{ "cells": [ { "cell_type": "markdown", "id": "f69832df", "metadata": { "editable": true }, "source": [ "\n", "\n", "# Demo - Integration of functions\n", "**Mikael Mortensen** (email: `mikaem@math.uio.no`), Department of Mathematics, University of Oslo.\n", "\n", "Date: **August 7, 2020**\n", "\n", "**Summary.** This is a demonstration of how the Python module [shenfun](https://github.com/spectralDNS/shenfun) can be used to\n", "integrate over 1D curves and 2D surfaces in 3D space.\n", "We make use of\n", "curvilinear coordinates, and reproduce some integrals\n", "performed by Behnam Hashemi with [Chebfun](http://www.chebfun.org/examples/approx3/SurfaceIntegral3D.html).\n", "\n", "**Notice.**\n", "\n", "For all the examples below we could just as well\n", "use Legendre polynomials instead of Chebyshev.\n", "Just replace 'C' with 'L' when creating function spaces.\n", "The accuracy ought to be similar." ] }, { "cell_type": "markdown", "id": "fc9ceef2", "metadata": { "editable": true }, "source": [ "## The inner product\n", "\n", "A lesser known fact about [shenfun](https://github.com/spectralDNS/shenfun) is\n", "that it can be used to perform regular, unweighted, integrals with\n", "spectral accuracy. With the newly added curvilinear coordinates\n", "feature, we can now also integrate over highly complex lines and surfaces\n", "embedded in a higher dimensional space.\n", "\n", "To integrate over a domain in shenfun we use the\n", "[inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner)\n", "function, with a constant test function. The [inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner)\n", "function in shenfun is defined as an integral over the\n", "entire domain $\\Omega$ in question" ] }, { "cell_type": "markdown", "id": "3ab8310f", "metadata": { "editable": true }, "source": [ "$$\n", "(u, v)_w = \\int_{\\Omega} u \\overline{v} w d\\Omega,\n", "$$" ] }, { "cell_type": "markdown", "id": "43bf0ffb", "metadata": { "editable": true }, "source": [ "for trial function $u$, test function $v$ and weight $w$.\n", "Also, $\\overline{v}$ represents the complex conjugate of $v$, in case\n", "we are working with complex functions (like Fourier exponentials).\n", "\n", "The functions and weights take on different form, but if\n", "the test function $v$ is chosen to be a constant, e.g., $v=1$,\n", "then the weight is also constant, $w=1$, and the inner product becomes\n", "an unweighted integral of $u$ over the domain" ] }, { "cell_type": "markdown", "id": "8cc37f68", "metadata": { "editable": true }, "source": [ "$$\n", "(u, 1)_w = \\int_{\\Omega} u d\\Omega.\n", "$$" ] }, { "cell_type": "markdown", "id": "0819d555", "metadata": { "editable": true }, "source": [ "Note that the integral in the inner product either can be computed\n", "exactly using Sympy, adaptively using Scipy or with quadrature\n", "using Shenfun. The quadrature can be computed with any fixed resolution,\n", "see [inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner)." ] }, { "cell_type": "markdown", "id": "dacc0e7d", "metadata": { "editable": true }, "source": [ "### Curve integrals\n", "\n", "For example, if we create some function space on the line from\n", "0 to 1, then we can get the length of this domain using `inner`" ] }, { "cell_type": "code", "execution_count": 1, "id": "239b46e9", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:47:58.556760Z", "iopub.status.busy": "2024-02-19T13:47:58.556479Z", "iopub.status.idle": "2024-02-19T13:47:59.390430Z", "shell.execute_reply": "2024-02-19T13:47:59.389349Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Length of domain = 1.0000000000000002\n" ] } ], "source": [ "from shenfun import *\n", "B = FunctionSpace(10, 'C', domain=(0, 1))\n", "u = Array(B, val=1)\n", "length = inner(u, 1)\n", "print('Length of domain =', length)" ] }, { "cell_type": "markdown", "id": "4e59fb61", "metadata": { "editable": true }, "source": [ "Note that we cannot simply do `inner(1, 1)`, because the\n", "`inner` function does not know about the domain, which is part\n", "of the [FunctionSpace](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.FunctionSpace). So to integrate `u=1`, we need to\n", "create `u` as an [Array](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.Array) with the constant value 1.