├── src ├── .DS_Store ├── __pycache__ │ ├── element.cpython-37.pyc │ ├── material.cpython-37.pyc │ ├── gmsh_parser.cpython-37.pyc │ ├── vtk_writer.cpython-37.pyc │ └── phase_field_model.cpython-37.pyc ├── material.py ├── vtk_writer.py ├── gmsh_parser.py ├── phase_field_model.py └── element.py ├── examples ├── .DS_Store ├── dogbone │ └── .DS_Store ├── dogbone-quad │ ├── dogbone.gif │ └── force-disp.gif └── dogbone-coarse-quad │ └── .DS_Store ├── mesh-files ├── .DS_Store ├── single-quad.msh ├── block.msh ├── block.geo ├── dogbone.geo ├── dogbone-coarse-quad.msh └── dogbone-quad.msh ├── __pycache__ ├── element.cpython-37.pyc ├── material.cpython-37.pyc ├── element_PF.cpython-37.pyc ├── gmsh_parser.cpython-37.pyc ├── material_PF.cpython-37.pyc ├── vtk_writer.cpython-37.pyc ├── phase_field_model.cpython-37.pyc └── solid_mechanics_model.cpython-37.pyc ├── input-PhaseField.in ├── README.md └── Main-PhaseField.py /src/.DS_Store: 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1 25 | 5 3 2 5 1 1 2 3 4 26 | $EndElements 27 | -------------------------------------------------------------------------------- /input-PhaseField.in: -------------------------------------------------------------------------------- 1 | # problem dimensions 2 | dim = 2 3 | two_dimensional_problem_type = plane_stress 4 | 5 | # material properties 6 | E = 210000 7 | nu = 0.3 8 | Gc = 2.7 9 | el = 0.05 10 | 11 | # boundary conditions 12 | # format: [value (space) direction (space) gmsh_physcial_surface_name] 13 | Dirichlet_BC = [0.01 1 'right'] 14 | Dirichlet_BC = [0.0 1 'left'] 15 | Dirichlet_BC = [0.0 2 'bottom'] 16 | 17 | # mesh file 18 | mesh_path = ./mesh-files/dogbone-coarse-quad.msh 19 | 20 | # specify analysis type 21 | analysis_type = static 22 | solver = Newton-Raphson 23 | 24 | 25 | -------------------------------------------------------------------------------- /mesh-files/block.msh: -------------------------------------------------------------------------------- 1 | $MeshFormat 2 | 2.2 0 8 3 | $EndMeshFormat 4 | $PhysicalNames 5 | 5 6 | 1 1 "top" 7 | 1 2 "bottom" 8 | 1 3 "left" 9 | 1 4 "right" 10 | 2 5 "block" 11 | $EndPhysicalNames 12 | $Nodes 13 | 9 14 | 1 0 0 0 15 | 2 1 0 0 16 | 3 1 1 0 17 | 4 0 1 0 18 | 5 0.499999999998694 0 0 19 | 6 1 0.499999999998694 0 20 | 7 0.5000000000020591 1 0 21 | 8 0 0.5000000000020591 0 22 | 9 0.5000000000003766 0.5000000000003766 0 23 | $EndNodes 24 | $Elements 25 | 12 26 | 1 1 2 2 1 1 5 27 | 2 1 2 2 1 5 2 28 | 3 1 2 4 2 2 6 29 | 4 1 2 4 2 6 3 30 | 5 1 2 1 3 3 7 31 | 6 1 2 1 3 7 4 32 | 7 1 2 3 4 4 8 33 | 8 1 2 3 4 8 1 34 | 9 3 2 5 1 1 5 9 8 35 | 10 3 2 5 1 8 9 7 4 36 | 11 3 2 5 1 5 2 6 9 37 | 12 3 2 5 1 9 6 3 7 38 | $EndElements 39 | -------------------------------------------------------------------------------- /mesh-files/block.geo: -------------------------------------------------------------------------------- 1 | Mesh.MshFileVersion = 2.2; 2 | 3 | Nb_elems = 1; 4 | 5 | x_size = 1; 6 | y_size = 1; 7 | 8 | h_e = x_size/Nb_elems; 9 | 10 | Point(1) = {0, 0, 0, h_e}; 11 | Point(2) = {x_size, 0, 0, h_e}; 12 | Point(3) = {x_size, y_size, 0, h_e}; 13 | Point(4) = {0, y_size, 0, h_e}; 14 | 15 | Line(1) = {1, 2}; 16 | Line(2) = {2, 3}; 17 | Line(3) = {3, 4}; 18 | Line(4) = {4, 1}; 19 | 20 | Line Loop(1) = {1, 2, 3, 4}; 21 | Plane Surface(1) = {1}; 22 | 23 | Physical Line("top") = {3}; 24 | Physical Line("bottom") = {1}; 25 | Physical Line("left") = {4}; 26 | Physical Line("right") = {2}; 27 | 28 | Physical Surface("block") = {1}; 29 | 30 | Transfinite Surface "*"; 31 | Recombine Surface "*"; 32 | -------------------------------------------------------------------------------- /mesh-files/dogbone.geo: -------------------------------------------------------------------------------- 1 | Mesh.MshFileVersion = 2.2; 2 | 3 | l = 0.5; 4 | h = 0.5; 5 | r = 0.25; 6 | 7 | E = 0.05; 8 | e = 0.01; 9 | 10 | Point(1) = { l, -h, 0, E}; 11 | Point(2) = { l, h, 0, E}; 12 | Point(3) = { r, h, 0, E}; 13 | Point(4) = { 0, r, 0, e}; 14 | Point(5) = {-r, h, 0, E}; 15 | Point(6) = {-l, h, 0, E}; 16 | Point(7) = {-l, -h, 0, E}; 17 | Point(8) = {-r, -h, 0, E}; 18 | Point(9) = { 0, -r, 0, e}; 19 | Point(10) = {r, -h, 0, E}; 20 | 21 | Point(11) = {0, h, 0, e}; 22 | Point(12) = {0, -h, 0, e}; 23 | 24 | Line(1) = {1, 2}; 25 | Line(2) = {2, 3}; 26 | Circle(3) = {4, 11, 3}; 27 | Circle(4) = {5, 11, 4}; 28 | Line(5) = {5, 6}; 29 | Line(6) = {6, 7}; 30 | Line(7) = {7, 8}; 31 | Circle(8) = {9, 12, 8}; 32 | Circle(9) = {10, 12, 9}; 33 | Line(10) = {10, 1}; 34 | Line(11) = {4, 9}; 35 | 36 | Line Loop(1) = {1, 2, -3, -4, 5, 6, 7, -8, -9, 10}; 37 | Plane Surface(1) = {1}; 38 | Line{11} In Surface {1}; 39 | 40 | Physical Surface("all") = {1}; 41 | Physical Line("left") = {6}; 42 | Physical Line("right") = {1}; 43 | 44 | //Recombine Surface "*"; 45 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # **Phase field modeling of fracture and fatigue in brittle materials** 2 | *** 3 | 4 | This is an open source object oriented Python project developed for the simulation of fracture and fatigue in brittle materials via phase field approch. 