123 lines
5.2 KiB
Python
123 lines
5.2 KiB
Python
import numpy as np
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def __ks(size,field,first_order=False):
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"""Get wave numbers operator."""
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grid = np.array(np.shape(field)[:3])
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k_sk = np.where(np.arange(grid[0])>grid[0]//2,np.arange(grid[0])-grid[0],np.arange(grid[0]))/size[0]
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if grid[0]%2 == 0 and first_order: k_sk[grid[0]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
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k_sj = np.where(np.arange(grid[1])>grid[1]//2,np.arange(grid[1])-grid[1],np.arange(grid[1]))/size[1]
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if grid[1]%2 == 0 and first_order: k_sj[grid[1]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
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k_si = np.arange(grid[2]//2+1)/size[2]
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kk, kj, ki = np.meshgrid(k_sk,k_sj,k_si,indexing = 'ij')
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return np.concatenate((ki[:,:,:,None],kj[:,:,:,None],kk[:,:,:,None]),axis = 3)
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def curl(size,field):
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"""Calculate curl of a vector or tensor field in Fourier space."""
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n = np.prod(field.shape[3:])
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k_s = __ks(size,field,True)
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e = np.zeros((3, 3, 3))
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e[0, 1, 2] = e[1, 2, 0] = e[2, 0, 1] = +1.0 # Levi-Civita symbol
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e[0, 2, 1] = e[2, 1, 0] = e[1, 0, 2] = -1.0
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field_fourier = np.fft.rfftn(field,axes=(0,1,2))
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curl = (np.einsum('slm,ijkl,ijkm ->ijks', e,k_s,field_fourier)*2.0j*np.pi if n == 3 else # vector, 3 -> 3
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np.einsum('slm,ijkl,ijknm->ijksn',e,k_s,field_fourier)*2.0j*np.pi) # tensor, 3x3 -> 3x3
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return np.fft.irfftn(curl,axes=(0,1,2),s=field.shape[:3])
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def divergence(size,field):
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"""Calculate divergence of a vector or tensor field in Fourier space."""
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n = np.prod(field.shape[3:])
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k_s = __ks(size,field,True)
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field_fourier = np.fft.rfftn(field,axes=(0,1,2))
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divergence = (np.einsum('ijkl,ijkl ->ijk', k_s,field_fourier)*2.0j*np.pi if n == 3 else # vector, 3 -> 1
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np.einsum('ijkm,ijklm->ijkl',k_s,field_fourier)*2.0j*np.pi) # tensor, 3x3 -> 3
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return np.fft.irfftn(divergence,axes=(0,1,2),s=field.shape[:3])
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def gradient(size,field):
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"""Calculate gradient of a vector or scalar field in Fourier space."""
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n = np.prod(field.shape[3:])
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k_s = __ks(size,field,True)
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field_fourier = np.fft.rfftn(field,axes=(0,1,2))
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gradient = (np.einsum('ijkl,ijkm->ijkm', field_fourier,k_s)*2.0j*np.pi if n == 1 else # scalar, 1 -> 3
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np.einsum('ijkl,ijkm->ijklm',field_fourier,k_s)*2.0j*np.pi) # vector, 3 -> 3x3
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return np.fft.irfftn(gradient,axes=(0,1,2),s=field.shape[:3])
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def coord0_cell(grid,size):
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"""Cell center positions (undeformed)."""
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delta = size/grid*0.5
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x, y, z = np.meshgrid(np.linspace(delta[2],size[2]-delta[2],grid[2]),
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np.linspace(delta[1],size[1]-delta[1],grid[1]),
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np.linspace(delta[0],size[0]-delta[0],grid[0]),
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indexing = 'ij')
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return np.concatenate((z[:,:,:,None],y[:,:,:,None],x[:,:,:,None]),axis = 3)
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def displacement_fluct_cell(size,F):
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"""Cell center displacement field from fluctuation part of the deformation gradient field."""
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integrator = 0.5j*size/np.pi
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k_s = __ks(size,F,False)
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k_s_squared = np.einsum('...l,...l',k_s,k_s)
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k_s_squared[0,0,0] = 1.0
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displacement = -np.einsum('ijkml,ijkl,l->ijkm',
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np.fft.rfftn(F,axes=(0,1,2)),
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k_s,
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integrator,
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) / k_s_squared[...,np.newaxis]
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return np.fft.irfftn(displacement,axes=(0,1,2),s=F.shape[:3])
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def displacement_avg_cell(size,F):
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"""Cell center displacement field from average part of the deformation gradient field."""
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F_avg = np.average(F,axis=(0,1,2))
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return np.einsum('ml,ijkl->ijkm',F_avg-np.eye(3),coord0_cell(F.shape[:3],size))
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def coord0_node(grid,size):
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"""Nodal positions (undeformed)."""
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x, y, z = np.meshgrid(np.linspace(0,size[2],1+grid[2]),
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np.linspace(0,size[1],1+grid[1]),
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np.linspace(0,size[0],1+grid[0]),
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indexing = 'ij')
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return np.concatenate((z[:,:,:,None],y[:,:,:,None],x[:,:,:,None]),axis = 3)
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def displacement_fluct_node(size,F):
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return cell_2_node(displacement_fluct_cell(size,F))
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def displacement_avg_node(size,F):
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F_avg = np.average(F,axis=(0,1,2))
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return np.einsum('ml,ijkl->ijkm',F_avg-np.eye(3),coord0_node(F.shape[:3],size))
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def cell_2_node(cell_data):
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"""Interpolate cell data to nodal data."""
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n = ( cell_data + np.roll(cell_data,1,(0,1,2))
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+ np.roll(cell_data,1,(0,)) + np.roll(cell_data,1,(1,)) + np.roll(cell_data,1,(2,))
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+ np.roll(cell_data,1,(0,1)) + np.roll(cell_data,1,(1,2)) + np.roll(cell_data,1,(2,0)))*0.125
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return np.pad(n,((0,1),(0,1),(0,1))+((0,0),)*len(cell_data.shape[3:]),mode='wrap')
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def node_2_cell(node_data):
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"""Interpolate nodal data to cell data."""
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c = ( node_data + np.roll(node_data,1,(0,1,2))
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+ np.roll(node_data,1,(0,)) + np.roll(node_data,1,(1,)) + np.roll(node_data,1,(2,))
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+ np.roll(node_data,1,(0,1)) + np.roll(node_data,1,(1,2)) + np.roll(node_data,1,(2,0)))*0.125
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return c[:-1,:-1,:-1]
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