Heat equation in 2DΒΆ

This tutorial simulates the stationary heat equation in 2D. The example is taken from the pyGIMLi paper (https://cg17.pygimli.org).

import pygimli as pg
import pygimli.meshtools as mt

Create geometry definition for the modelling domain.

world = mt.createWorld(start=[-20, 0], end=[20, -16], layers=[-2, -8],
# Create a heterogeneous block
block = mt.createRectangle(start=[-6, -3.5], end=[6, -6.0],
                           marker=4,  boundaryMarker=10, area=0.1)
# Merge geometrical entities
geom = mt.mergePLC([world, block])
pg.show(geom, boundaryMarker=True)


(<matplotlib.axes._subplots.AxesSubplot object at 0x7f87ff212b38>, None)

Create a mesh from based on the geometry definition.

mesh = mt.createMesh(geom, quality=33, area=0.2, smooth=[1, 10])


(<matplotlib.axes._subplots.AxesSubplot object at 0x7f87ff1fdac8>, None)

Call pygimli.solver.solveFiniteElements() to solve the heat diffusion equation \(\nabla\cdot(a\nabla T)=0\) with \(T(bottom)=1\) and \(T(top)=0\), where \(a\) is the thermal diffusivity and \(T\) is the temperature distribution.

T = pg.solver.solveFiniteElements(mesh,
                                  a=[[1, 1.0], [2, 2.0], [3, 3.0], [4, 0.1]],
                                  uB=[[8, 1.0], [4, 0.0]], verbose=True)

ax, _ = pg.show(mesh, data=T, label='Temperature $T$', cmap="hot_r")
pg.show(geom, ax=ax, fillRegion=False)

# just hold figure windows open if run outside from spyder, ipython or similar


Mesh:  Mesh: Nodes: 3011 Cells: 5832 Boundaries: 8842
('Asssemblation time: ', 0.034)
('Solving time: ', 0.019)

Gallery generated by Sphinx-Gallery

2019 - GIMLi Development Team
Created using Bootstrap, Sphinx and pyGIMLi 1.0.12+20.g65e71e67 on Mar 10, 2020.