Parametric domains - Revision history

Uduh at 11:26, 1 September 2020

2020-09-01T11:26:14Z

Uduh at 11:51, 31 August 2020

2020-08-31T11:51:08Z

Uduh at 10:55, 9 August 2019

2019-08-09T10:55:19Z

Uduh at 10:03, 9 August 2019

2019-08-09T10:03:30Z

Uduh at 10:00, 9 August 2019

2019-08-09T10:00:37Z

Uduh at 18:01, 8 August 2019

2019-08-08T18:01:52Z

Uduh at 18:01, 8 August 2019

2019-08-08T18:01:16Z

Jureslak at 17:17, 8 August 2019

2019-08-08T17:17:38Z

Jureslak at 17:17, 8 August 2019

2019-08-08T17:17:06Z

Jureslak: /* Variable node density and dirchlet boundary conditions in 2D */

2019-08-08T17:15:45Z

‎Variable node density and dirchlet boundary conditions in 2D

← Older revision		Revision as of 11:26, 1 September 2020
Line 104:		Line 104:

	The whole example can be found as [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.cpp parametric_domain_2D.cpp] along with the plot script that can be run by Matlab or Octave [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.m parametric_domain_2D.m].		The whole example can be found as [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.cpp parametric_domain_2D.cpp] along with the plot script that can be run by Matlab or Octave [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.m parametric_domain_2D.m].
−
−	~~See [[Positioning of computational nodes]] (TODO) for the theory behind our node placing algorithm.~~

@@ Line 103: / Line 103: @@
 [[File:Poisson_2D.png|800px]]
-The whole example can be found as [https://gitlab.com/e62Lab/medusa parametric_domain_2D.cpp] along with the plot script that can be run by Matlab or Octave [https://gitlab.com/e62Lab/medusa parametric_domain_2D.m].
+The whole example can be found as [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.cpp parametric_domain_2D.cpp] along with the plot script that can be run by Matlab or Octave [https://gitlab.com/e62Lab/medusa/-/blob/dev/examples/poisson_equation/parametric_domain_2D.m parametric_domain_2D.m].
 See [[Positioning of computational nodes]] (TODO) for the theory behind our node placing algorithm.

