├── solvers ├── heat_solver_1d.m ├── heat_solver_2d.m ├── navier_stokes_2d.m ├── possion_solver_2d.m ├── wave_solver_1d.m ├── wave_solver_2d.m ├── elasticity_solver_2d.m ├── convection_diffusion_reaction_1d.m ├── convection_diffusion_reaction_2d.m └── possion_solver_1d.m ├── fe_utils ├── assemble_module │ ├── assemble_matrix_steady_2d.m │ ├── assemble_vector_steady_2d.m │ ├── gauss_int_test_steady_2d.m │ ├── gauss_int_trial_test_steady_2d.m │ ├── gauss_int_test_steady_1d.m │ ├── assemble_vector_steady_1d.m │ ├── gauss_int_trial_test_steady_1d.m │ └── assemble_matrix_steady_1d.m ├── mesh │ ├── generate_P_T.m │ ├── generate_boundarynodes.m │ └── generate_Pb_Tb.m ├── error_compute │ ├── compute_error_on_nodes.m │ ├── compute_error_infinite_norm.m │ ├── select_evalution_points.m │ ├── FE_function_1d.m │ ├── guass_int_error_1d.m │ └── compute_error_Hs_norm.m ├── gauss_formula │ └── generate_gauss_formula_1d.m ├── treat_boundary │ └── treat_boundary_condition.m └── basis_function │ └── FE_local_basis.m ├── include.m ├── images ├── logo.png └── screenshot.png ├── plot └── plotConvergenceRate.m ├── example └── test_possion_example1.m ├── README.md └── LICENSE /solvers/heat_solver_1d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/heat_solver_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/navier_stokes_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/possion_solver_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/wave_solver_1d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/wave_solver_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/elasticity_solver_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/convection_diffusion_reaction_1d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /solvers/convection_diffusion_reaction_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/assemble_matrix_steady_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/assemble_vector_steady_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/gauss_int_test_steady_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/gauss_int_trial_test_steady_2d.m: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /include.m: -------------------------------------------------------------------------------- 1 | addpath(genpath(pwd), '-begin'); 2 | rmpath(genpath('./.git')); 3 | savepath; -------------------------------------------------------------------------------- /images/logo.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/images/logo.png -------------------------------------------------------------------------------- /images/screenshot.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/images/screenshot.png -------------------------------------------------------------------------------- /plot/plotConvergenceRate.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/plot/plotConvergenceRate.m -------------------------------------------------------------------------------- /fe_utils/mesh/generate_P_T.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/fe_utils/mesh/generate_P_T.m -------------------------------------------------------------------------------- /fe_utils/error_compute/compute_error_on_nodes.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/fe_utils/error_compute/compute_error_on_nodes.m -------------------------------------------------------------------------------- /fe_utils/gauss_formula/generate_gauss_formula_1d.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/fe_utils/gauss_formula/generate_gauss_formula_1d.m -------------------------------------------------------------------------------- /fe_utils/error_compute/compute_error_infinite_norm.