5 | 6 | ## Structure of the subdirectories 7 | * ./src 8 | - all source files are included in this folder 9 | * ./examples 10 | - benchmark problems 11 | * ./mesh-files 12 | - gmsh files for the benchmark problems 13 | 14 | ## Instruction 15 | To run any of the benchmark problems, place the related "Main.py" and "input.in" in the root directory, which contains above subdirectories. 16 | 17 | ## Structure of the input file 18 | Input file of the phase field analysis is structures as follows: 19 | 20 | ## Benchmark examples 21 | ### Tensile test on dogbone specimen 22 | 23 | -------------------------------------------------------------------------------- /src/material.py: -------------------------------------------------------------------------------- 1 | ############################################################################## # Author: Roozbeh Rezakhani # Email: rrezakhani@gmail.com # # Elastic material characterized by Young's modulus and Poisson's ratio # ############################################################################## import numpy as np class material: def __init__(self, E, nu, Gc, el, dim, twoD_type): self.E = E self.nu = nu self.Gc = Gc self.el = el if (twoD_type == 'plane_strain'): self.C = self.E/((1+self.nu)*(1-2*self.nu))* \ np.array([[(1-self.nu), self.nu, 0], [ self.nu, (1-self.nu), 0], [ 0, 0, (1-2*self.nu)/2]]) elif (twoD_type == 'plane_stress'): self.C = self.E/(1-self.nu**2)* \ np.array([[ 1, self.nu, 0], [ self.nu, 1, 0], [ 0, 0, (1-self.nu)/2]]) def get_C(self): return self.C -------------------------------------------------------------------------------- /src/vtk_writer.py: -------------------------------------------------------------------------------- 1 | ############################################################################## 2 | # Author: Roozbeh Rezakhani 3 | # Email: rrezakhani@gmail.com 4 | # 5 | # 2-dimensional mesh file generate by gmsh file (with version 2) 6 | # 7 | ############################################################################## 8 | 9 | import numpy as np 10 | 11 | def vtk_writer(path, l, mesh, U, phi, F_int): 12 | 13 | dim = mesh.get_dim() 14 | num_nodes = mesh.get_num_nodes() 15 | num_elems = mesh.get_num_elems() 16 | 17 | # get the node list coordinates 18 | node_list = np.array(mesh.get_nodes()).astype(np.float) 19 | # 1 is subtracted from elem_list because node indices start from 0 20 | elem_list = np.array(mesh.get_elems()).astype(np.int) - 1 21 | 22 | f = open(path+"/out-"+str(l)+".vtu", "w") 23 | 24 | ######## unstructured grids syntax ######## 25 | f.write('\n') 26 | f.write('\t\n') 27 | f.write('\t\t\n') 28 | 29 | ######## points ######## 30 | f.write('\t\t\t\n') 31 | f.write('\t\t\t\t\n') 32 | for row in node_list: 33 | f.write('\t\t\t\t\t'+' '.join(map(lambda x: "{:.3f}\t".format(x), row))+'\n') 34 | f.write('\t\t\t\t\n') 35 | f.write('\t\t\t\n') 36 | 37 | ######## cells ######## 38 | f.write('\t\t\t\n') 39 | f.write('\t\t\t\t\n') 40 | for row in elem_list: 41 | f.write('\t\t\t\t\t'+' '.join(map(lambda x: "{:d}\t".format(x), row[2:]))+'\n') 42 | f.write('\t\t\t\t\n') 43 | f.write('\t\t\t\t\n') 44 | cnt = 0 45 | for row in elem_list: 46 | cnt += len(row[2:]) 47 | f.write('\t\t\t\t\t'+str(cnt)+'\n') 48 | f.write('\t\t\t\t\n') 49 | f.write('\t\t\t\t\n') 50 | for row in elem_list: 51 | if (row[0]==2): # index of triangular element in gmsh 52 | f.write('\t\t\t\t\t'+str(5)+'\n') 53 | if (row[0]==3): # index of quadrilateral element in gmsh 54 | f.write('\t\t\t\t\t'+str(9)+'\n') 55 | f.write('\t\t\t\t\n') 56 | f.write('\t\t\t\n') 57 | 58 | ######## point data ######## 59 | f.write('\t\t\t\n'); 60 | f.write('\t\t\t\t\n'.format('disp')) 61 | for i in range(num_nodes): 62 | if (dim==2): 63 | f.write('\t\t\t\t\t'+str(U[2*i])+'\t'+str(U[2*i+1])+'\t'+'0.000\n') 64 | if (dim==3): 65 | f.write('\t\t\t\t\t'+str(U[3*i])+'\t'+str(U[3*i+1])+'\t'+str(U[3*i+2])+'n') 66 | f.write('\t\t\t\t\n') 67 | f.write('\t\t\t\t\n'.format('d')) 68 | for i in range(num_nodes): 69 | f.write('\t\t\t\t\t'+str(phi[i])+'\n') 70 | f.write('\t\t\t\t\n') 71 | f.write('\t\t\t\t\n'.format('Fint')) 72 | for i in range(num_nodes): 73 | if (dim==2): 74 | f.write('\t\t\t\t\t'+str(F_int[2*i])+'\t'+str(F_int[2*i+1])+'\t'+'0.000\n') 75 | if (dim==3): 76 | f.write('\t\t\t\t\t'+str(F_int[3*i])+'\t'+str(F_int[3*i+1])+'\t'+str(F_int[3*i+2])+'n') 77 | f.write('\t\t\t\t\n') 78 | f.write('\t\t\t\n') 79 | 80 | ######## close the file ######## 81 | f.write('\t\t\n') 82 | f.write('\t\n') 83 | f.write('') 84 | f.close() 85 | -------------------------------------------------------------------------------- /src/gmsh_parser.py: -------------------------------------------------------------------------------- 1 | ############################################################################## 2 | # Author: Roozbeh Rezakhani 3 | # Email: rrezakhani@gmail.com 4 | # 5 | # 2-dimensional mesh file generate by gmsh file (with version 2) 6 | # 7 | ############################################################################## 8 | 9 | class gmsh_parser: 10 | 11 | def __init__(self, path, dim): 12 | 13 | self.path = path 14 | self.dim = dim 15 | self.num_phys_names = 0 16 | self.phys_array = [] 17 | self.num_nodes = 0 18 | self.nodes = [] 19 | self.num_bndry_elems = 0 20 | self.bndry_elems = [] 21 | self.num_elems = 0 22 | self.elems = [] 23 | 24 | mesh_file = open(self.path, 'r') 25 | line = mesh_file.readline() 26 | while (line != ''): # breaks when EOF is reached 27 | line = mesh_file.readline() 28 | 29 | if("$PhysicalNames" in line): 30 | self.num_phys_names = mesh_file.readline() 31 | for i in range(int(self.num_phys_names)): 32 | line = mesh_file.readline() 33 | self.phys_array.append([line.split(' ')[0], 34 | line.split(' ')[1], 35 | line.split(' ')[2].split('\n')[0][1:-1]]) 36 | 37 | if("$Nodes" in line): 38 | self.num_nodes = mesh_file.readline() 39 | for i in range(int(self.num_nodes)): 40 | line = mesh_file.readline() 41 | if(self.dim == 2): 42 | self.nodes.append([line.split(' ')[1], 43 | line.split(' ')[2], 44 | line.split(' ')[3]]) 45 | elif(self.dim == 3): 46 | self.nodes.append([line.split(' ')[1], 47 | line.split(' ')[2], 48 | line.split(' ')[3]]) 49 | 50 | if("$Elements" in line): 51 | num_elems_total = mesh_file.readline() 52 | for i in range(int(num_elems_total)): 53 | line = mesh_file.readline() 54 | if(self.dim == 2): 55 | if (line.split(' ')[1] == '1'): 56 | self.bndry_elems.append([line.split(' ')[1], # gmsh element type 57 | line.split(' ')[3], # phys. tag 58 | line.split(' ')[-2], 59 | line.split(' ')[-1].split('\n')[0]]) 60 | elif (line.split(' ')[1] == '3'): 61 | self.elems.append([line.split(' ')[1], # gmsh element type 62 | line.split(' ')[3], # phys. tag 63 | line.split(' ')[-4], 64 | line.split(' ')[-3], 65 | line.split(' ')[-2], 66 | line.split(' ')[-1]]) 67 | self.num_bndry_elems = len(self.bndry_elems) 68 | self.num_elems = len(self.elems) 69 | 70 | 71 | def get_dim(self): 72 | return self.dim 73 | 74 | def get_nodes(self): 75 | return self.nodes 76 | 77 | def get_num_nodes(self): 78 | return int(self.num_nodes) 79 | 80 | def get_elems(self): 81 | return self.elems 82 | 83 | def get_num_elems(self): 84 | return int(self.num_elems) 85 | 86 | def get_bndry_elems(self): 87 | return self.bndry_elems 88 | 89 | def get_phys_array(self): 90 | return self.phys_array 91 | -------------------------------------------------------------------------------- /src/phase_field_model.py: -------------------------------------------------------------------------------- 1 | ############################################################################## 2 | # Author: Roozbeh Rezakhani 3 | # Email: rrezakhani@gmail.com 4 | # 5 | # This is the phase_field_model class. 6 | # 7 | ############################################################################## 8 | 9 | import numpy as np 10 | from src.element import element 11 | 12 | class phase_field_model: 13 | 14 | def __init__(self, mat, mesh, dim): 15 | print('Solid Mechanics Model object is constructed!') 16 | self.mat = mat 17 | self.mesh = mesh 18 | self.dim = dim 19 | 20 | # construct elements object list 21 | elem_list = np.array(self.mesh.get_elems()).astype(np.int) 22 | node_list = np.array(self.mesh.get_nodes()).astype(np.float) 23 | self.elem_obj_list = [] 24 | for e in range(len(elem_list)): 25 | nodes = node_list[elem_list[e][2:]-1][:,:self.dim] 26 | neN = len(nodes) 27 | element_gmsh_type = elem_list[e,0] 28 | 29 | Le = np.array([]) 30 | LePF = np.array([]) 31 | for i in range(elem_list.shape[1]-2): 32 | Le = np.concatenate((Le, [2*elem_list[e][i+2]-1, 2*elem_list[e][i+2]])) 33 | LePF = np.concatenate((LePF, [elem_list[e][i+2]])) 34 | Le = Le.astype(np.int) 35 | LePF = LePF.astype(np.int) 36 | 37 | elem_obj = element(element_gmsh_type, neN, self.dim, nodes, Le, LePF) 38 | self.elem_obj_list.append(elem_obj) 39 | 40 | def compute_disp_stiffness_matrix(self, phi): 41 | num_nodes_total = self.mesh.get_num_nodes() 42 | C = self.mat.get_C() 43 | K = np.zeros((num_nodes_total*self.dim, num_nodes_total*self.dim)) 44 | 45 | for elem_obj in self.elem_obj_list: 46 | LePF = elem_obj.get_elem_PF_connectivity() 47 | phi_elem = phi[LePF-1] 48 | K_elem = elem_obj.compute_element_stiffness(C, phi_elem) 49 | Le = elem_obj.get_elem_connectivity() 50 | 51 | for i in range(len(Le)): 52 | for j in range(len(Le)): 53 | K[Le[i]-1][Le[j]-1] = K[Le[i]-1][Le[j]-1] + K_elem[i][j] 54 | return K 55 | 56 | def compute_disp_internal_forces(self, U, dU, phi): 57 | num_nodes_total = self.mesh.get_num_nodes() 58 | C = self.mat.get_C() 59 | F_int = np.zeros(num_nodes_total*self.dim) 60 | 61 | for elem_obj in self.elem_obj_list: 62 | Le = elem_obj.get_elem_connectivity() 63 | u_elem = U[Le-1] 64 | du_elem = dU[Le-1] 65 | LePF = elem_obj.get_elem_PF_connectivity() 66 | phi_elem = phi[LePF-1] 67 | F_int_elem = elem_obj.compute_element_internal_forces(C, u_elem, du_elem, phi_elem) 68 | 69 | for i in range(len(Le)): 70 | F_int[Le[i]-1] = F_int[Le[i]-1] + F_int_elem[i] 71 | 72 | return F_int 73 | 74 | def compute_PF_stiffness_matrix(self): 75 | num_nodes_total = self.mesh.get_num_nodes() 76 | K_phi = np.zeros((num_nodes_total, num_nodes_total)) 77 | 78 | for elem_obj in self.elem_obj_list: 79 | LePF = elem_obj.get_elem_PF_connectivity() 80 | K_phi_elem = elem_obj.compute_element_PF_stiffness(self.mat) 81 | 82 | for i in range(len(LePF)): 83 | for j in range(len(LePF)): 84 | K_phi[LePF[i]-1][LePF[j]-1] = K_phi[LePF[i]-1][LePF[j]-1] + K_phi_elem[i][j] 85 | 86 | return K_phi 87 | 88 | def compute_PF_residual(self, phi): 89 | num_nodes_total = self.mesh.get_num_nodes() 90 | res_phi = np.zeros(num_nodes_total) 91 | 92 | for elem_obj in self.elem_obj_list: 93 | LePF = elem_obj.get_elem_PF_connectivity() 94 | phi_elem = phi[LePF-1] 95 | res_phi_elem = elem_obj.compute_element_PF_residual(phi_elem, self.mat) 96 | 97 | for i in range(len(LePF)): 98 | res_phi[LePF[i]-1] = res_phi[LePF[i]-1] + res_phi_elem[0,i] 99 | 100 | return res_phi 101 | 102 | 103 | -------------------------------------------------------------------------------- /Main-PhaseField.py: -------------------------------------------------------------------------------- 1 | ############################################################################## 2 | # Author: Roozbeh Rezakhani 3 | # Email: rrezakhani@gmail.com 4 | # 5 | # This is the main file, which serves as the driver of the finite element 6 | # simulation. 7 | # 8 | ############################################################################## 9 | import time 10 | import numpy as np 11 | from scipy.sparse import csc_matrix 12 | from scipy.sparse.linalg import spsolve 13 | 14 | from src.phase_field_model import phase_field_model 15 | from src.gmsh_parser import gmsh_parser 16 | from src.material import material 17 | from src.vtk_writer import vtk_writer 18 | from termcolor import colored 19 | 20 | ############################################################################## 21 | # Read the input file 22 | input_file = open("./input-PhaseField.in", 'r') 23 | line = input_file.readline() 24 | disp_BC = [] 25 | trac_BC = [] 26 | while (line != ''): # breaks when EOF is reached 27 | key = line.split(' ')[0] 28 | if (key == 'dim'): 29 | dim = int(line.split(' ')[-1]) 30 | if (key == 'two_dimensional_problem_type'): 31 | two_dimensional_problem_type = line.split(' ')[-1].split('\n')[0] 32 | if (key == 'E'): 33 | E = float(line.split(' ')[-1]) 34 | if (key == 'nu'): 35 | nu = float(line.split(' ')[-1]) 36 | if (key == 'Gc'): 37 | Gc = float(line.split(' ')[-1]) 38 | if (key == 'el'): 39 | el = float(line.split(' ')[-1]) 40 | if (key == 'mesh_path'): 41 | mesh_path = line.split(' ')[-1].split('\n')[0] 42 | if (key == 'Dirichlet_BC'): 43 | disp_BC.append([line.split(' ')[-3][1:], 44 | line.split(' ')[-2], 45 | line.split(' ')[-1].split('\n')[0][1:-2]]) 46 | if (key == 'analysis_type'): 47 | analysis_type = line.split(' ')[-1].split('\n')[0] 48 | if (key == 'solver'): 49 | solver = line.split(' ')[-1].split('\n')[0] 50 | line = input_file.readline() 51 | 52 | out_dir = "./examples/"+mesh_path.split('/')[-1][:-4] 53 | 54 | ############################################################################## 55 | # Create the mesh object 56 | mesh = gmsh_parser(mesh_path, dim) 57 | num_nodes = mesh.get_num_nodes() 58 | 59 | ############################################################################## 60 | # Instantiate material class and initialize material properties 61 | mat = material(E, nu, Gc, el, dim, two_dimensional_problem_type) 62 | C = mat.get_C() 63 | 64 | ############################################################################## 65 | # Instantiate model class to build the general framework 66 | phase_field_model = phase_field_model(mat, mesh, dim) 67 | 68 | ############################################################################## 69 | # Initialize global displacement vector 70 | U = np.zeros(num_nodes*dim) 71 | dU = np.zeros(num_nodes*dim) 72 | 73 | ############################################################################## 74 | # Boundary condition arrays 75 | blocked = np.zeros(num_nodes*dim) 76 | u_bar = np.zeros(num_nodes*dim) 77 | 78 | # Apply prescribed boundary conditions 79 | phys_array = mesh.get_phys_array() 80 | bndry_elems = np.array(mesh.get_bndry_elems()).astype(np.int) 81 | 82 | for d_BC in disp_BC: 83 | val = float(d_BC[0]) 84 | comp = int(d_BC[1]) 85 | phys_tag = d_BC[2] 86 | for i in range(len(phys_array)): 87 | if(phys_array[i][2] == phys_tag): 88 | phys_index = int(phys_array[i][1]) 89 | phys_elems = bndry_elems[bndry_elems[:,1]==phys_index][:,2:] 90 | blocked[2*(np.unique(phys_elems)-1)+comp-1] = 1 91 | u_bar[2*(np.unique(phys_elems)-1)+comp-1] = val 92 | 93 | ############################################################################## 94 | # External force vector 95 | F_ext = np.zeros(num_nodes*dim) 96 | 97 | # Reaction force vector 98 | R_ext = np.zeros(num_nodes*dim) 99 | 100 | # Internal force vector 101 | F_int = np.zeros(num_nodes*dim) 102 | 103 | ############################################################################## 104 | # Construct PF (phase field) related variables 105 | phi = np.zeros(num_nodes) 106 | K_phi = np.zeros((num_nodes, num_nodes)) 107 | res_phi = np.zeros(num_nodes) 108 | 109 | ############################################################################## 110 | # Iterative solve of the equilibruim equation 111 | # Newton Raphson Method 112 | num_load_steps = 30 113 | itr_max = 10 114 | itr_tol = 1E-5 115 | F_ext_ = np.zeros(num_nodes*dim) 116 | u_bar_ = np.zeros(num_nodes*dim) 117 | res = np.zeros(num_nodes*dim) 118 | 119 | vtk_writer(out_dir, '0', mesh, U, phi, F_int) 120 | 121 | # starting time 122 | start = time.time() 123 | 124 | for l in range(num_load_steps): 125 | 126 | print("Load step {}:".format(l+1)) 127 | 128 | #===================================================================== 129 | # update essential and neumann boundary conditions 130 | u_bar_ += 1/num_load_steps * u_bar 131 | F_ext_ += 1/num_load_steps * F_ext 132 | 133 | #===================================================================== 134 | # update stiffness matrix of the displacement governing equation 135 | K = phase_field_model.compute_disp_stiffness_matrix(phi) 136 | 137 | #===================================================================== 138 | # compute residual of the displacement governing equation 139 | res = F_ext_ + R_ext - F_int 140 | 141 | #===================================================================== 142 | # Impose essential boundary conditions 143 | K_temp = np.copy(K) 144 | for m in range(num_nodes*dim): 145 | if(blocked[m]==1): 146 | for n in range(num_nodes*dim): 147 | if(blocked[n]==0): 148 | res[n] -= K[n,m] * u_bar_[m] 149 | K_temp[m,:] = 0.0 150 | K_temp[:,m] = 0.0 151 | K_temp[m,m] = 1.0 152 | res[m] = u_bar_[m] 153 | 154 | #===================================================================== 155 | # Loop on iterations for the DISPLACEMENT governing equation 156 | max_itr_reached = False 157 | for k in range(itr_max): 158 | 159 | # calculate displacement vector increment 160 | #U = np.dot(inv(K_temp), res) 161 | 162 | # calculate displacement vector increment using scipy csc matrix 163 | K_temp_csc = csc_matrix(K_temp) 164 | U = spsolve(K_temp_csc, res) 165 | 166 | # update the reaction forces 167 | for m in range(num_nodes*dim): 168 | if(blocked[m]==1): 169 | R_ext[m] = np.dot(K[m,:], U) 170 | 171 | # compute internal forces vector 172 | F_int = phase_field_model.compute_disp_internal_forces(U, dU, phi) 173 | 174 | # update residual and check convergence 175 | res = F_ext_ + R_ext - F_int 176 | tol = np.linalg.norm(res)/np.linalg.norm(F_ext_ + R_ext) 177 | 178 | print("Displacement solve: Iteration {} - tolerance = {}".format(k+1, tol)) 179 | if (tol < itr_tol): 180 | print(colored("Displacement solve converged!",'green')) 181 | break 182 | if (k == itr_max-1): 183 | print(colored("Maximum number of iteration in displacement solve is reached!",'red')) 184 | print(colored("Displacement solve did NOT converge!",'red')) 185 | max_itr_reached = True 186 | break 187 | if(max_itr_reached): 188 | break 189 | 190 | #===================================================================== 191 | # compute residual of the PF (phase field) governing equation 192 | res_phi = phase_field_model.compute_PF_residual(phi) 193 | 194 | #===================================================================== 195 | # update stiffness matrix of the PF (phase field) governing equation 196 | K_phi = phase_field_model.compute_PF_stiffness_matrix() 197 | 198 | #===================================================================== 199 | # Loop on iterations for the PF (phase field) governing equation 200 | for k in range(itr_max): 201 | 202 | # calculate phase field vector increment 203 | #dphi = np.dot(inv(K_phi), -res_phi) 204 | 205 | # calculate phase field vector increment using scipy csc matrix 206 | K_phi_csc = csc_matrix(K_phi) 207 | dphi = spsolve(K_phi_csc, -res_phi) 208 | 209 | # update the phase field vector 210 | phi = phi + dphi 211 | 212 | # update residual and check convergence 213 | res_phi= phase_field_model.compute_PF_residual(phi) 214 | tol = np.linalg.norm(res_phi) 215 | 216 | print("Phase field solve: Iteration {} - tolerance = {}".format(k+1, tol)) 217 | if (tol < itr_tol): 218 | print(colored("Phase field solve converged!",'green')) 219 | break 220 | if (k == itr_max-1): 221 | print(colored("Maximum number of iteration in displacement solve is reached!",'red')) 222 | print(colored("Phase field solve did NOT converge!",'red')) 223 | max_itr_reached = True 224 | break 225 | if(max_itr_reached): 226 | break 227 | 228 | #===================================================================== 229 | # write vtk file 230 | vtk_writer(out_dir, l+1, mesh, U, phi, F_int) 231 | 232 | # end time 233 | end = time.time() 234 | 235 | # total time taken 236 | print(f"Runtime of the program is {end - start}") 237 | 238 | 239 | 240 | 241 | 242 | -------------------------------------------------------------------------------- /src/element.py: -------------------------------------------------------------------------------- 1 | ############################################################################## 2 | # Author: Roozbeh Rezakhani 3 | # Email: rrezakhani@gmail.com 4 | # 5 | # This is the Element class. 