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