\n", "\n", "Since the function space `B` is Cartesian the computed\n", "length is simply the domain length.\n", "Not very impressive, but the same goes for multidimensional\n", "tensor product domains" ] }, { "cell_type": "code", "execution_count": 2, "id": "c247086b", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:47:59.394012Z", "iopub.status.busy": "2024-02-19T13:47:59.393641Z", "iopub.status.idle": "2024-02-19T13:47:59.404456Z", "shell.execute_reply": "2024-02-19T13:47:59.403926Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Area of domain = 6.283185307179587\n" ] } ], "source": [ "F = FunctionSpace(10, 'F', domain=(0, 2*np.pi))\n", "T = TensorProductSpace(comm, (B, F))\n", "area = inner(1, Array(T, val=1))\n", "print('Area of domain =', area)" ] }, { "cell_type": "markdown", "id": "16e1cce2", "metadata": { "editable": true }, "source": [ "Still not very impressive, but moving to curvilinear coordinates\n", "it all starts to become more interesting. Lets\n", "look at a spiral $C$ embedded in $\\mathbb{R}^3$, parametrized\n", "by one single parameter $t$" ] }, { "cell_type": "markdown", "id": "0bea4158", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", "x(t) &= \\sin 2t \\\\ \n", "y(t) &= \\cos 2t \\\\ \n", "z(t) &= \\frac{t}{2} \\\\ \n", "0 \\le & t \\le 2\\pi\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "b13e1f89", "metadata": { "editable": true }, "source": [ "What is the length of this spiral? The spiral can be\n", "seen as the red curve in the figure a few cells below.\n", "\n", "The integral over the parametrized curve $C$ can\n", "be written as" ] }, { "cell_type": "markdown", "id": "158ad02c", "metadata": { "editable": true }, "source": [ "$$\n", "\\int_C ds = \\int_{t=0}^{2\\pi} \\sqrt{\\left(\\frac{d x}{d t}\\right)^2 + \\left(\\frac{d y}{d t}\\right)^2 + \\left(\\frac{d z}{d t}\\right)^2} dt.\n", "$$" ] }, { "cell_type": "markdown", "id": "e5871ca3", "metadata": { "editable": true }, "source": [ "We can find this integral easily using shenfun. Create\n", "a function space in curvilinear coordinates, providing\n", "the position vector $\\mathbf{r} = x(t)\\mathbf{i} + y(t) \\mathbf{j} + z(t) \\mathbf{k}$\n", "as input. Also, choose to work with covariant basis vectors, which\n", "is really not important unless you work with vector equations. The\n", "alternative is the default 'normal', where the basis vectors\n", "are normalized to unit length." ] }, { "cell_type": "code", "execution_count": 3, "id": "ab2ff2d7", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:47:59.407433Z", "iopub.status.busy": "2024-02-19T13:47:59.407241Z", "iopub.status.idle": "2024-02-19T13:47:59.714630Z", "shell.execute_reply": "2024-02-19T13:47:59.714005Z" } }, "outputs": [], "source": [ "import sympy as sp\n", "from shenfun import *\n", "config['basisvectors'] = 'covariant'\n", "t = sp.Symbol('x', real=True, positive=True)\n", "rv = (sp.sin(2*t), sp.cos(2*t), 0.5*t)\n", "C = FunctionSpace(100, 'C', domain=(0, 2*np.pi), coordinates=((t,), rv))" ] }, { "cell_type": "markdown", "id": "99693b79", "metadata": { "editable": true }, "source": [ "Then compute the arclength using [inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner), again by using a constant\n", "testfunction 1, and a constant [Array](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.Array) u=1" ] }, { "cell_type": "code", "execution_count": 4, "id": "b86162bf", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:47:59.717536Z", "iopub.status.busy": "2024-02-19T13:47:59.717319Z", "iopub.status.idle": "2024-02-19T13:47:59.768852Z", "shell.execute_reply": "2024-02-19T13:47:59.768382Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Length of spiral = 12.95311834341519\n" ] } ], "source": [ "length = inner(1, Array(C, val=1))\n", "print('Length of spiral =', length)" ] }, { "cell_type": "markdown", "id": "7c5ffdd2", "metadata": { "editable": true }, "source": [ "The arclength is found to be slightly longer than $4 \\pi$. Looking at the\n", "spiral below, the result looks reasonable." ] }, { "cell_type": "code", "execution_count": 5, "id": "0d9a6b65", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:47:59.771642Z", "iopub.status.busy": "2024-02-19T13:47:59.771416Z", "iopub.status.idle": "2024-02-19T13:48:00.230288Z", "shell.execute_reply": "2024-02-19T13:48:00.229812Z" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "\n", "import matplotlib.pyplot as plt\n", "fig = plt.figure(figsize=(4, 3))\n", "X = C.cartesian_mesh(kind='uniform')\n", "ax = fig.add_subplot(111, projection='3d')\n", "p = ax.plot(X[0], X[1], X[2], 'r')\n", "hx = ax.set_xticks(np.linspace(-1, 1, 5))\n", "hy = ax.set_yticks(np.linspace(-1, 1, 5))" ] }, { "cell_type": "markdown", "id": "5fb91f88", "metadata": { "editable": true }, "source": [ "The term $\\sqrt{\\left(\\frac{d x}{d t}\\right)^2 + \\left(\\frac{d y}{d t}\\right)^2 + \\left(\\frac{d z}{d t}\\right)^2}$\n", "is actually here a constant $\\sqrt{4.25}$, found in shenfun as" ] }, { "cell_type": "code", "execution_count": 6, "id": "b12c0f09", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.232856Z", "iopub.status.busy": "2024-02-19T13:48:00.232514Z", "iopub.status.idle": "2024-02-19T13:48:00.236257Z", "shell.execute_reply": "2024-02-19T13:48:00.235798Z" } }, "outputs": [ { "data": { "text/plain": [ "2.0615528128088303" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C.coors.sg" ] }, { "cell_type": "markdown", "id": "231ddeac", "metadata": { "editable": true }, "source": [ "We could also integrate a non-constant function over the spiral.\n", "For example, lets integrate the function $f(x, y, z)= \\sin^2 x$" ] }, { "cell_type": "markdown", "id": "d549c0c8", "metadata": { "editable": true }, "source": [ "$$\n", "\\int_C \\sin^2 x ds = \\int_{t=0}^{2\\pi} \\sin^2 (\\sin 2t) \\sqrt{\\left(\\frac{d x}{d t}\\right)^2 + \\left(\\frac{d y}{d t}\\right)^2 + \\left(\\frac{d z}{d t}\\right)^2} dt\n", "$$" ] }, { "cell_type": "code", "execution_count": 7, "id": "70f98c31", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.238860Z", "iopub.status.busy": "2024-02-19T13:48:00.238557Z", "iopub.status.idle": "2024-02-19T13:48:00.245883Z", "shell.execute_reply": "2024-02-19T13:48:00.244843Z" } }, "outputs": [ { "data": { "text/plain": [ "5.026517292599667" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "inner(1, Array(C, buffer=sp.sin(rv[0])**2))" ] }, { "cell_type": "markdown", "id": "79f71b48", "metadata": { "editable": true }, "source": [ "which can be easily verified using, e.g., Wolfram Alpha" ] }, { "cell_type": "code", "execution_count": 8, "id": "feb65bc7", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.248797Z", "iopub.status.busy": "2024-02-19T13:48:00.248568Z", "iopub.status.idle": "2024-02-19T13:48:00.253354Z", "shell.execute_reply": "2024-02-19T13:48:00.252755Z" } }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import IFrame\n", "IFrame(\"https://www.wolframalpha.com/input/?i=integrate+sin%5E2%28sin%282t%29%29+sqrt%284.25%29+from+t%3D0+to+2pi\", width=\"500px\", height=\"350px\")" ] }, { "cell_type": "markdown", "id": "f0402f37", "metadata": { "editable": true }, "source": [ "### Surface integrals\n", "\n", "Consider a 3D function $f(x,y,z) \\in \\mathbb{R}^3$ and\n", "a 2D surface (not neccessarily plane) $S(u, v)$,\n", "parametrized in two new coordinates $u$ and $v$. A position\n", "vector $\\mathbf{r}$ can be used to parametrize $S$" ] }, { "cell_type": "markdown", "id": "d2282d36", "metadata": { "editable": true }, "source": [ "$$\n", "\\mathbf{r} = x(u, v) \\,\\mathbf{i} + y(u, v) \\,\\mathbf{j} + z(u, v) \\,\\mathbf{k},\n", "$$" ] }, { "cell_type": "markdown", "id": "309a3f3f", "metadata": { "editable": true }, "source": [ "where $\\mathbf{i}, \\mathbf{j}, \\mathbf{k}$ are the Cartesian unit vectors.