├── .DS_Store ├── examples ├── .DS_Store ├── mindlin_plate │ ├── .DS_Store │ ├── modelSetup.m │ └── main.m └── zigzag_plate │ ├── modelSetup.m │ ├── main.m │ └── results │ ├── solution_1.vtk │ ├── solution_2.vtk │ ├── solution_3.vtk │ ├── solution_4.vtk │ ├── solution_5.vtk │ ├── solution_6.vtk │ └── solution_7.vtk ├── include ├── .DS_Store ├── postProcessing │ ├── .DS_Store │ └── matlab2vtk.m ├── FEM │ ├── computeGlobalStiffness.m │ ├── elementShapeFunctions.m │ ├── MindlinPlateElement.m │ ├── ZigZagPlateElement.m │ └── makeABDH2.m └── preProcessing │ └── makeMesh.m └── README.md /.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/tjdodwell/matLam/HEAD/.DS_Store -------------------------------------------------------------------------------- /examples/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/tjdodwell/matLam/HEAD/examples/.DS_Store -------------------------------------------------------------------------------- /include/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/tjdodwell/matLam/HEAD/include/.DS_Store -------------------------------------------------------------------------------- /examples/mindlin_plate/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/tjdodwell/matLam/HEAD/examples/mindlin_plate/.DS_Store -------------------------------------------------------------------------------- /include/postProcessing/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/tjdodwell/matLam/HEAD/include/postProcessing/.DS_Store -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # matLam 2 | Various finite element implementations for composite laminate plates in MatLab for teaching and small research projects 3 | 4 | Self-contained light weight implementation of finite element analysis of rectangular composite laminate using various shear deformation theories (inc. Resiner-Mindlin Plate, Refine Zig-Zag Plate theory). 5 | 6 | Visualisation is done using open-source code paraview (https://www.paraview.org) - mesh and solution variables in matlab are converted to .vtk format using script matlab2vtk (in 'include/postProcessing') 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /examples/mindlin_plate/modelSetup.m: -------------------------------------------------------------------------------- 1 | function model = modelSetup 2 | 3 | model.plotme = true; 4 | model.vtk_filename_base = 'results/solution_'; 5 | 6 | model.type = 'Mindlin'; 7 | 8 | model.Lx = 10.0; % Length (x dimension) 9 | model.Ly = 5.0; % Width (y dimension) 10 | 11 | 12 | 13 | model.numPly = 1; % number of plys 14 | model.t = [0.2]; 15 | model.ss = [0.0]; 16 | 17 | model.material.E1 = 130; % kN/mm^2 18 | model.material.E2 = 9.25; % kN/mm^2 19 | model.material.G12 = 5.1; % kN/mm^2 20 | model.material.G23 = 5.13; % kN/mm^2 21 | model.material.G13 = 5.13; % kN/mm^2 22 | model.material.nu12 = 0.36; 23 | model.material.SF = 5/6; % Shear Correction Factor 24 | model.material.density = 1584E-9; % probably consistent units re: the above 25 | 26 | model.timesteps = 10; 27 | model.dt = 0.1; 28 | model.A = 1.0; 29 | model.omega = 1.0; 30 | 31 | model.meshRefinement = 4; 32 | 33 | model.integrationOption = 'reduced'; % Type of Integration, if you use 'complete' you get shear locking 34 | 35 | end -------------------------------------------------------------------------------- /examples/zigzag_plate/modelSetup.m: -------------------------------------------------------------------------------- 1 | function model = modelSetup 2 | 3 | model.plotme = true; 4 | model.vtk_filename_base = 'results/solution_'; 5 | 6 | model.type = 'zigzag'; 7 | 8 | model.Lx = 10.0; % Length (x dimension) 9 | model.Ly = 5.0; % Width (y dimension) 10 | 11 | 12 | 13 | model.numPly = 1; % number of plys 14 | model.t = [0.2]; 15 | model.ss = [0.0]; 16 | 17 | model.material.G13i = 2.0; 18 | model.material.G23i = 2.0; 19 | 20 | model.material.E1 = 130; % kN/mm^2 21 | model.material.E2 = 9.25; % kN/mm^2 22 | model.material.G12 = 5.1; % kN/mm^2 23 | model.material.G23 = 5.13; % kN/mm^2 24 | model.material.G13 = 5.13; % kN/mm^2 25 | model.material.nu12 = 0.36; 26 | model.material.SF = 5/6; % Shear Correction Factor 27 | model.material.density = 1584E-9; % probably consistent units re: the above 28 | 29 | model.timesteps = 10; 30 | model.dt = 0.1; 31 | model.A = 1.0; 32 | model.omega = 1.0; 33 | 34 | model.meshRefinement = 4; 35 | 36 | model.integrationOption = 'reduced'; % Type of Integration, if you use 'complete' you get shear locking 37 | 38 | end -------------------------------------------------------------------------------- /examples/mindlin_plate/main.m: -------------------------------------------------------------------------------- 1 | close all 2 | clear all 3 | 4 | include_folder = '../../include'; 5 | 6 | addpath(genpath(include_folder)); 7 | 8 | model = modelSetup(); 9 | msh = makeMesh(model); 10 | 11 | K = computeGlobalStiffness(model,msh); 12 | 13 | % Lets do boundary conditions 14 | 15 | msh.lhs = find(msh.coords(:,1) < 1e-6); 16 | msh.rhs = find(msh.coords(:,1) > model.Lx - 1e-6); 17 | 18 | bnd_left = [msh.lhs; msh.lhs + msh.nnod; msh.lhs + 2 * msh.nnod; msh.lhs + 3 * msh.nnod; msh.lhs + 4 * msh.nnod]; 19 | bnd_right = [msh.rhs; msh.rhs + msh.nnod; msh.rhs + 2 * msh.nnod; msh.rhs + 3 * msh.nnod; msh.rhs + 4 * msh.nnod]; 20 | 21 | bnd = [bnd_left; bnd_right]; 22 | 23 | free = 1 : msh.tdof; free(bnd) = []; 24 | 25 | % Solve the problem 26 | 27 | U1 = zeros(msh.tdof,1); U0 = zeros(msh.tdof,1); 28 | 29 | t = 0.0; 30 | 31 | for i = 1 : model.timesteps 32 | 33 | t = t + model.dt; 34 | 35 | U1(msh.rhs + 2 * msh.nnod) = model.A * sin(model.omega * t); 36 | 37 | U1(free) = K(free,free) \ (-K(free,bnd) * U1(bnd)); 38 | 39 | % Plot Solution 40 | 41 | scalar_point.name = 'displacement'; 42 | scalar_point.data = U1(2*msh.nnod + 1: 3*msh.nnod); 43 | 44 | matlab2vtk(strcat(model.vtk_filename_base,int2str(i),'.vtk'),'DMTA', msh, 'quad', [], scalar_point, []); 45 | 46 | end 47 | 48 | -------------------------------------------------------------------------------- /include/FEM/computeGlobalStiffness.m: -------------------------------------------------------------------------------- 1 | function K = computeGlobalStiffness(model,msh) 2 | 3 | % computeGlobalStiffness.m ~ Dr. Tim Dodwell 4 | 5 | % Construct global stiffness matrices K for solution of linear system of equations 6 | % K * d = f 7 | % where d = (w_1, \phi_x1, \phi_y1, ..., w_N, \phi_xN, \phi_yN)'. 8 | % 9 | %% Construct ABD and H matrices for each element 10 | % Element-wise construction of plate thickness, ply angle interpolation, 11 | % and constitutive relation matrices 12 | 13 | mat = makeABDH2(model); 14 | 15 | %% Allocate storage for global stiffness matrices 16 | 17 | indx_j = repmat(1:msh.nedof,msh.nedof,1); 18 | indx_i = indx_j'; 19 | Kindx.i = msh.e2g(:,indx_i(:)); 20 | Kindx.j = msh.e2g(:,indx_j(:)); 21 | 22 | Ke_all = zeros(msh.nedof^2,msh.nel); 23 | 24 | %% Construct stiffness matrices for transverse calculation 25 | 26 | % Loop over elements: 27 | 28 | switch lower(model.type) 29 | 30 | case 'mindlin' 31 | for ie = 1:msh.nel 32 | Ke_all(:, ie) = MindlinPlateElement(ie,msh,mat); 33 | end 34 | 35 | case 'zigzag' 36 | for ie = 1:msh.nel 37 | Ke_all(:, ie) = ZigZagPlateElement(ie,msh,mat); 38 | end 39 | 40 | end 41 | 42 | %% Write data into global storage 43 | K = sparse(Kindx.i',Kindx.j',Ke_all); 44 | 45 | end 46 | 47 | 48 | 49 | -------------------------------------------------------------------------------- /examples/zigzag_plate/main.m: -------------------------------------------------------------------------------- 1 | close all 2 | clear all 3 | 4 | include_folder = '../../include'; 5 | addpath(genpath(include_folder)); 6 | 7 | model = modelSetup(); 8 | 9 | msh = makeMesh(model); 10 | 11 | K = computeGlobalStiffness(model,msh); 