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/chtld/Finite_Element_Method_Matlab/HEAD/fe_utils/error_compute/compute_error_infinite_norm.m -------------------------------------------------------------------------------- /fe_utils/error_compute/select_evalution_points.m: -------------------------------------------------------------------------------- 1 | function points = select_evalution_points(vertices) 2 | [~, gauss_points] = generate_gauss_formula_1d(vertices, 4); 3 | points = [vertices gauss_points]; 4 | end 5 | -------------------------------------------------------------------------------- /fe_utils/error_compute/FE_function_1d.m: -------------------------------------------------------------------------------- 1 | function result = FE_function_1d(x, uh_local, vertices, basis_type, der) 2 | number_of_local_basis = size(uh_local,1); 3 | result = 0; 4 | for k = 1: number_of_local_basis 5 | result = result + uh_local(k) * FE_local_basis(x, vertices, basis_type, k, der); 6 | end 7 | end -------------------------------------------------------------------------------- /fe_utils/error_compute/guass_int_error_1d.m: -------------------------------------------------------------------------------- 1 | function result = guass_int_error_1d (exact_solution_fun, gauss_weights, gauss_points, uh_local, vertices, basis_type, der) 2 | result = sum(gauss_weights .* ... 3 | (feval(exact_solution_fun, gauss_points)... 4 | - FE_function_1d(gauss_points, uh_local, vertices, basis_type, der)) .^ 2); 5 | end 6 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/gauss_int_test_steady_1d.m: -------------------------------------------------------------------------------- 1 | function result=gauss_int_test_steady_1d(right_hand_side, vertices, basis_type_test, basis_index_test,... 2 | der_test,gauss_type) 3 | [gauss_weight,gauss_point]=generate_gauss_formula_1d(vertices,gauss_type); 4 | result = sum(gauss_weight.*right_hand_side(gauss_point)... 5 | .*FE_local_basis(gauss_point,vertices,basis_type_test,basis_index_test,der_test)); 6 | end -------------------------------------------------------------------------------- /fe_utils/mesh/generate_boundarynodes.m: -------------------------------------------------------------------------------- 1 | function boundarynodes = generate_boundarynodes(pde, Pb) 2 | boundarynodes = zeros(2, 2); 3 | if pde.leftBoundaryType == "Dirichlet" 4 | boundarynodes(:, 1) = [-1; 1]; 5 | elseif pde.leftBoundaryType == "Neumann" 6 | boundarynodes(:, 1) = [-2; 1]; 7 | end 8 | if pde.rightBoundaryType == "Dirichlet" 9 | boundarynodes(:, 2) = [-1; size(Pb, 2)]; 10 | elseif pde.rightBoundaryType == "Neumann" 11 | boundarynodes(:, 2) = [-1; size(Pb, 2)]; 12 | end 13 | end -------------------------------------------------------------------------------- /fe_utils/assemble_module/assemble_vector_steady_1d.m: -------------------------------------------------------------------------------- 1 | function b=assemble_vector_steady_1d(right_hand_side,mesh,FE,basis_type_test,der_test,gauss_type) 2 | b=zeros(size(FE.Pb,2),1); 3 | number_of_elements=size(FE.Tb,2); 4 | number_of_local_basis=size(FE.Tb,1); 5 | for n=1:number_of_elements 6 | vertices=mesh.P(:,mesh.T(:,n)); 7 | for i=1:number_of_local_basis 8 | b(FE.Tb(i,n))=b(FE.Tb(i,n))+ gauss_int_test_steady_1d(right_hand_side, vertices, basis_type_test, i,... 