lecture-5.ipynb

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    "# Lecture 5: FEM, cont'd"
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    "## Linear Elasticity\n",
    "\n",
    "* Consider a domain $\\Omega$ as shown here:\n",
    "<img src=\"fig/2Delastic_domain_ext.png\">\n",
    "We assume it is a piece of elastic material (similar to a beam) of this shape. Fixed at the left end and the force $P$ is applied to its right end.\n",
    "\n",
    "* Boundary:\n",
    "    - leftmost: $\\Gamma_1$\n",
    "    - rightmost: $\\Gamma_3$\n",
    "    - rest: $\\Gamma_2$\n",
    "* Displacement: $(u,v)$ (in the $x$ and $y$ coord)\n",
    "\n",
    "* Elasticity tensors:\n",
    "$$\n",
    "\\begin{align*}\n",
    "C_{11} &= \\begin{pmatrix}\n",
    "\\frac{E (1-\\nu)}{(1-2\\nu)(1+\\nu)} & 0 \\\\ 0 & \\frac{E}{2(1+\\nu)} \\end{pmatrix},\n",
    "&\\qquad\n",
    "C_{12} &= \\begin{pmatrix} 0 & \\frac{E \\nu}{(1-2\\nu)(1+\\nu)}  \\\\ \\frac{E}{2(1+\\nu)} & 0 \\end{pmatrix},\n",
    "\\\\\n",
    "C_{21} &= C_{12}^{\\rm T},\n",
    "&\\qquad\n",
    "C_{22} &= \\begin{pmatrix} \\frac{E}{2(1+\\nu)} & 0 \\\\ 0 & \\frac{E (1-\\nu)}{(1-2\\nu)(1+\\nu)} \\end{pmatrix}\n",
    "\\end{align*}\n",
    "$$\n",
    "Here $E$ is called the modulus of elasticity and $\\nu$ is called the Poisson ratio.\n",
    "(In fact, $C$ is a four-dimensional tensor, denoted by $C$, most likely, after Cauchy.)\n",
    "\n",
    "* Equations:\n",
    "$$\n",
    "\\begin{align*}\n",
    "-{\\rm div} (C_{11} \\nabla u + C_{12} \\nabla v) &= 0\n",
    "\\qquad\\text{in }\\Omega\n",
    "\\\\\n",
    "-{\\rm div} (C_{21} \\nabla u + C_{22} \\nabla v) &= 0\n",
    "\\qquad\\text{in }\\Omega\n",
    "\\\\\n",
    "u = v &= 0 \\qquad\\text{on }\\Gamma_1\n",
    "\\\\\n",
    "C_{11} u_n + C_{12} v_n = C_{21} u_n + C_{22} v_n &= 0 \\qquad\\text{on }\\Gamma_2\n",
    "\\\\\n",
    "C_{11} u_n + C_{12} v_n &= 0 \\qquad\\text{on }\\Gamma_3\n",
    "\\\\\n",
    "C_{21} u_n + C_{22} v_n &= \\frac{P}{H W} \\qquad\\text{on }\\Gamma_3,\n",
    "\\end{align*}\n",
    "$$\n",
    "where $u_n$ and $v_n$ are normal derivatives of the solution.\n",
    "\n",
    "* Variational formulation:\n",
    "    - test functions $p=p(x,y)$ and $q = q(x,y)$\n",
    "    - Dirichet BCs: $p=q=0$ on $\\Gamma_1$\n",
    "    - Variational form:\n",
    "$$\n",
    "\\int_{\\Omega} \\big[\n",
    "    (C_{11} \\nabla u + C_{12} \\nabla v)\\cdot \\nabla p\n",
    "    +\n",
    "    (C_{21} \\nabla u + C_{22} \\nabla v)\\cdot \\nabla q\n",
    "\\big]\n",
    "= \\frac{P}{H W}  \\int_{\\Gamma_3} q\n",
    "\\qquad\\forall p, q\n",
    "$$\n",
    "\n",
    "* A good way to go about this problem is to think that your space consists of vector-valued functions $(u,v)\\in\\mathbb R^2$. Thus, at each node you have two basis functions: such that $(u,v) = (1,0)$ for that node, and $(u,v) = (0,1)$ for that node."
