├── .github └── workflows │ └── codeql-analysis.yml ├── .gitignore ├── CITATION.cff ├── LICENSE ├── README.md ├── analitical_solutions ├── analitical_solution_continuum_bar_vibration.py ├── analitical_solution_single_mass_vibration.py └── analitical_solution_wave_in_pile.py ├── modules ├── element.py ├── integration.py ├── interpolation.py ├── material.py ├── mesh.py ├── node.py ├── particle.py ├── setup.py ├── shape.py ├── solver.py └── update.py ├── tests ├── cpGIMP_interpolation_functions_test.png ├── cpgimp_interpolation_functions_test.py ├── damped_continuum_bar_vibration.png ├── interpolation_functions_test.py ├── linear_interpolation_functions_test.png ├── linear_interpolation_functions_test.py ├── mesh_elements_without_particles_test.png ├── mesh_elements_without_particles_test.py ├── mesh_test.png └── mesh_test.py └── verification_problems ├── mpm_continuum_bar_vibration.png ├── mpm_continuum_bar_vibration.py ├── mpm_continuum_bar_vibration_damping.png ├── mpm_continuum_bar_vibration_damping.py ├── mpm_single_mass_vibration.png ├── mpm_single_mass_vibration.py ├── mpm_single_mass_vibration_parametric_density.png ├── mpm_single_mass_vibration_parametric_density.py ├── mpm_viscous_bar_deformation.py ├── mpm_wave_in_pile.png ├── mpm_wave_in_pile.py ├── mpm_wave_in_pile_parametric_young.png └── mpm_wave_in_pile_parametric_young.py /.github/workflows/codeql-analysis.yml: -------------------------------------------------------------------------------- 1 | # For most projects, this workflow file will not need changing; you simply need 2 | # to commit it to your repository. 3 | # 4 | # You may wish to alter this file to override the set of languages analyzed, 5 | # or to provide custom queries or build logic. 6 | # 7 | # ******** NOTE ******** 8 | # We have attempted to detect the languages in your repository. Please check 9 | # the `language` matrix defined below to confirm you have the correct set of 10 | # supported CodeQL languages. 11 | # 12 | name: "CodeQL" 13 | 14 | on: 15 | push: 16 | branches: [ main ] 17 | pull_request: 18 | # The branches below must be a subset of the branches above 19 | branches: [ main ] 20 | schedule: 21 | - cron: '35 2 * * 3' 22 | 23 | jobs: 24 | analyze: 25 | name: Analyze 26 | runs-on: ubuntu-latest 27 | 28 | strategy: 29 | fail-fast: false 30 | matrix: 31 | language: [ 'python' ] 32 | # CodeQL supports [ 'cpp', 'csharp', 'go', 'java', 'javascript', 'python' ] 33 | # Learn more: 34 | # https://docs.github.com/en/free-pro-team@latest/github/finding-security-vulnerabilities-and-errors-in-your-code/configuring-code-scanning#changing-the-languages-that-are-analyzed 35 | 36 | steps: 37 | - name: Checkout repository 38 | uses: actions/checkout@v2 39 | 40 | # Initializes the CodeQL tools for scanning. 41 | - name: Initialize CodeQL 42 | uses: github/codeql-action/init@v1 43 | with: 44 | languages: ${{ matrix.language }} 45 | # If you wish to specify custom queries, you can do so here or in a config file. 46 | # By default, queries listed here will override any specified in a config file. 47 | # Prefix the list here with "+" to use these queries and those in the config file. 48 | # queries: ./path/to/local/query, your-org/your-repo/queries@main 49 | 50 | # Autobuild attempts to build any compiled languages (C/C++, C#, or Java). 51 | # If this step fails, then you should remove it and run the build manually (see below) 52 | - name: Autobuild 53 | uses: github/codeql-action/autobuild@v1 54 | 55 | # ℹ️ Command-line programs to run using the OS shell. 56 | # 📚 https://git.io/JvXDl 57 | 58 | # ✏️ If the Autobuild fails above, remove it and uncomment the following three lines 59 | # and modify them (or add more) to build your code if your project 60 | # uses a compiled language 61 | 62 | #- run: | 63 | # make bootstrap 64 | # make release 65 | 66 | - name: Perform CodeQL Analysis 67 | uses: github/codeql-action/analyze@v1 68 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | __pycache__/ 2 | *.html 3 | *.sublime-project 4 | *.sublime-workspace 5 | *.code-workspace -------------------------------------------------------------------------------- /CITATION.cff: -------------------------------------------------------------------------------- 1 | cff-version: 1.2.0 2 | message: "If you use this software, please cite it as below." 3 | authors: 4 | - family-names: "Fernández" 5 | given-names: "Fabricio" 6 | orcid: "https://orcid.org/0000-0001-7523-5961" 7 | title: "MPM-Py: A program for learning the Material Point Method using Python" 8 | version: 1.1.0 9 | doi: 0 10 | date-released: 2021-07-15 11 | url: "https://github.com/fabricix/MPM-Py" -------------------------------------------------------------------------------- /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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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 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # MPM-Py 2 | 3 | A Material Point Method implementation using Python: MPM-Py. This program uses objected-oriented programming paradigm to represent and modeling the elements and its interaction in the material point method context. 4 | 5 | ## Installation 6 | 7 | ```git 8 | git clone https://github.com/fabricix/MPM-Py.git 9 | ``` 10 | 11 | ## Documentation 12 | 13 | To create the documentation, install pdoc: 14 | 15 | ```bash 16 | pip3 install pdoc3 17 | ``` 18 | 19 | And then run 20 | 21 | ```bash 22 | pdoc --html -c latex_math=True modules/ 23 | ``` 24 | 25 | To read the documentation, open the file `/html/modules/index.html` using a web browser. 26 | 27 | Note that in Windows the module flag `-m` must be used to correctly use `pdoc`: 28 | 29 | ```bash 30 | python -m pdoc --html -c latex_math=True modules/ 31 | ``` 32 | ## Requirements 33 | 34 | * Python 3.7.4 or superior 35 | 36 | * Matplotlib 3.3.4 or superior 37 | 38 | ## Running tests and examples 39 | 40 | In the folders `verification_problems` and `tests` there are examples showing and testing the functionalities of the program. 41 | 42 | ### Mesh test 43 | 44 | #### All elements with particles 45 | 46 | The file `tests/mesh-test.py` tests the mesh generations module by plotting the mesh and showing the number of elements, nodes and material points. 47 | 48 | Run this example as: 49 | 50 | ```bash 51 | python mesh-test.py 52 | ``` 53 | ![Alt text](tests/mesh_test.png?raw=true "All elements with particles") 54 | 55 | #### Mesh with particles in some elements 56 | In this case the particles are distributed in some elements. 57 | 58 | Run this example as: 59 | 60 | ```bash 61 | python mesh_elements_without_particles_test.py 62 | ``` 63 | ![Alt text](tests/mesh_elements_without_particles_test.png?raw=true "Elements without particles") 64 | 65 | ### Interpolation function test 66 | 67 | The file `tests/interpolation_functions_test.py` shows the interpolation functions and its derivates over an one 1D element. 