Customization: Difference between revisions

Revision as of 12:26, 12 July 2024

Medusa library support users defining custom basis types, weights, operators and more, as long as they conform to the prescribed interfaces, given in the Concepts page.

Custom stencil selection

TODO

Custom RBF definition

TODO

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

@@ Line 7: / Line 7: @@
 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