pyramid Namespace Reference

Functions
	integrate_exact (f)
	integrate_approx (f, scheme)
	list_schemes ()
	generate_monomials (order)
	is_valid (scheme, tol=1e-6, relaxed=False)
	pick_scheme (all_schemes, order, relaxed=False)
	generate_cpp (selected_schemes)
	main ()

Variables
	b

Function Documentation

generate_cpp()

pyramid.generate_cpp

(

selected_schemes

)

Generate cpp code to fill points & weights.

Definition at line 155 of file pyramid.py.

Referenced by main().

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generate_monomials()

pyramid.generate_monomials

(

order

)

Generate trivariate monomials up to given order.

Definition at line 68 of file pyramid.py.

Referenced by pick_scheme().

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integrate_approx()

pyramid.integrate_approx	(		f,
			scheme
	)

Integrate numerically over the reference tetrahedron.

Definition at line 37 of file pyramid.py.

Referenced by pick_scheme().

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integrate_exact()

pyramid.integrate_exact

(

f

)

Integrate a function over the reference pyramid:
    0 <= z <= 1
    0 <= x <= (1 - z)
    0 <= y <= (1 - z)

Definition at line 17 of file pyramid.py.

Referenced by pick_scheme().

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is_valid()

pyramid.is_valid	(		scheme,
			tol = 1e-6,
			relaxed = False
	)

A scheme is valid if:
1. weights sum to 1
2. weights positive
3. points inside reference pyramid:
     0 <= z <= 1
     |x|, |y| <= 1 - z

Definition at line 82 of file pyramid.py.

Referenced by pick_scheme().

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list_schemes()

pyramid.list_schemes

(

)

List all existing schemes for tetrahedra.

Definition at line 51 of file pyramid.py.

Referenced by main().

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main()

pyramid.main

(

)

Definition at line 184 of file pyramid.py.

References generate_cpp(), list_schemes(), main(), and pick_scheme().

Referenced by main().

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pick_scheme()

pyramid.pick_scheme	(		all_schemes,
			order,
			relaxed = False
	)

Picks the best scheme for a given polynomial degree, following this strategy:
- Tries all schemes with the same order as required.
- Eliminate schemes with weights that do not sum up to 1, or points that lie
  outside the reference triangle.
- Among the remaining schemes, compare the integration error over all bivariate
  monomials of inferior order.
- Keeps only the scheme with total integration error lower than a certain threshold.
- Among those, keep the one with fewer number of integration points.
- Finally, break ties using the integration error accumulated over the monomials.

Definition at line 115 of file pyramid.py.

References generate_monomials(), integrate_approx(), integrate_exact(), and is_valid().

Referenced by main().

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Variable Documentation

b

pyramid.b

Initial value:

=  np.array([
    [0,0,0],
    [0,1,0],
    [1,1,0],
    [1,0,0],
    [0,0,1]
])

Definition at line 9 of file pyramid.py.