68 | 69 | In `test_interpolation_functions` function of the `shape` module, set `shape_type="linear"` for linear interpolation functions or `shape_type="cpGIMP"` for contiguous particle GIMP (generalized interpolation material point). 70 | 71 | Run this example as: 72 | 73 | ```bash 74 | python interpolation-functions-test.py 75 | ``` 76 | Linear interpolation functions: 77 | 78 | ![Alt text](tests/linear_interpolation_functions_test.png?raw=true "Linear interpolation functions") 79 | 80 | cpGIMP interpolation functions: 81 | 82 | ![Alt text](tests/cpGIMP_interpolation_functions_test.png?raw=true "Linear interpolation functions") 83 | 84 | ### Single mass vibration problem 85 | 86 | In this verification problem a single mass vibration is analyzed numerically and then the numerical solution is compared with the analytical one. 87 | 88 | Run this example as: 89 | 90 | ```bash 91 | python mpm-single-mass-bar-vibration.py 92 | ``` 93 | ![Alt text](verification_problems/mpm_single_mass_vibration.png?raw=true "Single mass vibration problem") 94 | 95 | ## Parametric analysis over material density 96 | 97 | ```bash 98 | python mpm-single-mass-bar-vibration_parametric_density.py 99 | ``` 100 | ![Alt text](verification_problems/mpm_single_mass_vibration_parametric_density.png?raw=true "Single mass vibration problem: Parametric analysis over density") 101 | 102 | 103 | ### Continuum bar vibration problem 104 | 105 | In this verification problem a continuum bar vibration is analyzed numerically and then the numerical solution is compared with the analytical one. 106 | 107 | Run this example as: 108 | 109 | ```bash 110 | python mpm-continuum-bar-vibration.py 111 | ``` 112 | 113 | ![Alt text](verification_problems/mpm_continuum_bar_vibration.png?raw=true "Continuum bar vibration problem") 114 | 115 | ### Wave traveling in a pile 116 | 117 | In this verification problem a wave traveling in a pile is analyzed numerically and then the numerical solution is compared with the analytical one. 118 | 119 | Run this example as: 120 | 121 | ```bash 122 | python mpm_wave_in_pile.py 123 | ``` 124 | 125 | ![Alt text](verification_problems/mpm_wave_in_pile.png?raw=true "Wave in pile vibration problem") 126 | 127 | ## Parametric analysis over Young modulus 128 | 129 | ```bash 130 | python mpm_wave_in_pile_parametric_young.py 131 | ``` 132 | 133 | ![Alt text](verification_problems/mpm_wave_in_pile_parametric_young.png?raw=true "Wave in pile vibration problem with parametric analysis over Young modulus") 134 | 135 | ### Local damping 136 | 137 | The local damping is a nodal force proportional to the unbalanced nodal total force, acting in opposite nodal velocity direction. 138 | 139 | The local damping must be setting up using the `model_setup` class. For example: 140 | 141 | ```bash 142 | msetup = setup.model_setup() 143 | msetup.damping_local_alpha=0.1 144 | ``` 145 | ## Parametric analysis over damping factor 146 | 147 | ```bash 148 | python mpm-continuum-bar-vibration_damping.py 149 | ``` 150 | 151 | ![Alt text](verification_problems/mpm_continuum_bar_vibration_damping.png?raw=true "Continuum bar vibration problem with parametric analysis over damping factor") 152 | -------------------------------------------------------------------------------- /analitical_solutions/analitical_solution_continuum_bar_vibration.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | Compute the analytical solution of a continuum bar vibration 7 | 8 | Arguments 9 | --------- 10 | L : float 11 | Bar length 12 | E : float 13 | Young's modulus 14 | rho : float 15 | density 16 | time : float 17 | total time 18 | dt : float 19 | time step 20 | vo : float 21 | initial velocity 22 | x_sol : float 23 | position to get the solution 24 | 25 | Data 26 | ---- 27 | Created on Sat Mar 20 20:26:08 2021 28 | 29 | Author 30 | ------ 31 | Fabricio Fernandez 32 | 33 | """ 34 | 35 | # import external modules 36 | import numpy as np 37 | 38 | def continuum_bar_vibration_solution(L,E,rho,time,dt,vo,x_sol): 39 | """calculate the continuum bar vibration solution (mode 1)""" 40 | 41 | # frequency of the system (mode 1) 42 | w1 = (np.pi/2.0/L)*((E/rho)**0.5) 43 | b1 = (np.pi/2.0/L) 44 | 45 | # position in time 46 | xt = [] 47 | vt = [] 48 | t = [] 49 | 50 | # initial time 51 | ti = 0 52 | 53 | # compute the exact solution 54 | while ti 32 | 33 | """ 34 | 35 | # import external modules 36 | import numpy as np 37 | 38 | def single_mass_point_vibration_solution(L,E,rho,time,dt,xo,vo): 39 | """ calculate the single mass point vibration solution """ 40 | 41 | # frequency of the system 42 | w = (1.0/L)*((E/rho)**0.5) 43 | 44 | # mass position in time 45 | xt = [] 46 | t = [] 47 | 48 | # initial time 49 | ti = 0 50 | 51 | # compute the exact solution 52 | while ti 34 | 35 | """ 36 | 37 | # import external modules 38 | import numpy as np 39 | 40 | def wave_in_pile_fixed_and_loaded(L,E,rho,time,dt,po,x,n_sum): 41 | """calculate the solution of the wave equation in a pile""" 42 | 43 | # Poisson's ratio in 1D is assumed to be equal to 0 44 | nu=0 45 | 46 | # bulk modulus 47 | K = E*(1-nu)/(1+nu)/(1-2*nu) 48 | 49 | # wave velocity 50 | c = np.sqrt(K/rho) 51 | 52 | # position an velocity in time 53 | xt = [] 54 | vt = [] 55 | t = [] 56 | 57 | # initial time 58 | ti = 0 59 | 60 | # compute the exact solution 61 | while ti0: 22 | 23 | # add damping force in nodes 24 | for node in msh.nodes: 25 | 26 | # damping factor 27 | alpha = msetup.damping_local_alpha 28 | 29 | # unbalanced nodal force magnitude 30 | unbalanced_force_mag = abs(node.f_int + node.f_ext) 31 | 32 | # nodal velocity 33 | nodal_vel = node.momentum/node.mass 34 | 35 | if abs(nodal_vel)!=0: 36 | 37 | # velocity direction 38 | vel_direction = nodal_vel/abs(nodal_vel) 39 | 40 | # damping force proportional to unbalanced forces and opposite to the nodal velocity 41 | node.f_damp = - alpha * unbalanced_force_mag * vel_direction 42 | 43 | # total nodal force 44 | for node in msh.nodes: 45 | node.f_tot = node.f_int + node.f_ext + node.f_damp 46 | 47 | def momentum_in_nodes(msh,dt): 48 | """ 49 | Calculate momentum in nodes 50 | 51 | Arguments 52 | --------- 53 | msh: mesh 54 | a mesh object 55 | 56 | dt: float 57 | time step 58 | """ 59 | for inode in msh.nodes: 60 | inode.momentum += inode.f_tot*dt -------------------------------------------------------------------------------- /modules/interpolation.