9 | der_test,gauss_type); 10 | end 11 | end 12 | end -------------------------------------------------------------------------------- /fe_utils/error_compute/compute_error_Hs_norm.m: -------------------------------------------------------------------------------- 1 | function error = compute_error_Hs_norm(exact_solution_fun, solution, P, T, Tb, basis_type, der, gauss_type) 2 | error = 0.0; 3 | number_of_elments = size(Tb,2); 4 | for n = 1: number_of_elments 5 | vertices = P(:,T(:,n)); 6 | uh_local = solution(Tb(:,n)); 7 | [gauss_weights, gauss_points] = generate_gauss_formula_1d(vertices, gauss_type); 8 | local_error = guass_int_error_1d(exact_solution_fun, gauss_weights, gauss_points, uh_local, vertices, basis_type, der); 9 | error=error + local_error; 10 | end 11 | error=sqrt(error); 12 | -------------------------------------------------------------------------------- /fe_utils/assemble_module/gauss_int_trial_test_steady_1d.m: -------------------------------------------------------------------------------- 1 | function result=gauss_int_trial_test_steady_1d(coef_fun, vertices, basis_type_trial,basis_type_test,basis_index_trial, basis_index_test,... 2 | der_trial,der_test,gauss_type) 3 | [gauss_weight,gauss_point]=generate_gauss_formula_1d(vertices,gauss_type); 4 | result = sum(gauss_weight.*coef_fun(gauss_point)... 5 | .*FE_local_basis(gauss_point,vertices,basis_type_trial,basis_index_trial,der_trial)... 6 | .*FE_local_basis(gauss_point,vertices,basis_type_test,basis_index_test,der_test)); 7 | end -------------------------------------------------------------------------------- /fe_utils/assemble_module/assemble_matrix_steady_1d.m: -------------------------------------------------------------------------------- 1 | function A = assemble_matrix_steady_1d(coef_fun,mesh,FE,basis_type_trial,basis_type_test,der_trial,der_test,gauss_type) 2 | A=zeros(size(FE.Pb,2),size(FE.Pb,2)); 3 | number_of_elements=size(FE.Tb,2); 4 | number_of_local_basis=size(FE.Tb,1); 5 | for n=1:number_of_elements 6 | vertices=mesh.P(:,mesh.T(:,n)); 7 | for j=1: number_of_local_basis 8 | for i=1:number_of_local_basis 9 | A(FE.Tb(i,n),FE.Tb(j,n))= A(FE.Tb(i,n),FE.Tb(j,n))... 10 | +gauss_int_trial_test_steady_1d(coef_fun, vertices, basis_type_trial,basis_type_test,... 11 | j, i,der_trial,der_test,gauss_type); 12 | end 13 | end 14 | end 15 | end 16 | -------------------------------------------------------------------------------- /fe_utils/treat_boundary/treat_boundary_condition.m: -------------------------------------------------------------------------------- 1 | function [A,b]=treat_boundary_condition(A,b,FE, pde) 2 | number_of_boundarynodes=size(FE.boundarynodes,2); 3 | for k=1:number_of_boundarynodes 4 | if FE.boundarynodes(1, k) == -1 5 | A(FE.boundarynodes(2,k),:) = 0; 6 | A(FE.boundarynodes(2,k), FE.boundarynodes(2,k)) = 1; 7 | b(FE.boundarynodes(2,k))= pde.Dirichlet(FE.Pb(FE.boundarynodes(2,k))); 8 | elseif FE.boundarynodes(1, k) == -2 9 | x = FE.Pb(FE.boundarynodes(2, k)); 10 | if pde.rightBoundaryType == "Neumann" 11 | b(FE.boundarynodes(2, k)) = b(FE.boundarynodes(2, k)) + pde.coefficient(x) .* pde.Neumann(x); 12 | end 13 | if pde.leftBoundaryType == "Neumann" 14 | b(FE.boundarynodes(2, k)) = b(FE.boundarynodes(2, k)) - pde.coefficient(x) .* pde.Neumann(x); 15 | end 16 | end 17 | end -------------------------------------------------------------------------------- /solvers/possion_solver_1d.m: -------------------------------------------------------------------------------- 1 | function [solution, error] = possion_solver_1d(pde, mesh, FE, basis_type_trial, basis_type_test) 2 | % possion_solver 3 | % 1-d possion_solver for possion equation 4 | % @author: chtld 5 | % @date: 20191013 6 | % @param pde: the pde struct, include the coefficient function, exact 7 | % solution, the right hand side and the boundary conditions of different 8 | % kind. 9 | A = assemble_matrix_steady_1d(pde.coefficient, mesh, FE, basis_type_trial, basis_type_test, 1, 1, 4); 10 | b = assemble_vector_steady_1d(pde.right_hand_side, mesh, FE, basis_type_test, 0, 4); 11 | [A, b] = treat_boundary_condition(A, b, FE, pde); 12 | solution = A \ b; 13 | error.l2_norm = compute_error_Hs_norm(pde.exact, solution, mesh.P, mesh.T, FE.Tb, basis_type_test, 0, 4); 14 | error.h1_semi_norm = compute_error_Hs_norm(pde.exact_x, solution, mesh.P, mesh.T, FE.Tb, basis_type_test, 1, 4); 15 | error.infinite_norm = compute_error_infinite_norm(pde.exact, solution, mesh.P, mesh.T, FE.Tb, basis_type_test, 0); 16 | end -------------------------------------------------------------------------------- /fe_utils/basis_function/FE_local_basis.m: -------------------------------------------------------------------------------- 1 | function result=FE_local_basis(x,vertices,basis_type,basis_index,der) 2 | h=(vertices(2)-vertices(1)); 3 | if basis_type==101 4 | if basis_index==1 5 | if der==0 6 | result=(vertices(2)-x)/h; 7 | elseif der==1 8 | result=-1/h; 9 | end 10 | elseif basis_index==2 11 | if der==0 12 | result=(x-vertices(1))/h; 13 | elseif der==1 14 | result=1/h; 15 | end 16 | end 17 | elseif basis_type == 102 18 | if basis_index == 1 19 | if der == 0 20 | result = 2 .* ((x - vertices(1)) ./ h) .^ 2 - 3 .* (x - vertices(1)) ./ h + 1; 21 | elseif der == 1 22 | result = 4 .* (x - vertices(1)) ./ (h.^2) - 3 / h; 23 | end 24 | elseif basis_index == 3 25 | if der == 0 26 | result = 2 .* ((x - vertices(1)) / h) .^ 2 - (x - vertices(1)) / h; 27 | elseif der == 1 28 | result = 4 .* (x - vertices(1)) ./ (h .^ 2) - 1 / h; 29 | end 30 | elseif basis_index == 2 31 | if der == 0 32 | result = -4 .* ((x - vertices(1)) / h) .^ 2 + 4 .* (x - vertices(1)) / h; 33 | elseif der == 1 34 | result = -8 .* (x - vertices(1)) / (h .^ 2) + 4 / h; 35 | end 36 | end 37 | elseif basis_type == 103 38 | 39 | end 40 | end -------------------------------------------------------------------------------- /fe_utils/mesh/generate_Pb_Tb.m: -------------------------------------------------------------------------------- 1 | function [Pb, Tb] = generate_Pb_Tb(P, T, basis_type) 2 | if basis_type == 101 3 | Pb = P; 4 | Tb = T; 5 | elseif basis_type == 102 6 | Pb = zeros(1, length(P) + length(T)); 7 | Tb = zeros(3, length(T)); 8 | PIndexToPbIndex = zeros(1, length(P)); 9 | k = 1; 10 | for i = 1: length(T) 11 | index = T(:, i); 12 | if PIndexToPbIndex(index(1)) == 0 13 | PIndexToPbIndex(index(1)) = k; 14 | Pb(:, k) = P(:, index(1)); 15 | k = k + 1; 16 | end 17 | vertices = P(:, index); 18 | Pb(:, k) = 0.5 * (vertices(1) + vertices(2)); k = k + 1; 19 | mid = k - 1; 20 | if PIndexToPbIndex(index(2)) == 0 21 | PIndexToPbIndex(index(2)) = k; 22 | Pb(:, k) = P(:, index(2)); 23 | k = k + 1; 24 | end 25 | Tb(:, i) = [PIndexToPbIndex(index(1)); mid; PIndexToPbIndex(index(2));]; 26 | end 27 | elseif basis_type == 103 28 | Pb = zeros(1, length(P) + length(T)); 29 | Tb = zeros(4, length(T)); 30 | PIndexToPbIndex = zeros(1, length(P)); 31 | k = 1; 32 | for i = 1: length(T) 33 | index = T(:, i); 34 | if PIndexToPbIndex(index(1)) == 0 35 | PIndexToPbIndex(index(1)) = k; 36 | Pb(:, k) = P(:, index(1)); 37 | k = k + 1; 38 | end 39 | vertices = P(:, index); 40 | Pb(:, k) = 1 / 3 * (vertices(1) + vertices(2)); k = k + 1; mid1 = k - 1; 41 | Pb(:, k) = 2 / 3 * (vertices(1) + vertices(2)); k = k + 1; mid2 = k - 1; 42 | if PIndexToPbIndex(index(2)) == 0 43 | PIndexToPbIndex(index(2)) = k; 44 | Pb(:, k) = P(:, index(2)); 45 | k = k + 1; 46 | end 47 | Tb(:, i) = [PIndexToPbIndex(index(1)); mid1; mid2; PIndexToPbIndex(index(2))]; 48 | end 49 | end 50 | end -------------------------------------------------------------------------------- /example/test_possion_example1.m: -------------------------------------------------------------------------------- 1 | % possion_test_example 2 | % This is the example for possion equation, sunch as the coefficient,right- 3 | % handside, the boundary condition, the equation like this 4 | % $-\frac{d}{dx}(c(x)\frac{du}{dx}) = f(x), a < x < b$. 