├── .github ├── dependabot.yml └── workflows │ └── tests.yml ├── .gitignore ├── COPYING ├── COPYING.LESSER ├── README.md ├── demo ├── conftest.py ├── demo_kirchhoff-love-clamped.py ├── demo_nonlinear-naghdi-clamped-semicylinder.py ├── demo_reissner-mindlin-clamped-tdnns.py ├── demo_reissner-mindlin-clamped.py ├── demo_reissner-mindlin-simply-supported.py ├── pytest.ini └── test_demos.py ├── doc ├── README.md └── source │ ├── .gitignore │ ├── _static │ └── .placeholder │ ├── conf.py │ ├── demos.rst │ ├── index.rst │ └── jupytext_process.py ├── launch-container.sh ├── pyproject.toml ├── src └── fenicsx_shells │ └── __init__.py └── test └── .placeholder /.github/dependabot.yml: -------------------------------------------------------------------------------- 1 | # To get started with Dependabot version updates, you'll need to specify which 2 | # package ecosystems to update and where the package manifests are located. 3 | # Please see the documentation for all configuration options: 4 | # https://docs.github.com/code-security/dependabot/dependabot-version-updates/configuration-options-for-the-dependabot.yml-file 5 | 6 | version: 2 7 | updates: 8 | - package-ecosystem: "github-actions" # See documentation for possible values 9 | directory: "/" # Location of package manifests 10 | schedule: 11 | interval: "weekly" 12 | - package-ecosystem: "pip" # See documentation for possible values 13 | directory: "/" 14 | schedule: 15 | interval: "weekly" 16 | -------------------------------------------------------------------------------- /.github/workflows/tests.yml: -------------------------------------------------------------------------------- 1 | name: tests 2 | on: 3 | push: 4 | branches: 5 | - "**" 6 | 7 | schedule: 8 | - cron: '0 0 * * 1' 9 | 10 | pull_request: 11 | branches: 12 | - main 13 | 14 | jobs: 15 | build-and-test: 16 | runs-on: ubuntu-latest 17 | container: ghcr.io/fenics/dolfinx/dolfinx:nightly 18 | steps: 19 | - name: Checkout 20 | uses: actions/checkout@v4 21 | 22 | - name: Install FEniCSx-Shells 23 | run: | 24 | python3 -m pip install --no-build-isolation --check-build-dependencies '.[ci]' 25 | 26 | - name: ruff checks 27 | run: | 28 | ruff check . 29 | ruff format --check . 30 | 31 | - name: Build documentation 32 | run: | 33 | cd doc 34 | python3 -m sphinx -W -b html source/ build/html/ 35 | 36 | - name: Run demos 37 | run: | 38 | python3 -m pytest demo 39 | 40 | - name: Create documentation artifact 41 | run: | 42 | tar \ 43 | --dereference --hard-dereference \ 44 | --directory doc/build/html \ 45 | -cvf artifact.tar \ 46 | --exclude=.git \ 47 | --exclude=.github \ 48 | . 49 | 50 | - name: Upload documentation artifact 51 | uses: actions/upload-artifact@v4 52 | with: 53 | name: github-pages 54 | path: artifact.tar 55 | retention-days: 1 56 | 57 | deploy: 58 | needs: build-and-test 59 | if: github.ref == 'refs/heads/main' 60 | # Sets permissions of the GITHUB_TOKEN to allow deployment to GitHub Pages 61 | permissions: 62 | pages: write 63 | id-token: write 64 | environment: 65 | name: github-pages 66 | url: ${{ steps.deployment.outputs.page_url }} 67 | 68 | runs-on: ubuntu-latest 69 | steps: 70 | - name: Setup Pages 71 | uses: actions/configure-pages@v5 72 | - name: Deploy to GitHub Pages 73 | id: deployment 74 | uses: actions/deploy-pages@v4 75 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Typical DOLFINx output 2 | *.xdmf 3 | *.h5 4 | 5 | # Byte-compiled / optimized / DLL files 6 | __pycache__/ 7 | *.py[cod] 8 | *$py.class 9 | 10 | # C extensions 11 | *.so 12 | 13 | # Distribution / packaging 14 | .Python 15 | build/ 16 | develop-eggs/ 17 | dist/ 18 | downloads/ 19 | eggs/ 20 | .eggs/ 21 | lib/ 22 | lib64/ 23 | parts/ 24 | sdist/ 25 | var/ 26 | wheels/ 27 | pip-wheel-metadata/ 28 | share/python-wheels/ 29 | *.egg-info/ 30 | .installed.cfg 31 | *.egg 32 | MANIFEST 33 | 34 | # PyInstaller 35 | # Usually these files are written by a python script from a template 36 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 37 | *.manifest 38 | *.spec 39 | 40 | # Installer logs 41 | pip-log.txt 42 | pip-delete-this-directory.txt 43 | 44 | # Unit test / coverage reports 45 | htmlcov/ 46 | .tox/ 47 | .nox/ 48 | .coverage 49 | .coverage.* 50 | .cache 51 | nosetests.xml 52 | coverage.xml 53 | *.cover 54 | *.py,cover 55 | .hypothesis/ 56 | .pytest_cache/ 57 | 58 | # Translations 59 | *.mo 60 | *.pot 61 | 62 | # Django stuff: 63 | *.log 64 | local_settings.py 65 | db.sqlite3 66 | db.sqlite3-journal 67 | 68 | # Flask stuff: 69 | instance/ 70 | .webassets-cache 71 | 72 | # Scrapy stuff: 73 | .scrapy 74 | 75 | # Sphinx documentation 76 | docs/_build/ 77 | 78 | # PyBuilder 79 | target/ 80 | 81 | # Jupyter Notebook 82 | .ipynb_checkpoints 83 | 84 | # IPython 85 | profile_default/ 86 | ipython_config.py 87 | 88 | # pyenv 89 | .python-version 90 | 91 | # pipenv 92 | # According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control. 93 | # However, in case of collaboration, if having platform-specific dependencies or dependencies 94 | # having no cross-platform support, pipenv may install dependencies that don't work, or not 95 | # install all needed dependencies. 96 | #Pipfile.lock 97 | 98 | # PEP 582; used by e.g. github.com/David-OConnor/pyflow 99 | __pypackages__/ 100 | 101 | # Celery stuff 102 | celerybeat-schedule 103 | celerybeat.pid 104 | 105 | # SageMath parsed files 106 | *.sage.py 107 | 108 | # Environments 109 | .env 110 | .venv 111 | env/ 112 | venv/ 113 | ENV/ 114 | env.bak/ 115 | venv.bak/ 116 | 117 | # Spyder project settings 118 | .spyderproject 119 | .spyproject 120 | 121 | # Rope project settings 122 | .ropeproject 123 | 124 | # mkdocs documentation 125 | /site 126 | 127 | # mypy 128 | .mypy_cache/ 129 | .dmypy.json 130 | dmypy.json 131 | 132 | # Pyre type checker 133 | .pyre/ 134 | -------------------------------------------------------------------------------- /COPYING: -------------------------------------------------------------------------------- 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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Revised Versions of the GNU Lesser General Public License. 145 | 146 | The Free Software Foundation may publish revised and/or new versions 147 | of the GNU Lesser General Public License from time to time. Such new 148 | versions will be similar in spirit to the present version, but may 149 | differ in detail to address new problems or concerns. 150 | 151 | Each version is given a distinguishing version number. If the 152 | Library as you received it specifies that a certain numbered version 153 | of the GNU Lesser General Public License "or any later version" 154 | applies to it, you have the option of following the terms and 155 | conditions either of that published version or of any later version 156 | published by the Free Software Foundation. If the Library as you 157 | received it does not specify a version number of the GNU Lesser 158 | General Public License, you may choose any version of the GNU Lesser 159 | General Public License ever published by the Free Software Foundation. 160 | 161 | If the Library as you received it specifies that a proxy can decide 162 | whether future versions of the GNU Lesser General Public License shall 163 | apply, that proxy's public statement of acceptance of any version is 164 | permanent authorization for you to choose that version for the 165 | Library. 166 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # FEniCSx-Shells 2 | 3 | A FEniCS Project-based library for simulating thin structures. 4 | 5 | [![tests](https://github.com/FEniCS-Shells/fenicsx-shells/actions/workflows/tests.yml/badge.svg?branch=main)](https://github.com/FEniCS-Shells/fenicsx-shells/actions/workflows/tests.yml) 6 | [![docs](https://img.shields.io/badge/docs-ready-success)](https://fenics-shells.github.io/fenicsx-shells) 7 | 8 | ## Description 9 | 10 | FEniCSx-Shells is an open-source library that provides finite element-based 11 | numerical methods for solving a wide range of thin structural models (beams, 12 | plates and shells) expressed in the Unified Form Language (UFL) of the FEniCS 13 | Project. 14 | 15 | *FEniCSx-Shells is an experimental version targeting version v0.10.0.dev0 of the new 16 | [DOLFINx solver](https://github.com/fenics/dolfinx).* 17 | 18 | The foundational aspects of the FEniCS-Shells project are described in the paper: 19 | 20 | Simple and extensible plate and shell finite element models through automatic 21 | code generation tools, J. S. Hale, M. Brunetti, S. P. A. Bordas, C. Maurini. 22 | Computers & Structures, 209, 163-181, 23 | [doi:10.1016/j.compstruc.2018.08.001](https://doi.org/10.1016/j.compstruc.2018.08.001). 24 | 25 | ## Documentation 26 | 27 | The documentation can be viewed [here](https://fenics-shells.github.io/fenicsx-shells). 28 | 29 | ## Features 30 | 31 | FEniCSx-Shells currently includes implementations of the following structural models: 32 | 33 | * Reissner-Mindlin plates. 34 | * Kirchhoff-Love plates. 35 | 36 | A roadmap for future developments will be shared soon. 37 | 38 | We are using a variety of numerical techniques for discretising the PDEs 39 | including: 40 | 41 | * Mixed Interpolation of Tensorial Component (MITC) reduction operators. 42 | * Tangential Displacement Normal-Normal Derivative (TDNNS) methods. 43 | * Hellman-Herrmann-Johnson (HHJ) finite elements. 44 | 45 | ## Citing 46 | 47 | Please consider citing the old FEniCS-Shells paper and code if you find this 48 | repository useful. 49 | 50 | ``` 51 | @article{hale_simple_2018, 52 | title = {Simple and extensible plate and shell finite element models through automatic code generation tools}, 53 | volume = {209}, 54 | issn = {0045-7949}, 55 | url = {http://www.sciencedirect.com/science/article/pii/S0045794918306126}, 56 | doi = {10.1016/j.compstruc.2018.08.001}, 57 | journal = {Computers \& Structures}, 58 | author = {Hale, Jack S. and Brunetti, Matteo and Bordas, Stéphane P. A. and Maurini, Corrado}, 59 | month = oct, 60 | year = {2018}, 61 | keywords = {Domain specific language, FEniCS, Finite element methods, Plates, Shells, Thin structures}, 62 | pages = {163--181}, 63 | } 64 | ``` 65 | along with the appropriate general [FEniCS citations](http://fenicsproject.org/citing). 66 | 67 | ## Authors 68 | 69 | - Jack S. Hale, University of Luxembourg, Luxembourg. 70 | - Tian Yang, EPFL, Switzerland. 71 | 72 | FEniCSx-Shells contains code and text adapted from the FEniCS-Shells project 73 | hosted on [BitBucket](https://bitbucket.org/unilucompmech/fenics-shells) 74 | authored by: 75 | 76 | - Matteo Brunetti, University of Udine, Udine. 77 | - Jack S. Hale, University of Luxembourg, Luxembourg. 78 | - Corrado Maurini, Sorbonne Université, France. 79 | 80 | ## Contributing 81 | 82 | We are always looking for contributions and help with FEniCSx-Shells. If you 83 | have ideas, nice applications or code contributions then we would be happy to 84 | help you get them included. We ask you to follow the FEniCS Project git 85 | workflow. 86 | 87 | ## Issues and Support 88 | 89 | Please use the GitHub issue tracker to report any issues. 90 | 91 | ## License 92 | 93 | FEniCSx-Shells is free software: you can redistribute it and/or 94 | modify it under the terms of the GNU Lesser General Public License as published 95 | by the Free Software Foundation, either version 3 of the License, or (at your 96 | option) any later version. 