12 | 13 | % Apply boundary conditions 14 | 15 | msh.lhs = find(msh.coords(:,1) < 1e-6); 16 | msh.rhs = find(msh.coords(:,1) > model.Lx - 1e-6); 17 | 18 | bnd_left = [msh.lhs; msh.lhs + msh.nnod; msh.lhs + 2 * msh.nnod; msh.lhs + 3 * msh.nnod; msh.lhs + 4 * msh.nnod; msh.lhs + 5 * msh.nnod; msh.lhs + 6 * msh.nnod]; 19 | bnd_right = [msh.rhs; msh.rhs + msh.nnod; msh.rhs + 2 * msh.nnod; msh.rhs + 3 * msh.nnod; msh.rhs + 4 * msh.nnod; msh.rhs + 5 * msh.nnod; msh.rhs + 6 * msh.nnod]; 20 | 21 | bnd = [bnd_left; bnd_right]; 22 | 23 | free = 1 : msh.tdof; free(bnd) = []; 24 | 25 | % Solve the problem 26 | 27 | U1 = zeros(msh.tdof,1); U0 = zeros(msh.tdof,1); 28 | 29 | t = 0.0; 30 | 31 | for i = 1 : model.timesteps 32 | 33 | t = t + model.dt; 34 | 35 | U1(msh.rhs + 2 * msh.nnod) = model.A * sin(model.omega * t); 36 | 37 | U1(free) = K(free,free) \ (-K(free,bnd) * U1(bnd)); 38 | 39 | % Plot Solution 40 | 41 | scalar_point.name = 'displacement'; 42 | scalar_point.data = U1(2*msh.nnod + 1: 3*msh.nnod); 43 | 44 | matlab2vtk(strcat(model.vtk_filename_base,int2str(i),'.vtk'),'DMTA', msh, 'quad', [], scalar_point, []); 45 | 46 | end 47 | 48 | -------------------------------------------------------------------------------- /include/FEM/elementShapeFunctions.m: -------------------------------------------------------------------------------- 1 | function [Ni,dNdX,detJ] = elementShapeFunctions(msh,ie,ip,integration_option) 2 | 3 | switch lower(integration_option); 4 | 5 | case 'full' 6 | 7 | [IP_X,IP_W] = ip_quad; 8 | 9 | [N, dNdu] = shapeFunctionQ4(IP_X); 10 | 11 | Ni = N{ip}; dNdui = dNdu{ip}; 12 | 13 | case 'reduced' 14 | 15 | Ni = msh.N{1}; dNdui = msh.dNdu{1}; 16 | 17 | end 18 | 19 | J = msh.coords(msh.elements(ie,:),:)'*dNdui'; 20 | detJ = det(J); 21 | 22 | dNdX = dNdui'*inv(J); 23 | 24 | end 25 | 26 | function [N, dNdu] = shapeFunctionQ4(IP_X) 27 | % TJD - June 2014 28 | nip = 4; 29 | N = cell(nip,1); 30 | dNdu = cell(nip,1); 31 | for i = 1:nip 32 | xi = IP_X(i,1); eta = IP_X(i,2); 33 | shp=0.25*[ (1-xi)*(1-eta); 34 | (1+xi)*(1-eta); 35 | (1+xi)*(1+eta); 36 | (1-xi)*(1+eta)]; 37 | deriv=0.25*[-(1-eta), -(1-xi); 38 | 1-eta, -(1+xi); 39 | 1+eta, 1+xi; 40 | -(1+eta), 1-xi]; 41 | N{i} = shp; 42 | dNdu{i} = deriv'; 43 | end 44 | end % end function shapeFunctionQ4 45 | 46 | function [IP_X,IP_W] = ip_quad 47 | % TJD - June 2014 48 | % Gauss quadrature for Q4 elements 49 | % option 'complete' (2x2) 50 | % option 'reduced' (1x1) 51 | % nip: Number of Integration Points 52 | % ipx: Gauss point locations 53 | % ipw: Gauss point weights 54 | IP_X=... 55 | [ -0.577350269189626 -0.577350269189626; 56 | 0.577350269189626 -0.577350269189626; 57 | 0.577350269189626 0.577350269189626; 58 | -0.577350269189626 0.577350269189626]; 59 | IP_W=[ 1;1;1;1]; 60 | end % end of function ip_quad 61 | 62 | -------------------------------------------------------------------------------- /include/FEM/MindlinPlateElement.m: -------------------------------------------------------------------------------- 1 | function Ke = MindlinPlateElement(ie,msh,mat) 2 | 3 | % Mindlin Plate - with selective integration to reduce shear locking 4 | 5 | % Degrees of freedom per element d = [u,v,w,tx,ty] - 5 dofs per node * 4 per element = 20 6 | 7 | K_elem = zeros(20); % Initialise Element Stiffness Matrix 8 | 9 | ne = msh.nnodel; % store nodes per element (4) to save writing out all the time 10 | 11 | % Loop over integration points 12 | 13 | for ip = 1:4 14 | 15 | % Load shape functions and their derivatives (wrt global coordinate 16 | % system) for integration point ip, as well as Jacobian determinant 17 | 18 | [~,dNdX,detJ] = elementShapeFunctions(msh,1,ip,'full'); 19 | 20 | % Construct [B], in plane matrix 21 | 22 | B = zeros(6,msh.nedof); 23 | 24 | B(1,1 : ne) = dNdX(:,1); % du/dx 25 | B(2,ne + 1 : 2 * ne) = dNdX(:,2); % dv/dx 26 | B(3,1 : ne) = dNdX(:,2); 27 | B(3,ne + 1 : 2 * ne) = dNdX(:,1); 28 | 29 | B(4,3*ne + 1 : 4*ne) = dNdX(:,1); % dtx / dx 30 | B(5,4*ne + 1 : 5*ne) = dNdX(:,2); % dty / dy 31 | 32 | B(6,3*ne + 1 : 4*ne) = dNdX(:,2); % dtx / dy 33 | B(6,4*ne + 1 : 5*ne) = dNdX(:,1); % dty / dx 34 | 35 | % Compute element stiffness matrices: 36 | 37 | % Construct stiffness matrix: 38 | % (Gauss integration weight is equal to one) 39 | 40 | C = [mat.A, mat.B; mat.B', mat.D]; 41 | 42 | 43 | K_elem = K_elem + (B' * C * B) * detJ; % Assumes weight is 1 44 | 45 | end 46 | 47 | % Reduced integration on shear terms to prevent shear locking 48 | 49 | ip = 1; 50 | 51 | % Load shape functions and their derivatives (wrt global coordinate 52 | % system) for integration point ip, as well as Jacobian determinant 53 | 54 | [Ni,dNdX,detJ] = elementShapeFunctions(msh,ie,ip,'reduced'); 55 | 56 | % Construct [B_s], matrix shear: 57 | 58 | B_s = zeros(2,msh.nedof); 59 | B_s(1,2*ne + 1 : 3*ne) = dNdX(:,1); % dw/dx 60 | B_s(2,2*ne + 1 : 3*ne) = dNdX(:,2); % dw/dy 61 | B_s(1,3*ne + 1 : 4*ne) = Ni; 62 | B_s(2,4*ne + 1 : 5*ne) = Ni; 63 | 64 | % Compute element stiffness matrices: 65 | 66 | K_elem = K_elem + (B_s'* mat.H *B_s) * 4.0 * detJ; % Reduced integration so 4 is integration weight 67 | 68 | % Store element stiffness matrices as columns 69 | 70 | Ke = K_elem(:); 71 | 72 | end -------------------------------------------------------------------------------- /include/postProcessing/matlab2vtk.m: -------------------------------------------------------------------------------- 1 | function matlab2vtk (filename,title, msh, elementType, scalar_point, vector_point, scalar_cell) 2 | 3 | output_unit = fopen(filename,'w+'); 4 | 5 | switch lower(elementType) 6 | 7 | case 'line' 8 | 9 | element_order = 2; 10 | cell_type = 3; 11 | 12 | case 'quad' 13 | 14 | element_order = 4; 15 | cell_type = 9; 16 | 17 | case 'hex' 18 | element_order = 8; 19 | cell_type = 12; 20 | 21 | case 'tet' 22 | 23 | element_order = 4; % Linear 24 | cell_type = 10; 25 | 26 | end 27 | 28 | fprintf ( output_unit, '# vtk DataFile Version 2.0\n' ); 29 | fprintf ( output_unit, '%s\n', title ); 30 | fprintf ( output_unit, 'ASCII\n' ); 31 | fprintf ( output_unit, '\n' ); 32 | fprintf ( output_unit, 'DATASET UNSTRUCTURED_GRID\n' ); 33 | fprintf ( output_unit, 'POINTS %d double\n', msh.nnod ); 34 | 35 | for i = 1 : msh.nnod 36 | fprintf ( output_unit, ' %f %f 0.0\n', msh.coords(i,:) ); 37 | end 38 | 39 | fprintf ( output_unit, '\n' ); 40 | fprintf ( output_unit, 'CELLS %d %d\n', msh.nel, (element_order+1)*msh.nel ); 41 | for ie = 1 : msh.nel 42 | fprintf ( output_unit, ' %d', element_order ); 43 | for j = 1 : element_order 44 | fprintf ( output_unit, ' %d', msh.elements(ie,j) - 1 ); % '-1' due to 0 node numbering 45 | end 46 | fprintf ( output_unit, '\n' ); 47 | end 48 | 49 | fprintf ( output_unit, '\n' ); 50 | fprintf ( output_unit, 'CELL_TYPES %d\n', msh.nel ); 51 | 52 | 53 | for i = 1 : msh.nel 54 | fprintf ( output_unit, '%d\n', cell_type); 55 | end 56 | 57 | 58 | if (isempty(scalar_point)==0) || (isempty(vector_point)==0) 59 | % POINT_DATA 60 | fprintf ( output_unit, '\n' ); 61 | fprintf ( output_unit, 'POINT_DATA %d\n', msh.nnod ); 62 | 63 | if isempty(scalar_point)==0 64 | % SCALAR 65 | fprintf ( output_unit, 'SCALARS %s double\n', scalar_point.name ); 66 | fprintf ( output_unit, 'LOOKUP_TABLE default\n' ); 67 | for i = 1 : msh.nnod 68 | fprintf ( output_unit, ' %f\n', scalar_point.data(i) ); 69 | end 70 | end 71 | 72 | if isempty(vector_point)==0 73 | % VECTOR 74 | fprintf ( output_unit, 'VECTORS %s double\n', vector_point.name); 75 | for i = 1 : msh.nnod 76 | fprintf ( output_unit, ' 0.0 0.0 %f\n', vector_point.data(i)); 77 | end 78 | end 79 | end 80 | 81 | % CELL_DATA 82 | if isempty(scalar_cell) == 0 83 | fprintf ( output_unit, '\n'); 84 | fprintf ( output_unit, 'CELL_DATA %d\n', msh.nel ); 85 | fprintf ( output_unit, 'SCALARS %s