6 | # 7 | ############################################################################## 8 | 9 | import numpy as np 10 | from numpy.linalg import inv 11 | from numpy.linalg import det 12 | 13 | class element: 14 | 15 | def __init__(self, element_gmsh_type, neN, dim, nodes, Le, LePF): 16 | if (element_gmsh_type == 1): 17 | self.elem_type = '2node-line' 18 | if (element_gmsh_type == 2): 19 | self.elem_type = '3node-triangular' 20 | self.nqp = 1 21 | if (element_gmsh_type == 3): 22 | self.elem_type = '4node-quadrilateral' 23 | self.nqp = 4 24 | self.neN = neN 25 | self.dim = dim 26 | self.nodes = nodes 27 | self.Le = Le 28 | self.K_elem = np.zeros((self.neN*self.dim, self.neN*self.dim)) 29 | self.F_int_elem = np.zeros(self.neN*self.dim) 30 | self.elas_str_ene = np.zeros(self.nqp) 31 | if (self.dim == 2): 32 | self.stress_total = np.zeros(self.nqp*3) 33 | self.strain_total = np.zeros(self.nqp*3) 34 | self.stress_incr = np.zeros(self.nqp*3) 35 | self.strain_incr = np.zeros(self.nqp*3) 36 | else: 37 | self.stress_total = np.zeros(self.nqp*6) 38 | self.strain_total = np.zeros(self.nqp*6) 39 | self.stress_incr = np.zeros(self.npp*6) 40 | self.strain_incr = np.zeros(self.nqp*6) 41 | self.LePF = LePF 42 | self.K_phi_elem = np.zeros((self.neN, self.neN)) 43 | self.res_phi_elem = np.zeros(self.neN) 44 | 45 | def compute_element_stiffness(self, C, phi_elem): 46 | self.K_elem = np.zeros((self.neN*self.dim, self.neN*self.dim)) 47 | w_qp, qp = self.get_w_qp() 48 | 49 | for p in range(self.nqp): 50 | xi = qp[p][0] 51 | eta = qp[p][1] 52 | 53 | J = self.compute_jacobian(xi, eta) 54 | B = self.compute_B_matrix(xi, eta) 55 | N = self.get_shape_function(xi, eta) 56 | 57 | # update material property matrix 58 | C_degraded = (1-np.dot(np.transpose(N),phi_elem))**2 * C 59 | 60 | K_elem_qp = np.dot(np.dot(np.transpose(B), C_degraded), B) * w_qp[p] * det(J) 61 | self.K_elem = self.K_elem + K_elem_qp 62 | 63 | return self.K_elem 64 | 65 | def compute_element_internal_forces(self, C, u_elem, du_elem, phi_elem): 66 | self.F_int_elem = np.zeros(self.neN*self.dim) 67 | w_qp, qp = self.get_w_qp() 68 | 69 | for p in range(self.nqp): 70 | xi = qp[p][0] 71 | eta = qp[p][1] 72 | 73 | J = self.compute_jacobian(xi, eta) 74 | B = self.compute_B_matrix(xi, eta) 75 | N = self.get_shape_function(xi, eta) 76 | 77 | # update material property matrix 78 | C_degraded = (1-np.dot(np.transpose(N),phi_elem))**2 * C 79 | 80 | # compute element total and incremental strain 81 | self.strain_incr[p*3:(p+1)*3] = np.dot(B, du_elem) 82 | self.strain_total[p*3:(p+1)*3] = np.dot(B, u_elem) 83 | 84 | # compute element total and incremental stress 85 | self.stress_incr[p*3:(p+1)*3] = np.dot(C_degraded, self.strain_incr[p*3:(p+1)*3]) 86 | self.stress_total[p*3:(p+1)*3] = np.dot(C_degraded, self.strain_total[p*3:(p+1)*3]) 87 | 88 | # compute elastic strain energy at the current qp 89 | self.elas_str_ene[p] = 0.5 * np.dot(np.dot(self.strain_total[p*3:(p+1)*3], C), 90 | self.strain_total[p*3:(p+1)*3]) 91 | 92 | # compute element internal forces at current qp 93 | F_int_elem_qp = np.dot(np.transpose(B), self.stress_total[p*3:(p+1)*3]) \ 94 | * w_qp[p] * np.linalg.det(J) 95 | self.F_int_elem = self.F_int_elem + F_int_elem_qp 96 | 97 | return self.F_int_elem 98 | 99 | def compute_jacobian(self, xi, eta): 100 | dNdxi = self.get_iso_shape_function_derivative(xi, eta) 101 | J = np.dot(np.transpose(dNdxi), self.nodes) 102 | 103 | return J 104 | 105 | def compute_shape_function_derivate(self, xi, eta): 106 | dNdxi = self.get_iso_shape_function_derivative(xi, eta) 107 | J = self.compute_jacobian(xi, eta) 108 | invJ = inv(J) 109 | dNdx = np.dot(invJ, np.transpose(dNdxi)) 110 | 111 | return dNdx 112 | 113 | def compute_B_matrix(self, xi, eta): 114 | dNdx = self.compute_shape_function_derivate(xi, eta) 115 | B = np.zeros((3, 2*self.neN)) 116 | B[0, 0:2*self.neN+1:2] = dNdx[0,:] 117 | B[1, 1:2*self.neN+1:2] = dNdx[1,:] 118 | B[2, 0:2*self.neN+1:2] = dNdx[1,:] 119 | B[2, 1:2*self.neN+1:2] = dNdx[0,:] 120 | 121 | return B 122 | 123 | def compute_B_PF_matrix(self, xi, eta): 124 | dNdx = self.compute_shape_function_derivate(xi, eta) 125 | B_PF = dNdx 126 | 127 | return B_PF 128 | 129 | def get_shape_function(self, xi, eta): 130 | # Isoparametric quadrilateral element shape functions 131 | N = 1/4 * np.array([(1-xi)*(1-eta), 132 | (1+xi)*(1-eta), 133 | (1+xi)*(1+eta), 134 | (1-xi)*(1+eta)]) 135 | 136 | return N 137 | 138 | def get_iso_shape_function_derivative(self, xi, eta): 139 | # Derivate of isoparametric quadrilateral element shape functions 140 | dNdxi = 1/4 * np.array([[-(1-eta), -(1-xi)], 141 | [ (1-eta), -(1+xi)], 142 | [ (1+eta), (1+xi)], 143 | [-(1+eta), (1-xi)]]) 144 | return dNdxi 145 | 146 | def get_w_qp(self): 147 | w_qp = np.array([1, 1, 1, 1]) 148 | qp = np.array([[ 0.5774, 0.5774], 149 | [ 0.5774, -0.5774], 150 | [-0.5774, 0.5774], 151 | [-0.5774, -0.5774]]) 152 | return w_qp, qp 153 | 154 | def compute_qp_stress(self, u_elem, du_elem): 155 | #RR link to material class goes here! 156 | return u_elem 157 | 158 | def get_elem_connectivity(self): 159 | return self.Le 160 | 161 | def get_elem_PF_connectivity(self): 162 | return self.LePF 163 | 164 | def compute_element_PF_residual(self, phi_elem, mat): 165 | self.res_phi_elem = np.zeros(self.neN) 166 | w_qp, qp = self.get_w_qp() 167 | 168 | Gc = mat.Gc 169 | el = mat.el 170 | 171 | for p in range(self.nqp): 172 | xi = qp[p][0] 173 | eta = qp[p][1] 174 | 175 | J = self.compute_jacobian(xi, eta) 176 | B_PF = self.compute_B_PF_matrix(xi, eta) 177 | N = self.get_shape_function(xi, eta) 178 | N = N[np.newaxis] 179 | 180 | # elastic strain energy at the current qp 181 | Eels = self.elas_str_ene[p] 