97 | 98 | This program is distributed in the hope that it will be useful, but WITHOUT ANY 99 | WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A 100 | PARTICULAR PURPOSE. See the GNU Lesser General Public License for more 101 | details. 102 | 103 | You should have received a copy of the GNU Lesser General Public License along 104 | with FEniCSx-Shells. If not, see http://www.gnu.org/licenses/. 105 | -------------------------------------------------------------------------------- /demo/conftest.py: -------------------------------------------------------------------------------- 1 | import pytest 2 | 3 | 4 | def pytest_addoption(parser): 5 | parser.addoption( 6 | "--mpiexec", 7 | action="store", 8 | default="mpirun", 9 | help="Name of program to run MPI, e.g. mpiexex", 10 | ) 11 | parser.addoption("--num-proc", action="store", default=1, help="Number of MPI processes to use") 12 | 13 | 14 | @pytest.fixture 15 | def mpiexec(request): 16 | return request.config.getoption("--mpiexec") 17 | 18 | 19 | @pytest.fixture 20 | def num_proc(request): 21 | return request.config.getoption("--num-proc") 22 | -------------------------------------------------------------------------------- /demo/demo_kirchhoff-love-clamped.py: -------------------------------------------------------------------------------- 1 | # --- 2 | # jupyter: 3 | # jupytext: 4 | # text_representation: 5 | # extension: .py 6 | # format_name: light 7 | # format_version: '1.5' 8 | # jupytext_version: 1.14.1 9 | # --- 10 | 11 | # # Clamped Kirchoff-Love plate under uniform load 12 | # 13 | # This demo program solves the out-of-plane Kirchoff-Love equations on the 14 | # unit square with uniform transverse loading and fully clamped boundary 15 | # conditions. 16 | # 17 | # It is assumed the reader understands most of the basic functionality of the 18 | # new FEniCSx Project. 19 | # 20 | # This demo illustrates how to: 21 | # 22 | # - Define the Kirchhoff-Love plate equations using UFL using the mixed finite 23 | # element formulation of Hellan-Herrmann-Johnson. 24 | # 25 | # A modern presentation of this approach can be found in the paper 26 | # 27 | # Arnold, D. N., Walker S. W., The Hellan--Herrmann--Johnson Method with Curved 28 | # Elements, SIAM Journal on Numerical Analysis 58:5, 2829-2855 (2020), 29 | # [doi:10.1137/19M1288723](https://doi.org/10.1137/19M1288723). 30 | # 31 | # We remark that this model can be recovered formally from the Reissner-Mindlin 32 | # models by taking the limit in the thickness $t \to 0$ and setting $\theta = 33 | # \grad w$. 34 | # 35 | # We begin by importing the necessary functionality from DOLFINx, UFL and 36 | # PETSc. 37 | 38 | from mpi4py import MPI 39 | 40 | import numpy as np 41 | 42 | import dolfinx 43 | import ufl 44 | from basix.ufl import element, mixed_element 45 | from dolfinx.fem import dirichletbc, functionspace 46 | from dolfinx.fem.petsc import LinearProblem 47 | from dolfinx.mesh import CellType, create_unit_square 48 | from ufl import FacetNormal, Identity, Measure, grad, inner, sym, tr 49 | 50 | # We then create a two-dimensional mesh of the mid-plane of the plate $\Omega = 51 | # [0, 1] \times [0, 1]$. 52 | 53 | mesh = create_unit_square(MPI.COMM_WORLD, 16, 16, CellType.triangle) 54 | 55 | # The Hellen-Herrmann-Johnson element for the Kirchhoff-Love plate problem 56 | # consists of: 57 | # 58 | # - $k + 1$-th order scalar-valued Lagrange element for the transverse displacement field 59 | # $w \in \mathrm{CG}_{k + 1}$ and, 60 | # - $k$-th order Hellan-Herrmann-Johnson finite elements for the bending moments, which 61 | # naturally discretise tensor-valued functions in $H(\mathrm{div}\;\mathrm{\bf{div}})$, 62 | # $M \in \mathrm{HHJ}_k$. 63 | # 64 | # The final element definition is 65 | 66 | # + 67 | k = 2 68 | U_el = mixed_element( 69 | [element("Lagrange", mesh.basix_cell(), k + 1), element("HHJ", mesh.basix_cell(), k)] 70 | ) 71 | U = functionspace(mesh, U_el) 72 | 73 | w, M = ufl.TrialFunctions(U) 74 | w_t, M_t = ufl.TestFunctions(U) 75 | 76 | # - 77 | 78 | # We assume constant material parameters; Young's modulus $E$, Poisson's ratio 79 | # $\nu$, shear-correction factor $\kappa$, and thickness $t$. 80 | 81 | # + 82 | E = 10920.0 83 | nu = 0.3 84 | t = 0.001 85 | 86 | # - 87 | 88 | # The weak form for the problem can be written as: 89 | # 90 | # Find $(w, M) \in \mathrm{CG}_{k + 1} \times \mathrm{HHJ}_k$ such that 91 | # 92 | # $$ 93 | # \left( k(M), \tilde{M} \right) + \left< \tilde{M}, \theta(w) \right> + \left< 94 | # M, \theta(\tilde{w}) \right> = -\left(f, \tilde{w} \right) \quad \forall 95 | # (\tilde{w}, \tilde{M}) \in \mathrm{CG}_{k + 1} \times \mathrm{HHJ}_k, 96 | # $$ 97 | # where $\left( \cdot, \cdot \right)$ is the usual $L^2$ inner product on the 98 | # mesh $\Omega$. 99 | # The rotations $\theta$ for the Kirchhoff-Love model can be written in 100 | # terms of the transverse displacements 101 | # 102 | # $$ 103 | # \theta(w) = \nabla w. 104 | # $$ 105 | # 106 | # The bending strain tensor $k$ for the Kirchoff-Love model can be expressed 107 | # in terms of the rotations 108 | # 109 | # $$ 110 | # k(\theta) = \dfrac{1}{2}(\nabla \theta + (\nabla \theta)^T). 111 | # $$ 112 | # 113 | # The bending strain tensor $k$ can also be written in terms of the bending 114 | # moments $M$ 115 | # 116 | # $$ 117 | # k(M) = \frac{12}{E t^3} \left[ (1 + \nu) M - \nu \mathrm{tr}\left( M \right) I \right], 118 | # $$ 119 | # with $\mathrm{tr}$ the trace operator and $I$ the identity tensor. 120 | # 121 | # The inner product $\left< \cdot, \cdot \right>$ is defined by 122 | # 123 | # $$ 124 | # \left< M, \theta \right> = -\left( M, k(\theta) \right) + \int_{\partial K} 125 | # M_{nn} \cdot [[ \theta ]]_n \; \mathrm{d}s, 126 | # $$ 127 | # where $M_{nn} = \left(Mn \right) \cdot n$ is the normal-normal component of 128 | # the bending moment, $\partial K$ are the facets of the mesh, $[[ \theta ]]$ is 129 | # the jump in the normal component of the rotations on the facets (reducing to 130 | # simply $\theta \cdot n$ on the exterior facets). 131 | # 132 | # The above equations can be written relatively straightforwardly in UFL as: 133 | 134 | # + 135 | dx = Measure("dx", mesh) 136 | dS = Measure("dS", mesh) 137 | ds = Measure("ds", mesh) 138 | 139 | 140 | def theta(w): 141 | """Rotations in terms of transverse displacements""" 142 | return grad(w) 143 | 144 | 145 | def k_theta(theta): 146 | """Bending strain tensor in terms of rotations""" 147 | return sym(grad(theta)) 148 | 149 | 150 | def k_M(M): 151 | """Bending strain tensor in terms of bending moments""" 152 | return (12.0 / (E * t**3)) * ((1.0 + nu) * M - nu * Identity(2) * tr(M)) 153 | 154 | 155 | def nn(M): 156 | """Normal-normal component of tensor""" 157 | n = FacetNormal(M.ufl_domain()) 158 | M_n = ufl.dot(M, n) 159 | M_nn = ufl.dot(M_n, n) 160 | return M_nn 161 | 162 | 163 | def inner_divdiv(M, theta): 164 | """Discrete div-div inner product""" 165 | n = FacetNormal(M.ufl_domain()) 166 | M_nn = nn(M) 167 | result = ( 168 | -inner(M, k_theta(theta)) * dx 169 | + inner(M_nn("+"), ufl.jump(theta, n)) * dS 170 | + inner(M_nn, ufl.dot(theta, n)) * ds 171 | ) 172 | return result 173 | 174 | 175 | a = inner(k_M(M), M_t) * dx + inner_divdiv(M_t, theta(w)) + inner_divdiv(M, theta(w_t)) 176 | L = -inner(t**3, w_t) * dx 177 | 178 | 179 | def all_boundary(x): 180 | return np.full(x.shape[1], True, dtype=bool) 181 | 182 | 183 | # - 184 | 185 | # We apply clamped boundary conditions on the entire boundary. The essential 186 | # boundary condition $w = 0$ is enforced directly in the finite element space, 187 | # while the condition $\nabla w \cdot n = 0$ is a natural condition that is 188 | # satisfied when the corresponding essential condition on $m_{nn}$ is dropped. 189 | 190 | # TODO: Add table like TDNNS example. 191 | 192 | # + 193 | boundary_entities = dolfinx.mesh.locate_entities_boundary(mesh, mesh.topology.dim - 1, all_boundary) 194 | 195 | bcs = [] 196 | # Transverse displacement 197 | boundary_dofs_displacement = dolfinx.fem.locate_dofs_topological( 198 | U.sub(0), mesh.topology.dim - 1, boundary_entities 199 | ) 200 | bcs.append(dirichletbc(np.array(0.0, dtype=np.float64), boundary_dofs_displacement, U.sub(0))) 201 | 202 | 203 | # - 204 | 205 | # Finally we solve the problem and output the transverse displacement at the 206 | # centre of the plate. 207 | 208 | # + 209 | problem = LinearProblem( 210 | a, 211 | L, 212 | bcs=bcs, 213 | petsc_options={"ksp_type": "preonly", "pc_type": "lu", "pc_factor_mat_solver_type": "mumps"}, 214 | ) 215 | u_h = problem.solve() 216 | 217 | bb_tree = dolfinx.geometry.bb_tree(mesh, 2) 218 | point = np.array([[0.5, 0.5, 0.0]], dtype=np.float64) 219 | cell_candidates = dolfinx.geometry.compute_collisions_points(bb_tree, point) 220 | cells = dolfinx.geometry.compute_colliding_cells(mesh, cell_candidates, point) 221 | 222 | w, M = u_h.split() 223 | 224 | if cells.array.shape[0] > 0: 225 | value = w.eval(point, cells.array[0]) 226 | print(value[0]) 227 | -------------------------------------------------------------------------------- /demo/demo_nonlinear-naghdi-clamped-semicylinder.py: -------------------------------------------------------------------------------- 1 | # %% [markdown] 2 | # # Clamped semi-cylindrical Naghdi shell under point load 3 | # 4 | # Authors: Tian Yang (FEniCSx-Shells), Matteo Brunetti (FEniCS-Shells) 5 | # 6 | # %% [markdown] 7 | # This demo program solves the nonlinear Naghdi shell equations for a 8 | # semi-cylindrical shell loaded by a point force. 9 | # 10 | # This problem is a standard reference for testing shell finite element 11 | # formulations, see [1]. The numerical locking issue is cured using enriched 12 | # finite element including cubic bubble shape functions and Partial Selective 13 | # Reduced Integration (PRSI) [2]. 14 | # 15 | # It is assumed the reader understands most of the basic functionality of the 16 | # new FEniCSx Project. 17 | # 18 | # This demo then illustrates how to: 19 | # 20 | # - Define and solve a nonlinear Naghdi shell problem with a curved stress-free 21 | # configuration given as analytical expression in terms of two curvilinear 22 | # coordinates. 23 | # - Use the PSRI approach to simultaneously cure shear- and membrane-locking 24 | # issues. 25 | # 26 | # We begin by importing the necessary functionality from DOLFINx, UFL and 27 | # PETSc. 28 | # 29 | # %% 30 | import typing 31 | from pathlib import Path 32 | 33 | # %% 34 | from mpi4py import MPI 35 | from petsc4py import PETSc 36 | 37 | import matplotlib.pyplot as plt 38 | import numpy as np 39 | 40 | import dolfinx 41 | import ufl 42 | 43 | # %% 44 | from basix.ufl import blocked_element, element, enriched_element, mixed_element 45 | from dolfinx.fem import Expression, Function, dirichletbc, functionspace, locate_dofs_topological 46 | from dolfinx.fem.bcs import DirichletBC 47 | from dolfinx.fem.function import Function as _Function 48 | from dolfinx.fem.petsc import NonlinearProblem, apply_lifting, assemble_vector, set_bc 49 | from dolfinx.mesh import CellType, create_rectangle, locate_entities_boundary 50 | from dolfinx.nls.petsc import NewtonSolver 51 | from ufl import grad, inner, split 52 | 53 | # %% [markdown] 54 | # We consider a semi-cylindrical shell of radius $r$ and axis length $L$. The 55 | # shell is made of a linear elastic isotropic homogeneous material with Young 56 | # modulus $E$ and Poisson ratio $\nu$. The (uniform) shell thickness is denoted 57 | # by $t$. The Lamé moduli $\lambda$, $\mu$ are introduced to write later the 2D 58 | # constitutive equation in plane-stress: 59 | # 60 | # %% 61 | r = 1.016 62 | L = 3.048 63 | E, nu = 2.0685e7, 0.3 64 | mu = E / (2.0 * (1.0 + nu)) 65 | lmbda = 2.0 * mu * nu / (1.0 - 2.0 * nu) 66 | t = 0.03 67 | # %% [markdown] 68 | # The midplane of the initial (stress-free) configuration $\vec{\phi_0}$ of the 69 | # shell is given in the form of an analytical expression: 70 | # 71 | # $$ 72 | # \vec{\phi}_0(\xi_1, \xi_2) \subset \mathbb{R}³ 73 | # $$ 74 | # 75 | # where $\xi_1 \in [-\pi/2, \pi/2]$ and $\xi_2 \in [0, L]$ are the curvilinear 76 | # coordinates. In this case, they represent the angular and axial coordinates, 77 | # respectively. 