5 | % $u(a) = u(b) = Dirichlet(x)$ 6 | % $u'(a) = u'(b) = Neumann(x)$ 7 | % $u(a) = u(b) = Robin(x)$ 8 | % @author: chtld 9 | % @date: 20191013 10 | pde.right_hand_side = @(x) -exp(x) .* (cos(x) - 2 * sin(x) - x .* cos(x) - x .* sin(x)); 11 | pde.coefficient = @(x) exp(x); 12 | pde.Dirichlet = @(x) x .* cos(x); 13 | pde.Neumann = @(x) cos(x) - x .* sin(x); 14 | pde.exact = @(x) x .* cos(x); 15 | pde.exact_x = @(x) cos(x) - x .* sin(x); 16 | pde.left = 0; 17 | pde.right = 1; 18 | 19 | % pde.leftBoundaryType = 'Dirichlet'; 20 | pde.leftBoundaryType = 'Neumann'; 21 | pde.rightBoundaryType = 'Dirichlet'; 22 | % pde.rightBoundaryType = "Neumann"; 23 | 24 | basis_type = 102; 25 | np = [8, 16, 32, 64, 128, 256, 512, 1024]; 26 | grid_size = zeros(length(np), 1); 27 | l2_error = zeros(length(np), 1); 28 | infinite_error = zeros(length(np), 1); 29 | h1_semi_error = zeros(length(np), 1); 30 | fprintf('error.l2_norm\t error.infinite_norm\t error.h1_semi_norm\t\n'); 31 | for i = 1: length(np) 32 | [mesh.P, mesh.T] = generate_P_T(pde.left, pde.right, np(i) ); 33 | [FE.Pb, FE.Tb] = generate_Pb_Tb(mesh.P, mesh.T, basis_type); 34 | FE.boundarynodes = generate_boundarynodes(pde, FE.Pb); 35 | [solution, error] = possion_solver_1d(pde, mesh, FE, basis_type, basis_type); 36 | fprintf('%e\t %e\t %e\t\n', error.l2_norm, error.infinite_norm, error.h1_semi_norm); 37 | grid_size(i) = 1 / np(i); 38 | l2_error(i) = error.l2_norm; 39 | infinite_error(i) = error.infinite_norm; 40 | h1_semi_error(i) = error.h1_semi_norm; 41 | end 42 | figure; 43 | plotConvergenceRate(grid_size, l2_error); hold on; 44 | plotConvergenceRate(grid_size, infinite_error); hold on; 45 | plotConvergenceRate(grid_size, h1_semi_error); hold on; 46 | lgd = legend({'$L_{2}$','$L_{\infty}$','$H_{1}$'}, 'Interpreter', 'latex', 'FontSize', 12, 'TextColor', 'blue'); -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | [![Contributors][contributors-shield]][contributors-url] 2 | [![Forks][forks-shield]][forks-url] 3 | [![Stargazers][stars-shield]][stars-url] 4 | [![Issues][issues-shield]][issues-url] 5 | [![MIT License][license-shield]][license-url] 6 | 7 | 8 | 9 | 10 |
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Finite_Element_Method_Matlab

18 | 19 |

20 | A Matlab implementation of the Finite Element Method 21 |
22 | Explore the docs » 23 |
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25 | View Demo 26 | · 27 | Report Bug 28 | · 29 | Request Feature 30 |

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32 | 33 | 34 | 35 | 36 | ## Table of Contents 37 | 38 | * [About the Project](#about-the-project) 39 | * [Built With](#built-with) 40 | * [Getting Started](#getting-started) 41 | * [Prerequisites](#prerequisites) 42 | * [Installation](#installation) 43 | * [Usage](#usage) 44 | * [Roadmap](#roadmap) 45 | * [Contributing](#contributing) 46 | * [License](#license) 47 | * [Contact](#contact) 48 | * [Acknowledgements](#acknowledgements) 49 | 50 | 51 | 52 | 53 | ## About The Project 54 | 57 | 58 | 59 | A simple, general, and easy-to-use Matlab implementation of the finite element method. 