fem.py

#!/usr/bin/env python

"""FEM library

Demonstrates some simple finite element definitions, and computes a mass
matrix

$ python fem.py
[  1/60,     0, -1/360,     0, -1/90, -1/360]
[     0,  4/45,      0,  2/45,  2/45,  -1/90]
[-1/360,     0,   1/60, -1/90,     0, -1/360]
[     0,  2/45,  -1/90,  4/45,  2/45,      0]
[ -1/90,  2/45,      0,  2/45,  4/45,      0]
[-1/360, -1/90, -1/360,     0,     0,   1/60]

"""

from sympy import symbols, Symbol, factorial, Rational, zeros, eye, \
    integrate, diff, pprint, reduced

x, y, z = symbols('x,y,z')


class ReferenceSimplex:
    def __init__(self, nsd):
        self.nsd = nsd
        if nsd <= 3:
            coords = symbols('x,y,z')[:nsd]
        else:
            coords = [Symbol("x_%d" % d) for d in range(nsd)]
        self.coords = coords

    def integrate(self, f):
        coords = self.coords
        nsd = self.nsd

        limit = 1
        for p in coords:
            limit -= p

        intf = f
        for d in range(0, nsd):
            p = coords[d]
            limit += p
            intf = integrate(intf, (p, 0, limit))
        return intf


def bernstein_space(order, nsd):
    if nsd > 3:
        raise RuntimeError("Bernstein only implemented in 1D, 2D, and 3D")
    sum = 0
    basis = []
    coeff = []

    if nsd == 1:
        b1, b2 = x, 1 - x
        for o1 in range(0, order + 1):
            for o2 in range(0, order + 1):
                if o1 + o2 == order:
                    aij = Symbol("a_%d_%d" % (o1, o2))
                    sum += aij*binomial(order, o1)*pow(b1, o1)*pow(b2, o2)
                    basis.append(binomial(order, o1)*pow(b1, o1)*pow(b2, o2))
                    coeff.append(aij)

    if nsd == 2:
        b1, b2, b3 = x, y, 1 - x - y
        for o1 in range(0, order + 1):
            for o2 in range(0, order + 1):
                for o3 in range(0, order + 1):
                    if o1 + o2 + o3 == order:
                        aij = Symbol("a_%d_%d_%d" % (o1, o2, o3))
                        fac = factorial(order) / (factorial(o1)*factorial(o2)*factorial(o3))
                        sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)
                        basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3))
                        coeff.append(aij)

    if nsd == 3:
        b1, b2, b3, b4 = x, y, z, 1 - x - y - z
        for o1 in range(0, order + 1):
            for o2 in range(0, order + 1):
                for o3 in range(0, order + 1):
                    for o4 in range(0, order + 1):
                        if o1 + o2 + o3 + o4 == order:
                            aij = Symbol("a_%d_%d_%d_%d" % (o1, o2, o3, o4))
                            fac = factorial(order)/(factorial(o1)*factorial(o2)*factorial(o3)*factorial(o4))
                            sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4)
                            basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4))
                            coeff.append(aij)

    return sum, coeff, basis


def create_point_set(order, nsd):
    h = Rational(1, order)
    set = []

    if nsd == 1:
        for i in range(0, order + 1):
            x = i*h
            if x <= 1:
                set.append((x, y))

    if nsd == 2:
        for i in range(0, order + 1):
            x = i*h
            for j in range(0, order + 1):
                y = j*h
                if x + y <= 1:
                    set.append((x, y))

    if nsd == 3:
        for i in range(0, order + 1):
            x = i*h
            for j in range(0, order + 1):
                y = j*h
                for k in range(0, order + 1):
                    z = k*h
                    if x + y + z <= 1:
                        set.append((x, y, z))

    return set


def create_matrix(equations, coeffs):
    A = zeros(len(equations))
    i = 0
    j = 0
    for j in range(0, len(coeffs)):
        c = coeffs[j]
        for i in range(0, len(equations)):
            e = equations[i]
            d, _ = reduced(e, [c])
            A[i, j] = d[0]
    return A


class Lagrange:
    def __init__(self, nsd, order):
        self.nsd = nsd
        self.order = order
        self.compute_basis()

    def nbf(self):
        return len(self.N)

    def compute_basis(self):
        order = self.order
        nsd = self.nsd
        N = []
        pol, coeffs, basis = bernstein_space(order, nsd)
        points = create_point_set(order, nsd)

        equations = []
        for p in points:
            ex = pol.subs(x, p[0])
            if nsd > 1:
                ex = ex.subs(y, p[1])
            if nsd > 2:
                ex = ex.subs(z, p[2])
            equations.append(ex)

        A = create_matrix(equations, coeffs)
        Ainv = A.inv()

        b = eye(len(equations))

        xx = Ainv*b

        for i in range(0, len(equations)):
            Ni = pol
            for j in range(0, len(coeffs)):
                Ni = Ni.subs(coeffs[j], xx[j, i])
            N.append(Ni)

        self.N = N


def main():
    t = ReferenceSimplex(2)
    fe = Lagrange(2, 2)

    u = 0
    # compute u = sum_i u_i N_i
    us = []
    for i in range(0, fe.nbf()):
        ui = Symbol("u_%d" % i)
        us.append(ui)
        u += ui*fe.N[i]

    J = zeros(fe.nbf())
    for i in range(0, fe.nbf()):
        Fi = u*fe.N[i]
        print(Fi)
        for j in range(0, fe.nbf()):
            uj = us[j]
            integrands = diff(Fi, uj)
            print(integrands)
            J[j, i] = t.integrate(integrands)

    pprint(J)


if __name__ == "__main__":
    main()