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   "source": [
    "## More advanced stuff"
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    "### Error analysis\n",
    "\n",
    "* If $u$ is the exact solution and $u_h$ is the approximate solution then one can prove that\n",
    "$$\n",
    "\\|\\nabla u_h - \\nabla u\\|_{L^2}\n",
    "\\leq\n",
    "C h \\|\\nabla^2 u\\|_{L^2},\n",
    "$$\n",
    "where $h$ is the **maximal** size of the triangle and $\\nabla^2 u$ is the <a href=\"http://en.wikipedia.org/wiki/Hessian_matrix\">Hessian matrix</a>.\n",
    "\n",
    "* Comes from a truly beautiful argument:\n",
    "    - One can show that $u_h$ is the minimizer of\n",
    "    $$\n",
    "    \\int \\nabla (v_h-u) \\cdot \\nabla (v_h-u) = \\|\\nabla v_h - \\nabla u\\|_{L^2}^2\n",
    "    $$\n",
    "    over all $v_h\\in \\mathcal P_1(\\mathcal T)$ (let's prove it).\n",
    "    - It remains to invoke a geometry-related argument that given $u$\n",
    "    there exists $v_h$ such that $\\|\\nabla v_h - \\nabla u\\| \\leq h \\|\\nabla^2 u\\|$.\n",
    "    This is achieved with, e.g., the Cl&eacute;ment interpolation (warning: google for that on your own risk!)\n",
    "    - **Exercise**: learn the difference in pronounciation between Clement and Cl&eacute;ment <img width=40 src=\"http://vignette2.wikia.nocookie.net/fallout/images/7/73/Trollface.png/revision/latest?cb=20120710173828\">\n",
    "    - Know <a href=\"https://www.linkedin.com/pub/cl%C3%A9ment-fortin/5/197/94a\">this guy</a>?\n"
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    "### Mesh refinement / Adaptivity\n",
    "\n",
    "<img src=\"fig/borehole_degen.png\">\n",
    "\n",
    "<img src=\"fig/Lshape_refin.png\">\n",
    "\n",
    "* Often done through the *a posteriori* error estimate, e.g.,\n",
    "$$\n",
    "\\|\\nabla u_h - \\nabla u\\|^2 \\leq \\sum_{{\\rm edge}} h_{{\\rm edge}} \\Big[\\frac{\\nabla u_h}{h_{{\\rm edge}}}\\Big]^2_{{\\rm edge}}\n",
    "$$\n",
    "where $[\\bullet]_{\\rm edge}$ is the jump across the edge (recall that $\\nabla u_h$ is piecewise constant)"
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    "### Mesh quality\n",
    "\n",
    "* Mesh should not have very sharp angles (otherwise the approximation will be bad)!\n",
    "\n",
    "<img src=\"fig/fig_mesh_quality.gif\">"
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    "### Collocation\n",
    "#### (this is an alternative to Galerkin)\n",
    "\n",
    "* If we have non-constant coefficients, we use the quadrature rules\n",
    "$$\n",
    "\\int_T k(x,y) \\nabla u\\cdot\\nabla v\n",
    "\\approx \\sum_{i} w_i k(x_i,y_i) \\nabla u(x_i,y_i)\\cdot\\nabla v(x_i,y_i)\n",
    "$$\n",
    "\n",
    "* must be careful to take the quadrature rule of sufficiently high order"
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   "source": [
    "### Quad(rilateral) elements:\n",
    "\n",
    "<img src=\"fig/FEM_quad_basis.png\">"
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    "### High-order elements\n",
    "\n",
    "* $\\mathcal P^p(\\mathcal T)$: polynomials of power $p$\n",
    "* $hp$-refinement\n",
    "\n",
    "A good source for these is <BR>\n",
    "C. Schwab, \"p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics\", Clarendon Press, 1998."
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    "### More stuff:\n",
    "\n",
    "* Discontinous Galerkin, $\\mathcal P^0$ elements\n",
    "* Other bases: <a href=\"http://www.ams.org/journals/mcom/2001-70-233/S0025-5718-00-01252-7/S0025-5718-00-01252-7.pdf\">wavelets</a>, radial basis functions\n",
    "* mixed FEM, generalized FEM, extended FEM, what-not FEM..."
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    "## Adv/Disadv of FEM\n",
    "\n",
    "* Good for simple physics, solid mechanics\n",
    "* Good energy (and other) conservation properties\n",
    "\n",
    "But...\n",
    "\n",
    "* Not easy for fluid flow (must work hard for stability)\n",
    "* sometimes non-monotonicity"
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    "##### Questions?\n"
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