78 | # 79 | # %% [markdown] 80 | # We generate a mesh in the $(\xi_1, \xi_2)$ space with quadrilateral cells 81 | # 82 | # %% 83 | mesh = create_rectangle( 84 | MPI.COMM_WORLD, np.array([[-np.pi / 2, 0], [np.pi / 2, L]]), [20, 20], CellType.triangle 85 | ) 86 | tdim = mesh.topology.dim # = 2 87 | 88 | # %% [markdown] 89 | # We provide the analytical expression of the initial shape as a `ufl` 90 | # expression 91 | # 92 | # %% 93 | x = ufl.SpatialCoordinate(mesh) 94 | phi0_ufl = ufl.as_vector([r * ufl.sin(x[0]), x[1], r * ufl.cos(x[0])]) 95 | 96 | 97 | # %% [markdown] 98 | # Given the analytical expression of midplane, we define the unit normal as 99 | # below: 100 | # 101 | # $$ 102 | # \vec{n} = \frac{\partial_1 \phi_0 \times \partial_2 \phi_0}{\| \partial_1 \phi_0 \times 103 | # \partial_2 \phi_0 \|} 104 | # $$ 105 | # 106 | # %% 107 | def unit_normal(phi): 108 | n = ufl.cross(phi.dx(0), phi.dx(1)) 109 | return n / ufl.sqrt(inner(n, n)) 110 | 111 | 112 | n0_ufl = unit_normal(phi0_ufl) 113 | 114 | 115 | # %% [markdown] 116 | # We define a local orthonormal frame $\{\vec{t}_{01}, \vec{t}_{02}, \vec{n}\}$ 117 | # of the initial configuration $\phi_0$ by rotating the global Cartesian basis 118 | # $\vec{e}_i$ with a rotation matrix $\mathbf{R}_0$: 119 | # 120 | # $$ 121 | # \vec{t}_{0i} = \mathbf{R}_0 \vec{e}_i , \quad \vec{n} = \vec{t}_{03}, 122 | # $$ 123 | # 124 | # A convienient choice of $\vec{t}_{01}$ and $\vec{t}_{02}$ (when $\vec{n} 125 | # \nparallel \vec{e}_2 $) could be: 126 | # 127 | # $$ 128 | # \vec{t}_{01} = \frac{\vec{e}_2 \times \vec{n}}{\| \vec{e}_2 \times \vec{n}\|} \\ 129 | # \vec{t}_{02} = \vec{n} \times \vec{t}_{01} 130 | # $$ 131 | # 132 | # The corresponding rotation matrix $\mathbf{R}_0$: 133 | # 134 | # $$ 135 | # \mathbf{R}_0 = [\vec{t}_{01}; \vec{t}_{02}; \vec{n}] 136 | # $$ 137 | # 138 | # %% 139 | def tangent_1(n): 140 | e2 = ufl.as_vector([0, 1, 0]) 141 | t1 = ufl.cross(e2, n) 142 | t1 = t1 / ufl.sqrt(inner(t1, t1)) 143 | return t1 144 | 145 | 146 | def tangent_2(n, t1): 147 | t2 = ufl.cross(n, t1) 148 | t2 = t2 / ufl.sqrt(inner(t2, t2)) 149 | return t2 150 | 151 | 152 | # The analytical expression of t1 and t2 153 | t1_ufl = tangent_1(n0_ufl) 154 | t2_ufl = tangent_2(n0_ufl, t1_ufl) 155 | 156 | 157 | # The analytical expression of R0 158 | def rotation_matrix(t1, t2, n): 159 | R = ufl.as_matrix([[t1[0], t2[0], n[0]], [t1[1], t2[1], n[1]], [t1[2], t2[2], n[2]]]) 160 | return R 161 | 162 | 163 | R0_ufl = rotation_matrix(t1_ufl, t2_ufl, n0_ufl) 164 | 165 | 166 | # %% [markdown] 167 | # The kinematics of the Nadghi shell model is defined by the following vector 168 | # fields: 169 | # - $\vec{\phi}$: the position of the midplane in the deformed configuration, 170 | # or equivalently, the displacement $\vec{u} = \vec{\phi} - \vec{\phi}_0$ 171 | # - $\vec{d}$: the director, a unit vector giving the orientation of fiber at 172 | # the midplane. (not necessarily normal to the midsplane because of shears) 173 | # 174 | # According to [3], the director $\vec{d}$ in the deformed configuration can be 175 | # parameterized with two successive rotation angles $\theta_1, \theta_2$ 176 | # 177 | # $$ 178 | # \vec{t}_i = \mathbf{R} \vec{e}_i, \quad \mathbf{R} = \text{exp}[\theta_1 179 | # \hat{\mathbf{t}}_1] \text{exp}[\theta_2 \hat{\mathbf{t}}_{02}] \mathbf{R}_0 180 | # $$ 181 | # 182 | # The rotation matrix $\mathbf{R}$ represents three successive rotations: 183 | # - First one: the initial rotation matrix $\mathbf{R}_0$ 184 | # - Second one :$\text{exp}[\theta_2 \hat{\mathbf{t}}_{02}]$ rotates a vector 185 | # about the axis $\vec{t}_{02}$ of $\theta_2$ angle; 186 | # - Third one : $\text{exp}[\theta_1 \hat{\mathbf{t}}_1]$ rotates a vector 187 | # about the axis $\vec{t}_{1}$ of $\theta_1$ angle, and $\vec{t}_1 = \text{exp} 188 | # [\theta_2\hat{\mathbf{t}}_{02}] \vec{t}_{01}$ 189 | # 190 | # The rotation matrix $\mathbf{R}$ on the other hand it is equivalent to rotate 191 | # around the fixed axis $\vec{e}_1$ and $\vec{e}_2$ (Proof see [3]): 192 | # 193 | # $$ 194 | # \mathbf{R} = \mathbf{R}_0 \text{exp}[\theta_2 \hat{\mathbf{e}}_{2}] 195 | # \text{exp}[\theta_1 \hat{\mathbf{e}}_1] 196 | # $$ 197 | # 198 | # Therefore, the director $\vec{d}$ is updated with $(\theta_1, \theta_2)$ by: 199 | # 200 | # $$ 201 | # \vec{d} =\mathbf{R} \vec{e}_3 = \mathbf{R}_0 \vec{\Lambda}_3, \quad \vec{\Lambda}_3 = 202 | # [\sin(\theta_2)\cos(\theta_1), -\sin(\theta_1), \cos(\theta_2)\cos(\theta_1)]^\text{T} 203 | # $$ 204 | # 205 | # Note: the above formular becomes singular when $\theta_1 = \pm \pi/2, ...$, (See Chapter 4.2.1 in 206 | # [3] for details) 207 | # %% 208 | def director(R0, theta): 209 | """Updates the director with two successive elementary rotations""" 210 | Lm3 = ufl.as_vector( 211 | [ 212 | ufl.sin(theta[1]) * ufl.cos(theta[0]), 213 | -ufl.sin(theta[0]), 214 | ufl.cos(theta[1]) * ufl.cos(theta[0]), 215 | ] 216 | ) 217 | d = ufl.dot(R0, Lm3) 218 | return d 219 | 220 | 221 | # %% [markdown] 222 | # In our 5-parameter Naghdi shell model the configuration of the shell is 223 | # assigned by: 224 | # - the 3-component vector field $\vec{u}$ representing the 225 | # displacement with respect to the initial configuration $\vec{\phi}_0$ 226 | # - the 2-component vector field $\vec{\theta}$ representing the angle 227 | # variation of the director $\vec{d}$ with respect to initial unit normal 228 | # $\vec{n}$ 229 | # 230 | # %% [markdown] 231 | # Following [1], we use a $[P_2 + B_3]^3$ element for $\vec{u}$ and a $[P_2]^2$ 232 | # element for $\vec{\theta}$ and collect them in the state vector $\vec{q} = 233 | # [\vec{u}, \vec{\theta}]$: 234 | # 235 | # %% 236 | cell = mesh.basix_cell() 237 | P2 = element("Lagrange", cell, degree=2) 238 | B3 = element("Bubble", cell, degree=3) 239 | P2B3 = enriched_element([P2, B3]) 240 | 241 | naghdi_shell_element = mixed_element( 242 | [blocked_element(P2B3, shape=(3,)), blocked_element(P2, shape=(2,))] 243 | ) 244 | naghdi_shell_FS = functionspace(mesh, naghdi_shell_element) 245 | 246 | # %% [markdown] 247 | # Then, we define `Function`, `TrialFunction` and `TestFunction` objects to 248 | # express the variational forms and we split the mixed function into two 249 | # subfunctions for displacement and rotation. 250 | 251 | # %% 252 | q_func = Function(naghdi_shell_FS) # current configuration 253 | q_trial = ufl.TrialFunction(naghdi_shell_FS) 254 | q_test = ufl.TestFunction(naghdi_shell_FS) 255 | 256 | u_func, theta_func = split(q_func) # current displacement and rotation 257 | 258 | # %% [markdown] 259 | # We calculate the deformation gradient and the first and second fundamental 260 | # forms: 261 | # 262 | # - Deformation gradient $\mathbf{F}$ 263 | # 264 | # $$ 265 | # \mathbf{F} = \nabla \vec{\phi} \quad (F_{ij} = \frac{\partial \phi_i}{\partial \xi_j}); 266 | # \quad \vec{\phi} = 267 | # \vec{\phi}_0 + 268 | # \vec{u} \quad i = 1,2,3; j = 1,2 269 | # $$ 270 | # 271 | # - Metric tensor $\mathbf{a} \in \mathbb{S}^2_+$ and curvature tensor 272 | # $\mathbf{b} \in \mathbb{S}^2$ (First and second fundamental form) 273 | # 274 | # $$ 275 | # \begin{aligned} 276 | # \mathbf{a} &= {\nabla \vec{\phi}} ^{T} \nabla \vec{\phi} \\ 277 | # \mathbf{b} &= -\frac{1}{2}({\nabla \vec{\phi}} ^{T} \nabla \vec{d} + {\nabla \vec{d}} ^{T} 278 | # \nabla \vec{\phi}) 279 | # \end{aligned} 280 | # $$ 281 | # 282 | # In the initial configuration, $\vec{d} = \vec{n}$, $\vec{\phi} = 283 | # \vec{\phi}_0$, the conresponding initial tensors are $\mathbf{a}_0$, 284 | # $\mathbf{b}_0$ 285 | 286 | # %% 287 | # Current deformation gradient 288 | F = grad(u_func) + grad(phi0_ufl) 289 | 290 | # current director 291 | d = director(R0_ufl, theta_func) 292 | 293 | # initial metric and curvature tensor a0 and b0 294 | a0_ufl = grad(phi0_ufl).T * grad(phi0_ufl) 295 | b0_ufl = -0.5 * (grad(phi0_ufl).T * grad(n0_ufl) + grad(n0_ufl).T * grad(phi0_ufl)) 296 | 297 | # %% [markdown] 298 | # We define strain measures of the Naghdi shell model: 299 | # - Membrane strain tensor $\boldsymbol{\varepsilon}(\vec{u})$ 300 | # 301 | # $$ 302 | # \boldsymbol{\varepsilon} (\vec{u})= \frac{1}{2} \left ( \mathbf{a}(\vec{u}) - \mathbf{a}_0 \right) 303 | # $$ 304 | # 305 | # - Bending strain tensor $\boldsymbol{\kappa}(\vec{u}, \vec{\theta})$ 306 | # 307 | # $$ 308 | # \boldsymbol{\kappa}(\vec{u}, \vec{\theta}) = \mathbf{b}(\vec{u}, \vec{\theta}) - \mathbf{b}_0 309 | # $$ 310 | # 311 | # - transverse shear strain vector $\vec{\gamma}(\vec{u}, \vec{\theta})$ 312 | # 313 | # $$ 314 | # \begin{aligned} 315 | # \vec{\gamma}(\vec{u}, \vec{\theta}) & = {\nabla \vec{\phi}(\vec{u})}^T \vec{d}(\vec{\theta}) 316 | # - {\nabla\vec{\phi}_0}^T 317 | # \vec{n} \\ 318 | # & = {\nabla \vec{\phi}(\vec{u})}^T \vec{d}(\vec{\theta}) \quad \text{if zero initial shears} 319 | # \end{aligned} 320 | # $$ 321 | # 322 | 323 | 324 | # %% 325 | def epsilon(F): 326 | """Membrane strain""" 327 | return 0.5 * (F.T * F - a0_ufl) 328 | 329 | 330 | def kappa(F, d): 331 | """Bending strain""" 332 | return -0.5 * (F.T * grad(d) + grad(d).T * F) - b0_ufl 333 | 334 | 335 | def gamma(F, d): 336 | """Transverse shear strain""" 337 | return F.T * d 338 | 339 | 340 | # %% [markdown] 341 | # In curvilinear coordinates, the stiffness modulus of linear isotropic 342 | # material is defined as: 343 | # 344 | # - Membrane and bending stiffness modulus $A^{\alpha\beta\sigma\tau}$, 345 | # $D^{\alpha\beta\sigma\tau}$ 346 | # (contravariant components) 347 | # 348 | # $$ 349 | # \frac{A^{\alpha\beta\sigma\tau}}t=12\frac{D^{\alpha\beta\sigma\tau}}{t^3}= 350 | # \frac{2\lambda\mu}{\lambda+2\mu} 351 | # a_0^{\alpha\beta}a_0^{\sigma\tau}+\mu(a_0^{\alpha\sigma}a_0^{\beta\tau}+a_0^{\alpha\tau} 352 | # a_0^{\beta\sigma}) 353 | # $$ 354 | # 355 | # - Shear stiffness modulus $S^{\alpha\beta}$ (contravariant components) 356 | # 357 | # $$ 358 | # \frac{S^{\alpha\beta}}t = \alpha_s \mu a_0^{\alpha\beta} , \quad \alpha_s = 359 | # \frac{5}{6}: \text{shear factor} 360 | # $$ 361 | # 362 | # where $a_0^{\alpha\beta}$ is the contravariant components of the initial 363 | # metric tensor $\mathbf{a}_0$ 364 | 365 | # %% 366 | a0_contra_ufl = ufl.inv(a0_ufl) 367 | j0_ufl = ufl.det(a0_ufl) 368 | 369 | i, j, l, m = ufl.indices(4) # noqa: E741 370 | A_contra_ufl = ufl.as_tensor( 371 | ( 372 | ((2.0 * lmbda * mu) / (lmbda + 2.0 * mu)) * a0_contra_ufl[i, j] * a0_contra_ufl[l, m] 373 | + 1.0 374 | * mu 375 | * (a0_contra_ufl[i, l] * a0_contra_ufl[j, m] + a0_contra_ufl[i, m] * a0_contra_ufl[j, l]) 376 | ), 377 | [i, j, l, m], 378 | ) 379 | 380 | # %% [markdown] 381 | # We define the resultant stress measures: 382 | # 383 | # - Membrane stress tensor $\mathbf{N}$ 384 | # 385 | # $$ 386 | # \mathbf{N} = \mathbf{A} : \boldsymbol{\varepsilon} 387 | # $$ 388 | # 389 | # - Bending stress tensor $\mathbf{M}$ 390 | # 391 | # $$ 392 | # \mathbf{M} = \mathbf{D} : \boldsymbol{\kappa} 393 | # $$ 394 | # 395 | # - Shear stress vector $\vec{T}$ 396 | # 397 | # $$ 398 | # \vec{T} = \mathbf{S} \cdot \vec{\gamma} 399 | # $$ 400 | # 401 | 402 | # %% 403 | N = ufl.as_tensor(t * A_contra_ufl[i, j, l, m] * epsilon(F)[l, m], [i, j]) 404 | M = ufl.as_tensor((t**3 / 12.0) * A_contra_ufl[i, j, l, m] * kappa(F, d)[l, m], [i, j]) 405 | T = ufl.as_tensor((t * mu * 5.0 / 6.0) * a0_contra_ufl[i, j] * gamma(F, d)[j], [i]) 406 | 407 | # %% [markdown] 408 | # We define elastic strain energy density $\psi_{m}$, $\psi_{b}$, $\psi_{s}$ for membrane, bending 409 | # and shear, 410 | # respectively. 