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | r""" 5 | 6 | This module defines interpolation functions to transfer quantities from particles to nodes 7 | 8 | These functions have the general form: 9 | 10 | .. math:: 11 | f_I = \sum_p N_I(x_p) f_p 12 | 13 | where, 14 | $$f_I$$: is a nodal quantity, 15 | $$f_p$$: is a particle quantity, and 16 | $$N_I(x_p)$$: is the weight or interpolation function of the node *I* evaluated at particle position $$x_p$$ 17 | 18 | """ 19 | 20 | def mass_to_nodes(msh): 21 | """ 22 | Interpolate mass from particles to nodes. 23 | 24 | Arguments 25 | --------- 26 | msh: mesh 27 | a mesh object 28 | """ 29 | for ie in msh.elements: 30 | for ip in ie.particles: 31 | 32 | ie.n1.mass+=ip.mass*ip.N1 33 | ie.n2.mass+=ip.mass*ip.N2 34 | 35 | 36 | def momentum_to_nodes(msh): 37 | """ 38 | Interpolate momentum from particles to nodes. 39 | 40 | Arguments 41 | --------- 42 | msh: mesh 43 | a mesh object 44 | """ 45 | for ie in msh.elements: 46 | for ip in ie.particles: 47 | 48 | ie.n1.momentum+=ip.mass*ip.velocity*ip.N1 49 | ie.n2.momentum+=ip.mass*ip.velocity*ip.N2 50 | 51 | 52 | def internal_force_to_nodes(msh): 53 | """ 54 | Interpolate internal forces from particles to nodes. 55 | 56 | Arguments 57 | --------- 58 | msh: mesh 59 | a mesh object 60 | """ 61 | for ie in msh.elements: 62 | for ip in ie.particles: 63 | 64 | ie.n1.f_int-=ip.dN1*ip.stress*ip.mass/ip.density 65 | ie.n2.f_int-=ip.dN2*ip.stress*ip.mass/ip.density 66 | 67 | def external_force_to_nodes(msh): 68 | """ 69 | Interpolate external forces from particles to nodes. 70 | 71 | Arguments 72 | --------- 73 | msh: mesh 74 | a mesh object 75 | """ 76 | for ie in msh.elements: 77 | for ip in ie.particles: 78 | 79 | ie.n1.f_ext+=ip.N1*ip.f_ext 80 | ie.n2.f_ext+=ip.N2*ip.f_ext -------------------------------------------------------------------------------- /modules/material.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | """ 5 | 6 | This module defines classes representing materials 7 | 8 | """ 9 | 10 | class linear_elastic: 11 | """ 12 | Represents a linear elastic material 13 | 14 | Arguments 15 | --------- 16 | E : float 17 | Young's modulus 18 | density : float 19 | density 20 | """ 21 | def __init__(self,E,density): 22 | 23 | self.E=E # Young's modulus 24 | self.density=density # density 25 | 26 | def update_stress(self,particle,dt): 27 | 28 | """ 29 | Update particle stress using linear elastic constitutive model 30 | 31 | Arguments 32 | --------- 33 | particle: particle 34 | a particle object 35 | """ 36 | 37 | particle.stress+=particle.dstrain*self.E 38 | 39 | class newtonian_fluid: 40 | """ 41 | Represents a Newtonian fluid material 42 | 43 | Arguments 44 | --------- 45 | 46 | mu : float 47 | Viscosity 48 | 49 | density : float 50 | density 51 | 52 | """ 53 | 54 | def __init__(self,mu,density): 55 | 56 | self.mu=mu # viscosity 57 | self.density=density # density 58 | 59 | def update_stress(self,particle,dt): 60 | 61 | """ 62 | Update particle stress 63 | 64 | Arguments 65 | --------- 66 | particle: particle 67 | a particle object 68 | """ 69 | 70 | particle.stress=self.mu*particle.dstrain/dt 71 | -------------------------------------------------------------------------------- /modules/mesh.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | """ 5 | 6 | This module defines classes representing a finite element mesh 7 | 8 | """ 9 | 10 | from modules import element 11 | from modules import node 12 | from modules import particle 13 | 14 | class mesh_1D: 15 | """ 16 | A class to represent a 1D Eulerian mesh containing elements, 17 | nodes and particles. 18 | 19 | Attributes 20 | ---------- 21 | elements : list 22 | elements forming the mesh 23 | 24 | particles : list 25 | particles in mesh 26 | 27 | nodes : list 28 | nodes in mesh 29 | 30 | nelem : int 31 | number of elements in mesh 32 | 33 | ppelem : int 34 | particles per element 35 | """ 36 | 37 | def __init__(self,L,nelem): 38 | 39 | self.elements=[] # mesh elements 40 | self.particles=[] # particles in mesh 41 | self.nodes=[] # nodes in mesh 42 | self.nelem=nelem # elements in mesh 43 | self.ppelem=0 # particles per element 44 | 45 | for i in range(nelem): 46 | 47 | if i==0: 48 | 49 | ielem = element.bar_1D() 50 | ielem.id=i 51 | 52 | ielem.n1=node.node_1D() 53 | ielem.n1.id=len(self.nodes) 54 | self.nodes.append(ielem.n1) 55 | 56 | ielem.n2=node.node_1D() 57 | ielem.n2.id=len(self.nodes) 58 | self.nodes.append(ielem.n2) 59 | 60 | else: 61 | ielem = element.bar_1D() 62 | ielem.id=i 63 | ielem.n1=self.elements[i-1].n2 64 | 65 | ielem.n2=node.node_1D() 66 | ielem.n2.id=len(self.nodes) 67 | self.nodes.append(ielem.n2) 68 | 69 | le = L/nelem 70 | ielem.L=le 71 | ielem.n1.x=i*le 72 | ielem.n2.x=ielem.n1.x+le 73 | 74 | self.elements.append(ielem) 75 | 76 | def put_particles_in_all_mesh_elements(self,ppelem,material): 77 | """ 78 | Distributes particles in elements mesh 79 | 80 | Arguments 81 | --------- 82 | ppelem: int 83 | number of particles per element 84 | 85 | material: material 86 | a material object 87 | 88 | """ 89 | self.ppelem=ppelem 90 | 91 | for ie in range(self.nelem): 92 | 93 | ie = self.elements[ie] 94 | le = ie.L 95 | 96 | for i in range(ppelem): 97 | 98 | # particle mass 99 | pmass = le*material.density/ppelem 100 | 101 | # particle position 102 | if(len(ie.particles)==0): 103 | xp=ie.n1.x+le/(2*ppelem) 104 | 105 | elif(len(ie.particles)==(ppelem-1)): 106 | xp=ie.n2.x-le/(2*ppelem) 107 | 108 | else: 109 | xp=ie.n1.x+le/(2*ppelem)+len(ie.particles)*(le/ppelem) 110 | 111 | # create particle 112 | ip = particle.material_point(pmass,material,xp) 113 | ip.id=len(self.particles) 114 | ip.size = ie.L/ppelem 115 | 116 | # set the element in the particle 117 | ip.element=ie 118 | 119 | # append in elements 120 | ie.particles.append(ip) 121 | 122 | # append in mesh 123 | self.particles.append(ip) 124 | 125 | def put_particles_in_mesh_by_elements_id(self,ppelem,material,elem_i,elem_f): 126 | """ 127 | Distributes particles in elements mesh from xi to xf 128 | 129 | Arguments 130 | --------- 131 | ppelem: int 132 | number of particles per element 133 | 134 | material: material 135 | a material object 136 | 137 | elem_i: int 138 | initial element id to distribute particles 139 | 140 | elem_f: int 141 | final element id to distribute particles 142 | 143 | """ 144 | self.ppelem=ppelem 145 | 146 | for ie in range(self.nelem): 147 | 148 | ie = self.elements[ie] 149 | le = ie.L 150 | 151 | # verify if the element is allow to get particles 152 | if ie.idelem_f: 153 | continue 154 | 155 | for i in range(ppelem): 156 | 157 | # particle mass 158 | pmass = le*material.density/ppelem 159 | 160 | # particle position 161 | if(len(ie.particles)==0): 162 | xp=ie.n1.x+le/(2*ppelem) 163 | 164 | elif(len(ie.particles)==(ppelem-1)): 165 | xp=ie.n2.x-le/(2*ppelem) 166 | 167 | else: 168 | xp=ie.n1.x+le/(2*ppelem)+len(ie.particles)*(le/ppelem) 169 | 170 | # create particle 171 | ip = particle.material_point(pmass,material,xp) 172 | ip.id=len(self.particles) 173 | 174 | # set the element in the particle 175 | ip.element=ie 176 | 177 | # append in elements 178 | ie.particles.append(ip) 179 | 180 | # append in mesh 181 | self.particles.append(ip) 182 | 183 | def print_mesh(self,print_labels=True): 184 | """ 185 | Function for print the mesh in a plot 186 | 187 | Arguments 188 | --------- 189 | print_labels: bool 190 | determines if the label of the mesh will be plotted 191 | """ 192 | import matplotlib.pyplot as plt 193 | plt.cla() 194 | 195 | for ie in self.elements: 196 | plt.plot([ie.n1.x,ie.n2.x],[0,0],'--sk') 197 | dy=0.005 198 | 199 | if(print_labels): 200 | x=(ie.n1.x+ie.n2.x)/2 201 | plt.annotate("e%d"%ie.id, xy=(x,-1.5*dy),fontsize=13) 202 | plt.annotate("n%d"%ie.n1.id, xy=(ie.n1.x,dy),fontsize=13) 203 | plt.annotate("n%d"%ie.n2.id, xy=(ie.n2.x,dy),fontsize=13) 204 | 205 | for ip in ie.particles: 206 | plt.plot(ip.position,0,'ob') 207 | if(print_labels): 208 | plt.annotate("p%d"%ip.id, xy=(ip.position,dy),fontsize=13) 209 | 210 | ie = self.elements[0] 211 | x = ie.n1.x 212 | y = -0.04 213 | plt.annotate("n = node\np = material point\ne = element", xy=(x,y),fontsize=13) 214 | plt.xlabel(r"Node position, $x_I$") 215 | plt.title("Mesh and Material Points") 216 | plt.show() 217 | 218 | def print_mesh_info(self): 219 | """ 220 | Function for print mesh informations 221 | """ 222 | 223 | print(20*'--') 224 | print('elements = %d'%self.nelem) 225 | print('particles per element = %d'%self.ppelem) 226 | print(20*'--') 227 | 228 | for ie in self.elements: 229 | 230 | print('element') 231 | print('id\tn1\txn1\tn2\txn2') 232 | print('%d\t%d\t%.2f\t%d\t%.2f'%(ie.id,ie.n1.id,ie.n1.x,ie.n2.id,ie.n2.x)) 233 | 234 | print('particles') 235 | print('id\txp') 236 | for ip in ie.particles: 237 | print('%d\t%.2f'%(ip.id,ip.position)) 238 | 239 | print(20*'--') -------------------------------------------------------------------------------- /modules/node.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | """ 5 | 6 | This module defines classes representing a node in a finite element mesh 7 | 8 | """ 9 | 10 | class node_1D: 11 | """ 12 | Represent a 1D node 13 | 14 | Attributes 15 | --------- 16 | id : int 17 | node identification 18 | 19 | x : float 20 | position 21 | 22 | velocity : float 23 | nodal velocity 24 | 25 | mass : float 26 | nodal mass 27 | 28 | momentum : float 29 | nodal momentum (mass*velocity) 30 | 31 | f_int : float 32 | nodal internal force 33 | 34 | f_ext : float 35 | nodal external force 36 | 37 | f_tot : float 38 | total force 39 | 40 | f_damp : float 41 | damping force 42 | """ 43 | def __init__(self): 44 | 45 | self.id = 0 46 | self.x = 0 47 | self.velocity = 0 48 | self.mass = 0 49 | self.momentum = 0 50 | self.f_int = 0 51 | self.f_ext = 0 52 | self.f_tot = 0 53 | self.f_damp = 0 -------------------------------------------------------------------------------- /modules/particle.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | """ 5 | 6 | This module defines classes representing a material point (or particle) 7 | 8 | """ 9 | 10 | class material_point: 11 | """ 12 | Represent a material point. 13 | 14 | Attributes 15 | ---------- 16 | 17 | mass : float 18 | particle mass 19 | 20 | position : float 21 | particle position 22 | 23 | material : material type 24 | material 25 | 26 | density : float 27 | particle density 28 | 29 | velocity : float 30 | particle velocity 31 | 32 | stress : float 33 | particle stress 34 | 35 | dstrain : float 36 | particle strain increment 37 | 38 | momentum : float 39 | particle momentum (mass*velocity) 40 | 41 | id : int 42 | particle identification 43 | 44 | f_ext : float 45 | external force in particle 46 | 47 | element : element type 48 | element containing the particle 49 | 50 | N1 : float 51 | value of the interpolation function of node 1 52 | 53 | N2 : float 54 | value of the interpolation function of node 2 55 | 56 | dN1 : float 57 | value of the interpolation function gradient of node 1 58 | 59 | dN2 : float 60 | value of the interpolation function gradient of node 2 61 | 62 | size: float 63 | particle size 64 | 65 | """ 66 | def __init__(self, mass, material,x): 67 | 68 | self.mass = mass 69 | self.position = x 70 | self.material = material 71 | self.density = material.density 72 | 73 | self.velocity = 0 74 | self.stress = 0 75 | self.dstrain = 0 76 | self.momentum = 0 77 | self.id = 0 78 | self.f_ext = 0 79 | self.element = 0 80 | self.N1 = 0 81 | self.N2 = 0 82 | self.dN1 = 0 83 | self.dN1 = 0 84 | self.size = 0 -------------------------------------------------------------------------------- /modules/setup.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | 5 | This module defines a class representing the configuration of the problem 6 | 7 | """ 8 | 9 | class model_setup: 10 | 11 | """ 12 | Represent a configuration object containing the model options 13 | 14 | Attributes 15 | ---------- 16 | interpolation_type : string 17 | interpolation function type, can be 'linear' or 'cpGIMP' 18 | 19 | integration_scheme : string 20 | material point method integration scheme, can be 'USL', 'USF' or 'MUSL' 21 | 22 | time : float 23 | simulation time 24 | 25 | dt : float 26 | time step 27 | 28 | solution_particle : integer 29 | index of the particle to get the solution 30 | 31 | solution_field : string 32 | the solution field, can be "velocity" or "position" 33 | 34 | solution_array : array 35 | array to store the solution in terms of time and field 36 | 37 | damping_local_alpha : float 38 | local damping factor proportional to the total nodal force 39 | 40 | """ 41 | def __init__(self): 42 | 43 | self.interpolation_type="linear" 44 | self.integration_scheme="MUSL" 45 | self.time=0 46 | self.dt=0 47 | self.solution_particle=0 48 | self.solution_field="position" 49 | self.solution_array=[[],[]] 50 | self.damping_local_alpha=0 -------------------------------------------------------------------------------- /modules/shape.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | 4 | """ 5 | 6 | This module defines nodal shape functions and its derivative 7 | 8 | """ 9 | 10 | def NiLinear(x,xI,L): 11 | """ 12 | Calculates the values of the linear interpolation function 13 | 14 | Arguments 15 | --------- 16 | x: float 17 | point to calculate the function 18 | xI: float 19 | nodal point position 20 | L: float 21 | grid cell spacing 22 | """ 23 | 24 | if(abs(x-xI)>=L): 25 | return 0 26 | 27 | if(-L<(x-xI) and (x-xI)<=0): 28 | return 1+(x-xI)/L 29 | 30 | if(0<(x-xI) and (x-xI)=L): 48 | return 0 49 | 50 | if(-L<(x-xI) and (x-xI)<=0): 51 | return 1/L 52 | 53 | if(0<(x-xI) and (x-xI)=(L+lp): 90 | return 0 91 | 92 | if (-L-lp)=(L+lp): 128 | return 0 129 | 130 | if (-L-lp)=xn1 and xp) 9 | 10 | Purpose 11 | ------- 12 | 13 | This examples tests the interpolation functions and its derivative. 