Diffusion µSpectre (feat/crystal_plasticity_finite) Browse

crystal_plasticity…

direct_comparison_small_strain.py
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			## Most of the following is copied from https://github.com/tdegeus/GooseFFT (MIT license)

			import numpy as np
			import scipy.sparse.linalg as sp
			import itertools

			import os
			import sys
			sys.path.append(os.path.join(os.getcwd(), "language_bindings/python"))
			import pyMuSpectre as µ


			# ----------------------------------- GRID ------------------------------------

			ndim = 2 # number of dimensions
			N = 99 #31 # number of voxels (assumed equal for all directions)
			offset = 3 #9
			ndof = ndim*2N**2 # number of degrees-of-freedom

			cell = µ.SystemFactory(µ.get_2d_cube(N),
			µ.get_2d_cube(1.),
			µ.Formulation.small_strain)

			# ---------------------- PROJECTION, TENSORS, OPERATIONS ----------------------

			# tensor operations/products: np.einsum enables index notation, avoiding loops
			# e.g. ddot42 performs $C_ij = A_ijkl B_lk$ for the entire grid
			trans2 = lambda A2 : np.einsum('ij... ->ji... ',A2 )
			ddot42 = lambda A4,B2: np.einsum('ijkl...,lk... ->ij... ',A4,B2)
			ddot44 = lambda A4,B4: np.einsum('ijkl...,lkmn...->ijmn...',A4,B4)
			dot22 = lambda A2,B2: np.einsum('ij... ,jk... ->ik... ',A2,B2)
			dot24 = lambda A2,B4: np.einsum('ij... ,jkmn...->ikmn...',A2,B4)
			dot42 = lambda A4,B2: np.einsum('ijkl...,lm... ->ijkm...',A4,B2)
			dyad22 = lambda A2,B2: np.einsum('ij... ,kl... ->ijkl...',A2,B2)

			shape = tuple((N for _ in range(ndim)))
			# identity tensor [single tensor]
			i = np.eye(ndim)
			def expand(arr):
			new_shape = (np.prod(arr.shape), np.prod(shape))
			ret_arr = np.zeros(new_shape)
			ret_arr[:] = arr.reshape(-1)[:, np.newaxis]
			return ret_arr.reshape((arr.shape, shape))

			# identity tensors [grid of tensors]
			I = expand(i)
			I4 = expand(np.einsum('il,jk',i,i))
			I4rt = expand(np.einsum('ik,jl',i,i))
			I4s = (I4+I4rt)/2.
			II = dyad22(I,I)

			# projection operator [grid of tensors]
			# NB can be vectorized (faster, less readable), see: "elasto-plasticity.py"
			# - support function / look-up list / zero initialize
			delta = lambda i,j: np.float(i==j) # Dirac delta function
			freq = np.arange(-(N-1)/2.,+(N+1)/2.) # coordinate axis -> freq. axis
			Ghat4 = np.zeros([ndim,ndim,ndim,ndim,N,N]) # zero initialize
			# - compute
			for i,j,l,m in itertools.product(range(ndim),repeat=4):
			for x,y in itertools.product(range(N), repeat=ndim):
			q = np.array([freq[x], freq[y]]) # frequency vector
			if not q.dot(q) == 0: # zero freq. -> mean
			Ghat4[i,j,l,m,x,y] = -(q[i]q[j]q[l]q[m])/(q.dot(q))*2+\
			(delta(j,l)q[i]q[m]+delta(j,m)q[i]q[l]+\
			delta(i,l)q[j]q[m]+delta(i,m)q[j]q[l])/(2.*q.dot(q))

			# (inverse) Fourier transform (for each tensor component in each direction)
			fft = lambda x: np.fft.fftshift(np.fft.fftn (np.fft.ifftshift(x),shape))
			ifft = lambda x: np.fft.fftshift(np.fft.ifftn(np.fft.ifftshift(x),shape))