60 | The motivation for this project: 61 | * A universal finite element package is often too complex 62 | * The fundamental principles of the finite element method are general 63 | * As a student majoring in computational mathematics, I hope to reduce technical barriers in scientific computing 64 | * This project is based on the lectures of Professor He Xiaoming. Through his course, I learned some finite element basics, and I am very grateful to him. 65 | 66 | Some commonly used resources that I find helpful are listed in the acknowledgements. 67 | 68 | ### Built With 69 | 70 | This project is developed with: 71 | * [matlab](https://www.mathworks.com) 72 | 73 | 74 | ## Getting Started 75 | 76 | Below is how to set up your environment to use this project. 77 | 78 | ### Prerequisites 79 | 80 | Software that needs to be installed in advance: 81 | * matlab 82 | ``` 83 | Please install Matlab by yourself. For installation instructions, please refer to Baidu. 84 | ``` 85 | 86 | ### Installation 87 | 88 | 1. Clone the repository 89 | ```sh 90 | git clone https://github.com/chtld/Finite_Element_Method_Matlab.git 91 | ``` 92 | 2. Add this project to Matlab’s search path 93 | ``` 94 | Run include.m 95 | ``` 96 | 3. Enjoy! 97 | 98 | 99 | ## Usage 100 | 101 | Examples for different equations can be found in the example directory. Please check the source code for details. 102 | 103 | 104 | 105 | ## Roadmap 106 | 107 | For a list of proposed features (and known issues), please refer to the[open issues](https://github.com/chtld/Finite_Element_Method_Matlab/issues) . 108 | See the [open issues](https://github.com/chtld/Finite_Element_Method_Matlab/issues) for a list of proposed features (and known issues). 109 | 110 | 111 | ## Contributing 112 | 113 | Contributions are welcome! 114 | 115 | 1. Fork the Project 116 | 2. Create your Feature Branch (`git checkout -b feature/AmazingFeature`) 117 | 3. Commit your Changes (`git commit -m 'Add some AmazingFeature'`) 118 | 4. Push to the Branch (`git push origin feature/AmazingFeature`) 119 | 5. Open a Pull Request 120 | 121 | 122 | ## License 123 | 124 | Distributed under the GPL-3.0 License. See `LICENSE` for more information. 125 | 126 | 127 | 128 | 129 | ## Contact 130 | 131 | chtld - contact by issue 132 | 133 | Project Link: [https://github.com/chtld/Finite_Element_Method_Matlab](https://github.com/chtld/Finite_Element_Method_Matlab) 134 | 135 | 136 | 137 | 138 | ## Acknowledgements 139 | * [GitHub Emoji Cheat Sheet](https://www.webpagefx.com/tools/emoji-cheat-sheet) 140 | * [Img Shields](https://shields.io) 141 | * [Choose an Open Source License](https://choosealicense.com) 142 | * [Best-README-Template](https://github.com/othneildrew/Best-README-Template) 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | [contributors-shield]: https://img.shields.io/github/contributors/chtld/Finite_Element_Method_Matlab.svg?style=flat-square 152 | [contributors-url]: https://github.com/chtld/Finite_Element_Method_Matlab/graphs/contributors 153 | [forks-shield]: https://img.shields.io/github/forks/chtld/Finite_Element_Method_Matlab.svg?style=flat-square 154 | [forks-url]: https://github.com/chtld/Finite_Element_Method_Matlab/network/members 155 | [stars-shield]: https://img.shields.io/github/stars/chtld/Finite_Element_Method_Matlab.svg?style=flat-square 156 | [stars-url]: https://github.com/chtld/Finite_Element_Method_Matlab/stargazers 157 | [issues-shield]: https://img.shields.io/github/issues/chtld/Finite_Element_Method_Matlab?style=flat-square 158 | [issues-url]: https://github.com/chtld/Finite_Element_Method_Matlab/issues 159 | [license-shield]: https://img.shields.io/github/license/chtld/Finite_Element_Method_Matlab.svg?style=flat-square 160 | [license-url]: https://github.com/chtld/Finite_Element_Method_Matlab/blob/master/LICENSE.txt 161 | [product-screenshot]: images/screenshot.png 162 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------