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ratnaniaratnania
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adding applications and sphinx documentation of psydac and sympde
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‎.gitignore

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.env
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.iga-python
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*_autosummary*

‎_config.yml

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sphinx:
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extra_extensions:
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- sphinx_proof
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- sphinx.ext.autodoc
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- sphinx.ext.autosummary
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config:
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autosummary_generate: True
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exclude_patterns: [README.md, .iga-python]

‎_toc.yml

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chapters:
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- file: chapter2/poisson
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- file: chapter2/poisson-nitsche
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- file: chapter2/elliptic-general-form
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- file: chapter2/biharmonic
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- file: chapter2/vector-poisson
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- file: chapter2/advection-diffusion
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- file: chapter4/subdomains
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sections:
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- file: chapter4/poisson-nitsche
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- caption: Applications
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chapters:
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- file: chapter5/cfd
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sections:
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- file: chapter5/buoyancy-driven_natural_convection
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- file: chapter5/compressible_flow_in_a_nozzle
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- file: chapter5/convection-diffusion_equation_in_a_channel
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- file: chapter5/free_surface_flow
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- file: chapter5/heat_conduction_in_a_solid
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- file: chapter5/incompressible_flow_past_a_cylinder
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- file: chapter5/magnetohydrodynamics_flow
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- file: chapter5/particle-laden_flow
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- file: chapter5/stokes_flow_in_a_lid-driven_cavity
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- file: chapter5/two-phase_flow
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- file: chapter5/non-newtonian-fluids
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sections:
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- file: chapter5/power_law_fluid_flow_in_a_channel
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- file: chapter5/bingham_plastic_flow_in_a_pipe
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- file: chapter5/oldroyd-b_fluid_flow_in_a_channel
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- file: chapter5/herschel-bulkley_fluid_flow_in_a_pipe
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- file: chapter5/cross_power_law_fluid_flow_in_a_channel
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- file: chapter5/casson_fluid_flow_in_a_channel
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- file: chapter5/papanastasiou_fluid_flow_in_a_channel
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- file: chapter5/fsi
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- file: chapter5/electromagnetics
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- file: chapter5/mhd
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- file: chapter5/material-science
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- file: chapter5/multiphysics
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- caption: Documentation
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chapters:
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- file: documentation/sympde
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- file: documentation/psydac

