{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Component-wise Dirichlet BC\n", "Author: Jørgen S. Dokken\n", "\n", "In this section, we will learn how to prescribe Dirichlet boundary conditions on a component of your unknown $u_h$.\n", "We will illustrate the problem using a `VectorElement`. However, the method generalizes to any `MixedElement`.\n", "\n", "We will use a slightly modified version of [the linear elasticity demo](./../chapter2/linearelasticity_code), namely\n", "$$\n", "-\\nabla \\cdot \\sigma (u) = f\\quad \\text{in } \\Omega,\n", "$$\n", "\n", "$$\n", "\\sigma \\cdot n = 0 \\quad \\text{on } \\partial \\Omega_N,\n", "$$\n", "\n", "$$\n", "u= 0\\quad \\text{at } \\partial\\Omega_{D},\n", "$$\n", "\n", "$$\n", "u_x=0 \\quad \\text{at } \\partial\\Omega_{Dx},\n", "$$\n", "\n", "$$\n", "\\sigma(u)= \\lambda \\mathrm{tr}(\\epsilon(u))I + 2 \\mu \\epsilon(u), \\qquad \\epsilon(u) = \\frac{1}{2}\\left(\\nabla u + (\\nabla u )^T\\right).\n", "$$\n", "We will consider a two dimensional box spanning $[0,L]\\times[0,H]$, where\n", "$\\partial\\Omega_N$ is the left and right side of the beam, $\\partial\\Omega_D$ the bottom of the beam, while $\\partial\\Omega_{Dx}$ is the right side of the beam.\n", "We will prescribe a displacement $u_x=0$ on the right side of the beam, while the beam is being deformed under its own weight. The sides of the box is traction free." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "L = 1\n", "H = 1.3\n", "lambda_ = 1.25 \n", "mu = 1\n", "rho = 1\n", "g = 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As in the previous demos, we define our mesh and function space. We will create a `ufl.VectorElement` to create a two dimensional vector space." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "INFO:root:running build_ext\n", "INFO:root:building 'libffcx_elements_79d1ca1b50dce9bf871b096ebf68bead53fe62a9' extension\n", "INFO:root:x86_64-linux-gnu-gcc -Wno-unused-result -Wsign-compare -DNDEBUG -g -fwrapv -O2 -Wall -g -fstack-protector-strong -Wformat -Werror=format-security -g -fwrapv -O2 -fPIC -I/usr/local/lib/python3.10/dist-packages/ffcx/codegeneration -I/usr/include/python3.10 -c libffcx_elements_79d1ca1b50dce9bf871b096ebf68bead53fe62a9.c -o ./libffcx_elements_79d1ca1b50dce9bf871b096ebf68bead53fe62a9.o -O2\n", "INFO:root:x86_64-linux-gnu-gcc -shared -Wl,-O1 -Wl,-Bsymbolic-functions -Wl,-Bsymbolic-functions -g -fwrapv -O2 ./libffcx_elements_79d1ca1b50dce9bf871b096ebf68bead53fe62a9.o -L/usr/lib/x86_64-linux-gnu -o ./libffcx_elements_79d1ca1b50dce9bf871b096ebf68bead53fe62a9.cpython-310-x86_64-linux-gnu.so\n" ] } ], "source": [ "from dolfinx.fem import (Constant, dirichletbc, Function, FunctionSpace, locate_dofs_geometrical,\n", " locate_dofs_topological)\n", "from dolfinx.fem.petsc import LinearProblem\n", "from dolfinx.mesh import CellType, create_rectangle, locate_entities_boundary\n", "from ufl import Identity, Measure, TestFunction, TrialFunction, VectorElement, dot, dx, inner, grad, nabla_div, sym\n", "from mpi4py import MPI\n", "from petsc4py.PETSc import ScalarType\n", "\n", "import numpy as np\n", "import pyvista\n", "\n", "mesh = create_rectangle(MPI.COMM_WORLD, np.array([[0,0],[L, H]]), [30,30], cell_type=CellType.triangle)\n", "element = VectorElement(\"CG\", mesh.ufl_cell(), 1)\n", "V = FunctionSpace(mesh, element)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Boundary conditions\n", "As we would like to clamp the boundary at $x=0$, we do this by using a marker function, we use `dolfinx.fem.locate_dofs_geometrical` to identify the relevant degrees of freedom." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def clamped_boundary(x):\n", " return np.isclose(x[1], 0)\n", "\n", "u_zero = np.array((0,)*mesh.geometry.dim, dtype=ScalarType)\n", "bc = dirichletbc(u_zero, locate_dofs_geometrical(V, clamped_boundary), V)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we would like to constrain the $x$-component of our solution at $x=L$ to $0$. We start by creating the sub space only containing the $x$\n", "-component." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next, we locate the degrees of freedom on the top boundary. However, as the boundary condition is in a sub space of our solution, we need to supply both the parent space $V$ and the sub space $V_0$ to `dolfinx.locate_dofs_topological`." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "def right(x):\n", " return np.logical_and(np.isclose(x[0], L), x[1] < H)\n", "boundary_facets = locate_entities_boundary(mesh, mesh.topology.dim-1, right)\n", "boundary_dofs_x = locate_dofs_topological(V.sub(0), mesh.topology.dim-1, boundary_facets)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now create our Dirichlet condition" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "bcx = dirichletbc(ScalarType(0), boundary_dofs_x, V.sub(0))\n", "bcs = [bc, bcx]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.Constant`" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "T = Constant(mesh, ScalarType((0, 0)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also want to specify the integration measure $\\mathrm{d}s$, which should be the integral over the boundary of our domain. We do this by using `ufl`, and its built in integration measures" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "ds = Measure(\"ds\", domain=mesh)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Variational formulation\n", "We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "def epsilon(u):\n", " return sym(grad(u)) \n", "def sigma(u):\n", " return lambda_ * nabla_div(u) * Identity(len(u)) + 2*mu*epsilon(u)\n", "\n", "u = TrialFunction(V)\n", "v = TestFunction(V)\n", "f = Constant(mesh, ScalarType((0, -rho*g)))\n", "a = inner(sigma(u), epsilon(v)) * dx\n", "L = dot(f, v) * dx + dot(T, v) * ds" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solve the linear variational problem\n", "As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "INFO:root:running build_ext\n", "INFO:root:building 'libffcx_forms_e3d2a7da58068b924efb208149d620a4032938c8' extension\n", "INFO:root:x86_64-linux-gnu-gcc -Wno-unused-result -Wsign-compare -DNDEBUG -g -fwrapv -O2 -Wall -g -fstack-protector-strong -Wformat -Werror=format-security -g -fwrapv -O2 -fPIC -I/usr/local/lib/python3.10/dist-packages/ffcx/codegeneration -I/usr/include/python3.10 -c libffcx_forms_e3d2a7da58068b924efb208149d620a4032938c8.c -o ./libffcx_forms_e3d2a7da58068b924efb208149d620a4032938c8.o -O2\n", "INFO:root:x86_64-linux-gnu-gcc -shared -Wl,-O1 -Wl,-Bsymbolic-functions -Wl,-Bsymbolic-functions -g -fwrapv -O2 ./libffcx_forms_e3d2a7da58068b924efb208149d620a4032938c8.o -L/usr/lib/x86_64-linux-gnu -o ./libffcx_forms_e3d2a7da58068b924efb208149d620a4032938c8.cpython-310-x86_64-linux-gnu.so\n", "INFO:root:running build_ext\n", "INFO:root:building 'libffcx_forms_5abf2a91856d4dfd9a5fd1eec567ddd24166c138' extension\n", "INFO:root:x86_64-linux-gnu-gcc -Wno-unused-result -Wsign-compare -DNDEBUG -g -fwrapv -O2 -Wall -g -fstack-protector-strong -Wformat -Werror=format-security -g -fwrapv -O2 -fPIC -I/usr/local/lib/python3.10/dist-packages/ffcx/codegeneration -I/usr/include/python3.10 -c libffcx_forms_5abf2a91856d4dfd9a5fd1eec567ddd24166c138.c -o ./libffcx_forms_5abf2a91856d4dfd9a5fd1eec567ddd24166c138.o -O2\n", "INFO:root:x86_64-linux-gnu-gcc -shared -Wl,-O1 -Wl,-Bsymbolic-functions -Wl,-Bsymbolic-functions -g -fwrapv -O2 ./libffcx_forms_5abf2a91856d4dfd9a5fd1eec567ddd24166c138.o -L/usr/lib/x86_64-linux-gnu -o ./libffcx_forms_5abf2a91856d4dfd9a5fd1eec567ddd24166c138.cpython-310-x86_64-linux-gnu.so\n" ] } ], "source": [ "problem = LinearProblem(a, L, bcs=bcs, petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"})\n", "uh = problem.solve()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Visualization" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "lines_to_next_cell": 0 }, "outputs": [ { "data": { "application/javascript": [ "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " var force = true;\n", "\n", " if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n", " root._bokeh_onload_callbacks = [];\n", " root._bokeh_is_loading = undefined;\n", " }\n", "\n", " if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n", " root._bokeh_timeout = Date.now() + 5000;\n", " root._bokeh_failed_load = false;\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks\n", " }\n", " 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