\n", "The two new coordinates $u$ and $v$ are functions of $x, y, z$,\n", "and they each have a one-dimensional domain" ] }, { "cell_type": "markdown", "id": "9e1f19ca", "metadata": { "editable": true }, "source": [ "$$\n", "u \\in D_u \\quad v \\in D_v.\n", "$$" ] }, { "cell_type": "markdown", "id": "73f360d4", "metadata": { "editable": true }, "source": [ "The exact size of the domain depends on the problem at hand. The computational\n", "domain of the surface $S$ is $D=D_u \\times D_v$.\n", "\n", "A surface integral of $f$ over $S$ can now be written" ] }, { "cell_type": "markdown", "id": "a9a5dbe0", "metadata": { "editable": true }, "source": [ "$$\n", "\\int_S f(x, y, z) dS = \\int_D f(x(u, v), y(u, v), z(u, v)) \\left|\\frac{\\partial \\mathbf{r}}{\\partial u} \\times \\frac{\\partial \\mathbf{r}}{\\partial v} \\right| dudv,\n", "$$" ] }, { "cell_type": "markdown", "id": "d4873da3", "metadata": { "editable": true }, "source": [ "where $dS$ is a surface area element. With shenfun such integrals\n", "are trivial, even for highly complex domains." ] }, { "cell_type": "markdown", "id": "c6775cbe", "metadata": { "editable": true }, "source": [ "## Example 1\n", "\n", "Consider first the surface integral of $f(x,y,z)=x^2$\n", "over the unit sphere. We use regular spherical coordinates," ] }, { "cell_type": "markdown", "id": "82604644", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", "0 &\\le \\theta \\le \\pi \\\\ \n", "0 &\\le \\phi \\le 2\\pi \\\\ \n", "x(\\theta, \\phi) &= \\sin \\theta \\cos \\phi \\\\ \n", "y(\\theta, \\phi) &= \\sin \\theta \\sin \\phi \\\\ \n", "z(\\theta, \\phi) &= \\cos \\theta\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "bcf630fd", "metadata": { "editable": true }, "source": [ "The straight forward implementation of a function space for\n", "the unit sphere reads" ] }, { "cell_type": "code", "execution_count": 9, "id": "5a4c60c4", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.256487Z", "iopub.status.busy": "2024-02-19T13:48:00.256287Z", "iopub.status.idle": "2024-02-19T13:48:00.611349Z", "shell.execute_reply": "2024-02-19T13:48:00.610708Z" } }, "outputs": [], "source": [ "import sympy as sp\n", "\n", "theta, phi = psi =sp.symbols('x,y', real=True, positive=True)\n", "rv = (sp.sin(theta)*sp.cos(phi), sp.sin(theta)*sp.sin(phi), sp.cos(theta))\n", "\n", "B0 = FunctionSpace(0, 'C', domain=(0, np.pi))\n", "B1 = FunctionSpace(0, 'F', dtype='d')\n", "T = TensorProductSpace(comm, (B0, B1), coordinates=(psi, rv, sp.Q.positive(sp.sin(theta))))" ] }, { "cell_type": "markdown", "id": "39dc8175", "metadata": { "editable": true }, "source": [ "where `sp.Q.positive(sp.sin(theta))` is a restriction that\n", "helps `Sympy` in computing the Jacobian required for the integral.\n", "We can now approximate the function $f$ on this surface" ] }, { "cell_type": "code", "execution_count": 10, "id": "8bd8dca7", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.614134Z", "iopub.status.busy": "2024-02-19T13:48:00.613915Z", "iopub.status.idle": "2024-02-19T13:48:00.847512Z", "shell.execute_reply": "2024-02-19T13:48:00.847006Z" } }, "outputs": [], "source": [ "f = Array(T, buffer=rv[0]**2)" ] }, { "cell_type": "markdown", "id": "c1b22719", "metadata": { "editable": true }, "source": [ "and we can integrate over $S$" ] }, { "cell_type": "code", "execution_count": 11, "id": "10a07f12", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.850388Z", "iopub.status.busy": "2024-02-19T13:48:00.850179Z", "iopub.status.idle": "2024-02-19T13:48:00.854894Z", "shell.execute_reply": "2024-02-19T13:48:00.854354Z" } }, "outputs": [], "source": [ "I = inner(1, f)" ] }, { "cell_type": "markdown", "id": "007dc278", "metadata": { "editable": true }, "source": [ "and finally compare to the exact result, which is $4 \\pi / 3$" ] }, { "cell_type": "code", "execution_count": 