411 | # 412 | # $$ 413 | # \psi_m = \frac{1}{2} \mathbf{N} : \boldsymbol{\varepsilon}; \quad 414 | # \psi_b = \frac{1}{2} \mathbf{M} : \boldsymbol{\kappa}; \quad 415 | # \psi_s = \frac{1}{2} \vec{T} \cdot \vec{\gamma} 416 | # $$ 417 | # 418 | # They are per unit surface in the initial configuration: 419 | 420 | # %% 421 | psi_m = 0.5 * inner(N, epsilon(F)) 422 | psi_b = 0.5 * inner(M, kappa(F, d)) 423 | psi_s = 0.5 * inner(T, gamma(F, d)) 424 | 425 | # %% [markdown] 426 | # Shear and membrane locking is treated using the partial reduced selective 427 | # integration proposed in Arnold and Brezzi [2]. 428 | # 429 | # We introduce a parameter $\alpha \in \mathbb{R}$ that splits the membrane and 430 | # shear energy in the energy functional into a weighted sum of two parts: 431 | # 432 | # $$ 433 | # \begin{aligned}\Pi_{N}(u,\theta)&=\Pi^b(u_h,\theta_h)+\alpha\Pi^m(u_h)+(1-\alpha)\Pi^m(u_h)\\&+ 434 | # \alpha\Pi^s(u_h,\theta_h) 435 | # +(1-\alpha)\Pi^s(u_h,\theta_h)-W_{\mathrm{ext}},\end{aligned} 436 | # $$ 437 | # 438 | # We apply reduced integration to the parts weighted by the factor $(1-\alpha)$ 439 | # 440 | # More details: 441 | # - Optimal choice $\alpha = \frac{t^2}{h^2}$, $h$ is the diameter of the cell 442 | # - Full integration : Gauss quadrature of degree 4 (6 integral points for triangle) 443 | # - Reduced integration : Gauss quadrature of degree 2 (3 integral points for triangle). 444 | # - While [1] suggests a 1-point reduced integration, we observed that this 445 | # leads to spurious modes in the present case. 446 | # 447 | 448 | # %% 449 | # Full integration of order 4 450 | dx_f = ufl.Measure("dx", domain=mesh, metadata={"quadrature_degree": 4}) 451 | 452 | # Reduced integration of order 2 453 | dx_r = ufl.Measure("dx", domain=mesh, metadata={"quadrature_degree": 2}) 454 | 455 | # Calculate the factor alpha as a function of the mesh size h 456 | h = ufl.CellDiameter(mesh) 457 | alpha_FS = functionspace(mesh, element("DG", cell, 0)) 458 | alpha_expr = Expression(t**2 / h**2, alpha_FS.element.interpolation_points) 459 | alpha = Function(alpha_FS) 460 | alpha.interpolate(alpha_expr) 461 | 462 | # Full integration part of the total elastic energy 463 | Pi_PSRI = psi_b * ufl.sqrt(j0_ufl) * dx_f 464 | Pi_PSRI += alpha * psi_m * ufl.sqrt(j0_ufl) * dx_f 465 | Pi_PSRI += alpha * psi_s * ufl.sqrt(j0_ufl) * dx_f 466 | 467 | # Reduced integration part of the total elastic energy 468 | Pi_PSRI += (1.0 - alpha) * psi_m * ufl.sqrt(j0_ufl) * dx_r 469 | Pi_PSRI += (1.0 - alpha) * psi_s * ufl.sqrt(j0_ufl) * dx_r 470 | 471 | # External work part (zero in this case) 472 | W_ext = 0.0 473 | Pi_PSRI -= W_ext 474 | 475 | # %% [markdown] 476 | # The residual and jacobian are the first and second order derivatives of the 477 | # total potential energy, respectively 478 | 479 | # %% 480 | F = ufl.derivative(Pi_PSRI, q_func, q_test) 481 | J = ufl.derivative(F, q_func, q_trial) 482 | 483 | 484 | # %% [markdown] 485 | # Next, we prescribe the dirichlet boundary conditions: 486 | # - fully clamped boundary conditions on the top boundary ($\xi_2 = 0$): 487 | # - $u_{1,2,3} = \theta_{1,2} = 0$ 488 | # %% 489 | def clamped_boundary(x): 490 | return np.isclose(x[1], 0.0) 491 | 492 | 493 | fdim = tdim - 1 494 | clamped_facets = locate_entities_boundary(mesh, fdim, clamped_boundary) 495 | 496 | u_FS, _ = naghdi_shell_FS.sub(0).collapse() 497 | theta_FS, _ = naghdi_shell_FS.sub(1).collapse() 498 | 499 | # u1, u2, u3 = 0 on the clamped boundary 500 | u_clamped = Function(u_FS) 501 | clamped_dofs_u = locate_dofs_topological((naghdi_shell_FS.sub(0), u_FS), fdim, clamped_facets) 502 | bc_clamped_u = dirichletbc(u_clamped, clamped_dofs_u, naghdi_shell_FS.sub(0)) 503 | 504 | # theta1, theta2 = 0 on the clamped boundary 505 | theta_clamped = Function(theta_FS) 506 | clamped_dofs_theta = locate_dofs_topological( 507 | (naghdi_shell_FS.sub(1), theta_FS), fdim, clamped_facets 508 | ) 509 | bc_clamped_theta = dirichletbc(theta_clamped, clamped_dofs_theta, naghdi_shell_FS.sub(1)) 510 | 511 | # %% [markdown] 512 | # - symmetry boundary conditions on the left and right side ($\xi_1 = \pm 513 | # \pi/2$): 514 | # - $u_3 = \theta_2 = 0$ 515 | 516 | 517 | # %% 518 | def symm_boundary(x): 519 | return np.isclose(abs(x[0]), np.pi / 2) 520 | 521 | 522 | symm_facets = locate_entities_boundary(mesh, fdim, symm_boundary) 523 | 524 | symm_dofs_u = locate_dofs_topological( 525 | (naghdi_shell_FS.sub(0).sub(2), u_FS.sub(2)), fdim, symm_facets 526 | ) 527 | bc_symm_u = dirichletbc(u_clamped, symm_dofs_u, naghdi_shell_FS.sub(0).sub(2)) 528 | 529 | symm_dofs_theta = locate_dofs_topological( 530 | (naghdi_shell_FS.sub(1).sub(1), theta_FS.sub(1)), fdim, symm_facets 531 | ) 532 | bc_symm_theta = dirichletbc(theta_clamped, symm_dofs_theta, naghdi_shell_FS.sub(1).sub(1)) 533 | 534 | bcs = [bc_clamped_u, bc_clamped_theta, bc_symm_u, bc_symm_theta] 535 | 536 | 537 | # %% [markdown] 538 | # The loading is exerted by a point force along the $z$ direction applied at 539 | # the midpoint of the bottom boundary. 540 | # Since `PointSource` function is not available by far in new FEniCSx, we 541 | # achieve the same functionality according to the method detailed in [4]. 542 | # %% 543 | def compute_cell_contributions(V, points): 544 | """Returns the cell containing points and the values of the basis functions 545 | at that point""" 546 | # Determine what process owns a point and what cells it lies within 547 | mesh = V.mesh 548 | point_ownership_data = dolfinx.cpp.geometry.determine_point_ownership( 549 | mesh._cpp_object, points, 1e-6 550 | ) 551 | 552 | owning_points = np.asarray(point_ownership_data.dest_points).reshape(-1, 3) 553 | cells = point_ownership_data.dest_cells 554 | 555 | # Pull owning points back to reference cell 556 | mesh_nodes = mesh.geometry.x 557 | cmap = mesh.geometry.cmap 558 | ref_x = np.zeros((len(cells), mesh.geometry.dim), dtype=mesh.geometry.x.dtype) 559 | for i, (point, cell) in enumerate(zip(owning_points, cells)): 560 | geom_dofs = mesh.geometry.dofmap[cell] 561 | ref_x[i] = cmap.pull_back(point.reshape(-1, 3), mesh_nodes[geom_dofs]) 562 | 563 | # Create expression evaluating a trial function (i.e. just the basis function) 564 | u = ufl.TrialFunction(V.sub(0).sub(2)) 565 | num_dofs = V.sub(0).sub(2).dofmap.dof_layout.num_dofs * V.sub(0).sub(2).dofmap.bs 566 | if len(cells) > 0: 567 | # NOTE: Expression lives on only this communicator rank 568 | expr = dolfinx.fem.Expression(u, ref_x, comm=MPI.COMM_SELF) 569 | values = expr.eval(mesh, np.asarray(cells, dtype=np.int32)) 570 | 571 | # Strip out basis function values per cell 572 | basis_values = values[: num_dofs : num_dofs * len(cells)] 573 | else: 574 | basis_values = np.zeros((0, num_dofs), dtype=dolfinx.default_scalar_type) 575 | return cells, basis_values 576 | 577 | 578 | # %% 579 | # Point source 580 | if mesh.comm.rank == 0: 581 | points = np.array([[0.0, L, 0.0]], dtype=mesh.geometry.x.dtype) 582 | else: 583 | points = np.zeros((0, 3), dtype=mesh.geometry.x.dtype) 584 | 585 | cells, basis_values = compute_cell_contributions(naghdi_shell_FS, points) 586 | 587 | 588 | # %% [markdown] 589 | # We define a custom `NonlinearProblem` which is able to include the point 590 | # force. 591 | # %% 592 | class NonlinearProblemPointSource(NonlinearProblem): 593 | def __init__( 594 | self, 595 | F: ufl.form.Form, 596 | u: _Function, 597 | bcs: typing.List[DirichletBC] = [], # noqa: UP006 598 | J: ufl.form.Form = None, 599 | cells=[], 600 | basis_values=[], 601 | PS: float = 0.0, 602 | ): 603 | super().__init__(F, u, bcs, J) 604 | self.PS = PS 605 | self.cells = cells 606 | self.basis_values = basis_values 607 | self.function_space = u.function_space 608 | 609 | def F(self, x: PETSc.Vec, b: PETSc.Vec) -> None: 610 | with b.localForm() as b_local: 611 | b_local.set(0.0) 612 | assemble_vector(b, self._L) 613 | 614 | # Add point source 615 | if len(self.cells) > 0: 616 | for cell, basis_value in zip(self.cells, self.basis_values): 617 | dofs = self.function_space.sub(0).sub(2).dofmap.cell_dofs(cell) 618 | with b.localForm() as b_local: 619 | b_local.setValuesLocal( 620 | dofs, basis_value * self.PS, addv=PETSc.InsertMode.ADD_VALUES 621 | ) 622 | 623 | apply_lifting(b, [self._a], bcs=[self.bcs], x0=[x], alpha=-1.0) 624 | b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) 625 | set_bc(b, self.bcs, x, -1.0) 626 | 627 | 628 | # %% [markdown] 629 | # We use the standard Newton iteration. 630 | 631 | # %% 632 | problem = NonlinearProblemPointSource(F, q_func, bcs, J, cells, basis_values) 633 | 634 | solver = NewtonSolver(mesh.comm, problem) 635 | 636 | # Set Newton solver options 637 | solver.rtol = 1e-6 638 | solver.atol = 1e-6 639 | solver.max_it = 30 640 | solver.convergence_criterion = "incremental" 641 | solver.report = True 642 | 643 | # Modify the linear solver in each Newton iteration 644 | ksp = solver.krylov_solver 645 | opts = PETSc.Options() 646 | option_prefix = ksp.getOptionsPrefix() 647 | opts[f"{option_prefix}ksp_type"] = "preonly" 648 | opts[f"{option_prefix}pc_factor_mat_solver_type"] = "mumps" 649 | ksp.setFromOptions() 650 | 651 | # %% [markdown] 652 | # Finally, we can solve the quasi-static problem, incrementally increasing the 653 | # loading from $0$N to $2000$N 654 | # %% 655 | PS_diff = 50.0 656 | n_step = 40 657 | 658 | # Store the displacement at the point load 659 | if mesh.comm.rank == 0: 660 | u3_list = np.zeros(n_step + 1) 661 | PS_list = np.arange(0, PS_diff * (n_step + 1), PS_diff) 662 | 663 | q_func.x.array[:] = 0.0 664 | 665 | bb_point = np.array([[0.0, L, 0.0]], dtype=np.float64) 666 | 667 | for i in range(1, n_step + 1): 668 | problem.PS = PS_diff * i 669 | n, converged = solver.solve(q_func) 670 | assert converged 671 | q_func.x.scatter_forward() 672 | if mesh.comm.rank == 0: 673 | print(f"Load step {i:d}, Number of iterations: {n:d}, Load: {problem.PS:.2f}", flush=True) 674 | # Calculate u3 at the point load 675 | u3_bb = None 676 | u3_func = q_func.sub(0).sub(2).collapse() 677 | if len(cells) > 0: 678 | u3_bb = u3_func.eval(bb_point, cells[0])[0] 679 | u3_bb = mesh.comm.gather(u3_bb, root=0) 680 | if mesh.comm.rank == 0: 681 | for u3 in u3_bb: 682 | if u3 is not None: 683 | u3_list[i] = u3 684 | break 685 | 686 | # %% [markdown] 687 | # We write the outputs of $\vec{u}$, $\vec{\theta}$, and $\vec{\phi}$ in the 688 | # second order Lagrange space. 