122 lines
3.6 KiB
Python
122 lines
3.6 KiB
Python
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"""
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integrate.py
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--------------
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Utilities for integrating functions over meshes surfaces.
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"""
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import sympy as sp
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from sympy.parsing.sympy_parser import parse_expr as sympy_parse
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from .constants import log
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from . import util
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def symbolic_barycentric(function):
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"""
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Symbolically integrate a function(x,y,z) across a triangle or mesh.
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Parameters
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----------
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function: string or sympy expression
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x, y, z will be replaced with a barycentric representation
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and the the function is integrated across the triangle.
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Returns
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----------
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evaluator: numpy lambda function of result which takes a mesh
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expr: sympy expression of result
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Examples
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-----------
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In [1]: function = '1'
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In [2]: integrator, expr = integrate_barycentric(function)
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In [3]: integrator
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Out[3]: <__main__.evaluator instance at 0x7f66cd2a6200>
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In [4]: expr
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Out[4]: 1/2
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In [5]: result = integrator(mesh)
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In [6]: mesh.area
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Out[6]: 34.641016151377542
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In [7]: result.sum()
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Out[7]: 34.641016151377542
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"""
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class evaluator:
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def __init__(self, expr, expr_args):
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self.lambdified = sp.lambdify(args=expr_args,
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expr=expr,
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modules=['numpy', 'sympy'])
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def __call__(self, mesh, *args):
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"""
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Quickly evaluate the surface integral across a mesh
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Parameters
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----------
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mesh: Trimesh object
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Returns
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----------
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integrated: (len(faces),) float, integral evaluated for each face
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"""
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integrated = self.lambdified(*mesh.triangles.reshape((-1, 9)).T)
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integrated *= 2 * mesh.area_faces
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return integrated
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function, symbols = substitute_barycentric(function)
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b1, b2, x1, x2, x3, y1, y2, y3, z1, z2, z3 = symbols
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# do the first integral for b1
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integrated_1 = sp.integrate(function, b1)
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integrated_1 = (integrated_1.subs({b1: 1 - b2}) -
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integrated_1.subs({b1: 0}))
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integrated_2 = sp.integrate(integrated_1, b2)
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integrated_2 = (integrated_2.subs({b2: 1}) -
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integrated_2.subs({b2: 0}))
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lambdified = evaluator(expr=integrated_2,
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expr_args=[x1, y1, z1, x2, y2, z2, x3, y3, z3])
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return lambdified, integrated_2
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def substitute_barycentric(function):
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if util.is_string(function):
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function = sympy_parse(function)
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# barycentric coordinates
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b1, b2 = sp.symbols('b1 b2',
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real=True,
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positive=True)
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# vertices of the triangles
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x1, x2, x3, y1, y2, y3, z1, z2, z3 = sp.symbols(
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'x1,x2,x3,y1,y2,y3,z1,z2,z3', real=True)
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# generate the substitution dictionary to convert from cartesian to barycentric
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# since the input could have been a sympy expression or a string
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# that we parsed substitute based on name to avoid id(x) issues
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substitutions = {}
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for symbol in function.free_symbols:
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if symbol.name == 'x':
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substitutions[symbol] = b1 * x1 + b2 * x2 + (1 - b1 - b2) * x3
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elif symbol.name == 'y':
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substitutions[symbol] = b1 * y1 + b2 * y2 + (1 - b1 - b2) * y3
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elif symbol.name == 'z':
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substitutions[symbol] = b1 * z1 + b2 * z2 + (1 - b1 - b2) * z3
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# apply the conversion to barycentric
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function = function.subs(substitutions)
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log.debug('converted function to barycentric: %s', str(function))
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symbols = (b1, b2, x1, x2, x3, y1, y2, y3, z1, z2, z3)
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return function, symbols
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