i=1}^n c_i u_i.$$		which enable us to approximate operators as $$(\mathcal{L}u)(p) = \sum_{i=1}^n c_i u_i.$$

	Refer to [[Weighted Least Squares (WLS)]] and [[Computation of shape functions]] for details about numerical calculation of shape functions.		Refer to [[Weighted Least Squares (WLS)]] and [[Computation of shape functions]] for details about numerical calculation of shape functions / stencil weights $\b c = (c_i)_i$.

	The presented formulation is convenient for implementation since most of the complex operations, i.e. finding support nodes and building shape functions, are performed only when nodal topology changes. To evaluate the desired operator in the main simulation, only a dot product of the pre-computed shape functions with the vector of field values in the support needs to be computed. The presented approach is even easier to handle than in FDM, however, despite its simplicity it offers many possibilities for treating challenging cases, e.g. nodal adaptivity to address regions with sharp discontinuities or $p$-adaptivity to treat obscure anomalies in physical field, furthermore, the stability versus computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled during the simulation by re computing the shape functions with different setups.		The presented formulation is convenient for implementation since most of the complex operations, i.e. finding support nodes and building shape functions, are performed only when nodal topology changes. To evaluate the desired operator in the main simulation, only a dot product of the pre-computed shape functions with the vector of field values in the support needs to be computed. The presented approach is even easier to handle than in FDM, however, despite its simplicity it offers many possibilities for treating challenging cases, e.g. nodal adaptivity to address regions with sharp discontinuities or $p$-adaptivity to treat obscure anomalies in physical field, furthermore, the stability versus computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled during the simulation by re computing the shape functions with different setups.

Jureslak at 16:39, 8 September 2019

2019-09-08T16:39:07Z

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 The elegance of MLSM is its simplicity and generality. The presented methodology can be also easily upgraded or altered, e.g. with nodal adaptation, basis augmentation, conditioning of the approximation, etc., to treat anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations. In MLSM, the type of approximation, the size of support domain, and the type and number of basis function is general.
-For example, minimal support size for 2D transport problem is five, however higher support domains can be used to stabilize computations on scattered nodes at the cost of computational complexity. Various types of basis functions might appear in the approximation, however, the most commonly used are multiquadrics, Gaussians and monomials. Some authors also enrich the radial basis with monomials to improve performance of the method (see [[]]). All these features can be controlled on the fly during the simulation. From the computation point of view, the localization of the method reduces inter-processor communication, which is often a bottleneck of parallel algorithms.
+For example, minimal support size for 2D transport problem is five, however higher support domains can be used to stabilize computations on scattered nodes at the cost of computational complexity. Various types of basis functions might appear in the approximation, however, the most commonly used are multiquadrics, Gaussians and monomials. Some authors also enrich the radial basis with monomials to improve performance of the method (see [[Radial basis function-generated finite differences (RBF-FD)]]). All these features can be controlled during the simulation. From the computation point of view, the localization of the method reduces inter-processor communication, which is often a bottleneck of parallel algorithms.
 <figure id="fig:implementation">
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 </figure>
-computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity  of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$  stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a [[Kd Tree|kD-tree]] data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.
+The core of the spatial discretization is a local [[Weighted Least Squares (WLS)]] approximation of a considered field over the overlapping local support domains,
 == References ==

Jureslak at 16:28, 8 September 2019

2019-09-08T16:28:02Z

← Older revision		Revision as of 16:28, 8 September 2019
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	The Meshless Local Strong Form Method (MLSM) is a generalization of methods which are in literature also known as Diffuse Approximate Method (DAM), Local Radial Basis Function Collocation Methods (LRBFCM), Generalized FDM, Collocated discrete least squares (CDLS) meshless, etc. Although each of the named methods ~~pose~~ some unique properties, the basic concept of all local strong form methods is similar, namely to approximate ~~treated fields with~~		The Meshless Local Strong Form Method (MLSM) is a generalization of methods which are in literature also known as Diffuse Approximate Method (DAM), Local Radial Basis Function Collocation Methods (LRBFCM), Generalized FDM (GFDM), Collocated discrete least squares (CDLS) meshless, etc. Although each of the named methods has some unique properties, the basic concept of all local strong form methods is similar, namely to approximate differential operators as

	<figure id="fig:meshless1">		<figure id="fig:meshless1">
	[[File:image_1avhoje8u10tg1cip1atj1qo51j8b9.png\|500px\|thumb\|upright=2\|alt=The scheme of local meshless principle.\|<caption>The scheme of local meshless principle. </caption>]]		[[File:image_1avhoje8u10tg1cip1atj1qo51j8b9.png\|500px\|thumb\|upright=2\|alt=The scheme of local meshless principle.\|<caption>The scheme of local meshless principle. </caption>]]
	</figure>		</figure>

	~~nodal trial functions~~ over the local support domain. ~~The nodal trial function is then used to evaluate various operators, e.g. derivation, integration, and after all, approximation of a considered field in arbitrary position. The~~ MLSM ~~could easily~~ be understood as a ~~meshless~~ generalization of ~~the~~ FDM, however much more powerful~~. MLSM has an ambition to avoid using pre-defined relations between nodes and shift this task into the solution procedure. The final goal of such an approach is higher flexibility in complex domains~~.		weighted sums of function values over the local support domain. MLSM can be understood as a generalization of FDM to scattered nodes, however much more powerful.