int\n', scalar_cell.name); 86 | fprintf ( output_unit, 'LOOKUP_TABLE default\n'); 87 | for i = 1 : msh.nel 88 | fprintf ( output_unit, ' %d\n', scalar_cell.data(i) ); 89 | end 90 | end 91 | 92 | 93 | return 94 | end 95 | -------------------------------------------------------------------------------- /include/FEM/ZigZagPlateElement.m: -------------------------------------------------------------------------------- 1 | function Ke = ZigZagPlateElement(ie,msh,mat) 2 | 3 | % ZigZag Plate - with selective integration to reduce shear locking 4 | 5 | % Degrees of freedom per element d = [u,v,w,tx,ty,phix,phiy] - 7 dofs per node * 4 per element = 28 6 | 7 | K_elem = zeros(28); % Initialise Element Stiffness Matrix 8 | 9 | ne = msh.nnodel; % store nodes per element (4) to save writing out all the time 10 | 11 | % Loop over integration points 12 | 13 | for ip = 1:4 14 | 15 | % Load shape functions and their derivatives (wrt global coordinate 16 | % system) for integration point ip, as well as Jacobian determinant 17 | 18 | [~,dNdX,detJ] = elementShapeFunctions(msh,1,ip,'full'); 19 | 20 | % Construct [B], in plane matrix 21 | 22 | B = zeros(10,msh.nedof); 23 | 24 | % In plane strains of mid-plane 25 | 26 | B(1,1 : ne) = dNdX(:,1); % du/dx 27 | B(2,ne + 1 : 2 * ne) = dNdX(:,2); % dv/dx 28 | B(3,1 : ne) = dNdX(:,2); 29 | B(3,ne + 1 : 2 * ne) = dNdX(:,1); 30 | 31 | B(4,3*ne + 1 : 4*ne) = dNdX(:,1); % dtx / dx 32 | B(5,5*ne + 1 : 6*ne) = dNdX(:,1); % dpsi_1/dx 33 | 34 | B(6,4*ne + 1 : 5*ne) = dNdX(:,2); % dty / dy 35 | B(7,6*ne + 1 : 7*ne) = dNdX(:,2); % dpsi_2/dy 36 | 37 | B(8,3*ne + 1 : 4*ne) = dNdX(:,2); % dtx / dy 38 | B(8,4*ne + 1 : 5*ne) = dNdX(:,1); % dty / dx 39 | 40 | B(9,5*ne + 1 : 6*ne) = dNdX(:,2); 41 | B(10,6*ne + 1 : 7*ne) = dNdX(:,1); 42 | 43 | % Compute element stiffness matrices: 44 | 45 | % Construct stiffness matrix: 46 | % (Gauss integration weight is equal to one) 47 | 48 | C = [mat.A, mat.B; mat.B', mat.D]; 49 | 50 | K_elem = K_elem + (B' * C * B) * detJ; % Assumes weight is 1 51 | 52 | end 53 | 54 | % Reduced integration on shear terms to prevent shear locking 55 | 56 | ip = 1; 57 | 58 | % Load shape functions and their derivatives (wrt global coordinate 59 | % system) for integration point ip, as well as Jacobian determinant 60 | 61 | [Ni,dNdX,detJ] = elementShapeFunctions(msh,ie,ip,'reduced'); 62 | 63 | % Construct [B_s], matrix shear: 64 | % Note that some on the shear terms are the other way around to Mindlin Plate formulation - just from papers on zigzag - has no effect though 65 | B_s = zeros(4,msh.nedof); 66 | B_s(1,2*ne + 1 : 3*ne) = dNdX(:,2); % dw/dy + theta_2 67 | B_s(1,4*ne + 1 : 5*ne) = Ni; 68 | B_s(2,6*ne + 1 : 7*ne) = Ni; % psi_2 69 | 70 | B_s(3,2*ne + 1 : 3*ne) = dNdX(:,1); % dw/dx + theta_1 71 | B_s(3,3*ne + 1 : 4*ne) = Ni; 72 | B_s(3,5*ne + 1 : 6*ne) = Ni; % psi_1 73 | 74 | % Compute element stiffness matrices: 75 | 76 | K_elem = K_elem + (B_s' * mat.H * B_s) * 4.0 * detJ; % Reduced integration so 4 is integration weight 77 | 78 | % Store element stiffness matrices as columns 79 | 80 | Ke = K_elem(:); 81 | 82 | end -------------------------------------------------------------------------------- /include/preProcessing/makeMesh.m: -------------------------------------------------------------------------------- 1 | function msh = makeMesh(model) 2 | % ----------------------------------------------------------------------- 3 | % This code is released under GNU LESSER GENERAL PUBLIC LICENSE v3 (LGPL) 4 | % 5 | % Details are provided in license.txt file in the main directory 6 | % 7 | % 1/8/14 - Dr T. J. Dodwell - University of Bath - tjd20@bath.ac.uk 8 | % ----------------------------------------------------------------------- 9 | 10 | % makemsh.m - Written (TJD - 3/6/2014) 11 | % 12 | % Creates Coarse Quadrilateral msh on [0,Lx] by [0,Ly] and refines uniformly to desired msh size 13 | % 14 | 15 | % -------------------------------- 16 | % (1) Set up Coarse Rectangle 17 | % -------------------------------- 18 | msh.coords = [0 0; 19 | model.Lx 0; 20 | model.Lx model.Ly; 21 | 0 model.Ly]; 22 | msh.elements = 1:4; 23 | % 24 | nodesOfRefinement = zeros(9,1); 25 | 26 | edgeTable = zeros(0,3); % Create an empty matrix with two columns 27 | 28 | 29 | for ii = 1:model.meshRefinement % For each refinement 30 | 31 | visitedEdges = 0; 32 | nelem = 0; 33 | inode = 0; 34 | newcoords = []; 35 | nodesPreviousRefinement = size(msh.coords(:,1),1); 36 | 37 | for ie = 1:size(msh.elements,1); % Each Element 38 | 39 | nodesOfRefinement(1:4) = msh.elements(ie,:); 40 | 41 | % First Add Mid Point 42 | inode=inode+1; 43 | newcoords(inode,:) = 0.5*(msh.coords(msh.elements(ie,1),:) + msh.coords(msh.elements(ie,3),:)); 44 | nodesOfRefinement(5)=inode + nodesPreviousRefinement; 45 | 46 | for edge = 1:4 % For Each Edge 47 | 48 | n1 = msh.elements(ie,edge); n2 = msh.elements(ie,mod(edge,4)+1); 49 | 50 | % Has Edge been visited before - if so return id = 1 and the node 51 | [id,oldNode] = edgeVisited(edgeTable,n1,n2); 52 | 53 | if id == 0 % If new edge add midpoint as new node 54 | inode=inode+1; 55 | nodesOfRefinement(5+edge) = inode + nodesPreviousRefinement; 56 | newcoords(inode,:) = 0.5*(msh.coords(n1,:)+msh.coords(n2,:)); 57 | edgeTable(visitedEdges+1,1:2) = [n1,n2]; 58 | edgeTable(visitedEdges+1,3) = inode + nodesPreviousRefinement; 59 | visitedEdges=visitedEdges+1; 60 | else 61 | nodesOfRefinement(5+edge) = oldNode; 62 | end 63 | 64 | end % For each edge 65 | 66 | newelements(nelem+1,:)=nodesOfRefinement([1,6,5,9]); 67 | newelements(nelem+2,:)=nodesOfRefinement([6,2,7,5]); 68 | newelements(nelem+3,:)=nodesOfRefinement([5,7,3,8]); 69 | newelements(nelem+4,:)=nodesOfRefinement([9,5,8,4]); 70 | nelem=nelem+4; 71 | 72 | end 73 | 74 | msh.elements = newelements; 75 | msh.coords = [msh.coords;newcoords]; 76 | 77 | end % for each refinelement 78 | 79 | % Mesh Constructed 80 | 81 | msh.nnod = size(msh.coords,1); 82 | msh.nel = size(msh.elements,1); 83 | msh.ndim = 2; 84 | 85 | switch lower(model.type) 86 | case 'mindlin' 87 | msh.dof = 5; 88 | case 'zigzag' 89 | msh.dof = 7; 90 | end 91 | 92 | msh.nnodel = 4; % Nodes per Element 93 | msh.nedof = msh.nnodel*msh.dof; 94 | msh.tdof = msh.dof*msh.nnod; 95 | 96 | 97 | % Setup local to global number of general element formulation with msh.dof degrees of freedom per node. 98 | msh.e2g = zeros(msh.nel,msh.nedof); 99 | for ie = 1 : msh.nel 100 | ne = msh.elements(ie,:); 101 | for j = 1 : msh.dof 102 | msh.e2g(ie,(j-1)*length(ne) + 1: j * length(ne)) = (j-1) * msh.nnod + ne; 103 | end 104 | end 105 | 106 | [msh.nip,msh.IP_X, msh.IP_w] = ip_quad(model.integrationOption); 107 | [msh.N, msh.dNdu] = shapeFunctionQ4(msh.IP_X); 108 | 109 | 110 | end 111 | 112 | function [id,node] = edgeVisited(edgeTable,n1,n2) 113 | 114 | id = 0; node = 0; 115 | 116 | numEdgesVisited = size(edgeTable,1); 117 | 118 | temp = find(edgeTable(:,1) == n1); 119 | for i = 1:length(temp) 120 | if edgeTable(temp(i),2) == n2 121 | id = 1; 122 | node = edgeTable(temp(i),3); 123 | end 124 | end 125 | if id == 0 126 | temp = find(edgeTable(:,2) == n1); 127 | for i = 1:length(temp) 128 | if edgeTable(temp(i),1) == n2 129 | id = 1; 130 | node = edgeTable(temp(i),3); 131 | end 132 | end 133 | end 134 | 135 | end 136 | 137 | function [N, dNdu] = shapeFunctionQ4(IP_X,nnodel) 138 | % TJD - June 2014 139 | nip = size(IP_X,1); 140 | N = cell(nip,1); 141 | dNdu = cell(nip,1); 142 | for i = 1:nip 143 | xi = IP_X(i,1); eta = IP_X(i,2); 144 | shp=0.25*[ (1-xi)*(1-eta); 145 | (1+xi)*(1-eta); 146 | (1+xi)*(1+eta); 147 | (1-xi)*(1+eta)]; 148 | deriv=0.25*[-(1-eta), -(1-xi); 149 | 1-eta, -(1+xi); 150 | 1+eta, 1+xi; 151 | -(1+eta), 1-xi]; 152 | N{i} = shp; 153 | dNdu{i} = deriv'; 154 | end 155 | end % end function shapeFunctionQ4 156 | 157 | function [nip,IP_X,IP_W] = ip_quad(option) 158 | % TJD - June 2014 159 | % Gauss quadrature for Q4 elements 160 | % option 'complete' (2x2) 161 | % option 'reduced' (1x1) 162 | % nip: Number of Integration Points 163 | % ipx: Gauss point locations 164 | % ipw: Gauss point weights 165 | switch option 166 | case 'complete' 167 | nip = 4; 168 | IP_X=... 