‎chapter2/elliptic-general-form.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "b5e0474c",
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"metadata": {},
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"source": [
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"# Elliptic equation in the general form\n",
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"\n",
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"We consider here, the following general form of an elliptic partial differential equation,\n",
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"\n",
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"$$\n",
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"\\begin{align}\n",
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"- \\nabla \\cdot \\left( A \\nabla u \\right) + \\mathbf{b} \\cdot \\nabla u + c u &= f, \\quad \\Omega \\\\\n",
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"u &= 0, \\quad \\partial \\Omega\n",
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"\\end{align}\n",
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"$$\n",
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"\n",
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"## Variational Formulation\n",
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"\n",
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"An $H^1$-conforming variational formulation of reads\n",
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"\n",
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"$$\n",
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"\\begin{align}\n",
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" \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
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"\\end{align}\n",
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"$$\n",
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"\n",
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"where \n",
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"\n",
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"- $V \\subset H^1(\\Omega)$, \n",
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"- $a(u,v) := \\int_{\\Omega} \\left( \\left( A \\nabla u \\right) \\cdot \\nabla v + \\left( \\mathbf{b} \\cdot \\nabla u \\right) v + cuv \\right) ~ d\\Omega$,\n",
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"- $l(v) := \\int_{\\Omega} f v ~ d\\Omega$.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c5c944a1",
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"metadata": {},
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"source": [
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"## Formal Model"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "c2cee2d9",
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"metadata": {},
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"outputs": [],
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"source": [
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"from sympde.expr import BilinearForm, LinearForm, integral\n",
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"from sympde.expr import find, EssentialBC, Norm, SemiNorm\n",
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"from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
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"from sympde.calculus import grad, dot, div\n",
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"from sympde.core import Vector, Matrix\n",
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"\n",
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"from sympy import pi, sin\n",
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"\n",
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"from psydac.api.discretization import discretize\n",
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"\n",
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"domain = Square()\n",
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"\n",
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"V = ScalarFunctionSpace('V', domain)\n",
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"\n",
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"x,y = domain.coordinates\n",
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"\n",
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"u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
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"\n",
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"c = x*y\n",
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"b = Vector([1e-2, 1e-1], name='b')\n",
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"A = Matrix([[1,1], [0,1]], name='A')\n",
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"\n",
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"# bilinear form\n",
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"expr = dot(grad(v), A * grad(u)) + dot(b, grad(u))*v + c*u*v\n",
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"a = BilinearForm((u,v), integral(domain, expr))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "366c2d67",
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"metadata": {},
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"source": [
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"### Manifactured solution"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "2b05c79f",
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"metadata": {},
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"outputs": [],
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"source": [
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"# the analytical solution and its rhs\n",
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"ue = sin(pi * x) * sin(pi * y)\n",
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"\n",
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"L = lambda u: - div(A*grad(u)) + dot(b,grad(u)) + c*u\n",
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"f = L(ue)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "66b50430",
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"metadata": {},
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"source": [
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"### Formal Equation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "d0d386a2",
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"metadata": {},
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"outputs": [],
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"source": [
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"l = LinearForm(v, integral(domain, f*v))\n",
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"\n",
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"# Dirichlet boundary conditions\n",
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"bc = [EssentialBC(u, 0, domain.boundary)]\n",
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"\n",
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"# Variational problem\n",
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"equation = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0614c740",
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"metadata": {},
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"source": [
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"## Discretization"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"id": "96c29a27",
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"metadata": {},
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"outputs": [],
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"source": [
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"degree = [2,2]\n",
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"ncells = [8,8]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"id": "ef6a9709",
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
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"\n",
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"non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
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"list of lists, TableForm or some other data structure instead. See\n",
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"https://github.com/sympy/sympy/issues/21497 for more info.\n",
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"\n",
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" ).warn()\n"
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]
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}
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],
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"source": [
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"# Create computational domain from topological domain\n",
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"domain_h = discretize(domain, ncells=ncells, comm=None)\n",
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"\n",
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"# Create discrete spline space\n",
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"Vh = discretize(V, domain_h, degree=degree)\n",
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"\n",
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"# Discretize equation\n",
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"equation_h = discretize(equation, domain_h, [Vh, Vh])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c286e572",
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"metadata": {},
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"source": [
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"### Solving the PDE"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"id": "3bafe9f5",
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"metadata": {},
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"outputs": [],
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"source": [
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"equation_h.set_solver('gmres', info=False, tol=1e-8)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"id": "b88daf50",
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"metadata": {},
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"outputs": [],
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"source": [
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"uh = equation_h.solve()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d865a17f",
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"metadata": {},
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"source": [
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"## Computing the error norm"
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]
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},
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{
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"cell_type": "markdown",
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"id": "be88e26c",
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"metadata": {},
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"source": [
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"### Computing the $L^2$ norm"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"id": "3440e74a",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.00021948770944141364\n"
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]
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}
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],
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"source": [
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"u = element_of(V, name='u')\n",
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"\n",
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"# create the formal Norm object\n",
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"l2norm = Norm(u - ue, domain, kind='l2')\n",
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"\n",
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"# discretize the norm\n",
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"l2norm_h = discretize(l2norm, domain_h, Vh)\n",
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"\n",
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"# assemble the norm\n",
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"l2_error = l2norm_h.assemble(u=uh)\n",
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"\n",
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"# print the result\n",
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"print(l2_error)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "5d7d9180",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": ".iga-python",
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"language": "python",
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"name": ".iga-python"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.12"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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