Customization - Revision history

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T13:31:24Z

Custom operators: Hyperviscosity operator

← Older revision		Revision as of 13:31, 12 July 2024
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	== Custom operators: Hyperviscosity operator ==		== Custom operators: Hyperviscosity operator ==
	Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,		Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are <code>apply</code> and <code>applyAt0</code>, the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation to the operator applied to function $u$ is,
	$$		$$
	\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
	$$		$$
	where $\Psi_i (u)$ are cardinal functions. The evaluated cardinal functions are called weights. The weights are obtained using a system of RBF coupled with monomial augmentation. Therefore, our operator must have a definition of the function ''apply'' for monomial and RBF basis.		where $\Psi_i (u)$ are cardinal functions and $\alpha$ is the order of hyperviscosity. The evaluated cardinal functions are called weights. The weights are obtained by using a system of RBF coupled with monomial augmentation. Therefore, our operator must have a definition of the function <code>apply</code> or <code> applyAt0 </code> for monomial and RBF basis. We define the struct, as

			<syntaxhighlight lang="cpp">
			template <int dim, int order>
			struct Hyperviscosity : public mm::Operator<Hyperviscosity <dim, order>>
			</syntaxhighlight>

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T13:25:33Z

Custom operators: Hyperviscosity operator

← Older revision		Revision as of 13:25, 12 July 2024
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	== Custom operators: Hyperviscosity operator ==		== Custom operators: Hyperviscosity operator ==

	Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,		Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,
	$$		$$
	\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
	$$		$$
	where $\Psi_i (u)$ are cardinal functions		where $\Psi_i (u)$ are cardinal functions. The evaluated cardinal functions are called weights. The weights are obtained using a system of RBF coupled with monomial augmentation. Therefore, our operator must have a definition of the function ''apply'' for monomial and RBF basis.

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:27:59Z

Custom operators: Hyperviscosity operator

← Older revision		Revision as of 12:27, 12 July 2024
Line 11:		Line 11:
	Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,		Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,
	$$		$$
	\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) ~~+ \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i~~,		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
	$$		$$
	where $\Psi_i (u)$ are cardinal functions ~~and $\beta_i(u_i)$ are the monomial weights~~		where $\Psi_i (u)$ are cardinal functions

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:27:06Z

Custom operators: Hyperviscosity operator

← Older revision		Revision as of 12:27, 12 July 2024
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	\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i,		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i,
	$$		$$
	where $\Psi_i (u)$ are cardinal functions and $\~~beta~~$ are the monomial weights		where $\Psi_i (u)$ are cardinal functions and $\beta_i(u_i)$ are the monomial weights

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:26:56Z

Custom operators: Hyperviscosity operator

← Older revision		Revision as of 12:26, 12 July 2024
Line 11:		Line 11:
	Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,		Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,
	$$		$$
	\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i,
	$$		$$
			where $\Psi_i (u)$ are cardinal functions and $\beta$ are the monomial weights

Zigavaupotic: /* Custom operators: Biharmonic equation */

2024-07-12T12:26:23Z

Custom operators: Biharmonic equation

← Older revision		Revision as of 12:26, 12 July 2024
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	TODO		TODO

	== Custom operators: ~~Biharmonic equation~~ ==		== Custom operators: Hyperviscosity operator ==
	~~Biharmonic operator $\nabla^4$ is not included in Medusa by default, but it is easy to define.~~

	~~We solve the problem~~		Medusa library also allows custom linear operators $ \mathcal{L} $ to be used. This functionality is achieved by adding a child struct that inherits the struct Operator. The virtual functions of struct Operator that need to be defined are ''apply'' and ''applyAt0'', the functions require one to also specify the basis to which the operator is applied. For example if one was to define the hyperviscosity operator denoted as $\nabla^{2\alpha}$, we would say that the RBF-FD approximation is,
			$$
	$		\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i
	\~~begin~~{~~align~~}		$$
	\nabla^~~4 u &= f &&~~\~~text{in~~ } ~~\Omega~~, \\
	u ~~&= g_d &&~~\~~text~~{on } \~~partial~~ \~~Omega,~~\\
	~~\frac~~{~~\partial u~~}{\~~partial~~ \~~vec n} &= g_n &&~~\~~text{on~~ } \~~partial \Omega,~~
	~~\end~~{~~align~~}
	$
	~~where $u = TODO~~$ ~~and $f$, $g_d$ and $g_n$ are computed from $u$.~~

	~~TODO: JureMB~~

Jureslak at 15:37, 10 June 2020

2020-06-10T15:37:24Z

← Older revision		Revision as of 15:37, 10 June 2020
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	Medusa library support users defining custom basis types, weights, operators and more, as long as they conform to the prescribed interfaces, given in the [http://e6.ijs.si/medusa/docs/html/concepts.html Concepts page].		Medusa library support users defining custom basis types, weights, operators and more, as long as they conform to the prescribed interfaces, given in the [http://e6.ijs.si/medusa/docs/html/concepts.html Concepts page].
	~~Here we show this on an example of the biharmonic equation.~~
			== Custom stencil selection ==
			TODO

			== Custom RBF definition ==
			TODO

	== Custom operators: Biharmonic equation ==		== Custom operators: Biharmonic equation ==

Jureslak at 14:09, 4 August 2019

2019-08-04T14:09:57Z

← Older revision		Revision as of 14:09, 4 August 2019
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	== Custom operators: Biharmonic equation ==		== Custom operators: Biharmonic equation ==
			Biharmonic operator $\nabla^4$ is not included in Medusa by default, but it is easy to define.

	We solve the problem		We solve the problem

Jureslak at 14:08, 4 August 2019

2019-08-04T14:08:38Z

← Older revision		Revision as of 14:08, 4 August 2019
Line 2:		Line 2:
	Here we show this on an example of the biharmonic equation.		Here we show this on an example of the biharmonic equation.

	== Biharmonic equation ==		== Custom operators: Biharmonic equation ==
	We solve the problem		We solve the problem

Jureslak: /* Biharmonic equation */

2019-07-19T14:04:19Z

Biharmonic equation

← Older revision		Revision as of 14:04, 19 July 2019
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	\end{align}		\end{align}
	$		$
			where $u = TODO$ and $f$, $g_d$ and $g_n$ are computed from $u$.

	TODO: JureMB		TODO: JureMB