689 | # %% 690 | # Interpolate phi_ufl into CG2 Space 691 | u_P2B3 = q_func.sub(0).collapse() 692 | theta_P2 = q_func.sub(1).collapse() 693 | 694 | # Interpolate phi in the [P2]³ Space 695 | phi_FS = functionspace(mesh, blocked_element(P2, shape=(3,))) 696 | phi_expr = Expression(phi0_ufl + u_P2B3, phi_FS.element.interpolation_points) 697 | phi_func = Function(phi_FS) 698 | phi_func.interpolate(phi_expr) 699 | 700 | # Interpolate u in the [P2]³ Space 701 | u_P2 = Function(phi_FS) 702 | u_P2.interpolate(u_P2B3) 703 | 704 | results_folder = Path("results/nonlinear_naghdi/semi_cylinder") 705 | results_folder.mkdir(exist_ok=True, parents=True) 706 | 707 | with dolfinx.io.VTXWriter(mesh.comm, results_folder / "u_naghdi.bp", [u_P2]) as vtx: 708 | vtx.write(0) 709 | 710 | with dolfinx.io.VTXWriter(mesh.comm, results_folder / "theta_naghdi.bp", [theta_P2]) as vtx: 711 | vtx.write(0) 712 | 713 | with dolfinx.io.VTXWriter(mesh.comm, results_folder / "phi_naghdi.bp", [phi_func]) as vtx: 714 | vtx.write(0) 715 | 716 | # %% [markdown] 717 | # The results for the transverse displacement at the point of application of 718 | # the force are validated against a standard reference from the literature, 719 | # obtained using Abaqus S4R element and a structured mesh of 40 times 40 720 | # elements, see [1]: 721 | 722 | # %% 723 | if mesh.comm.rank == 0: 724 | fig = plt.figure() 725 | reference_u3 = 1.0e-2 * np.array( 726 | [ 727 | 0.0, 728 | 5.421, 729 | 16.1, 730 | 22.195, 731 | 27.657, 732 | 32.7, 733 | 37.582, 734 | 42.633, 735 | 48.537, 736 | 56.355, 737 | 66.410, 738 | 79.810, 739 | 94.669, 740 | 113.704, 741 | 124.751, 742 | 132.653, 743 | 138.920, 744 | 144.185, 745 | 148.770, 746 | 152.863, 747 | 156.584, 748 | 160.015, 749 | 163.211, 750 | 166.200, 751 | 168.973, 752 | 171.505, 753 | ] 754 | ) 755 | reference_P = 2000.0 * np.array( 756 | [ 757 | 0.0, 758 | 0.05, 759 | 0.1, 760 | 0.125, 761 | 0.15, 762 | 0.175, 763 | 0.2, 764 | 0.225, 765 | 0.25, 766 | 0.275, 767 | 0.3, 768 | 0.325, 769 | 0.35, 770 | 0.4, 771 | 0.45, 772 | 0.5, 773 | 0.55, 774 | 0.6, 775 | 0.65, 776 | 0.7, 777 | 0.75, 778 | 0.8, 779 | 0.85, 780 | 0.9, 781 | 0.95, 782 | 1.0, 783 | ] 784 | ) 785 | plt.plot(-u3_list, PS_list, label="FEniCSx-Shells 20 x 20") 786 | plt.plot(reference_u3, reference_P, "or", label="Sze (Abaqus S4R)") 787 | plt.xlabel("Displacement (mm)") 788 | plt.ylabel("Load (N)") 789 | plt.legend() 790 | plt.grid() 791 | plt.tight_layout() 792 | plt.savefig(results_folder / "comparisons.pdf") 793 | 794 | # %% [markdown] 795 | # References: 796 | # 797 | # [1] K. Sze, X. Liu, and S. Lo. Popular benchmark problems for geometric 798 | # nonlinear analysis of shells. Finite Elements in Analysis and Design, 799 | # 40(11):1551 - 1569, 2004. 800 | # 801 | # [2] D. Arnold and F.Brezzi, Mathematics of Computation, 66(217): 1-14, 1997. 802 | # https://www.ima.umn.edu/~arnold//papers/shellelt.pdf 803 | # 804 | # [3] P. Betsch, A. Menzel, and E. Stein. On the parametrization of finite 805 | # rotations in computational mechanics: A classification of concepts with 806 | # application to smooth shells. Computer Methods in Applied Mechanics and 807 | # Engineering, 155(3):273 - 305, 1998. 808 | # 809 | # [4] https://fenicsproject.discourse.group/t/point-sources-redux/13496/4 810 | -------------------------------------------------------------------------------- /demo/demo_reissner-mindlin-clamped-tdnns.py: -------------------------------------------------------------------------------- 1 | # --- 2 | # jupyter: 3 | # jupytext: 4 | # text_representation: 5 | # extension: .py 6 | # format_name: light 7 | # format_version: '1.5' 8 | # jupytext_version: 1.14.1 9 | # --- 10 | 11 | # # Clamped Reissner-Mindlin plate under uniform load using TDNNS element 12 | # 13 | # This demo program solves the out-of-plane Reissner-Mindlin equations on the 14 | # unit square with uniform transverse loading and fully clamped boundary 15 | # conditions using the TDNNS (tangential displacement normal-normal stress) 16 | # element developed in: 17 | # 18 | # Pechstein, A. S., Schöberl, J. The TDNNS method for Reissner-Mindlin plates. 19 | # Numer. Math. 137, 713-740 (2017). 20 | # [doi:10.1007/s00211-017-0883-9](https://doi.org/10.1007/s00211-017-0883-9) 21 | # 22 | # The idea behind this element construction is that the rotations, transverse 23 | # displacement and bending moments are discretised separately. The finite 24 | # element space for the transverse displacement is chosen such that the 25 | # gradient is a subset of the rotation space and therefore the Kirchhoff 26 | # constraint as the thickness parameter tends to zero is exactly satisfied by 27 | # construction. 28 | # 29 | # Mathematically the forms in this work follow exactly that shown in Pechstein 30 | # and Schöberl except for a few minor notational changes to match the rest of 31 | # FEniCSx-Shells. 32 | # 33 | # It is assumed the reader understands most of the basic functionality of the 34 | # new FEniCSx Project. 35 | # 36 | # We begin by importing the necessary functionality from DOLFINx, UFL and 37 | # PETSc. 38 | 39 | # + 40 | from mpi4py import MPI 41 | 42 | import numpy as np 43 | 44 | import dolfinx 45 | import ufl 46 | from basix.ufl import element, mixed_element 47 | from dolfinx.fem import Function, dirichletbc, functionspace 48 | from dolfinx.fem.petsc import LinearProblem 49 | from dolfinx.mesh import CellType, create_unit_square 50 | from ufl import FacetNormal, Identity, Measure, grad, inner, split, sym, tr 51 | 52 | # - 53 | 54 | # We then create a two-dimensional mesh of the mid-plane of the plate $\Omega = 55 | # [0, 1] \times [0, 1]$. 56 | 57 | # + 58 | mesh = create_unit_square(MPI.COMM_WORLD, 16, 16, CellType.triangle) 59 | 60 | # - 61 | 62 | # The Pechstein-Schöberl element of order $k$ for the Reissner-Mindlin plate problem 63 | # consists of: 64 | # 65 | # - $k$-th order vector-valued Nédélec elements of the second kind for the 66 | # rotations $\theta \in \mathrm{NED}_k$ and, 67 | # - $k + 1$-th order scalar-valued Lagrange element for the transverse displacement field 68 | # $w \in \mathrm{CG}_{k + 1}$ and, 69 | # - $k$-th order Hellan-Herrmann-Johnson finite elements for the bending moments, which 70 | # naturally discretise tensor-valued functions in $H(\mathrm{div}\;\mathrm{\bf{div}})$, 71 | # $M \in \mathrm{HHJ}_k$. 72 | # 73 | # The final element definition with $k = 3$ is: 74 | 75 | # + 76 | k = 3 77 | cell = mesh.basix_cell() 78 | U_el = mixed_element( 79 | [element("N2curl", cell, k), element("Lagrange", cell, k + 1), element("HHJ", cell, k)] 80 | ) 81 | U = functionspace(mesh, U_el) 82 | 83 | u = ufl.TrialFunction(U) 84 | u_t = ufl.TestFunction(U) 85 | 86 | theta, w, M = split(u) 87 | theta_t, w_t, M_t = split(u_t) 88 | # - 89 | 90 | # We assume constant material parameters; Young's modulus $E$, Poisson's ratio 91 | # $\nu$, shear-correction factor $\kappa$, and thickness $t$. 92 | 93 | # + 94 | E = 10920.0 95 | nu = 0.3 96 | kappa = 5.0 / 6.0 97 | t = 0.001 98 | # - 99 | 100 | # The overall weak form for the problem can be written as: 101 | # 102 | # Find $(\theta, w, M) \in \mathrm{NED}_k \times \mathrm{CG}_1 \times \mathrm{HHJ}_k$ 103 | # such that 104 | # 105 | # $$ 106 | # \left( k(M), \tilde{M} \right) + \left< \tilde{M}, \theta \right> + \left< M, 107 | # \tilde{\theta} \right> - \left( \mu \gamma(\theta, w), \gamma(\tilde{\theta}, 108 | # \tilde{w}) \right) \\ = -\left(t^3, \tilde{w} \right) \quad \forall (\theta, 109 | # w, M) \in \mathrm{NED}_k \times \mathrm{CG}_1 \times \mathrm{HHJ}_k, 110 | # $$ 111 | # where $\left( \cdot, \cdot \right)$ is the usual $L^2$ inner product on the 112 | # mesh $\Omega$, $\gamma(\theta, w) = \nabla w - \theta \in H(\mathrm{rot})$ is 113 | # the shear-strain, $\mu = E \kappa t/(2(1 + \nu))$ the shear modulus. $k(M)$ 114 | # are the bending strains $k(\theta) = \mathrm{sym}(\nabla \theta)$ written in terms of 115 | # the bending moments (stresses) 116 | # 117 | # $$ 118 | # k(M) = \frac{12}{E t^3} \left[ (1 + \nu) M - \nu \mathrm{tr}\left( M \right) I \right], 119 | # $$ 120 | # with $\mathrm{tr}$ the trace operator and $I$ the identity tensor. 121 | # 122 | # The inner product $\left< \cdot, \cdot \right>$ is defined by 123 | # 124 | # $$ 125 | # \left< M, \theta \right> = -\left( M, k(\theta) \right) + \int_{\partial K} 126 | # M_{nn} \cdot [[ \theta ]]_n \; \mathrm{d}s, 127 | # $$ 128 | # where $M_{nn} = \left(Mn \right) \cdot n$ is the normal-normal component of 129 | # the bending moment, $\partial K$ are the facets of the mesh, $[[ \theta ]]$ is 130 | # the jump in the normal component of the rotations on the facets (reducing to 131 | # simply $\theta \cdot n$ on the exterior facets). 132 | # 133 | # The above equations can be written relatively straightforwardly in UFL as: 134 | 135 | # + 136 | dx = Measure("dx", mesh) 137 | dS = Measure("dS", mesh) 138 | ds = Measure("ds", mesh) 139 | 140 | 141 | def k_theta(theta): 142 | """Bending strain tensor in terms of rotations""" 143 | return sym(grad(theta)) 144 | 145 | 146 | def k_M(M): 147 | """Bending strain tensor in terms of bending moments""" 148 | return (12.0 / (E * t**3)) * ((1.0 + nu) * M - nu * Identity(2) * tr(M)) 149 | 150 | 151 | def nn(M): 152 | """Normal-normal component of tensor""" 153 | n = FacetNormal(M.ufl_domain()) 154 | M_n = M * n 155 | M_nn = ufl.dot(M_n, n) 156 | return M_nn 157 | 158 | 159 | def inner_divdiv(M, theta): 160 | """Discrete div-div inner product""" 161 | n = FacetNormal(M.ufl_domain()) 162 | M_nn = nn(M) 163 | result = ( 164 | -inner(M, k_theta(theta)) * dx 165 | + inner(M_nn("+"), ufl.jump(theta, n)) * dS 166 | + inner(M_nn, ufl.dot(theta, n)) * ds 167 | ) 168 | return result 169 | 170 | 171 | def gamma(theta, w): 172 | """Shear strain""" 173 | return grad(w) - theta 174 | 175 | 176 | a = ( 177 | inner(k_M(M), M_t) * dx 178 | + inner_divdiv(M_t, theta) 179 | + inner_divdiv(M, theta_t) 180 | - ((E * kappa * t) / (2.0 * (1.0 + nu))) * inner(gamma(theta, w), gamma(theta_t, w_t)) * dx 181 | ) 182 | L = -inner(1.0 * t**3, w_t) * dx 183 | 184 | # - 185 | 186 | # Imposition of boundary conditions requires some care. We reproduce the table 187 | # from Pechstein and Schöberl specifying the different types of boundary condition. 188 | # 189 | # | Essential | Natural | Non-homogeneous term | # noqa: E501 190 | # | ------------------------------------ | ------------------------------------ | ------------------------------------------------------------------- | # noqa: E501 191 | # | $w = \bar{w}$ | $\mu(\partial_n w - \theta_n) = g_w$ | $\int_{\Gamma} g_w \tilde{w} \; \mathrm{d}s$ | # noqa: E501 192 | # | $\theta_\tau = \bar{\theta}_{\tau} $ | $m_{n\tau} = g_{\theta_\tau}$ | $\int_{\Gamma} g_{\theta_\tau} \cdot \tilde{\theta} \; \mathrm{d}s$ | # noqa: E501 193 | # | $m_{nn} = \bar{m}_{nn}$ | $\theta_n = g_{\theta_n}$ | $\int_{\Gamma} g_{\theta_n} \tilde{m}_{nn} \; \mathrm{d}s$ | # noqa: E501 194 | # 195 | # where $\theta_{n} = \theta_n$ is the normal component of the rotation, 196 | # $\theta_{\tau} = \theta \cdot \tau $ is the tangential component of the 197 | # rotation, $m_{n\tau} = m \cdot n - \sigma_{nn} n$ is the normal-tangential 198 | # component of $n$, and $g_{w}$ etc. are known natural boundary data and 199 | # $\bar{w}$ etc. are known essential boundary data. 200 | # 201 | # In the case of an essential boundary condition the values are enforced 202 | # directly in the finite element space using `dolfinx.dirichletbc`. In the case 203 | # of a homogeneous natural boundary condition the corresponding essential 204 | # boundary condition should be dropped. In the case of a non-homogeneous 205 | # condition an extra term must be added to the weak formulation. 206 | # 207 | # For a fully clamped plate we have on the entire boundary $\bar{w} = 0$ 208 | # (homogeneous essential), $\bar{\theta_\tau} = 0$ (homogeneous essential), and 209 | # $g_{\theta_n} = 0$ (homogeneous natural). 