169 | [ -0.577350269189626 -0.577350269189626; 170 | 0.577350269189626 -0.577350269189626; 171 | 0.577350269189626 0.577350269189626; 172 | -0.577350269189626 0.577350269189626]; 173 | IP_W=[ 1;1;1;1]; 174 | case 'reduced' 175 | nip = 1; 176 | IP_X=[0 0]; 177 | IP_W=[4]; 178 | end % end of switch 'option' 179 | end % end of function ip_quad 180 | 181 | -------------------------------------------------------------------------------- /include/FEM/makeABDH2.m: -------------------------------------------------------------------------------- 1 | function mat = makeABDH2(model) 2 | 3 | switch lower(model.type) 4 | 5 | 6 | case 'mindlin' 7 | 8 | % upper and lower coordinates 9 | z = zeros(1,model.numPly+1); 10 | z(1) = 0; 11 | for i = 2:model.numPly+1 12 | z(i) = z(i-1) + model.t(i-1); 13 | end 14 | z = z - mean(z); 15 | 16 | model.material.nu21=model.material.nu12*(model.material.E2/model.material.E1); 17 | factor=1-model.material.nu12*model.material.nu21; 18 | 19 | Q = zeros(5); 20 | Q(1,1)=model.material.E1/factor; 21 | Q(1,2)=model.material.nu12*model.material.E2/factor; 22 | Q(2,1)=Q(1,2); 23 | Q(2,2)=model.material.E2/factor; 24 | Q(3,3)=model.material.G12; 25 | Q(4,4)=model.material.SF*model.material.G23; 26 | Q(5,5)=model.material.SF*model.material.G13; 27 | 28 | %______________________________________________ 29 | A = zeros(5); B = zeros(5); D = zeros(5); H = zeros(5); T = zeros(5); 30 | for k=1:model.numPly 31 | 32 | phi = model.ss(k); 33 | 34 | % Transformation Matrix 35 | c = cos(phi); s = sin(phi); 36 | T(1,1) = c^2; T(1,2) = s^2; T(1,3) = 2*c*s; 37 | T(2,1) = s^2; T(2,2) = c^2; T(2,3) = -2*c*s; 38 | T(3,1) = -c*s; T(3,2) = c*s; T(3,3) = c^2-s^2; 39 | T(4,4) = c; T(4,5) = s; 40 | T(5,4) = -s; T(5,5) = c; 41 | 42 | % [Q] in structural axes 43 | invT = inv(T); 44 | Qbar= invT*Q*(invT'); 45 | 46 | A= A + Qbar*(z(k+1)-z(k)); 47 | B= B + Qbar*(z(k+1)^2-z(k)^2)/2; 48 | D= D + Qbar*(z(k+1)^3-z(k)^3)/3; 49 | H= H + Qbar*(z(k+1)-z(k)); 50 | 51 | end 52 | 53 | A = A(1:3,1:3); B = B(1:3,1:3); D = D(1:3,1:3); 54 | H = H(4:5,4:5); 55 | 56 | mat.A = A; mat.B = B; mat.D = D; mat.H = H; 57 | 58 | case 'zigzag' 59 | 60 | % Compute the interfaces 61 | 62 | % upper and lower coordinates 63 | z = zeros(1,model.numPly+1); 64 | z(1) = 0; 65 | for i = 2:model.numPly+1 66 | z(i) = z(i-1) + model.t(i-1); 67 | end 68 | z = z - mean(z); 69 | 70 | % Compute Q - composite matrix in local axis 71 | 72 | model.material.nu21=model.material.nu12*(model.material.E2/model.material.E1); 73 | factor=1-model.material.nu12*model.material.nu21; 74 | 75 | Q = zeros(5); 76 | Q(1,1)=model.material.E1/factor; 77 | Q(1,2)=model.material.nu12*model.material.E2/factor; 78 | Q(2,1)=Q(1,2); 79 | Q(2,2)=model.material.E2/factor; 80 | Q(3,3)=model.material.G12; 81 | 82 | G23 = model.material.G23; 83 | G13 = model.material.G13; 84 | 85 | G13i = model.material.G13i; 86 | G23i = model.material.G23i; 87 | 88 | E_int = model.material.E2; 89 | nu12_int = model.material.nu12; 90 | 91 | 92 | % Compute Zig-Zag Matrices 93 | 94 | [G1, G2] = computeGs(model,G13,G23,G13i,G23i); % Compute laminate shear moduli 95 | 96 | A = zeros(3); B = zeros(3,7); D = zeros(7); H = zeros(4); 97 | 98 | for k = 1 : model.numPly 99 | 100 | phi = model.ss(k); 101 | 102 | if (phi > 0) % This is a composite ply 103 | 104 | % Transformation Matrix 105 | c = cos(phi); s = sin(phi); 106 | T = zeros(3); 107 | T(1,1) = c^2; T(1,2) = s^2; T(1,3) = 2*c*s; 108 | T(2,1) = s^2; T(2,2) = c^2; T(2,3) = -2*c*s; 109 | T(3,1) = -c*s; T(3,2) = c*s; T(3,3) = c^2-s^2; 110 | 111 | % Rotate [Q] to structural axes 112 | invT = inv(T); 113 | Qk = invT * Q * (invT'); 114 | 115 | % Shear Matrix 116 | 117 | Hk = [cos(phi)^2 * G23 + sin(phi)^2 * G13, sin(phi) * cos(phi) * (G13 - G23); 118 | sin(phi) * cos(phi) * (G13 - G23), (cos(phi) ^2) * G13 + (sin(phi) ^ 2) * G23]; 119 | 120 | else % This is an interface layer 121 | 122 | % In this case notation is required. 