14 | 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # local modules 22 | from modules import mesh # for mesh generation 23 | from modules import shape as shape# for interpolation 24 | 25 | # domain 26 | L = 2 27 | 28 | # elements 29 | nelem = 2 30 | 31 | # create an 1D mesh 32 | msh=mesh.mesh_1D(L,nelem) 33 | 34 | #show mesh 35 | # msh.print_mesh() 36 | 37 | #interpolation function type 38 | interpolation_type = 'cpGIMP' #'linear' or 'cpGIMP' 39 | 40 | # print the interpolation functions and its derivative 41 | shape.test_interpolation_functions(x1=0-0.5,x2=L+0.5,xI=L/2,L=L/nelem,shape_type=interpolation_type) -------------------------------------------------------------------------------- /tests/damped_continuum_bar_vibration.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/tests/damped_continuum_bar_vibration.png -------------------------------------------------------------------------------- /tests/interpolation_functions_test.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Created on Wed Mar 17 08:52:30 2021 5 | 6 | Author 7 | ------ 8 | Fabricio Fernandez () 9 | 10 | Purpose 11 | ------- 12 | 13 | This examples tests the interpolation functions and its derivative. 14 | 15 | """ 16 | 17 | import matplotlib.pyplot as plt 18 | 19 | # include the modules' path to the current path 20 | import sys 21 | sys.path.append("..") 22 | 23 | # local modules 24 | from modules import shape as shape# for interpolation 25 | 26 | # domain 27 | L = 2 28 | 29 | # elements 30 | nelem = 2 31 | 32 | # cell dimension 33 | le = L/nelem 34 | 35 | # get linear interpolation functions values in nodes 36 | [x1,n1_linear,dn1_linear] = shape.get_interpolation_functions(x1=0,x2=L,xI=0,L=L/nelem,shape_type="linear") 37 | [x2,n2_linear,dn2_linear] = shape.get_interpolation_functions(x1=0,x2=L,xI=L/2,L=L/nelem,shape_type="linear") 38 | [x3,n3_linear,dn3_linear] = shape.get_interpolation_functions(x1=0,x2=L,xI=L,L=L/nelem,shape_type="linear") 39 | 40 | plt.plot(x1,n1_linear,"--",label="n1-linear") 41 | plt.plot(x2,n2_linear,"--",label="n2-linear") 42 | plt.plot(x3,n3_linear,"--",label="n3-linear") 43 | plt.plot(x1,n1_linear+n2_linear+n3_linear,"--",label="n1+n2+n3-linear") 44 | plt.legend() 45 | 46 | # get cpgimp interpolation functions values 47 | [x1,n1_cpgimp,dn1_cpgimp] = shape.get_interpolation_functions(x1=0,x2=L,xI=0,L=L/nelem,shape_type="cpGIMP") 48 | [x2,n2_cpgimp,dn2_cpgimp] = shape.get_interpolation_functions(x1=0,x2=L,xI=L/2,L=L/nelem,shape_type="cpGIMP") 49 | [x3,n3_cpgimp,dn3_cpgimp] = shape.get_interpolation_functions(x1=0,x2=L,xI=L,L=L/nelem,shape_type="cpGIMP") 50 | 51 | plt.plot(x1,n1_cpgimp, label="n1-cpGIMP") 52 | plt.plot(x2,n2_cpgimp, label="n2-cpGIMP") 53 | plt.plot(x3,n3_cpgimp, label="n3-cpGIMP") 54 | plt.plot(x1,n1_cpgimp+n2_cpgimp+n3_cpgimp, label="n1+n2+n3-cpGIMP") 55 | plt.legend() -------------------------------------------------------------------------------- /tests/linear_interpolation_functions_test.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/tests/linear_interpolation_functions_test.png -------------------------------------------------------------------------------- /tests/linear_interpolation_functions_test.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Created on Wed Mar 17 08:52:30 2021 5 | 6 | Author 7 | ------ 8 | Fabricio Fernandez () 9 | 10 | Purpose 11 | ------- 12 | 13 | This examples tests the interpolation functions and its derivative. 14 | 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # local modules 22 | from modules import mesh # for mesh generation 23 | from modules import shape as shape# for interpolation 24 | 25 | # domain 26 | L = 2 27 | 28 | # elements 29 | nelem = 2 30 | 31 | # create an 1D mesh 32 | msh=mesh.mesh_1D(L,nelem) 33 | 34 | #show mesh 35 | # msh.print_mesh() 36 | 37 | #interpolation function type 38 | interpolation_type = 'linear' #'linear' or 'cpGIMP' 39 | 40 | # print the interpolation functions and its derivative 41 | shape.test_interpolation_functions(x1=0,x2=L,xI=L/2,L=L/nelem,shape_type=interpolation_type) -------------------------------------------------------------------------------- /tests/mesh_elements_without_particles_test.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/tests/mesh_elements_without_particles_test.png -------------------------------------------------------------------------------- /tests/mesh_elements_without_particles_test.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Created on Sat Mar 20 13:39:50 2021 5 | 6 | Author 7 | ------ 8 | Fabricio Fernandez () 9 | 10 | Purpose 11 | ------- 12 | 13 | This examples tests the mesh creation function. 14 | 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # local modules 22 | from modules import mesh # for mesh generation 23 | from modules import material # for material definition 24 | 25 | # domain 26 | L = 5 27 | 28 | # elements 29 | nelem = 6 30 | 31 | # particles per element 32 | p_per_elem = 1 33 | 34 | # create an 1D mesh 35 | msh = mesh.mesh_1D(L,nelem) 36 | 37 | # define a linear material 38 | elastic = material.linear_elastic(E=50,density=1) 39 | 40 | # put particles in some elements, from elem_i to elem_f 41 | msh.put_particles_in_mesh_by_elements_id(ppelem=p_per_elem,material=elastic, elem_i=0,elem_f=3) 42 | 43 | #show mesh 44 | msh.print_mesh() -------------------------------------------------------------------------------- /tests/mesh_test.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/tests/mesh_test.png -------------------------------------------------------------------------------- /tests/mesh_test.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Created on Sat Mar 20 13:39:50 2021 5 | 6 | Author 7 | ------ 8 | Fabricio Fernandez () 9 | 10 | Purpose 11 | ------- 12 | 13 | This examples tests the mesh creation function. 14 | 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # local modules 22 | from modules import mesh # for mesh generation 23 | from modules import material # for material definition 24 | 25 | # domain 26 | L = 4 27 | 28 | # elements 29 | nelem = 2 30 | 31 | # particles per element 32 | p_per_elem = 2 33 | 34 | # create an 1D mesh 35 | msh = mesh.mesh_1D(L,nelem) 36 | 37 | # define a linear material 38 | elastic = material.linear_elastic(E=50,density=1) 39 | 40 | # put particles in all elements 41 | msh.put_particles_in_all_mesh_elements(ppelem=p_per_elem,material=elastic) 42 | 43 | #show mesh 44 | msh.print_mesh() -------------------------------------------------------------------------------- /verification_problems/mpm_continuum_bar_vibration.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_continuum_bar_vibration.png -------------------------------------------------------------------------------- /verification_problems/mpm_continuum_bar_vibration.