210 | 211 | # + 212 | 213 | 214 | def all_boundary(x): 215 | return np.full(x.shape[1], True, dtype=bool) 216 | 217 | 218 | boundary_entities = dolfinx.mesh.locate_entities_boundary(mesh, mesh.topology.dim - 1, all_boundary) 219 | 220 | bcs = [] 221 | # Transverse displacement 222 | boundary_dofs = dolfinx.fem.locate_dofs_topological( 223 | U.sub(1), mesh.topology.dim - 1, boundary_entities 224 | ) 225 | bcs.append(dirichletbc(np.array(0.0, dtype=np.float64), boundary_dofs, U.sub(1))) 226 | 227 | # Fix tangential component of rotation 228 | R = U.sub(0).collapse()[0] 229 | boundary_dofs = dolfinx.fem.locate_dofs_topological( 230 | (U.sub(0), R), mesh.topology.dim - 1, boundary_entities 231 | ) 232 | 233 | theta_bc = Function(R) 234 | bcs.append(dirichletbc(theta_bc, boundary_dofs, U.sub(0))) 235 | 236 | # - 237 | 238 | # Finally we solve the problem and output the transverse displacement at the 239 | # centre of the plate. 240 | 241 | # + 242 | problem = LinearProblem( 243 | a, 244 | L, 245 | bcs=bcs, 246 | petsc_options={"ksp_type": "preonly", "pc_type": "lu", "pc_factor_mat_solver_type": "mumps"}, 247 | ) 248 | u_h = problem.solve() 249 | 250 | bb_tree = dolfinx.geometry.bb_tree(mesh, 2) 251 | point = np.array([[0.5, 0.5, 0.0]], dtype=np.float64) 252 | cell_candidates = dolfinx.geometry.compute_collisions_points(bb_tree, point) 253 | cells = dolfinx.geometry.compute_colliding_cells(mesh, cell_candidates, point) 254 | 255 | theta, w, M = u_h.split() 256 | 257 | if cells.array.shape[0] > 0: 258 | value = w.eval(point, cells.array[0]) 259 | print(value[0]) 260 | -------------------------------------------------------------------------------- /demo/demo_reissner-mindlin-clamped.py: -------------------------------------------------------------------------------- 1 | # --- 2 | # jupyter: 3 | # jupytext: 4 | # text_representation: 5 | # extension: .py 6 | # format_name: light 7 | # format_version: '1.5' 8 | # jupytext_version: 1.14.1 9 | # --- 10 | 11 | # # Clamped Reissner-Mindlin plate under uniform load 12 | # 13 | # This demo program solves the out-of-plane Reissner-Mindlin equations on the 14 | # unit square with uniform transverse loading with fully clamped boundary 15 | # conditions. This version does not use the special projected assembly routines 16 | # in FEniCSx-Shells. 17 | # 18 | # It is assumed the reader understands most of the basic functionality of the 19 | # new FEniCSx Project. 20 | # 21 | # This demo illustrates how to: 22 | # 23 | # - Define the Reissner-Mindlin plate equations using UFL. 24 | # - Define the Durán-Liberman (MITC) reduction operator using UFL. This 25 | # procedure eliminates the shear-locking problem. 26 | # 27 | # We begin by importing the necessary functionality from DOLFINx, UFL and 28 | # PETSc. 29 | 30 | from mpi4py import MPI 31 | 32 | import numpy as np 33 | 34 | import dolfinx 35 | import ufl 36 | from basix.ufl import element, mixed_element 37 | from dolfinx.fem import Function, dirichletbc, functionspace 38 | from dolfinx.fem.petsc import LinearProblem 39 | from dolfinx.io.utils import XDMFFile 40 | from dolfinx.mesh import CellType, create_unit_square 41 | from ufl import dx, grad, inner, split, sym, tr 42 | 43 | # We then create a two-dimensional mesh of the mid-plane of the plate $\Omega = 44 | # [0, 1] \times [0, 1]$. `GhostMode.shared_facet` is required as the Form will 45 | # use Nédélec elements and DG-type restrictions. 46 | 47 | mesh = create_unit_square(MPI.COMM_WORLD, 32, 32, CellType.triangle) 48 | 49 | # The Durán-Liberman element [1] for the Reissner-Mindlin plate problem 50 | # consists of: 51 | # 52 | # - second-order vector-valued Lagrange element for the rotation field $\theta 53 | # \in [\mathrm{CG}_2]^2$ and, 54 | # - a first-order scalar valued Lagrange element for the transverse 55 | # displacement field $w \in \mathrm{CG}_1$ and, 56 | # - the reduced shear strain $\gamma_R \in \mathrm{NED}_1$ the vector-valued 57 | # Nédélec elements of the first kind, and 58 | # - a Lagrange multiplier field $p$ that ties together the shear strain 59 | # calculated from the primal variables $\gamma = \nabla w - \theta$ and the 60 | # reduced shear strain $\gamma_R$. Both $p$ and $\gamma_R$ are are discretised 61 | # in the space $\mathrm{NED}_1$, the vector-valued Nédélec elements of the 62 | # first kind. 63 | # 64 | # The final element definition is 65 | 66 | # + 67 | cell = mesh.basix_cell() 68 | U_el = mixed_element( 69 | [ 70 | element("Lagrange", cell, 2, shape=(2,)), 71 | element("Lagrange", cell, 1), 72 | element("N1curl", cell, 1), 73 | element("N1curl", cell, 1), 74 | ] 75 | ) 76 | U = functionspace(mesh, U_el) 77 | 78 | u_ = Function(U) 79 | u = ufl.TrialFunction(U) 80 | u_t = ufl.TestFunction(U) 81 | 82 | theta_, w_, R_gamma_, p_ = split(u_) 83 | # - 84 | 85 | # We assume constant material parameters; Young's modulus $E$, Poisson's ratio 86 | # $\nu$, shear-correction factor $\kappa$, and thickness $t$. 87 | 88 | E = 10920.0 89 | nu = 0.3 90 | kappa = 5.0 / 6.0 91 | t = 0.001 92 | 93 | # The bending strain tensor $k$ for the Reissner-Mindlin model can be expressed 94 | # in terms of the rotation field $\theta$ 95 | # 96 | # $$ 97 | # k(\theta) = \dfrac{1}{2}(\nabla \theta + (\nabla \theta)^T) 98 | # $$ 99 | # 100 | # The bending energy density $\psi_b$ for the Reissner-Mindlin model is a 101 | # function of the bending strain tensor $k$ 102 | # 103 | # $$ 104 | # \psi_b(k) = \frac{1}{2} D \left( (1 - \nu) \, \mathrm{tr}\,(k^2) 105 | # + \nu \, (\mathrm{tr} \,k)^2 \right) \qquad 106 | # D = \frac{Et^3}{12(1 - \nu^2)} 107 | # $$ 108 | # 109 | # which can be expressed in UFL as 110 | 111 | D = (E * t**3) / (24.0 * (1.0 - nu**2)) 112 | k = sym(grad(theta_)) 113 | psi_b = D * ((1.0 - nu) * tr(k * k) + nu * (tr(k)) ** 2) 114 | 115 | # Because we are using a mixed variational formulation, we choose to write the 116 | # shear energy density $\psi_s$ is a function of the reduced shear strain 117 | # vector 118 | # 119 | # $$\psi_s(\gamma_R) = \frac{E \kappa t}{4(1 + \nu)}\gamma_R^2,$$ 120 | # 121 | # or in UFL: 122 | 123 | psi_s = ((E * kappa * t) / (4.0 * (1.0 + nu))) * inner(R_gamma_, R_gamma_) 124 | 125 | # Finally, we can write out external work due to the uniform loading in the out-of-plane direction 126 | # 127 | # $$ 128 | # W_{\mathrm{ext}} = \int_{\Omega} ft^3 \cdot w \; \mathrm{d}x. 129 | # $$ 130 | 131 | # where $f = 1$ and $\mathrm{d}x$ is a measure on the whole domain. 132 | # The scaling by $t^3$ is included to ensure a correct limit solution as 133 | # $t \to 0$. 134 | # 135 | # In UFL this can be expressed as 136 | 137 | W_ext = inner(1.0 * t**3, w_) * dx 138 | 139 | # With all of the standard mechanical terms defined, we can turn to defining 140 | # the Duran-Liberman reduction operator. This operator 'ties' our reduced shear 141 | # strain field to the shear strain calculated in the primal space. A partial 142 | # explanation of the thinking behind this approach is given in the Appendix at 143 | # the bottom of this notebook. 144 | # 145 | # The shear strain vector $\gamma$ can be expressed in terms of the rotation 146 | # and transverse displacement field 147 | # 148 | # $$\gamma(\theta, w) = \nabla w - \theta$$ 149 | # 150 | # or in UFL 151 | 152 | gamma = grad(w_) - theta_ 153 | 154 | # We require that the shear strain calculated using the displacement unknowns 155 | # $\gamma = \nabla w - \theta$ be equal, in a weak sense, to the conforming 156 | # shear strain field $\gamma_R \in \mathrm{NED}_1$ that we used to define the 157 | # shear energy above. We enforce this constraint using a Lagrange multiplier 158 | # field $p \in \mathrm{NED}_1$. We can write the Lagrangian functional of this 159 | # constraint as: 160 | # 161 | # $$\Pi_R(\gamma, \gamma_R, p) = 162 | # \int_{e} \left( \left\lbrace \gamma_R - \gamma \right\rbrace \cdot t \right) 163 | # \cdot \left( p \cdot t \right) \; \mathrm{d}s$$ 164 | # 165 | # where $e$ are all of edges of the cells in the mesh and $t$ is the tangent 166 | # vector on each edge. 167 | # 168 | # Writing this operator out in UFL is quite verbose, so `fenicsx_shells` 169 | # includes a special inner product function `inner_e` to help. However, we 170 | # choose to write this function in full here. 171 | 172 | # + 173 | dSp = ufl.Measure("dS", metadata={"quadrature_degree": 1}) 174 | dsp = ufl.Measure("ds", metadata={"quadrature_degree": 1}) 175 | 176 | n = ufl.FacetNormal(mesh) 177 | t = ufl.as_vector((-n[1], n[0])) 178 | 179 | 180 | def inner_e(x, y): 181 | return (inner(x, t) * inner(y, t))("+") * dSp + (inner(x, t) * inner(y, t)) * dsp 182 | 183 | 184 | Pi_R = inner_e(gamma - R_gamma_, p_) 185 | # - 186 | 187 | # We can now define our Lagrangian for the complete system and derive the 188 | # residual and Jacobian automatically using the standard UFL `derivative` 189 | # function 190 | 191 | Pi = psi_b * dx + psi_s * dx + Pi_R - W_ext 192 | F = ufl.derivative(Pi, u_, u_t) 193 | J = ufl.derivative(F, u_, u) 194 | 195 | # In the following we use standard from `dolfinx` to apply boundary conditions, 196 | # assemble, solve and output the solution. 197 | # 198 | # Note that for a fully clamped plate $\gamma_R \in H_0(\mathrm{rot}; \Omega)$ 199 | # which is the Sobolev space of vector-valued functions with square integrable 200 | # whose tangential component $\gamma_R \cdot t$ vanishes on the boundary. So, 201 | # it is also necessary to apply boundary conditions on this space. 202 | # 203 | # For simplicity of implementation we also apply boundary conditions on the 204 | # Lagrange multiplier space but this is not strictly necessary as the Lagrange 205 | # multiplier simply constrains $\gamma_R \cdot t$ to $\gamma = \nabla w - 206 | # \theta$, all of which are enforced to be zero by definition. 207 | 208 | # + 209 | 210 | 211 | def all_boundary(x): 212 | return np.full(x.shape[1], True, dtype=bool) 213 | 214 | 215 | u_boundary = Function(U) 216 | boundary_entities = dolfinx.mesh.locate_entities_boundary(mesh, mesh.topology.dim - 1, all_boundary) 217 | boundary_dofs = dolfinx.fem.locate_dofs_topological(U, mesh.topology.dim - 1, boundary_entities) 218 | bcs = [dirichletbc(u_boundary, boundary_dofs)] 219 | 220 | problem = LinearProblem( 221 | J, 222 | -F, 223 | bcs=bcs, 224 | petsc_options={"ksp_type": "preonly", "pc_type": "lu", "pc_factor_mat_solver_type": "mumps"}, 225 | ) 226 | u_ = problem.solve() 227 | 228 | bb_tree = dolfinx.geometry.bb_tree(mesh, 2) 229 | point = np.array([[0.5, 0.5, 0.0]], dtype=np.float64) 230 | cell_candidates = dolfinx.geometry.compute_collisions_points(bb_tree, point) 231 | cells = dolfinx.geometry.compute_colliding_cells(mesh, cell_candidates, point) 232 | 233 | theta, w, R_gamma, p = u_.split() 234 | 235 | if cells.array.shape[0] > 0: 236 | value = w.eval(point, cells.array[0]) 237 | print(value[0]) 238 | # NOTE: FEniCS-Shells (old dolfin) `demo/documented/reissner-mindlin-clamped` 239 | # gives 1.28506469462e-06 on a 32 x 32 mesh and 1.2703580973e-06 on a 64 x 64 240 | # mesh. 241 | 242 | def test_center_displacement(): 243 | assert np.isclose(value[0], 1.285e-6, atol=1e-3, rtol=1e-3) 244 | 245 | 246 | with XDMFFile(MPI.COMM_WORLD, "w.xdmf", "w") as f: 247 | f.write_mesh(mesh) 248 | f.write_function(w) 249 | 250 | # - 251 | 252 | # ## Appendix 253 | 254 | # For the clamped problem we have the following regularity for our two fields, 255 | # $\theta \in [H^1_0(\Omega)]^2$ and $w \in [H^1_0(\Omega)]^2$ where 256 | # $H^1_0(\Omega)$ is the usual Sobolev space of functions with square 257 | # integrable first derivatives that vanish on the boundary. If we then take 258 | # $\nabla w$ we have the result $\nabla w \in H_0(\mathrm{rot}; \Omega)$ which 259 | # is the Sobolev space of vector-valued functions with square integrable 260 | # $\mathrm{rot}$ whose tangential component $\nabla w \cdot t$ vanishes on the 261 | # boundary. 