Development tools (panels.dev)

Tabulation of the approximation functions (panels.dev.tabulate)

Tabulation of the approximation functions in the numerical kernels

The numerically integrated kernels, e.g. fkC_num of panels/models/<model>_num.pyx or fkCSB11_dmg of panels/multidomain/connections/kCSB_dmg.pyx, evaluate the approximation functions inside their loops over the terms, e.g. fBu = f(k, xi, x1u, x1ur, x2u, x2ur) in the innermost loop over the pairs of terms of a stiffness matrix. tabulate_basis() rewrites a kernel such that each function is evaluated once per integration point and term, in tables such as ftab[k, i] along \(x\) and gtab[k, j] along \(y\), filled at the start of the loop over the integration points and read in the loops over the terms. The arithmetic of the integrands is unchanged, so are the results, bit for bit.

A call NAME = f(IDX, COORD, x1F, x1Fr, x2F, x2Fr), with f, fp or fpp, the boundary flags x1F... along \(x\) or y1F... along \(y\) of the field F and an optional suffix of the panel, e.g. x1u1, x1ur1, x2u1, x2ur1, becomes NAME = ftab1[k, IDX], with one table per direction, suffix and coordinate COORD, whose number of columns is that of the loop for IDX in range(...) of the kernel, and which is filled with IDX as loop variable. Without a loop over the integration points enclosing the calls, e.g. along a line of constant \(\xi\) of the connections, the tables are filled once, before the outermost loop over the terms. A kernel is left unchanged when it has no such call, or when it is already tabulated. A call in another form raises a ValueError.

The scripts that generate the kernel modules from the symbolic integrands apply it to their output; python -m panels.dev.tabulate <file.pyx> ... applies it to the modules written by hand.

Notes

An entry of a numerically integrated stiffness matrix between the terms \((i, j)\) and \((k, l)\) of the Ritz series is a sum over the Gauss-Legendre points of products of the approximation functions \(P_i(\xi)\), \(P_k(\xi)\), \(P_j(\eta)\) and \(P_l(\eta)\), and of their derivatives, see panels.bardell. In the innermost loop over the pairs of terms each function is evaluated \(m^2 n^2\) times per integration point, although it depends on one index and one coordinate only. Tabulated, it is evaluated \(m\) (or \(n\)) times per point, and the cost of the functions becomes negligible against the \(m^2 n^2\) products of the assembly, see Approximation functions tabulated at the integration points. Since only where the values are computed changes, and not the operations on them, the results are identical.

panels.dev.tabulate.tabulate_basis(source)

Tabulate the approximation functions of every kernel of the module source, see the module docstring

The source is split before each line starting with def, cdef or cpdef; only the parts that are def functions are rewritten, the others are kept as they are. A ValueError is raised for a kernel whose calls are not in the expected form.

Parameters:

sourcestr

The text of a .pyx module.

Returns:

sourcestr

The rewritten text, equal to source when no kernel is changed.