Difference between revisions of "Parametric domains"
From Medusa: Coordinate Free Mehless Method implementation
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Go back to [[Medusa#Examples|Examples]]. | Go back to [[Medusa#Examples|Examples]]. | ||
| − | == Variable node density and | + | == Variable node density and dirchlet boundary conditions in 2D == |
| + | With medusa, we can also solve partial differential equations on parametric domains. Consider the solution of a simple 2D Poisson equation with Dirichlet boundary conditions: | ||
| + | <math> | ||
| + | \begin{align*} | ||
| + | \Delta u &= 0.5 &&\text{in } \Omega, \\ | ||
| + | u &= 0 &&\text{on } \partial \Omega, | ||
| + | \end{align*} | ||
| + | </math> | ||
| + | where $u(x,y)$ is the solution to the problem. Let's define $\Omega$ to be the interior of the parametrically given curve $f(t)$: | ||
| − | + | <math> | |
| + | \begin{align*} | ||
| + | r(t) &=& |\cos(1.5 t)| ^ {\sin(3t)} | ||
| + | f(t) &=& (r \cos(t), r \sin(t)) | ||
| + | \end{align*} | ||
| + | </math> | ||
See [[Positioning of computational nodes]] TODO. | See [[Positioning of computational nodes]] TODO. | ||
Revision as of 16:24, 8 August 2019
Go back to Examples.
Variable node density and dirchlet boundary conditions in 2D
With medusa, we can also solve partial differential equations on parametric domains. Consider the solution of a simple 2D Poisson equation with Dirichlet boundary conditions\[ \begin{align*} \Delta u &= 0.5 &&\text{in } \Omega, \\ u &= 0 &&\text{on } \partial \Omega, \end{align*} \] where $u(x,y)$ is the solution to the problem. Let's define $\Omega$ to be the interior of the parametrically given curve $f(t)$\[ \begin{align*} r(t) &=& |\cos(1.5 t)| ^ {\sin(3t)} f(t) &=& (r \cos(t), r \sin(t)) \end{align*} \]
See Positioning of computational nodes TODO.
