Heat Equation, Dirichlet boundary condi

.gitignore

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+.ipynb_checkpoints

Readme.md

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+
+# Introduction
+The Poisson's equation is a second-order partial differential equation that stats the negative Laplacian $-\Delta u$ of an unknown field $u=u(x)$ is equal to a given function $f=f(x)$ on a domain $\Omega \subset \mathbb{R}^d$, most probably defined by a set of boundary conditions for the solution $u$ on the boundary $\partial \Omega$ of $\Omega$:
+
+$$-\Delta u =f \quad \text{in } \Omega\text{,}$$
+$$u=u_0 \quad \text{on } \Gamma_D \subset \partial\Omega \text{,}$$
+
+here the Dirichlet's boundary condition $u=u_0$ signifies a prescribed values for the unknown $u$ on the boundary.
+
+The Poisson's equation is the simplest model for gravity, electromagnetism, heat transfer, among others.
+
+The specific case of $f=0$ and a negative $k$ value, leaves to the Fourier's Law.
+
+## Comparative analysis
+Along this example, the fenics platfomr is used to compare results obtained by solving the heat equation (Laplace equation) in 2-D:
+
+$$\frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2}=0$$
+
+the problem is defined by the next geometry considerations:
+
+![](physicalproblem.png)
+
+The resulting contour of temperature, solving using finite diferences, is shown next:
+
+![](resulteq.png)
+
+# Solving by Finite Element Method with Varational Problem formulation
+
+
+```python
+#1 Loading functions  and modules
+from fenics import *
+import matplotlib.pyplot as plt
+```
+
+
+```python
+#2 Create mesh and define function space
+mesh = RectangleMesh(Point(0,0),Point(20,20),10, 10,'left')
+V = FunctionSpace(mesh, 'Lagrange', 1) #Lagrange are triangular elements
+plot(mesh)
+plt.show()
+```
+
+
+![png](output_6_0.png)
+
+
+
+```python
+#3 Defining boundary conditions (Dirichlet)
+tol = 1E-14 # tolerance for coordinate comparisons
+#at y=20
+def Dirichlet_boundary1(x, on_boundary):
+    return on_boundary and abs(x[1] - 20) < tol
+#at y=0
+def Dirichlet_boundary0(x, on_boundary):
+    return on_boundary and abs(x[1] - 0) < tol
+#at x=0
+def Dirichlet_boundarx0(x, on_boundary):
+    return on_boundary and abs(x[0] - 0) < tol
+#at x=20
+def Dirichlet_boundarx1(x, on_boundary):
+    return on_boundary and abs(x[0] - 20) < tol
+
+bc0 = DirichletBC(V, Constant(0), Dirichlet_boundary0)
+bc1 = DirichletBC(V, Constant(100), Dirichlet_boundary1) #100C
+bc2 = DirichletBC(V, Constant(0), Dirichlet_boundarx0)
+bc3 = DirichletBC(V, Constant(0), Dirichlet_boundarx1)
+bcs = [bc0,bc1, bc2,bc3]
+```
+
+
+```python
+#4 Defining variational problem and its solution
+k =1
+u = TrialFunction(V)
+v = TestFunction(V)
+f = Constant(0)
+a = dot(k*grad(u), grad(v))*dx
+L = f*v*dx
+
+# Compute solution
+u = Function(V)
+solve(a == L, u, bcs)
+
+# Plot solution and mesh
+plot(u)
+plot(mesh)
+
+# Save solution to file in VTK format
+vtkfile = File('solution.pvd')
+vtkfile << u
+```
+
+
+![png](output_8_0.png)
+
+
+# Results after editing color-map on paraview
+![](paraview-results.png)

solution.pvd

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+<?xml version="1.0"?>
+<VTKFile type="Collection" version="0.1">
+  <Collection>
+    <DataSet timestep="0" part="0" file="solution000000.vtu" />
+  </Collection>
+</VTKFile>