182 | 183 | # compute element internal forces at current qp 184 | res_phi_elem_qp = (Gc/el * np.dot(np.dot(np.transpose(N), N), phi_elem) + \ 185 | Gc*el * np.dot(np.dot(np.transpose(B_PF), B_PF), phi_elem) + \ 186 | 2*Eels * np.dot(np.dot(np.transpose(N), N), phi_elem) - \ 187 | 2*Eels * N) \ 188 | * w_qp[p] * np.linalg.det(J) 189 | 190 | self.res_phi_elem = self.res_phi_elem + res_phi_elem_qp 191 | 192 | return self.res_phi_elem 193 | 194 | def compute_element_PF_stiffness(self, mat): 195 | self.K_phi_elem = np.zeros((self.neN, self.neN)) 196 | w_qp, qp = self.get_w_qp() 197 | 198 | Gc = mat.Gc 199 | el = mat.el 200 | 201 | for p in range(self.nqp): 202 | xi = qp[p][0] 203 | eta = qp[p][1] 204 | 205 | J = self.compute_jacobian(xi, eta) 206 | B_PF = self.compute_B_PF_matrix(xi, eta) 207 | N = self.get_shape_function(xi, eta) 208 | N = N[np.newaxis] 209 | 210 | # elastic strain energy at the current qp 211 | Eels = self.elas_str_ene[p] 212 | 213 | K_phi_elem_qp = (Gc/el * np.dot(np.transpose(N), N) + \ 214 | Gc*el * np.dot(np.transpose(B_PF), B_PF) + \ 215 | 2*Eels * np.dot(np.transpose(N), N)) \ 216 | * w_qp[p] * np.linalg.det(J) 217 | self.K_phi_elem = self.K_phi_elem + K_phi_elem_qp 218 | 219 | return self.K_phi_elem -------------------------------------------------------------------------------- /mesh-files/dogbone-coarse-quad.msh: -------------------------------------------------------------------------------- 1 | $MeshFormat 2 | 2.2 0 8 3 | $EndMeshFormat 4 | $PhysicalNames 5 | 3 6 | 1 2 "left" 7 | 1 3 "right" 8 | 2 1 "all" 9 | $EndPhysicalNames 10 | $Nodes 11 | 471 12 | 1 0.5 -0.5 0 13 | 2 0.5 0.5 0 14 | 3 0.25 0.5 0 15 | 4 0 0.25 0 16 | 5 -0.25 0.5 0 17 | 6 -0.5 0.5 0 18 | 7 -0.5 -0.5 0 19 | 8 -0.25 -0.5 0 20 | 9 0 -0.25 0 21 | 10 0.25 -0.5 0 22 | 11 0.5 -0.4500000000001387 0 23 | 12 0.5 -0.4000000000002774 0 24 | 13 0.5 -0.3500000000004161 0 25 | 14 0.5 -0.3000000000005548 0 26 | 15 0.5 -0.2500000000006935 0 27 | 16 0.5 -0.2000000000008322 0 28 | 17 0.5 -0.1500000000009709 0 29 | 18 0.5 -0.1000000000011096 0 30 | 19 0.5 -0.05000000000124821 0 31 | 20 0.5 -1.376121439022882e-12 0 32 | 21 0.5 0.04999999999875171 0 33 | 22 0.5 0.09999999999889042 0 34 | 23 0.5 0.1499999999990291 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2645 | 1060 3 2 1 1 631 228 1355 931 2646 | 1061 3 2 1 1 431 1134 103 102 2647 | 1062 3 2 1 1 753 988 44 43 2648 | 1063 3 2 1 1 225 1429 1304 371 2649 | 1064 3 2 1 1 370 395 1086 378 2650 | 1065 3 2 1 1 152 153 872 799 2651 | 1066 3 2 1 1 154 153 932 971 2652 | 1067 3 2 1 1 1237 107 106 835 2653 | 1068 3 2 1 1 359 48 47 1334 2654 | 1069 3 2 1 1 620 739 1527 1062 2655 | 1070 3 2 1 1 639 1275 1516 1007 2656 | 1071 3 2 1 1 1121 440 1419 267 2657 | 1072 3 2 1 1 1192 701 1157 528 2658 | 1073 3 2 1 1 1571 1134 330 281 2659 | 1074 3 2 1 1 1570 988 1380 237 2660 | 1075 3 2 1 1 1093 1526 1289 655 2661 | 1076 3 2 1 1 1309 1500 1291 376 2662 | 1077 3 2 1 1 820 833 72 73 2663 | 1078 3 2 1 1 680 1369 13 14 2664 | 1079 3 2 1 1 123 122 1110 1446 2665 | 1080 3 2 1 1 64 63 1112 1445 2666 | 1081 3 2 1 1 1349 266 932 760 2667 | 1082 3 2 1 1 1261 664 872 1168 2668 | 1083 3 2 1 1 1507 447 370 457 2669 | 1084 3 2 1 1 1474 518 225 900 2670 | 1085 3 2 1 1 374 1173 997 214 2671 | 1086 3 2 1 1 674 924 1559 221 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1201 1477 2752 | 1167 3 2 1 1 392 875 414 1090 2753 | 1168 3 2 1 1 665 1441 563 1018 2754 | 1169 3 2 1 1 816 605 423 460 2755 | 1170 3 2 1 1 731 439 1204 234 2756 | 1171 3 2 1 1 498 978 539 1235 2757 | 1172 3 2 1 1 252 1444 733 788 2758 | 1173 3 2 1 1 215 1175 1457 1146 2759 | 1174 3 2 1 1 199 1560 1148 678 2760 | 1175 3 2 1 1 1195 843 1323 338 2761 | 1176 3 2 1 1 240 671 1295 340 2762 | 1177 3 2 1 1 675 981 460 1434 2763 | 1178 3 2 1 1 672 956 234 496 2764 | 1179 3 2 1 1 1189 906 1089 381 2765 | 1180 3 2 1 1 1436 902 1228 258 2766 | 1181 3 2 1 1 1468 1321 357 582 2767 | 1182 3 2 1 1 191 1135 581 522 2768 | 1183 3 2 1 1 87 88 754 914 2769 | 1184 3 2 1 1 28 29 565 927 2770 | 1185 3 2 1 1 620 1416 346 739 2771 | 1186 3 2 1 1 639 1552 351 1275 2772 | 1187 3 2 1 1 1253 732 710 278 2773 | 1188 3 2 1 1 1399 961 407 286 2774 | 1189 3 2 1 1 1180 346 1416 1091 2775 | 1190 3 2 1 1 1414 351 1552 348 2776 | 1191 3 2 1 1 1066 174 175 1313 2777 | 1192 3 2 1 1 624 132 131 1417 2778 | 1193 3 2 1 1 411 427 537 777 2779 | 1194 3 2 1 1 463 929 280 536 2780 | 1195 3 2 1 1 67 1240 727 66 2781 | 1196 3 2 1 1 126 1008 1149 125 2782 | 1197 3 2 1 1 1181 609 832 1145 2783 | 1198 3 2 1 1 1433 420 1458 712 2784 | 1199 3 2 1 1 372 655 251 1200 2785 | 1200 3 2 1 1 462 376 255 1442 2786 | 1201 3 2 1 1 1010 454 881 880 2787 | 1202 3 2 1 1 794 382 913 921 2788 | 1203 3 2 1 1 175 176 1302 1231 2789 | 1204 3 2 1 1 131 130 1315 1232 2790 | 1205 3 2 1 1 1419 440 1409 1481 2791 | 1206 3 2 1 1 1157 701 1156 1518 2792 | 1207 3 2 1 1 1443 1299 510 185 2793 | 1208 3 2 1 1 1196 1021 1300 182 2794 | 1209 3 2 1 1 1145 175 174 1181 2795 | 1210 3 2 1 1 712 131 132 1433 2796 | 1211 3 2 1 1 1462 450 1051 1055 2797 | 1212 3 2 1 1 1549 200 1030 1569 2798 | 1213 3 2 1 1 766 1323 669 265 2799 | 1214 3 2 1 1 430 1295 667 850 2800 | 1215 3 2 1 1 440 1121 1024 227 2801 | 1216 3 2 1 1 