			# functions for the projection 'G', and the product 'G : K : eps'
			G = lambda A2 : np.real( ifft( ddot42(Ghat4,fft(A2)) ) ).reshape(-1)
			K_deps = lambda depsm: ddot42(K4,depsm.reshape(ndim,ndim,N,N))
			G_K_deps = lambda depsm: G(K_deps(depsm))

			# ------------------- PROBLEM DEFINITION / CONSTITIVE MODEL -------------------

			# phase indicator: cubical inclusion of volume fraction (93)/(313)

			phase = np.zeros(shape); phase[-offset:,:offset,] = 1.
			phase = np.zeros(shape); phase[0,:] = 1.
			# material parameters + function to convert to grid of scalars
			param = lambda M0,M1: M0np.ones(shape)(1.-phase)+M1np.ones(shape)phase
			# K = param(0.833,8.33) # bulk modulus [grid of scalars]
			# mu = param(0.386,3.86) # shear modulus [grid of scalars]
			E2, E1 = 210e9, 70e9
			poisson = .33
			K2, K1 = (E/(3(1-2poisson)) for E in (E2, E1))
			m2, m1 = (E/(2*(1+poisson)) for E in (E2, E1))
			K = param(K1, K2)
			mu = param(m1, m2)

			# stiffness tensor [grid of tensors]
			K4 = KII+2.mu(I4s-1./3.II)

			hard = µ.material.MaterialHooke2d.make(cell, "hard",
			E2, poisson)
			soft = µ.material.MaterialHooke2d.make(cell, "soft",
			E1, poisson)

			for pixel in cell:
			if ((pixel[0] >= N-offset) and
			(pixel[1] < offset)):
			hard.add_pixel(pixel)
			else:
			soft.add_pixel(pixel)



			# ----------------------------- NEWTON ITERATIONS -----------------------------

			# initialize stress and strain tensor [grid of tensors]
			sig = np.zeros([ndim,ndim,N,N])
			eps = np.zeros([ndim,ndim,N,N])

			# set macroscopic loading
			DE = np.zeros([ndim,ndim,N,N])
			DE[0,1] += 0.01
			DE[1,0] += 0.01

			µDE = np.zeros([ndim**2, cell.size()])
			cell.evaluate_stress_tangent(µDE);
			µDE[:,:] = DE[:,:,0,0].reshape([-1, 1])


			# initial residual: distribute "DE" over grid using "K4"
			b = -G_K_deps(DE)
			G_K_deps2 = lambda de: cell.directional_stiffness(de)
			b2 = -G_K_deps2(µDE)

			eps += DE
			En = np.linalg.norm(eps)
			iiter = 0

			# iterate as long as the iterative update does not vanish
			class accumul(object):
			def __init__(self):
			self.counter = 0
			def __call__(self, dummy):
			self.counter += 1

			acc = accumul()
			while True:
			depsm,_ = sp.cg(tol=1.e-8,
			A = sp.LinearOperator(shape=(ndof,ndof),matvec=G_K_deps,dtype='float'),
			b = b,
			callback=acc
			) # solve linear system using CG
			eps += depsm.reshape(ndim,ndim,N,N) # update DOFs (array -> tens.grid)
			sig = ddot42(K4,eps) # new residual stress
			b = -G(sig) # convert residual stress to residual
			print('%10.2e'%(np.max(depsm)/En)) # print residual to the screen
			if np.linalg.norm(depsm)/En<1.e-5 and iiter>0: break # check convergence
			iiter += 1

			print("nb_cg: {0}".format(acc.counter))

			import matplotlib.pyplot as plt

			plt.pcolormesh(eps[0,0,:,:])
			plt.show()

muSpectre/bin/direct_comparison_small_strain.py5d0691f92a3ffeat/crystal_plasticity…

direct_comparison_small_strain.pyView Options

muSpectre/bin/direct_comparison_small_strain.py
5d0691f92a3ffeat/crystal_plasticity…

direct_comparison_small_strain.py
View Options