# Source code for pymor.analyticalproblems.thermalblock

# This file is part of the pyMOR project (http://www.pymor.org).

from itertools import product

from pymor.analyticalproblems.domaindescriptions import RectDomain
from pymor.analyticalproblems.elliptic import StationaryProblem
from pymor.analyticalproblems.functions import ConstantFunction, ExpressionFunction, LincombFunction
from pymor.parameters.functionals import ProjectionParameterFunctional

[docs]def thermal_block_problem(num_blocks=(3, 3), parameter_range=(0.1, 1)):
"""Analytical description of a 2D 'thermal block' diffusion problem.

The problem is to solve the elliptic equation ::

- ∇ ⋅ [ d(x, μ) ∇ u(x, μ) ] = f(x, μ)

on the domain [0,1]^2 with Dirichlet zero boundary values. The domain is
partitioned into nx x ny blocks and the diffusion function d(x, μ) is
constant on each such block i with value μ_i. ::

----------------------------
|        |        |        |
|  μ_4   |  μ_5   |  μ_6   |
|        |        |        |
|---------------------------
|        |        |        |
|  μ_1   |  μ_2   |  μ_3   |
|        |        |        |
----------------------------

Parameters
----------
num_blocks
The tuple (nx, ny)
parameter_range
A tuple (μ_min, μ_max). Each |Parameter| component μ_i is allowed
to lie in the interval [μ_min, μ_max].
"""

def parameter_functional_factory(ix, iy):
return ProjectionParameterFunctional('diffusion',
size=num_blocks[0]*num_blocks[1],
index=ix + iy*num_blocks[0],
name=f'diffusion_{ix}_{iy}')

def diffusion_function_factory(ix, iy):
if ix + 1 < num_blocks[0]:
X = '(x[..., 0] >= ix * dx) * (x[..., 0] < (ix + 1) * dx)'
else:
X = '(x[..., 0] >= ix * dx)'
if iy + 1 < num_blocks[1]:
Y = '(x[..., 1] >= iy * dy) * (x[..., 1] < (iy + 1) * dy)'
else:
Y = '(x[..., 1] >= iy * dy)'
return ExpressionFunction(f'{X} * {Y} * 1.',
2, (), {}, {'ix': ix, 'iy': iy, 'dx': 1. / num_blocks[0], 'dy': 1. / num_blocks[1]},
name=f'diffusion_{ix}_{iy}')

return StationaryProblem(

domain=RectDomain(),

rhs=ConstantFunction(dim_domain=2, value=1.),

diffusion=LincombFunction([diffusion_function_factory(ix, iy)
for iy, ix in product(range(num_blocks[1]), range(num_blocks[0]))],
[parameter_functional_factory(ix, iy)
for iy, ix in product(range(num_blocks[1]), range(num_blocks[0]))],
name='diffusion'),

parameter_ranges=parameter_range,

name=f'ThermalBlock({num_blocks})'

)