	The elegance of MLSM is its simplicity and generality. The presented methodology can be also easily upgraded or altered, e.g. with nodal adaptation, basis augmentation, conditioning of the approximation, etc., to treat anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations. In ~~the~~ MLSM, the type of approximation, the size of support domain, and the type and number of basis function is general. For example minimal support size for 2D transport problem ~~(system of PDEs of second order)~~ is five, however higher support domains can be used to stabilize computations on scattered nodes at the cost of computational complexity. Various types of basis functions might appear in the ~~calculation of the trial function~~, however, the most commonly used are multiquadrics, Gaussians and monomials. Some authors also enrich the radial basis with monomials to improve performance of the method. All these features can be controlled on the fly during the simulation. From the computation point of view, the localization of the method reduces inter-processor communication, which is often a bottleneck of parallel algorithms.		The elegance of MLSM is its simplicity and generality. The presented methodology can be also easily upgraded or altered, e.g. with nodal adaptation, basis augmentation, conditioning of the approximation, etc., to treat anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations. In MLSM, the type of approximation, the size of support domain, and the type and number of basis function is general.
			For example, minimal support size for 2D transport problem is five, however higher support domains can be used to stabilize computations on scattered nodes at the cost of computational complexity. Various types of basis functions might appear in the approximation, however, the most commonly used are multiquadrics, Gaussians and monomials. Some authors also enrich the radial basis with monomials to improve performance of the method (see [[]]). All these features can be controlled on the fly during the simulation. From the computation point of view, the localization of the method reduces inter-processor communication, which is often a bottleneck of parallel algorithms.

	The core of the spatial discretization is a local [[Moving Least Squares (MLS)]] approximation of a considered field over the overlapping local support domains, i.e. in each node we use approximation over a small local sub-set of neighbouring $n$ nodes. The trial function is thus introduced as		The core of the spatial discretization is a local [[Moving Least Squares (MLS)]] approximation of a considered field over the overlapping local support domains, i.e. in each node we use approximation over a small local sub-set of neighbouring $n$ nodes. The trial function is thus introduced as

Jureslak: /* Shape functions */

2019-07-12T11:20:07Z

Shape functions

← Older revision		Revision as of 11:20, 12 July 2019
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	== Shape functions ==		== Shape functions ==
	Refer to ~~'''~~[[Moving Least Squares (MLS)]]~~'''~~ for details about numerical calculation of shape functions.		Refer to [[Moving Least Squares (MLS)]] and [[Computation of shape functions]] for details about numerical calculation of shape functions.

	By explicit expressing the coefficients $\alpha$ into the trial function		By explicit expressing the coefficients $\alpha$ into the trial function

Gkosec: /* Shape functions */

2017-07-07T05:22:59Z

Shape functions

← Older revision		Revision as of 05:22, 7 July 2017
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	== Shape functions ==		== Shape functions ==
	Refer to [[Moving Least Squares (MLS)]] for details about numerical calculation of shape functions.		Refer to '''[[Moving Least Squares (MLS)]]''' for details about numerical calculation of shape functions.

	By explicit expressing the coefficients $\alpha$ into the trial function		By explicit expressing the coefficients $\alpha$ into the trial function

Gkosec: /* Shape functions */

2017-07-06T11:58:05Z

Shape functions

← Older revision		Revision as of 11:58, 6 July 2017
Line 18:		Line 18:

	== Shape functions ==		== Shape functions ==
	Refer to [[~~Numerical calculation of the shape functions~~]] for details about numerical calculation of shape functions.		Refer to [[Moving Least Squares (MLS)]] for details about numerical calculation of shape functions.

	By explicit expressing the coefficients $\alpha$ into the trial function		By explicit expressing the coefficients $\alpha$ into the trial function

Gkosec: /* Shape functions */

2017-07-06T11:57:19Z

Shape functions

← Older revision		Revision as of 11:57, 6 July 2017
Line 18:		Line 18:

	== Shape functions ==		== Shape functions ==
			Refer to [[Numerical calculation of the shape functions]] for details about numerical calculation of shape functions.

	By explicit expressing the coefficients $\alpha$ into the trial function		By explicit expressing the coefficients $\alpha$ into the trial function
	\[~\hat{u}\left( \mathbf{p} \right)=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{u}=\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\]		\[~\hat{u}\left( \mathbf{p} \right)=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{u}=\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\]

Gkosec: /* Shape functions */

2017-07-06T11:56:09Z

Shape functions

← Older revision		Revision as of 11:56, 6 July 2017
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	computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$ stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a [[Kd Tree\|kD-tree]] data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.		computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$ stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a [[Kd Tree\|kD-tree]] data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.



	== References ==		== References ==

Gkosec at 12:48, 22 February 2017

2017-02-22T12:48:42Z

← Older revision		Revision as of 12:48, 22 February 2017
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	== Shape functions ==		== Shape functions ==
	By explicit ~~expressiong~~ the coefficients $\alpha$ into the trial function		By explicit expressing the coefficients $\alpha$ into the trial function
	\[~\hat{u}\left( \mathbf{p} \right)=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{u}=\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\]		\[~\hat{u}\left( \mathbf{p} \right)=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{u}=\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\]
	is obtained, where $\mathbf{\chi}$ stand for the shape functions. Now, we can apply partial differential operator, which is our goal, on the trial function		is obtained, where $\mathbf{\chi}$ stand for the shape functions. Now, we can apply partial differential operator, which is our goal, on the trial function

Anja at 15:56, 24 November 2016

2016-11-24T15:56:40Z

← Older revision		Revision as of 15:56, 24 November 2016
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	</figure>		</figure>

	computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$ stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a kD-tree ~~[20~~] data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.		computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$ stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a [[Kd Tree\|kD-tree]] data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.