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the continuum 1D bar vibration problem using MPM. 7 | 8 | Data 9 | ---- 10 | Created on Sat Mar 20 20:25:53 2021 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | import numpy as np # for sin 24 | 25 | # local modules 26 | from modules import mesh # for mesh definition 27 | from modules import material # for material definition 28 | from modules import setup # for setup the problem 29 | from modules import solver # for solving the problem in time 30 | 31 | # bar length 32 | L=25 33 | 34 | # number of elements 35 | nelements=15 36 | 37 | # create an 1D mesh 38 | msh = mesh.mesh_1D(L=L,nelem=nelements) 39 | 40 | # define a linear material 41 | elastic = material.linear_elastic(E=100,density=1) 42 | 43 | # put particles in mesh element and set the material 44 | msh.put_particles_in_all_mesh_elements(ppelem=2,material=elastic) 45 | 46 | # setup the model 47 | msetup = setup.model_setup() 48 | msetup.interpolation_type="linear" 49 | msetup.integration_scheme="MUSL" 50 | msetup.time=60 51 | msetup.dt=0.1 52 | msetup.solution_particle=-1 53 | msetup.solution_field='velocity' 54 | msetup.damping_local_alpha=0.0 55 | 56 | # verify time step 57 | dt_critical=msh.elements[0].L/(elastic.E/elastic.density)**0.5 58 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 59 | 60 | # impose initial condition in particles 61 | vo=0.1 62 | b1=np.pi/2.0/L 63 | for ip in msh.particles: 64 | ip.velocity=vo*np.sin(b1*ip.position) 65 | 66 | # solve the problem in time 67 | solver.explicit_solution(msh,msetup) 68 | 69 | # subset for numerical solution 70 | n_values = 250 71 | indices = np.linspace(0, len(msetup.solution_array[0])-1, n_values, dtype=int) 72 | 73 | y_subset = [] 74 | x_subset = [] 75 | 76 | for i_index in range(len(indices)): 77 | x_subset.append(msetup.solution_array[0][indices[i_index]]) 78 | y_subset.append(msetup.solution_array[1][indices[i_index]]) 79 | 80 | # plot mpm solution 81 | plt.plot(x_subset,y_subset,' ',color='r',marker='s',markerfacecolor='none',label='MPM') 82 | 83 | 84 | # plot the analytical solution 85 | from analitical_solutions import analitical_solution_continuum_bar_vibration as cbv 86 | [anal_xt,anal_vt, anal_t] = cbv.continuum_bar_vibration_solution(L,elastic.E,elastic.density,msetup.time,msetup.dt,vo,msh.particles[msetup.solution_particle].position) 87 | plt.plot(anal_t,anal_vt,'b',linewidth=2,label='Analytical') 88 | 89 | # configure axis, legends and show plot 90 | plt.xlabel('Time (s)') 91 | plt.ylabel('Velocity (m/s)') 92 | plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5)) 93 | plt.show() -------------------------------------------------------------------------------- /verification_problems/mpm_continuum_bar_vibration_damping.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_continuum_bar_vibration_damping.png -------------------------------------------------------------------------------- /verification_problems/mpm_continuum_bar_vibration_damping.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the continuum 1D damped bar vibration problem using MPM. 7 | 8 | Data 9 | ---- 10 | Created on Sat Mar 30 14:45:07 2023 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | import numpy as np # for sin 24 | 25 | # local modules 26 | from modules import mesh # for mesh definition 27 | from modules import material # for material definition 28 | from modules import setup # for setup the problem 29 | from modules import solver # for solving the problem in time 30 | 31 | damping_serie = np.linspace(0, 0.2, 10) 32 | 33 | for i in range(len(damping_serie)): 34 | 35 | # bar length 36 | L=25 37 | 38 | # number of elements 39 | nelements=15 40 | 41 | # create an 1D mesh 42 | msh = mesh.mesh_1D(L=L,nelem=nelements) 43 | 44 | # define a linear material 45 | elastic = material.linear_elastic(E=100,density=1) 46 | 47 | # put particles in mesh element and set the material 48 | msh.put_particles_in_all_mesh_elements(ppelem=2,material=elastic) 49 | 50 | # setup the model 51 | msetup = setup.model_setup() 52 | msetup.interpolation_type="linear" 53 | msetup.integration_scheme="MUSL" 54 | msetup.time=120 55 | msetup.dt=0.35 56 | msetup.solution_particle=-1 57 | msetup.solution_field='velocity' 58 | msetup.damping_local_alpha=msetup.damping_local_alpha=damping_serie[i] 59 | 60 | # verify time step 61 | dt_critical=msh.elements[0].L/(elastic.E/elastic.density)**0.5 62 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 63 | 64 | # impose initial condition in particles 65 | vo=0.1 66 | b1=np.pi/2.0/L 67 | for ip in msh.particles: 68 | ip.velocity=vo*np.sin(b1*ip.position) 69 | 70 | # solve the problem in time 71 | solver.explicit_solution(msh,msetup) 72 | 73 | # plot mpm solution 74 | plt.plot(msetup.solution_array[0],msetup.solution_array[1],label=str('Damping={:.2f}-MPM'.format(damping_serie[i]))) 75 | 76 | # plot the analytical solution 77 | from analitical_solutions import analitical_solution_continuum_bar_vibration as cbv 78 | [anal_xt,anal_vt, anal_t] = cbv.continuum_bar_vibration_solution(L,elastic.E,elastic.density,msetup.time,msetup.dt,vo,msh.particles[msetup.solution_particle].position) 79 | plt.plot(anal_t,anal_vt,'k',linewidth=1.5,label='Damping={:.2f}-Analytical'.format(0)) 80 | 81 | # configure axis, legends and show plot 82 | plt.xlabel('Time (s)') 83 | plt.ylabel('Velocity (m/s)') 84 | plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5)) 85 | plt.show() -------------------------------------------------------------------------------- /verification_problems/mpm_single_mass_vibration.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_single_mass_vibration.png -------------------------------------------------------------------------------- /verification_problems/mpm_single_mass_vibration.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the 1D single mass bar vibration problem using MPM. 7 | 8 | Data 9 | ---- 10 | Created on Mon Mar 15 13:53:24 2021 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | import numpy as np 24 | 25 | # local modules 26 | from modules import mesh # for mesh definition 27 | from modules import material # for material definition 28 | from modules import setup # for setup the problem 29 | from modules import solver # for solving the problem in time 30 | 31 | # bar length 32 | L=1 33 | 34 | # number of elements 35 | nelements=1 36 | 37 | # create an 1D mesh 38 | msh = mesh.mesh_1D(L,nelements) 39 | 40 | # define a linear material 41 | elastic = material.linear_elastic(E=50,density=1) 42 | 43 | # put particles in mesh element and set the material 44 | msh.put_particles_in_all_mesh_elements(ppelem=1,material=elastic) 45 | 46 | # setup the model 47 | msetup = setup.model_setup() 48 | msetup.interpolation_type="linear" 49 | msetup.integration_scheme="MUSL" 50 | msetup.time=10 51 | msetup.dt=0.001 52 | msetup.solution_particle=-1 53 | msetup.solution_field="position" 54 | msetup.damping_local_alpha=0.0 55 | 56 | # verify time step 57 | dt_critical=msh.elements[0].L/(elastic.E/elastic.density)**0.5 58 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 59 | 60 | # impose initial condition in particle 61 | vo = 0.1 62 | msh.particles[-1].velocity=vo 63 | 64 | # solve the problem in time 65 | solver.explicit_solution(msh,msetup) 66 | 67 | n_values = 400 68 | indices = np.linspace(0, len(msetup.solution_array[0])-1, n_values, dtype=int) 69 | 70 | y_subset = [] 71 | x_subset = [] 72 | 73 | for i_index in range(len(indices)): 74 | x_subset.append(msetup.solution_array[0][indices[i_index]]) 75 | y_subset.append(msetup.solution_array[1][indices[i_index]]) 76 | 77 | # plot mpm solution 78 | plt.plot(x_subset,y_subset,' ',color='r',marker='s',markerfacecolor='none',label='MPM') 79 | 80 | # plot the analytical solution 81 | from analitical_solutions import analitical_solution_single_mass_vibration as smpv 82 | [anal_xt, anal_t] = smpv.single_mass_point_vibration_solution(L,elastic.E,elastic.density,msetup.time,msetup.dt,L/2,vo) 83 | 84 | plt.plot(anal_t,anal_xt,'-',color='b',label='Analytical') 85 | 86 | # configure axis, legends and show plot 87 | plt.xlabel('Time (s)') 88 | plt.ylabel('Position (m)') 89 | plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5)) 90 | plt.show() -------------------------------------------------------------------------------- /verification_problems/mpm_single_mass_vibration_parametric_density.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_single_mass_vibration_parametric_density.png -------------------------------------------------------------------------------- /verification_problems/mpm_single_mass_vibration_parametric_density.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the 1D single mass bar vibration problem using MPM. 7 | In this example, a parametric study over the density of the mass is performed 8 | 9 | Data 10 | ---- 11 | Created on Mon Mar 15 13:53:24 2021 12 | 13 | Author 14 | ------ 15 | Fabricio Fernandez () 16 | """ 17 | 18 | # include the modules' path to the current path 19 | import sys 20 | sys.path.append("..") 21 | 22 | # external modules 23 | import matplotlib.pyplot as plt # for plot 24 | import matplotlib.colors as mcolors # for colors 25 | import numpy as np # numpy 26 | 27 | # local modules 28 | from modules import mesh # for mesh definition 29 | from modules import material # for material definition 30 | from modules import setup # for setup the problem 31 | from modules import solver # for solving the problem in time 32 | 33 | # density values 34 | density_serie = np.linspace(1,15,6) 35 | 36 | 37 | # get color palete for plot 38 | tableau_colors = mcolors.TABLEAU_COLORS 39 | color_list = list(tableau_colors.values())[:len(density_serie)] 40 | 41 | # main loop 42 | for i in range(len(density_serie)): 43 | 44 | # bar length 45 | L=1 46 | 47 | # number of elements 48 | nelements=1 49 | 50 | # create an 1D mesh 51 | msh = mesh.mesh_1D(L,nelements) 52 | 53 | # define a linear material 54 | elastic = material.linear_elastic(E=50,density=density_serie[i]) 55 | 56 | # put particles in mesh element and set the material 57 | msh.put_particles_in_all_mesh_elements(ppelem=1,material=elastic) 58 | 59 | # setup the model 60 | msetup = setup.model_setup() 61 | msetup.interpolation_type="linear" 62 | msetup.integration_scheme="MUSL" 63 | msetup.time=10 64 | msetup.dt=0.001 65 | msetup.solution_particle=-1 66 | msetup.solution_field="position" 67 | msetup.damping_local_alpha=0.0 68 | 69 | # verify time step 70 | dt_critical=msh.elements[0].L/(elastic.E/elastic.density)**0.5 71 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 72 | 73 | # impose initial condition in particle 74 | vo = 0.1 75 | msh.particles[-1].velocity=vo 76 | 77 | # solve the problem in time 78 | solver.explicit_solution(msh,msetup) 79 | 80 | n_values = 100 81 | indices = np.linspace(0, len(msetup.solution_array[0])-1, n_values, dtype=int) 82 | 83 | y_subset = [] 84 | x_subset = [] 85 | 86 | for i_index in range(len(indices)): 87 | x_subset.append(msetup.solution_array[0][indices[i_index]]) 88 | y_subset.append(msetup.solution_array[1][indices[i_index]]) 89 | 90 | # plot mpm solution 91 | plt.plot(x_subset,y_subset,' ',color=color_list[i],marker='s',markerfacecolor='none',label='Density={:.2f}-MPM'.format(density_serie[i])) 92 | 93 | # plot the analytical solution 94 | from analitical_solutions import analitical_solution_single_mass_vibration as smpv 95 | [anal_xt, anal_t] = smpv.single_mass_point_vibration_solution(L,elastic.E,elastic.density,msetup.time,msetup.dt,L/2,vo) 96 | 97 | plt.plot(anal_t,anal_xt,'-',color=color_list[i],label='Density={:.2f}-Analytical'.format(density_serie[i])) 98 | 99 | # configure axis, legends and show plot 100 | plt.xlabel('Time (s)') 101 | plt.ylabel('Position (m)') 102 | plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5)) 103 | plt.show() -------------------------------------------------------------------------------- /verification_problems/mpm_viscous_bar_deformation.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example demostrate the use of the viscous constitutive model. 7 | 8 | Data 9 | ---- 10 | Created on Wed Apr 27 14:05:43 2022 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | 24 | # local modules 25 | from modules import mesh # for mesh definition 26 | from modules import material # for material definition 27 | from modules import setup # for setup the problem 28 | from modules import solver # for solving the problem in time 29 | 30 | # mesh definition 31 | L=10 32 | 33 | # number of elements 34 | nelements=20 35 | 36 | # particles per element 37 | p_per_elem = 4 38 | 39 | # create an 1D mesh 40 | msh = mesh.mesh_1D(L,nelements) 41 | 42 | # define a viscous material 43 | material = material.newtonian_fluid(mu=100000,density=2000) 44 | 45 | # put particles in some elements, from elem_i to elem_f 46 | msh.put_particles_in_mesh_by_elements_id(ppelem=p_per_elem,material=material, elem_i=0,elem_f=5) 47 | 48 | #show mesh 49 | #msh.print_mesh() 50 | 51 | # setup the model 52 | msetup = setup.model_setup() 53 | msetup.interpolation_type="linear" 54 | msetup.integration_scheme="MUSL" 55 | msetup.time=1 56 | msetup.dt=0.001 57 | msetup.solution_particle=-1 58 | msetup.solution_field="position" 59 | msetup.damping_local_alpha=0.0 60 | 61 | # set gravity load in particles 62 | for ip in msh.particles: 63 | ip.f_ext += ip.mass*9.81 64 | 65 | # solve the problem in time 66 | solver.explicit_solution(msh,msetup) 67 | 68 | # plot mpm solution 69 | plt.plot(msetup.solution_array[0],msetup.solution_array[1],'-',markersize=2,label=r'mpm, $\mu$='+str(material.mu)) 70 | 71 | # configure axis, legends and show plot 72 | plt.xlabel('time (s)') 73 | plt.ylabel('displacement (m)') 74 | plt.legend() 75 | plt.show() 76 | 77 | #msh.print_mesh() -------------------------------------------------------------------------------- /verification_problems/mpm_wave_in_pile.