def curl_norm(e, Y): """ it calculates curl-based norm, it controls that the fields are curl-free with zero mean as it is required of electric fields Parameters ---------- e - electric field Y - the size of periodic unit cell Returns ------- curlnorm - curl-based norm """ N = np.array(np.shape(e[0])) d = np.size(N) xil = Grid.get_xil(N, Y) xiM = [] Fe = [] for m in np.arange(d): Nshape = np.ones(d) Nshape[m] = N[m] Nrep = np.copy(N) Nrep[m] = 1 xiM.append(np.tile(np.reshape(xil[m], Nshape), Nrep)) Fe.append(DFT.fftnc(e[m], N)/np.prod(N)) if d == 2: Fe.append(np.zeros(N)) xiM.append(np.zeros(N)) ind_mean = tuple(np.fix(N/2)) curl = [] e0 = [] for m in np.arange(3): j = (m+1) % 3 k = (j+1) % 3 curl.append(xiM[j]*Fe[k]-xiM[k]*Fe[j]) e0.append(np.real(Fe[m][ind_mean])) curl = np.array(curl) curlnorm = np.real(np.sum(curl[:]*np.conj(curl[:]))) curlnorm = (curlnorm/np.prod(N))**0.5 norm_e0 = np.linalg.norm(e0) if norm_e0 > 1e-10: curlnorm = curlnorm/norm_e0 return curlnorm
def scalar(N, Y, centered=True, NyqNul=True): """ Assembly of discrete kernels in Fourier space for scalar elliptic problems. Parameters ---------- N : numpy.ndarray no. of discretization points Y : numpy.ndarray size of periodic unit cell Returns ------- G1l : numpy.ndarray discrete kernel in Fourier space; provides projection on curl-free fields with zero mean G2l : numpy.ndarray discrete kernel in Fourier space; provides projection on divergence-free fields with zero mean """ d = np.size(N) N = np.array(N) if NyqNul: Nred = get_Nodd(N) else: Nred = N xi = Grid.get_xil(Nred, Y) xi2 = [] for m in np.arange(d): xi2.append(xi[m]**2) G0l = np.zeros(np.hstack([d, d, Nred])) G1l = np.zeros(np.hstack([d, d, Nred])) G2l = np.zeros(np.hstack([d, d, Nred])) num = np.zeros(np.hstack([d, d, Nred])) denom = np.zeros(Nred) ind_center = tuple(np.fix(np.array(Nred)/2)) for m in np.arange(d): # diagonal components Nshape = np.ones(d) Nshape[m] = Nred[m] Nrep = np.copy(Nred) Nrep[m] = 1 a = np.reshape(xi2[m], Nshape) num[m][m] = np.tile(a, Nrep) # numerator denom = denom + num[m][m] G0l[m, m][ind_center] = 1 for m in np.arange(d): # upper diagonal components for n in np.arange(m+1, d): NshapeM = np.ones(d) NshapeM[m] = Nred[m] NrepM = np.copy(Nred) NrepM[m] = 1 NshapeN = np.ones(d) NshapeN[n] = Nred[n] NrepN = np.copy(Nred) NrepN[n] = 1 num[m][n] = np.tile(np.reshape(xi[m], NshapeM), NrepM) \ * np.tile(np.reshape(xi[n], NshapeN), NrepN) # avoiding a division by zero denom[ind_center] = 1 # calculation of projections for m in np.arange(d): for n in np.arange(m, d): G1l[m][n] = num[m][n]/denom G2l[m][n] = (m == n)*np.ones(Nred) - G1l[m][n] G2l[m][n][ind_center] = 0 # symmetrization for m in np.arange(1, d): for n in np.arange(m): G1l[m][n] = G1l[n][m] G2l[m][n] = G2l[n][m] if not centered: for m in np.arange(d): for n in np.arange(d): G1l[m][n] = np.fft.ifftshift(G1l[m][n]) G2l[m][n] = np.fft.ifftshift(G2l[m][n]) G0l = Matrix(name='hG0', val=G0l, Fourier=True) G1l = Matrix(name='hG1', val=G1l, Fourier=True) G2l = Matrix(name='hG2', val=G2l, Fourier=True) if NyqNul: G0l = G0l.enlarge(N) G1l = G1l.enlarge(N) G2l = G2l.enlarge(N) return G0l, G1l, G2l
