adding applications and sphinx documentation of psydac and sympde

.gitignore

@@ -18,4 +18,4 @@ usr
 
 .env
 .iga-python
-
+*_autosummary*

_config.yml

@@ -38,5 +38,9 @@ html:
 sphinx:
   extra_extensions:
     - sphinx_proof
+    - sphinx.ext.autodoc
+    - sphinx.ext.autosummary
+  config:
+    autosummary_generate: True
 
 exclude_patterns: [README.md, .iga-python]

_toc.yml

@@ -37,6 +37,7 @@ parts:
     chapters:
       - file: chapter2/poisson
       - file: chapter2/poisson-nitsche
+      - file: chapter2/elliptic-general-form
       - file: chapter2/biharmonic
       - file: chapter2/vector-poisson
       - file: chapter2/advection-diffusion
@@ -65,4 +66,35 @@ parts:
       - file: chapter4/subdomains
         sections:
           - file: chapter4/poisson-nitsche
-
+  - caption: Applications 
+    chapters:
+      - file: chapter5/cfd
+        sections:
+          - file: chapter5/buoyancy-driven_natural_convection         
+          - file: chapter5/compressible_flow_in_a_nozzle              
+          - file: chapter5/convection-diffusion_equation_in_a_channel 
+          - file: chapter5/free_surface_flow                          
+          - file: chapter5/heat_conduction_in_a_solid                 
+          - file: chapter5/incompressible_flow_past_a_cylinder        
+          - file: chapter5/magnetohydrodynamics_flow                  
+          - file: chapter5/particle-laden_flow                        
+          - file: chapter5/stokes_flow_in_a_lid-driven_cavity         
+          - file: chapter5/two-phase_flow                             
+          - file: chapter5/non-newtonian-fluids
+            sections:
+              - file: chapter5/power_law_fluid_flow_in_a_channel          
+              - file: chapter5/bingham_plastic_flow_in_a_pipe             
+              - file: chapter5/oldroyd-b_fluid_flow_in_a_channel          
+              - file: chapter5/herschel-bulkley_fluid_flow_in_a_pipe      
+              - file: chapter5/cross_power_law_fluid_flow_in_a_channel    
+              - file: chapter5/casson_fluid_flow_in_a_channel             
+              - file: chapter5/papanastasiou_fluid_flow_in_a_channel      
+      - file: chapter5/fsi
+      - file: chapter5/electromagnetics
+      - file: chapter5/mhd
+      - file: chapter5/material-science
+      - file: chapter5/multiphysics
+  - caption: Documentation
+    chapters:
+      - file: documentation/sympde
+      - file: documentation/psydac

chapter2/elliptic-general-form.ipynb

@@ -0,0 +1,278 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b5e0474c",
+   "metadata": {},
+   "source": [
+    "# Elliptic equation in the general form\n",
+    "\n",
+    "We consider here, the following general form of an elliptic partial differential equation,\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "- \\nabla \\cdot \\left( A \\nabla u \\right) + \\mathbf{b} \\cdot \\nabla u + c u &= f, \\quad \\Omega \\\\\n",
+    "u &= 0, \\quad \\partial \\Omega\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## Variational Formulation\n",
+    "\n",
+    "An $H^1$-conforming variational formulation of reads\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^1(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\left( \\left( A \\nabla u \\right) \\cdot \\nabla v + \\left( \\mathbf{b} \\cdot \\nabla u \\right) v + cuv \\right) ~ d\\Omega$,\n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega$.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5c944a1",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "c2cee2d9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot, div\n",
+    "from sympde.core import Vector, Matrix\n",
+    "\n",
+    "from sympy import pi, sin\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "c = x*y\n",
+    "b = Vector([1e-2, 1e-1], name='b')\n",
+    "A = Matrix([[1,1], [0,1]], name='A')\n",
+    "\n",
+    "# bilinear form\n",
+    "expr = dot(grad(v), A * grad(u)) + dot(b, grad(u))*v + c*u*v\n",
+    "a = BilinearForm((u,v), integral(domain, expr))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "366c2d67",
+   "metadata": {},
+   "source": [
+    "### Manifactured solution"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "2b05c79f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# the analytical solution and its rhs\n",
+    "ue = sin(pi * x) * sin(pi * y)\n",
+    "\n",
+    "L = lambda u: - div(A*grad(u)) + dot(b,grad(u)) + c*u\n",
+    "f  = L(ue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66b50430",
+   "metadata": {},
+   "source": [
+    "### Formal Equation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "d0d386a2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0614c740",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "96c29a27",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "ef6a9709",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c286e572",
+   "metadata": {},
+   "source": [
+    "### Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "3bafe9f5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "b88daf50",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d865a17f",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "be88e26c",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "3440e74a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.00021948770944141364\n"
+     ]
+    }
+   ],
+   "source": [
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5d7d9180",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}