262 | # 263 | # Let's look at our expression for the shear strain vector in light of these 264 | # new results. In the thin-plate limit $t \to 0$, we would like to recover our 265 | # the standard Kirchhoff-Love problem where we do not have transverse shear 266 | # strains $\gamma \to 0$ at all. In a finite element context, where we have 267 | # discretised fields $w_h$ and $\theta_h$ we then would like 268 | # 269 | # $$\gamma(\theta_h, w_h) := \nabla w_h - \theta_h = 0 \quad t \to 0 \; \forall x \in \Omega$$ 270 | # 271 | # If we think about using first-order piecewise linear polynomial finite 272 | # elements for both fields, then we are requiring that piecewise constant 273 | # functions ($\nabla w_h$) are equal to piecewise linear functions ($\theta_h$) 274 | # # ! This is strong requirement, and is the root of the famous shear-locking 275 | # problem. The trick of the Durán-Liberman approach is recognising that by 276 | # modifying the rotation field at the discrete level by applying a special 277 | # operator $R_h$ that takes the rotations to the conforming space 278 | # $\mathrm{NED}_1 \subset H_0(\mathrm{rot}; \Omega)$ for the shear strains that 279 | # we previously identified: 280 | # 281 | # $$R_h : H_0^1(\Omega) \to H_0(\mathrm{rot}; \Omega)$$ 282 | # 283 | # we can 'unlock' the element. With this reduction operator applied as follows: 284 | # 285 | # $$\gamma(\theta_h, w_h) := R_h(\nabla w_h - \theta_h = 0) \quad t \to 0 \; \forall x \in \Omega$$ 286 | # 287 | # our requirement of vanishing shear strains can actually hold. This is the 288 | # basic mathematical idea behind all MITC approaches, of which the 289 | # Durán-Liberman approach is a specific implementation. 290 | -------------------------------------------------------------------------------- /demo/demo_reissner-mindlin-simply-supported.py: -------------------------------------------------------------------------------- 1 | # --- 2 | # jupyter: 3 | # jupytext: 4 | # text_representation: 5 | # extension: .py 6 | # format_name: light 7 | # format_version: '1.5' 8 | # jupytext_version: 1.14.1 9 | # --- 10 | 11 | # # Simply-supported Reissner-Mindlin plate under uniform load 12 | # 13 | # This demo program solves the out-of-plane Reissner-Mindlin equations on the 14 | # unit square with uniform transverse loading and simply supported boundary 15 | # conditions. This version does not use the special projected assembly routines 16 | # in FEniCSx-Shells. 17 | # 18 | # It is assumed the reader understands most of the basic functionality of the 19 | # new FEniCSx Project. 20 | # 21 | # This demo illustrates how to: 22 | # 23 | # - Define the Reissner-Mindlin plate equations using UFL. 24 | # - Define the MITC4 reduction operator using UFL. This procedure eliminates 25 | # the shear-locking problem. 26 | # 27 | # We begin by importing the necessary functionality from DOLFINx, UFL and 28 | # PETSc. 29 | 30 | from mpi4py import MPI 31 | 32 | import numpy as np 33 | 34 | import dolfinx 35 | import ufl 36 | from basix.ufl import element, mixed_element 37 | from dolfinx.fem import Function, dirichletbc, functionspace 38 | from dolfinx.fem.petsc import LinearProblem 39 | from dolfinx.mesh import CellType, create_unit_square 40 | from ufl import dx, grad, inner, split, sym, tr 41 | 42 | # We then create a two-dimensional mesh of the mid-plane of the plate $\Omega = 43 | # [0, 1] \times [0, 1]$. `GhostMode.shared_facet` is required as the Form will 44 | # use Nédélec elements and DG-type restrictions. 45 | 46 | mesh = create_unit_square(MPI.COMM_WORLD, 32, 32, CellType.quadrilateral) 47 | 48 | # The MITC4 element [1] for the Reissner-Mindlin plate problem 49 | # consists of: 50 | # 51 | # - first-order vector-valued Lagrange element for the rotation field $\theta 52 | # \in [\mathrm{CG}_1]^2$ and, 53 | # - a first-order scalar valued Lagrange element for the transverse 54 | # displacement field $w \in \mathrm{CG}_1$ and, 55 | # - the reduced shear strain $\gamma_R \in \mathrm{RTCE}_1$ the vector-valued 56 | # Nédélec elements of the first kind, and 57 | # - a Lagrange multiplier field $p$ that ties together the shear strain 58 | # calculated from the primal variables $\gamma = \nabla w - \theta$ and the 59 | # reduced shear strain $\gamma_R$. Both $p$ and $\gamma_R$ are are discretised 60 | # in the space $\mathrm{RTCE}_1$, the vector-valued Nédélec elements of the 61 | # first kind. 62 | # 63 | # The final element definition is 64 | 65 | # + 66 | cell = mesh.basix_cell() 67 | U_el = mixed_element( 68 | [ 69 | element("Lagrange", cell, 1, shape=(2,)), 70 | element("Lagrange", cell, 1), 71 | element("RTCE", cell, 1), 72 | element("RTCE", cell, 1), 73 | ] 74 | ) 75 | U = functionspace(mesh, U_el) 76 | 77 | u_ = Function(U) 78 | u = ufl.TrialFunction(U) 79 | u_t = ufl.TestFunction(U) 80 | 81 | theta_, w_, R_gamma_, p_ = split(u_) 82 | # - 83 | 84 | # We assume constant material parameters; Young's modulus $E$, Poisson's ratio 85 | # $\nu$, shear-correction factor $\kappa$, and thickness $t$. 86 | 87 | E = 10920.0 88 | nu = 0.3 89 | kappa = 5.0 / 6.0 90 | t = 0.001 91 | 92 | # The bending strain tensor $k$ for the Reissner-Mindlin model can be expressed 93 | # in terms of the rotation field $\theta$ 94 | # 95 | # $$ 96 | # k(\theta) = \dfrac{1}{2}(\nabla \theta + (\nabla \theta)^T) 97 | # $$ 98 | # 99 | # The bending energy density $\psi_b$ for the Reissner-Mindlin model is a 100 | # function of the bending strain tensor $k$ 101 | # 102 | # $$ 103 | # \psi_b(k) = \frac{1}{2} D \left( (1 - \nu) \, \mathrm{tr}\,(k^2) 104 | # + \nu \, (\mathrm{tr} \,k)^2 \right) \qquad 105 | # D = \frac{Et^3}{12(1 - \nu^2)} 106 | # $$ 107 | # 108 | # which can be expressed in UFL as 109 | 110 | D = (E * t**3) / (24.0 * (1.0 - nu**2)) 111 | k = sym(grad(theta_)) 112 | psi_b = D * ((1.0 - nu) * tr(k * k) + nu * (tr(k)) ** 2) 113 | 114 | # Because we are using a mixed variational formulation, we choose to write the 115 | # shear energy density $\psi_s$ is a function of the reduced shear strain 116 | # vector 117 | # 118 | # $$\psi_s(\gamma_R) = \frac{E \kappa t}{4(1 + \nu)}\gamma_R^2,$$ 119 | # 120 | # or in UFL: 121 | 122 | psi_s = ((E * kappa * t) / (4.0 * (1.0 + nu))) * inner(R_gamma_, R_gamma_) 123 | 124 | # Finally, we can write out external work due to the uniform loading in the out-of-plane direction 125 | # 126 | # $$ 127 | # W_{\mathrm{ext}} = \int_{\Omega} ft^3 \cdot w \; \mathrm{d}x. 128 | # $$ 129 | 130 | # where $f = 1$ and $\mathrm{d}x$ is a measure on the whole domain. 131 | # The scaling by $t^3$ is included to ensure a correct limit solution as 132 | # $t \to 0$. 133 | # 134 | # In UFL this can be expressed as 135 | 136 | W_ext = inner(1.0 * t**3, w_) * dx 137 | 138 | # With all of the standard mechanical terms defined, we can turn to defining 139 | # the Duran-Liberman reduction operator. This operator 'ties' our reduced shear 140 | # strain field to the shear strain calculated in the primal space. A partial 141 | # explanation of the thinking behind this approach is given in the Appendix at 142 | # the bottom of this notebook. 143 | # 144 | # The shear strain vector $\gamma$ can be expressed in terms of the rotation 145 | # and transverse displacement field 146 | # 147 | # $$\gamma(\theta, w) = \nabla w - \theta$$ 148 | # 149 | # or in UFL 150 | 151 | gamma = grad(w_) - theta_ 152 | 153 | # We require that the shear strain calculated using the displacement unknowns 154 | # $\gamma = \nabla w - \theta$ be equal, in a weak sense, to the conforming 155 | # shear strain field $\gamma_R \in \mathrm{NED}_1$ that we used to define the 156 | # shear energy above. We enforce this constraint using a Lagrange multiplier 157 | # field $p \in \mathrm{NED}_1$. We can write the Lagrangian functional of this 158 | # constraint as: 159 | # 160 | # $$\Pi_R(\gamma, \gamma_R, p) = 161 | # \int_{e} \left( \left\lbrace \gamma_R - \gamma \right\rbrace \cdot t \right) 162 | # \cdot \left( p \cdot t \right) \; \mathrm{d}s$$ 163 | # 164 | # where $e$ are all of edges of the cells in the mesh and $t$ is the tangent 165 | # vector on each edge. 166 | # 167 | # Writing this operator out in UFL is quite verbose, so `fenicsx_shells` 168 | # includes a special inner product function `inner_e` to help. However, we 169 | # choose to write this function in full here. 170 | 171 | # + 172 | dSp = ufl.Measure("dS", metadata={"quadrature_degree": 1}) 173 | dsp = ufl.Measure("ds", metadata={"quadrature_degree": 1}) 174 | 175 | n = ufl.FacetNormal(mesh) 176 | t = ufl.as_vector((-n[1], n[0])) 177 | 178 | 179 | def inner_e(x, y): 180 | return (inner(x, t) * inner(y, t))("+") * dSp + (inner(x, t) * inner(y, t)) * dsp 181 | 182 | 183 | Pi_R = inner_e(gamma - R_gamma_, p_) 184 | # - 185 | 186 | # We can now define our Lagrangian for the complete system and derive the 187 | # residual and Jacobian automatically using the standard UFL `derivative` 188 | # function 189 | 190 | Pi = psi_b * dx + psi_s * dx + Pi_R - W_ext 191 | F = ufl.derivative(Pi, u_, u_t) 192 | J = ufl.derivative(F, u_, u) 193 | 194 | # In the following we use standard from `dolfinx` to apply boundary conditions, 195 | # assemble, solve and output the solution. 196 | # 197 | # For simplicity of implementation we also apply boundary conditions on the 198 | # Lagrange multiplier space but this is not strictly necessary as the Lagrange 199 | # multiplier simply constrains $\gamma_R \cdot t$ to $\gamma = \nabla w - 200 | # \theta$, all of which are enforced to be zero by definition. 