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_wave_in_pile.png -------------------------------------------------------------------------------- /verification_problems/mpm_wave_in_pile.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the continuum 1D bar vibration problem using MPM. 7 | 8 | Data 9 | ---- 10 | Created on Sat Mar 20 20:25:53 2021 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | 24 | # local modules 25 | from modules import mesh # for mesh definition 26 | from modules import material # for material definition 27 | from modules import setup # for setup the problem 28 | from modules import solver # for solving the problem in time 29 | from analitical_solutions import analitical_solution_wave_in_pile as wip 30 | 31 | # pile length 32 | L=15 33 | 34 | # number of elements 35 | nelements=150 36 | 37 | # create an 1D mesh 38 | msh = mesh.mesh_1D(L=L,nelem=nelements) 39 | 40 | # define a linear material 41 | elastic = material.linear_elastic(E=100e6,density=2500) 42 | 43 | # put particles in mesh element and set the material 44 | msh.put_particles_in_all_mesh_elements(ppelem=2,material=elastic) 45 | 46 | # setup the model 47 | msetup = setup.model_setup() 48 | msetup.interpolation_type="linear" 49 | msetup.integration_scheme="USF" 50 | msetup.time=0.2 51 | msetup.dt=0.01 52 | msetup.solution_particle=0 53 | msetup.solution_field='position' 54 | msetup.damping_local_alpha=0.0 55 | 56 | # verify time step 57 | dt_critical=msh.elements[0].L/(elastic.E/elastic.density)**0.5 58 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 59 | 60 | # external force 61 | po =-10e3 62 | 63 | # impose intial condition in particle 64 | msh.particles[-1].f_ext=po 65 | 66 | # initial particle position to calculate analytical solution 67 | pos_initial=msh.particles[0].position 68 | 69 | # solve the problem in time 70 | solver.explicit_solution(msh,msetup) 71 | 72 | # figure to plot 73 | fig,ax = plt.subplots() 74 | 75 | # plot mpm solution 76 | ax.plot(msetup.solution_array[0],msetup.solution_array[1],linestyle='solid',linewidth=1.5,color="blue",marker='o',markersize=0,markerfacecolor='none',label='MPM') 77 | 78 | # plot the analytical solution 79 | [anal_xt,anal_vt, anal_t] = wip.wave_in_pile_fixed_and_loaded(L=L,E=elastic.E,rho=elastic.density,time=msetup.time,dt=msetup.dt/2,po=po,x=pos_initial,n_sum=1000) 80 | ax.plot(anal_t,anal_xt,'r',linewidth=1.5,label='Analytical solution') 81 | 82 | # vertical line for the theorical arrival time 83 | plt.axvline(x = (L-pos_initial)/(elastic.E/elastic.density)**0.5,linestyle="dashed",color='k',label='Analitical arrival time') 84 | 85 | # configure axis, legends and show plot 86 | ax.set_xlabel('Time (s)') 87 | ax.set_ylabel('Position (m)') 88 | ax.legend() 89 | plt.show() -------------------------------------------------------------------------------- /verification_problems/mpm_wave_in_pile_parametric_young.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fabricix/MPM-Py/b9933488ee033bc9eea8fba4e7084c824a5087bc/verification_problems/mpm_wave_in_pile_parametric_young.png -------------------------------------------------------------------------------- /verification_problems/mpm_wave_in_pile_parametric_young.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python3 2 | # -*- coding: utf-8 -*- 3 | """ 4 | Purpose 5 | ------- 6 | This example approximates the continuum 1D bar vibration problem using MPM. 7 | 8 | Data 9 | ---- 10 | Created on Sat Mar 20 20:25:53 2021 11 | 12 | Author 13 | ------ 14 | Fabricio Fernandez () 15 | """ 16 | 17 | # include the modules' path to the current path 18 | import sys 19 | sys.path.append("..") 20 | 21 | # external modules 22 | import matplotlib.pyplot as plt # for plot 23 | import numpy as np 24 | 25 | # local modules 26 | from modules import mesh # for mesh definition 27 | from modules import material # for material definition 28 | from modules import setup # for setup the problem 29 | from modules import solver # for solving the problem in time 30 | from analitical_solutions import analitical_solution_wave_in_pile as wip 31 | 32 | 33 | # density values 34 | young_serie = np.linspace(100e6,200e6,5) 35 | 36 | # get color palete for plot 37 | color_list = ['b', 'g', 'r', 'c', 'm', 'y', 'k', 'w'] 38 | 39 | for i in range(len(young_serie)): 40 | 41 | # pile length 42 | L=15 43 | 44 | # number of elements 45 | nelements=150 46 | 47 | # create an 1D mesh 48 | msh = mesh.mesh_1D(L=L,nelem=nelements) 49 | 50 | # define a linear material 51 | elastic = material.linear_elastic(E=young_serie[i],density=2500) 52 | 53 | # put particles in mesh element and set the material 54 | msh.put_particles_in_all_mesh_elements(ppelem=2,material=elastic) 55 | 56 | # setup the model 57 | msetup = setup.model_setup() 58 | msetup.interpolation_type="linear" 59 | msetup.integration_scheme="USF" 60 | msetup.time=0.14 61 | msetup.dt=0.01 62 | msetup.solution_particle=0 63 | msetup.solution_field='position' 64 | 65 | # verify time step 66 | dt_critical=msh.elements[0].L/(young_serie[i]/elastic.density)**0.5 67 | msetup.dt = msetup.dt if msetup.dt < dt_critical else dt_critical 68 | 69 | # external force 70 | po =-10e3 71 | 72 | # impose intial condition in particle 73 | msh.particles[-1].f_ext=po 74 | 75 | # initial particle position to calculate analytical solution 76 | pos_initial=msh.particles[0].position 77 | 78 | # solve the problem in time 79 | solver.explicit_solution(msh,msetup) 80 | 81 | # subset data for plot 82 | n_values = 100 83 | indices = np.linspace(0, len(msetup.solution_array[0])-1, n_values, dtype=int) 84 | 85 | y_subset = [] 86 | x_subset = [] 87 | 88 | for i_index in range(len(indices)): 89 | x_subset.append(msetup.solution_array[0][indices[i_index]]) 90 | y_subset.append(msetup.solution_array[1][indices[i_index]]) 91 | 92 | # plot mpm solution 93 | plt.plot(x_subset,y_subset,linestyle='solid',linewidth=1,color=color_list[i],marker='o',markersize=3,markerfacecolor='none',label='Young='+'{:.1f}e6-MPM'.format(young_serie[i]/1e6)) 94 | 95 | # plot the analytical solution 96 | [anal_xt,anal_vt, anal_t] = wip.wave_in_pile_fixed_and_loaded(L=L,E=young_serie[i],rho=elastic.density,time=msetup.time,dt=msetup.dt/2,po=po,x=pos_initial) 97 | plt.plot(anal_t,anal_xt,color=color_list[i],linewidth=1,label='Young='+'{:.1f}e6-Analytical'.format(young_serie[i]/1e6)) 98 | 99 | # configure axis, legends and show plot 100 | plt.gca().set_xlabel('Time (s)') 101 | plt.gca().set_ylabel('Position (m)') 102 | plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5)) 103 | plt.show() --------------------------------------------------------------------------------