701 1192 1430 526 2802 | 1217 3 2 1 1 1424 942 1158 292 2803 | 1218 3 2 1 1 1367 561 1120 598 2804 | 1219 3 2 1 1 18 19 1162 790 2805 | 1220 3 2 1 1 77 78 910 841 2806 | 1221 3 2 1 1 1480 1268 691 804 2807 | 1222 3 2 1 1 184 1408 566 839 2808 | 1223 3 2 1 1 699 898 1273 335 2809 | 1224 3 2 1 1 1071 433 1201 654 2810 | 1225 3 2 1 1 729 504 1364 1073 2811 | 1226 3 2 1 1 1329 254 1170 888 2812 | 1227 3 2 1 1 385 789 793 1123 2813 | 1228 3 2 1 1 384 1016 734 1174 2814 | 1229 3 2 1 1 1533 662 281 592 2815 | 1230 3 2 1 1 1534 1098 237 663 2816 | 1231 3 2 1 1 1093 1311 1558 572 2817 | 1232 3 2 1 1 1309 331 1208 792 2818 | 1233 3 2 1 1 172 969 629 171 2819 | 1234 3 2 1 1 134 714 1305 135 2820 | 1235 3 2 1 1 951 260 1215 982 2821 | 1236 3 2 1 1 1107 507 1218 1217 2822 | 1237 3 2 1 1 1496 1227 381 1089 2823 | 1238 3 2 1 1 1546 495 258 1228 2824 | 1239 3 2 1 1 1062 201 1341 756 2825 | 1240 3 2 1 1 1007 638 1402 1288 2826 | 1241 3 2 1 1 768 1155 1542 1183 2827 | 1242 3 2 1 1 764 1153 1513 1154 2828 | 1243 3 2 1 1 1532 844 1412 241 2829 | 1244 3 2 1 1 1535 829 1413 304 2830 | 1245 3 2 1 1 967 994 236 1163 2831 | 1246 3 2 1 1 586 1113 588 1262 2832 | 1247 3 2 1 1 948 860 1562 619 2833 | 1248 3 2 1 1 950 706 1203 409 2834 | 1249 3 2 1 1 1171 402 1197 925 2835 | 1250 3 2 1 1 896 403 1136 379 2836 | 1251 3 2 1 1 801 577 1046 1027 2837 | 1252 3 2 1 1 803 642 1206 1205 2838 | 1253 3 2 1 1 648 287 974 471 2839 | 1254 3 2 1 1 271 244 1370 877 2840 | 1255 3 2 1 1 1442 255 423 605 2841 | 1256 3 2 1 1 1200 251 1204 439 2842 | 1257 3 2 1 1 728 978 1449 681 2843 | 1258 3 2 1 1 633 1444 1238 687 2844 | 1259 3 2 1 1 116 601 1353 117 2845 | 1260 3 2 1 1 57 931 1355 58 2846 | 1261 3 2 1 1 294 1229 1453 1242 2847 | 1262 3 2 1 1 315 787 1454 1244 2848 | 1263 3 2 1 1 969 172 173 1185 2849 | 1264 3 2 1 1 714 134 133 962 2850 | 1265 3 2 1 1 289 1101 691 1268 2851 | 1266 3 2 1 1 291 1250 566 1408 2852 | 1267 3 2 1 1 506 1034 849 1531 2853 | 1268 3 2 1 1 1190 1394 741 1537 2854 | 1269 3 2 1 1 106 520 320 835 2855 | 1270 3 2 1 1 47 1241 513 1334 2856 | 1271 3 2 1 1 543 707 1437 1083 2857 | 1272 3 2 1 1 546 721 1230 1106 2858 | 1273 3 2 1 1 1063 1550 172 171 2859 | 1274 3 2 1 1 1065 1551 134 135 2860 | 1275 3 2 1 1 761 445 1132 966 2861 | 1276 3 2 1 1 1448 197 219 1267 2862 | 1277 3 2 1 1 575 1092 165 166 2863 | 1278 3 2 1 1 861 1553 141 140 2864 | 1279 3 2 1 1 1532 891 873 1055 2865 | 1280 3 2 1 1 1535 1568 1213 1569 2866 | 1281 3 2 1 1 223 1555 264 1067 2867 | 1282 3 2 1 1 270 890 257 484 2868 | 1283 3 2 1 1 296 1361 337 1088 2869 | 1284 3 2 1 1 302 1265 783 926 2870 | 1285 3 2 1 1 249 878 1548 1276 2871 | 1286 3 2 1 1 649 874 1333 1223 2872 | 1287 3 2 1 1 885 1186 390 1354 2873 | 1288 3 2 1 1 1193 1102 261 1564 2874 | 1289 3 2 1 1 240 340 1252 975 2875 | 1290 3 2 1 1 1195 338 938 698 2876 | 1291 3 2 1 1 95 94 295 1404 2877 | 1292 3 2 1 1 36 35 205 933 2878 | 1293 3 2 1 1 1278 1540 1451 1116 2879 | 1294 3 2 1 1 868 1528 1452 1247 2880 | 1295 3 2 1 1 1269 405 1299 1443 2881 | 1296 3 2 1 1 1567 870 1021 1196 2882 | 1297 3 2 1 1 628 201 1062 1212 2883 | 1298 3 2 1 1 1078 638 1007 1351 2884 | 1299 3 2 1 1 1338 532 1149 1008 2885 | 1300 3 2 1 1 531 449 727 1240 2886 | 1301 3 2 1 1 622 357 1321 332 2887 | 1302 3 2 1 1 677 581 1135 621 2888 | 1303 3 2 1 1 1004 192 1198 755 2889 | 1304 3 2 1 1 726 1482 1420 892 2890 | 1305 3 2 1 1 1327 167 166 1438 2891 | 1306 3 2 1 1 1328 139 140 508 2892 | 1307 3 2 1 1 237 928 45 1570 2893 | 1308 3 2 1 1 281 224 104 1571 2894 | 1309 3 2 1 1 1476 493 510 708 2895 | 1310 3 2 1 1 1487 313 1300 569 2896 | 1311 3 2 1 1 629 969 1253 640 2897 | 1312 3 2 1 1 1305 714 1399 282 2898 | 1313 3 2 1 1 873 891 821 660 2899 | 1314 3 2 1 1 1213 1568 1282 594 2900 | 1315 3 2 1 1 824 890 270 1248 2901 | 1316 3 2 1 1 740 1555 223 1249 2902 | 1317 3 2 1 1 1104 723 1284 1327 2903 | 1318 3 2 1 1 1447 399 1330 1328 2904 | 1319 3 2 1 1 1191 971 932 266 2905 | 1320 3 2 1 1 1166 799 872 664 2906 | 1321 3 2 1 1 493 1476 1072 955 2907 | 1322 3 2 1 1 313 1487 1388 919 2908 | 1323 3 2 1 1 171 629 801 170 2909 | 1324 3 2 1 1 135 1305 803 136 2910 | 1325 3 2 1 1 175 1145 882 176 2911 | 1326 3 2 1 1 131 712 1133 130 2912 | 1327 3 2 1 1 1178 364 240 975 2913 | 1328 3 2 1 1 1398 659 1195 698 2914 | 1329 3 2 1 1 982 1215 246 1475 2915 | 1330 3 2 1 1 1217 1218 1216 1467 2916 | 1331 3 2 1 1 230 875 392 915 2917 | 1332 3 2 1 1 334 1441 665 1292 2918 | 1333 3 2 1 1 1398 698 938 284 2919 | 1334 3 2 1 1 1178 975 1252 456 2920 | 1335 3 2 1 1 877 1083 1437 1505 2921 | 1336 3 2 1 1 471 1106 1230 1479 2922 | 1337 3 2 1 1 779 1404 492 1287 2923 | 1338 3 2 1 1 478 933 494 1119 2924 | 1339 3 2 1 1 655 372 1311 1093 2925 | 1340 3 2 1 1 376 462 331 1309 2926 | 1341 3 2 1 1 1334 513 1137 359 2927 | 1342 3 2 1 1 835 320 1566 1237 2928 | 1343 3 2 1 1 728 895 346 1359 2929 | 1344 3 2 1 1 633 984 351 1415 2930 | 1345 3 2 1 1 325 1532 1055 1051 2931 | 1346 3 2 1 1 400 1535 1569 1030 2932 | 1347 3 2 1 1 706 406 506 1203 2933 | 1348 3 2 1 1 860 408 1190 1562 2934 | 1349 3 2 1 1 339 1259 1530 749 2935 | 1350 3 2 1 1 568 1389 1536 905 2936 | 1351 3 2 1 1 284 428 423 255 2937 | 1352 3 2 1 1 456 239 1204 251 2938 | 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