201 | 202 | # + 203 | 204 | 205 | def all_boundary(x): 206 | return np.full(x.shape[1], True, dtype=bool) 207 | 208 | 209 | def left_or_right(x): 210 | return np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)) 211 | 212 | 213 | def top_or_bottom(x): 214 | return np.logical_or(np.isclose(x[1], 0.0), np.isclose(x[1], 1.0)) 215 | 216 | 217 | def make_bc(value, V, on_boundary): 218 | boundary_entities = dolfinx.mesh.locate_entities_boundary( 219 | mesh, mesh.topology.dim - 1, on_boundary 220 | ) 221 | boundary_dofs = dolfinx.fem.locate_dofs_topological(V, mesh.topology.dim - 1, boundary_entities) 222 | bc = dirichletbc(value, boundary_dofs, V) 223 | return bc 224 | 225 | 226 | bcs = [] 227 | # Transverse displacements fixed everywhere 228 | bcs.append(make_bc(np.array(0.0, dtype=np.float64), U.sub(1), all_boundary)) 229 | 230 | # First component of rotation fixed on top and bottom 231 | bcs.append(make_bc(np.array(0.0, dtype=np.float64), U.sub(0).sub(0), top_or_bottom)) 232 | 233 | # Second component of rotation fixed on left and right 234 | bcs.append(make_bc(np.array(0.0, dtype=np.float64), U.sub(0).sub(1), left_or_right)) 235 | 236 | 237 | problem = LinearProblem( 238 | J, 239 | -F, 240 | bcs=bcs, 241 | petsc_options={"ksp_type": "preonly", "pc_type": "lu", "pc_factor_mat_solver_type": "mumps"}, 242 | ) 243 | u_ = problem.solve() 244 | 245 | bb_tree = dolfinx.geometry.bb_tree(mesh, 2) 246 | point = np.array([[0.5, 0.5, 0.0]], dtype=np.float64) 247 | cell_candidates = dolfinx.geometry.compute_collisions_points(bb_tree, point) 248 | cells = dolfinx.geometry.compute_colliding_cells(mesh, cell_candidates, point) 249 | 250 | theta, w, R_gamma, p = u_.split() 251 | 252 | if cells.array.shape[0] > 0: 253 | value = w.eval(point, cells.array[0]) 254 | print(value[0]) 255 | # - 256 | -------------------------------------------------------------------------------- /demo/pytest.ini: -------------------------------------------------------------------------------- 1 | [pytest] 2 | addopts = --strict-markers 3 | markers = 4 | serial 5 | mpi 6 | -------------------------------------------------------------------------------- /demo/test_demos.py: -------------------------------------------------------------------------------- 1 | # Copyright (C) 2016-2016 Garth N. Wells 2 | # 3 | # This file is part of DOLFINx (https://www.fenicsproject.org) 4 | # 5 | # SPDX-License-Identifier: LGPL-3.0-or-later 6 | 7 | import pathlib 8 | import subprocess 9 | import sys 10 | 11 | import pytest 12 | 13 | # Get directory of this file 14 | path = pathlib.Path(__file__).resolve().parent 15 | 16 | # Build list of demo programs 17 | demos = [] 18 | demo_files = list(path.glob("**/demo_*.py")) 19 | for f in demo_files: 20 | demos.append((f.parent, f.name)) 21 | 22 | print(demos) 23 | 24 | 25 | @pytest.mark.serial 26 | @pytest.mark.parametrize("path,name", demos) 27 | def test_demos(path, name): 28 | ret = subprocess.run([sys.executable, name], cwd=str(path), check=True) 29 | assert ret.returncode == 0 30 | 31 | 32 | @pytest.mark.mpi 33 | @pytest.mark.parametrize("path,name", demos) 34 | def test_demos_mpi(num_proc, mpiexec, path, name): 35 | cmd = [mpiexec, "-np", str(num_proc), sys.executable, name] 36 | print(cmd) 37 | ret = subprocess.run(cmd, cwd=str(path), check=True) 38 | assert ret.returncode == 0 39 | -------------------------------------------------------------------------------- /doc/README.md: -------------------------------------------------------------------------------- 1 | # Building documentation 2 | 3 | To build the documentation install Sphinx and run 4 | 5 | python3 -m sphinx -W -b html source/ build/html/ 6 | -------------------------------------------------------------------------------- /doc/source/.gitignore: -------------------------------------------------------------------------------- 1 | demo/ 2 | -------------------------------------------------------------------------------- /doc/source/_static/.placeholder: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/FEniCS-Shells/fenicsx-shells/e1bbfd920ed42843720273eebf35aa940ba422c7/doc/source/_static/.placeholder -------------------------------------------------------------------------------- /doc/source/conf.py: -------------------------------------------------------------------------------- 1 | # Configuration file for the Sphinx documentation builder. 2 | # 3 | # For the full list of built-in configuration values, see the documentation: 4 | # https://www.sphinx-doc.org/en/master/usage/configuration.html 5 | 6 | # -- Project information ----------------------------------------------------- 7 | # https://www.sphinx-doc.org/en/master/usage/configuration.html#project-information 8 | 9 | import os 10 | import sys 11 | 12 | sys.path.insert(0, os.path.abspath(".")) 13 | import jupytext_process 14 | 15 | jupytext_process.process() 16 | 17 | project = "FEniCSx-Shells" 18 | copyright = "2022-2024, FEniCSx-Shells Authors" 19 | author = "FEniCSx-Shells Authors" 20 | release = "0.10.0.dev0" 21 | 22 | # -- General configuration --------------------------------------------------- 23 | # https://www.sphinx-doc.org/en/master/usage/configuration.html#general-configuration 24 | 25 | extensions = [ 26 | "sphinx.ext.autodoc", 27 | "sphinx.ext.autosummary", 28 | "sphinx.ext.mathjax", 29 | "sphinx.ext.napoleon", 30 | "sphinx.ext.todo", 31 | "sphinx.ext.viewcode", 32 | "myst_parser", 33 | ] 34 | 35 | templates_path = ["_templates"] 36 | exclude_patterns = [] 37 | 38 | source_suffix = [".rst", ".md"] 39 | 40 | html_theme = "sphinx_rtd_theme" 41 | 42 | myst_enable_extensions = [ 43 | "dollarmath", 44 | ] 45 | 46 | autodoc_default_options = { 47 | "members": True, 48 | "show-inheritance": True, 49 | "imported-members": True, 50 | "undoc-members": True, 51 | } 52 | autosummary_generate = True 53 | autoclass_content = "both" 54 | 55 | # -- Options for HTML output ------------------------------------------------- 56 | # https://www.sphinx-doc.org/en/master/usage/configuration.html#options-for-html-output 57 | 58 | html_static_path = ["_static"] 59 | -------------------------------------------------------------------------------- /doc/source/demos.rst: -------------------------------------------------------------------------------- 1 | Documented demos 2 | ================ 3 | 4 | Clamped linear Reissner-Mindlin plate problem using the Duran-Liberman reduction 5 | operator (MITC) to cure shear-locking: 6 | 7 | .. toctree:: 8 | :titlesonly: 9 | :maxdepth: 1 10 | 11 | demo/demo_reissner-mindlin-clamped.md 12 | 13 | Clamped linear Reissner-Mindlin plate problem using the Pechstein-Schöberl 14 | TDNNS (tangential displacement normal-normal stress) element to cure 15 | shear-locking: 16 | 17 | .. toctree:: 18 | :titlesonly: 19 | :maxdepth: 1 20 | 21 | demo/demo_reissner-mindlin-clamped-tdnns.md 22 | 23 | Simply-supported linear Reissner-Mindlin plate problem using the MITC4 24 | reduction operator to cure shear-locking: 25 | 26 | .. toctree:: 27 | :titlesonly: 28 | :maxdepth: 1 29 | 30 | demo/demo_reissner-mindlin-simply-supported.md 31 | 32 | Clamped linear Kirchhoff-Love plate problem using the Hellan-Herrmann-Johnson 33 | finite element. 34 | 35 | .. toctree:: 36 | :titlesonly: 37 | :maxdepth: 1 38 | 39 | demo/demo_kirchhoff-love-clamped.md 40 | 41 | Clamped nonlinear Naghdi semi-cylindrical shell under a point load using Partial 42 | Selective Reduced Integration (PSRI) to cure shear and membrane locking: 43 | 44 | .. toctree:: 45 | :titlesonly: 46 | :maxdepth: 1 47 | 48 | demo/demo_nonlinear-naghdi-clamped-semicylinder.md 49 | -------------------------------------------------------------------------------- /doc/source/index.rst: -------------------------------------------------------------------------------- 1 | .. FEniCSx-Shells documentation master file, created by 2 | sphinx-quickstart on Fri Aug 26 15:38:02 2022. 3 | You can adapt this file completely to your liking, but it should at least 4 | contain the root `toctree` directive. 5 | 6 | .. toctree:: 7 | :maxdepth: 1 8 | :titlesonly: 9 | 10 | Demos 11 | 12 | .. include:: ../../README.md 13 | :parser: myst_parser.sphinx_ 14 | 15 | .. 16 | Indices and tables 17 | ================== 18 | 19 | * :ref:`genindex` 20 | * :ref:`modindex` 21 | * :ref:`search` 22 | 23 | 24 | -------------------------------------------------------------------------------- /doc/source/jupytext_process.py: -------------------------------------------------------------------------------- 1 | # Copyright (C) 2017-2022 Garth N. Wells, Jack S. Hale 2 | # 3 | # This file is part of DOLFINx (https://www.fenicsproject.org) 4 | # 5 | # SPDX-License-Identifier: LGPL-3.0-or-later 6 | 7 | import os 8 | import pathlib 9 | import shutil 10 | 11 | import jupytext 12 | 13 | 14 | def process(): 15 | """Convert light format demo Python files into MyST flavoured markdown and 16 | ipynb using Jupytext. These files can then be included in Sphinx 17 | documentation""" 18 | # Directories to scan 19 | subdirs = [pathlib.Path("../../demo")] 20 | 21 | # Iterate over subdirectories containing demos 22 | for subdir in subdirs: 23 | # Make demo doc directory 24 | demo_dir = pathlib.Path("./demo") 25 | demo_dir.mkdir(parents=True, exist_ok=True) 26 | 27 | # Process each demo using jupytext/myst 28 | for demo in subdir.glob("**/demo*.py"): 29 | # If demo saves matplotlib images, run the demo 30 | if "savefig" in demo.read_text(): 31 | here = os.getcwd() 32 | os.chdir(demo.parent) 33 | os.system(f"python3 {demo.name}") 34 | os.chdir(here) 35 | 36 | python_demo = jupytext.read(demo) 37 | myst_text = jupytext.writes(python_demo, fmt="myst") 38 | 39 | # myst-parser does not process blocks with {code-cell} 40 | myst_text = myst_text.replace("{code-cell}", "python") 41 | myst_file = (demo_dir / demo.name).with_suffix(".md") 42 | with open(myst_file, "w") as fw: 43 | fw.write(myst_text) 44 | 45 | ipynb_file = (demo_dir / demo.name).with_suffix(".ipynb") 46 | jupytext.write(python_demo, ipynb_file, fmt="ipynb") 47 | 48 | # Copy python demo files into documentation demo directory 49 | shutil.copy(demo, demo_dir) 50 | 51 | # Copy images used in demos 52 | for file in subdir.glob("**/*.png"): 53 | shutil.copy(file, demo_dir) 54 | 55 | 56 | if __name__ == "__main__": 57 | process() 58 | -------------------------------------------------------------------------------- /launch-container.sh: -------------------------------------------------------------------------------- 1 | #!/bin/bash 2 | CONTAINER_ENGINE="podman" 3 | ${CONTAINER_ENGINE} run -ti -v $(pwd):/shared -w /shared dolfinx/dolfinx:nightly 4 | -------------------------------------------------------------------------------- /pyproject.toml: -------------------------------------------------------------------------------- 1 | [build-system] 2 | requires = ["setuptools", "wheel"] 3 | build-backend = "setuptools.build_meta" 4 | 5 | [project] 6 | name = "fenicsx-shells" 7 | version = "0.10.0.dev0" 8 | description = "A FEniCSx library for simulating thin structures" 9 | readme = "README.md" 10 | requires-python = ">=3.9.0" 11 | license = { file = "COPYING.LESSER" } 12 | authors = [ 13 | { name = "Jack S. Hale", email = "mail@jackhale.co.uk" }, 14 | { name = "Tian Yang" } 15 | ] 16 | dependencies = [ 17 | "fenics-dolfinx>=0.10.0.dev0,<0.11.0", 18 | ] 19 | 20 | [project.optional-dependencies] 21 | doc = ["jupytext", "myst_parser", "sphinx_rtd_theme", "sphinx"] 22 | demo = ["matplotlib"] 23 | test = ["pytest"] 24 | lint = ["ruff"] 25 | ci = [ 26 | "fenicsx-shells[doc]", 27 | "fenicsx-shells[demo]", 28 | "fenicsx-shells[test]", 29 | "fenicsx-shells[lint]", 30 | ] 31 | 32 | [tool.ruff] 33 | line-length = 100 34 | indent-width = 4 35 | 36 | [tool.ruff.lint] 37 | select = [ 38 | "E", # pycodestyle 39 | "W", # pycodestyle 40 | "F", # pyflakes 41 | "I", # isort - use standalone isort 42 | "RUF", # Ruff-specific rules 43 | "UP", # pyupgrade 44 | "ICN", # flake8-import-conventions 45 | "NPY", # numpy-specific rules 46 | "FLY", # use f-string not static joins 47 | ] 48 | ignore = ["UP007", "RUF012"] 49 | allowed-confusables = ["σ"] 50 | 51 | [tool.ruff.lint.isort] 52 | known-first-party = ["basix", "dolfinx", "ffcx", "ufl"] 53 | known-third-party = ["gmsh", "numba", "numpy", "pytest", "pyvista"] 54 | section-order = [ 55 | "future", 56 | "standard-library", 57 | "mpi", 58 | "third-party", 59 | "first-party", 60 | "local-folder", 61 | ] 62 | 63 | [tool.ruff.lint.isort.sections] 64 | "mpi" = ["mpi4py", "petsc4py"] 65 | -------------------------------------------------------------------------------- /src/fenicsx_shells/__init__.py: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/FEniCS-Shells/fenicsx-shells/e1bbfd920ed42843720273eebf35aa940ba422c7/src/fenicsx_shells/__init__.py -------------------------------------------------------------------------------- /test/.placeholder: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/FEniCS-Shells/fenicsx-shells/e1bbfd920ed42843720273eebf35aa940ba422c7/test/.placeholder --------------------------------------------------------------------------------