@@ Line 6: / Line 6: @@
 == Variable node density and Dirichlet 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>
@@ Line 24: / Line 23: @@
 Using tools such as Wolfram Mathematica, we can find the Jacobian matrix ($\textbf{J}$) of $r$. Now, we can translate the definition of $r$ into code:
 <syntaxhighlight lang="c++" line>
-auto example_r = [](Vec1d t) {
+auto example_r = [](Vec<double, 1> t) {
-    t(0) = -t(0);
+     double r = pow(abs(cos(1.5 * t(0))), sin(3 * t(0)));
-     double f = pow(abs(cos(1.5 * t(0))), sin(3 * t(0)));
+     return Vec2d(r * cos(t(0)), r * sin(t(0)));
 };
-auto der_example_r = [](Vec1d t) {
+auto der_example_r = [](Vec<double, 1> t) {
-    t(0) = -t(0);
+     double r = pow(abs(cos(1.5 * t(0))), sin(3 * t(0)));
-     double f = pow(abs(cos(1.5 * t(0))), sin(3 * t(0)));
+     double der_r = (-1.5 * pow(abs(cos(1.5 * t(0))),
-     double der_f = (-1.5 * pow(abs(cos(1.5 * t(0))),
+             sin(3 * t(0))) * sin(3 * t(0)) * sin(1.5 * t(0)) +
-             sin(3 * t(0))) * sin(3 * t(0)) * sin(1.5 * t(0))  +  3 * pow(abs(cos(1.5 * t(0))),
+* pow(abs(cos(1.5 * t(0))),
-                    sin(3 * t(0))) * cos(3 * t(0)) * cos(1.5 * t(0)) *
+                            sin(3 * t(0))) * cos(3 * t(0)) * cos(1.5 * t(0))
-                             log(abs(cos(1.5 * t(0))))) / cos(1.5 * t(0));
+                             * log(abs(cos(1.5 * t(0))))) / cos(1.5 * t(0));
      Eigen::Matrix<double, 2, 1> jm;
-     jm.col(0) << -(der_f * cos(t(0)) - f * sin(t(0))), -(der_f * sin(t(0)) + f * cos(t(0)));
+     jm.col(0) << der_r * cos(t(0)) - r * sin(t(0)), der_r * sin(t(0)) + r * cos(t(0));
      return jm;
@@ Line 45: / Line 43: @@
 // Define parametric curve's domain.
-BoxShape<Vec1d> bs(Vec<double, 1>{- 2 * PI}, Vec<double, 1>{0.0});
+BoxShape<Vec1d> param_bs(Vec<double, 1>{0.0}, Vec<double, 1>{2 * PI});
 </syntaxhighlight>
-We implemented $r$ and $\textbf{J}$ as lambda expressions and $r$'s domain as an instance of <code>DomainDiscretization</code> class. Now we can fill the target domain with nodes given by $r$ and the density function $h$ by using <code>GeneralSurfaceFill</code>. For the sake of this example, we choose a gradient-like density function <code>gradient_h</code>.
+We implemented $r$ and $\textbf{J}$ as lambda expressions and $r$'s domain as an instance of <code>BoxShape</code> class. Now we can fill the target domain with nodes given by $r$ and the density function $h$ by using <code>GeneralSurfaceFill</code>. For the sake of this example, we choose a gradient-like density function <code>gradient_h</code>.
 <syntaxhighlight lang="c++" line>
 UnknownShape<Vec2d> shape;
@@ Line 62: / Line 59: @@
 GeneralSurfaceFill<Vec2d, Vec1d> gsf;
-domain.fill(gsf, param_d, example_r, der_example_r, gradient_h);
+domain.fill(gsf, param_bs, example_r, der_example_r, gradient_h);
 </syntaxhighlight>

@@ Line 14: / Line 14: @@
 \end{align*}
 </math>
-Let's define $\Omega$ to be the interior of the parametrically given curve $r(t): [- 2 \pi, 0] \to \mathbb{R}^2$:
+Let's define $\Omega$ to be the interior of the parametrically given curve $r(t): [0, 2 \pi] \to \mathbb{R}^2$:
 <math>
 \begin{align*}
-     	f(t) &= |\cos(-1.5 t)| ^ {\sin(-3t)} \\
+     	f(t) &= |\cos(1.5 t)| ^ {\sin(3t)} \\
-         r(t) &= (f(t) \cos(-t), f(t) \sin(-t))
+         r(t) &= (f(t) \cos(t), f(t) \sin(t))
 \end{align*}
 </math>

@@ Line 6: / Line 6: @@
 == Variable node density and Dirichlet 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>

@@ Line 2: / Line 2: @@
 Medusa supports creating domains bounded by parametric curves and (hyper-)surfaces and filling them and their interiors with node distributions of arbitrary densities.
-See [[TODO]] for details on node positioning algorithms.
 == Variable node density and Dirichlet boundary conditions in 2D ==

← Older revision		Revision as of 17:17, 8 August 2019
Line 1:		Line 1:
	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].
		+
		+	Medusa supports creating domains bounded by parametric curves and (hyper-)surfaces and filling them and their interiors with node distributions of arbitrary densities.
		+	See [[TODO]] for details on node positioning algorithms.

	== Variable node density and Dirichlet boundary conditions in 2D ==		== Variable node density and Dirichlet boundary conditions in 2D ==

@@ Line 1: / Line 1: @@
 Go back to [[Medusa#Examples|Examples]].
-== Variable node density and dirchlet boundary conditions in 2D ==
+== Variable node density and Dirichlet 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:
+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*}