123 | 124 | factor = 1.0 - nu12_int * nu12_int; 125 | 126 | Qk = [E_int/factor, nu12_int * E_int/factor, 0.0; nu12_int * E_int/factor, E_int/factor, 0.0; 0.0, 0.0, E_int / (2.0 * (1.0 + nu12_int))]; 127 | 128 | Hk = [G23i,0.0;0.0,G13i]; 129 | 130 | end 131 | 132 | % Compute A matrix - constant within each layer 133 | 134 | A = A + Qk * model.t(k); 135 | 136 | % Compute B matrix - linear in each layer - compute exactly with trapezoidal rule 137 | 138 | Bk0 = calB_phi(z(k),k,G13,G23,G13i,G23i,G1,G2,model); 139 | Bk1 = calB_phi(z(k+1),k,G13,G23,G13i,G23i,G1,G2,model); 140 | 141 | B = B + 0.5 * model.t(k) * Qk * (Bk1 + Bk0); 142 | 143 | % Compute D matrix - since quadratic in each layer - compute exactly with simpsons rule 144 | 145 | Bkhalf = calB_phi(0.5*(z(k)+z(k+1)),k,G13,G23,G13i,G23i,G1,G2,model); 146 | 147 | D = D + (1.0 / 6.0) * model.t(k) * (Bk0' * Qk * Bk0 + 4.0 * Bkhalf' * Qk * Bkhalf + Bk1' * Qk * Bk1); 148 | 149 | % Compute G matrix 150 | 151 | [beta1, beta2] = computeBetas(k,G1,G2,G13,G23,G13i,G23i,model); 152 | 153 | Bb = [1.0, beta2, 0.0, 0.0; 0.0, 0.0, 1.0, beta1]; 154 | 155 | H = H + model.t(k) * (Bb' * Hk * Bb) * model.t(k); 156 | 157 | end % end for each ply 158 | 159 | 160 | mat.A = A; mat.B = B; mat.D = D; mat.H = H; 161 | 162 | end 163 | 164 | end 165 | 166 | 167 | % model.t - contains layer thickness 168 | % model.ss - contains stacking sequence 169 | 170 | 171 | % Need a function which calculates Bphi 172 | 173 | function [G1, G2] = computeGs(model,G13,G23,G13i,G23i) 174 | G1 = 0.0; G2 = 0.0; 175 | for k = 1 : 2 : model.numPly 176 | phi = model.ss(k); 177 | Q11k = (cos(phi) ^ 2) * G13 + (sin(phi) ^ 2) * G23; 178 | Q22k = (cos(phi) ^ 2) * G23 + (sin(phi) ^ 2) * G13; 179 | G1 = G1 + model.t(k) / Q11k; 180 | G2 = G2 + model.t(k) / Q22k; 181 | end 182 | for k = 2 : 2 : model.numPly % For the interfaces 183 | G1 = G1 + model.t(k) / G13i; 184 | G2 = G2 + model.t(k) / G23i; 185 | end 186 | G1 = G1 / sum(model.t); G2 = G2 / sum(model.t); 187 | G1 = 1 / G1; G2 = 1 / G2; 188 | end 189 | 190 | function [beta1, beta2] = computeBetas(k,G1,G2,G13,G23,G13i,G23i,model) 191 | if (model.ss(k) < 0.0) % It is an interface 192 | beta1 = G1 / G13i - 1.0; 193 | beta2 = G2 / G23i - 1.0; 194 | else 195 | phi = model.ss(k); 196 | Q11k = (cos(phi) ^ 2) * G13 + (sin(phi) ^ 2) * G23; 197 | Q22k = (cos(phi) ^ 2) * G23 + (sin(phi) ^ 2) * G13; 198 | beta1 = G1 / Q11k - 1.0; 199 | beta2 = G2 / Q22k - 1.0; 200 | end 201 | end 202 | 203 | 204 | function B = calB_phi(z,k,G13,G23,G13i,G23i,G1,G2,model) 205 | B = zeros(3,7); 206 | [phi1, phi2] = calPhi(z,k,G13,G23,G13i,G23i,G1,G2,model); 207 | B(1,1) = z; B(1,2) = phi1; 208 | B(2,3) = z; B(2,4) = phi2; 209 | B(3,5) = z; B(3,6) = phi1; B(3,7) = phi2; 210 | end 211 | 212 | function [phi1, phi2] = calPhi(z,k,G13,G23,G13i,G23i,G1,G2,model) 213 | % Note that this is a linear function in z 214 | h = 0.5 * sum(model.t); 215 | phi = model.ss(k); 216 | Q11k = (cos(phi) ^ 2) * G13 + (sin(phi) ^ 2) * G23; 217 | Q22k = (cos(phi) ^ 2) * G23 + (sin(phi) ^ 2) * G13; 218 | phi1 = (z + h) * (G1 / Q11k - 1.0); 219 | phi2 = (z + h) * (G2 / Q22k - 1.0); 220 | if (k > 0) 221 | for i = 2 : model.numPly 222 | phi = model.ss(i); 223 | if(phi < 0.0) 224 | Q11i = G13i; 225 | Q22i = G23i; 226 | else 227 | Q11i = (cos(phi) ^ 2 * G13) + (sin(phi) ^ 2) * G23; 228 | Q22i = (cos(phi) ^ 2 * G23) + (sin(phi) ^ 2) * G13; 229 | end 230 | phi1 = phi1 + model.t(i-1) * (G1 / Q11i - G1 / Q11k); 231 | 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