def elasticity(N, Y, centered=True, NyqNul=True): """ Projection matrix on a space of admissible strain fields INPUT = N : ndarray of e.g. stiffness coefficients d : dimension; d = 2 D : dimension in engineering notation; D = 3 Y : the size of periodic unit cell OUTPUT = G1h,G1s,G2h,G2s : projection matrices of size DxDxN """ xi = Grid.get_xil(N, Y) N = np.array(N) d = N.size D = d*(d+1)/2 if NyqNul: Nred = get_Nodd(N) else: Nred = N xi2 = [] for ii in np.arange(d): xi2.append(xi[ii]**2) num = np.zeros(np.hstack([d, d, Nred])) norm2_xi = np.zeros(Nred) for mm in np.arange(d): # diagonal components Nshape = np.ones(d) Nshape[mm] = Nred[mm] Nrep = np.copy(Nred) Nrep[mm] = 1 num[mm][mm] = np.tile(np.reshape(xi2[mm], Nshape), Nrep) # numerator norm2_xi += num[mm][mm] norm4_xi = norm2_xi**2 ind_center = tuple(Nred/2) # avoid division by zero norm2_xi[ind_center] = 1 norm4_xi[ind_center] = 1 for m in np.arange(d): # upper diagonal components for n in np.arange(m+1, d): NshapeM = np.ones(d) NshapeM[m] = Nred[m] NrepM = np.copy(Nred) NrepM[m] = 1 NshapeN = np.ones(d) NshapeN[n] = Nred[n] NrepN = np.copy(Nred) NrepN[n] = 1 num[m][n] = np.tile(np.reshape(xi[m], NshapeM), NrepM) \ * np.tile(np.reshape(xi[n], NshapeN), NrepN) # G1h = np.zeros([D,D]).tolist() G1h = np.zeros(np.hstack([D, D, Nred])) G1s = np.zeros(np.hstack([D, D, Nred])) IS0 = np.zeros(np.hstack([D, D, Nred])) mean = np.zeros(np.hstack([D, D, Nred])) Lamh = np.zeros(np.hstack([D, D, Nred])) S = np.zeros(np.hstack([D, D, Nred])) W = np.zeros(np.hstack([D, D, Nred])) WT = np.zeros(np.hstack([D, D, Nred])) for m in np.arange(d): S[m][m] = 2*num[m][m]/norm2_xi for n in np.arange(d): G1h[m][n] = num[m][m]*num[n][n]/norm4_xi Lamh[m][n] = np.ones(Nred)/d Lamh[m][n][ind_center] = 0 for m in np.arange(D): IS0[m][m] = np.ones(Nred) IS0[m][m][ind_center] = 0 mean[m][m][ind_center] = 1 if d == 2: S[0][2] = 2**0.5*num[0][1]/norm2_xi S[1][2] = 2**0.5*num[0][1]/norm2_xi S[2][2] = np.ones(Nred) S[2][2][ind_center] = 0 G1h[0][2] = 2**0.5*num[0][0]*num[0][1]/norm4_xi G1h[1][2] = 2**0.5*num[0][1]*num[1][1]/norm4_xi G1h[2][2] = 2*num[0][0]*num[1][1]/norm4_xi for m in np.arange(d): for n in np.arange(d): W[m][n] = num[m][m]/norm2_xi W[2][m] = 2**.5*num[0][1]/norm2_xi elif d == 3: for m in np.arange(d): S[m+3][m+3] = 1 - num[m][m]/norm2_xi S[m+3][m+3][ind_center] = 0 for m in np.arange(d): for n in np.arange(m+1, d): S[m+3][n+3] = num[m][n]/norm2_xi G1h[m+3][n+3] = num[m][m]*num[n][n]/norm4_xi for m in np.arange(d): for n in np.arange(d): ind = sp.setdiff1d(np.arange(d), [n]) S[m][n+3] = (0 == (m == n))*2**.5*num[ind[0]][ind[1]]/norm2_xi G1h[m][n+3] = 2**.5*num[m][m]*num[ind[0]][ind[1]]/norm4_xi W[m][n] = num[m][m]/norm2_xi W[n+3][m] = 2**.5*num[ind[0]][ind[1]]/norm2_xi for m in np.arange(d): for n in np.arange(d): ind_m = sp.setdiff1d(np.arange(d), [m]) ind_n = sp.setdiff1d(np.arange(d), [n]) G1h[m+3][n+3] = 2*num[ind_m[0]][ind_m[1]] \ * num[ind_n[0]][ind_n[1]] / norm4_xi # symmetrization for n in np.arange(D): for m in np.arange(n+1, D): S[m][n] = S[n][m] G1h[m][n] = G1h[n][m] for m in np.arange(D): for n in np.arange(D): G1s[m][n] = S[m][n] - 2*G1h[m][n] WT[m][n] = W[n][m] G2h = 1./(d-1)*(d*Lamh + G1h - W - WT) G2s = IS0 - G1h - G1s - G2h if not centered: for m in np.arange(d): for n in np.arange(d): G1h[m][n] = np.fft.ifftshift(G1h[m][n]) G1s[m][n] = np.fft.ifftshift(G1s[m][n]) G2h[m][n] = np.fft.ifftshift(G2h[m][n]) G2s[m][n] = np.fft.ifftshift(G2s[m][n]) G0 = Matrix(name='hG1', val=mean, Fourier=True) G1h = Matrix(name='hG1', val=G1h, Fourier=True) G1s = Matrix(name='hG1', val=G1s, Fourier=True) G2h = Matrix(name='hG1', val=G2h, Fourier=True) G2s = Matrix(name='hG1', val=G2s, Fourier=True) if NyqNul: G0 = G0.enlarge(N) G1h = G1h.enlarge(N) G1s = G1s.enlarge(N) G2h = G2h.enlarge(N) G2s = G2s.enlarge(N) return mean, G1h, G1s, G2h, G2s