Customization - Revision history

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T13:31:24Z

‎Custom operators: Hyperviscosity operator

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T13:25:33Z

‎Custom operators: Hyperviscosity operator

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:27:59Z

‎Custom operators: Hyperviscosity operator

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:27:06Z

‎Custom operators: Hyperviscosity operator

Zigavaupotic: /* Custom operators: Hyperviscosity operator */

2024-07-12T12:26:56Z

‎Custom operators: Hyperviscosity operator

Zigavaupotic: /* Custom operators: Biharmonic equation */

2024-07-12T12:26:23Z

‎Custom operators: Biharmonic equation

Jureslak at 15:37, 10 June 2020

2020-06-10T15:37:24Z

Jureslak at 14:09, 4 August 2019

2019-08-04T14:09:57Z

Jureslak at 14:08, 4 August 2019

2019-08-04T14:08:38Z

Jureslak: /* Biharmonic equation */

2019-07-19T14:04:19Z

‎Biharmonic equation

@@ Line 8: / Line 8: @@
 == 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,
 $$
 \nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
 $$
-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

@@ Line 8: / Line 8: @@
 == 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,
 $$
 \nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
 $$
-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.

@@ 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,
 $$
-\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i,
+\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u),
 $$
-where $\Psi_i (u)$ are cardinal functions and $\beta_i(u_i)$ are the monomial weights
+where $\Psi_i (u)$ are cardinal functions

@@ Line 13: / Line 13: @@
 \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

@@ 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,
 $$
-\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i
+\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i,
 $$

@@ Line 7: / Line 7: @@
 TODO
-== Custom operators: Biharmonic equation ==
+== Custom operators: Hyperviscosity operator ==
-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,
+$$
+\nabla^{2\alpha} u \approx \sum_{i=0}^n u_i \nabla^{2\alpha} \Psi_i (u) + \sum_{j=0}^\ell \nabla^{2\alpha}\beta_{i} u_i
-\begin{align}
+$$

@@ Line 1: / Line 1: @@
 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 operators: Biharmonic equation ==

← Older revision		Revision as of 14:09, 4 August 2019
Line 3:		Line 3:

	== 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

@@ Line 12: / Line 12: @@
 \end{align}
+$
 TODO: JureMB