12, "id": "7b8f9179", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.857162Z", "iopub.status.busy": "2024-02-19T13:48:00.856972Z", "iopub.status.idle": "2024-02-19T13:48:00.860297Z", "shell.execute_reply": "2024-02-19T13:48:00.859510Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error = 7.459207029825166e-08\n" ] } ], "source": [ "print('Error =', abs(I-4*np.pi/3))" ] }, { "cell_type": "markdown", "id": "1382ae60", "metadata": { "editable": true }, "source": [ "Note that we can here achieve better accuracy by using\n", "more quadrature points. For example by refining `f`" ] }, { "cell_type": "code", "execution_count": 13, "id": "6ac0a284", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:00.862540Z", "iopub.status.busy": "2024-02-19T13:48:00.862354Z", "iopub.status.idle": "2024-02-19T13:48:01.077411Z", "shell.execute_reply": "2024-02-19T13:48:01.076913Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error = 0.0\n" ] } ], "source": [ "T = T.get_refined(2*np.array(f.global_shape))\n", "f = Array(T, buffer=rv[0]**2)\n", "print('Error =', abs(inner(1, f)-4*np.pi/3))" ] }, { "cell_type": "markdown", "id": "8b74e9dc", "metadata": { "editable": true }, "source": [ "Not bad at all:-)\n", "\n", "To go a little deeper into the integral, we can get the\n", "term $\\left|\\frac{\\partial \\mathbf{r}}{\\partial u} \\times \\frac{\\partial \\mathbf{r}}{\\partial v} \\right|$\n", "as" ] }, { "cell_type": "code", "execution_count": 14, "id": "dc0793f7", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:01.080297Z", "iopub.status.busy": "2024-02-19T13:48:01.080067Z", "iopub.status.idle": "2024-02-19T13:48:01.083469Z", "shell.execute_reply": "2024-02-19T13:48:01.082900Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sin(x)\n" ] } ], "source": [ "print(T.coors.sg)" ] }, { "cell_type": "markdown", "id": "0b5dcced", "metadata": { "editable": true }, "source": [ "Here the printed variable is `x`, but this is because `theta`\n", "is named `x` internally by `Sympy`. This is because of the definition\n", "used above: `theta, phi = sp.symbols('x,y', real=True, positive=True)`.\n", "\n", "Note that $\\mathbf{b}_u = \\frac{\\partial \\mathbf{r}}{\\partial u}$ and\n", "$\\mathbf{b}_v = \\frac{\\partial \\mathbf{r}}{\\partial v}$ are the two\n", "basis vectors used by shenfun for the surface $S$. The basis\n", "vectors are obtainable as `T.coors.b`, and can also be printed\n", "in latex using:" ] }, { "cell_type": "code", "execution_count": 15, "id": "985cee68", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:01.086216Z", "iopub.status.busy": "2024-02-19T13:48:01.086038Z", "iopub.status.idle": "2024-02-19T13:48:01.091770Z", "shell.execute_reply": "2024-02-19T13:48:01.090924Z" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\mathbf{b}_{\\theta} =\\cos{\\left(\\theta \\right)} \\cos{\\left(\\phi \\right)}\\,\\mathbf{i}+\\sin{\\left(\\phi \\right)} \\cos{\\left(\\theta \\right)}\\,\\mathbf{j}- \\sin{\\left(\\theta \\right)}\\,\\mathbf{k} \\\\ \\mathbf{b}_{\\phi} =- \\sin{\\left(\\theta \\right)} \\sin{\\left(\\phi \\right)}\\,\\mathbf{i}+\\sin{\\left(\\theta \\right)} \\cos{\\left(\\phi \\right)}\\,\\mathbf{j} \\\\ $" ], "text/plain": [ "" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import Math\n", "Math(T.coors.latex_basis_vectors(symbol_names={theta: '\\\\theta', phi: '\\\\phi'}))" ] }, { "cell_type": "markdown", "id": "29294d79", "metadata": { "editable": true }, "source": [ "where we tell latex to print `theta` as $\\theta$, and not `x`:-)\n", "\n", "From the basis vectors it should be easy to see that $\\left| \\mathbf{b}_{\\theta} \\times \\mathbf{b}_{\\phi} \\right| = \\sin \\theta$." ] }, { "cell_type": "markdown", "id": "4b3a42b8", "metadata": { "editable": true }, "source": [ "## Example 2\n", "\n", "Next, we solve [Example 5](http://www.math24.net/surface-integrals-of-first-kind.html)\n", "from the online resources at math24.net. Here" ] }, { "cell_type": "markdown", "id": "11067437", "metadata": { "editable": true }, "source": [ "$$\n", "f = \\sqrt{1+x^2+y^2}\n", "$$" ] }, { "cell_type": "markdown", "id": "e3d6ffcd", "metadata": { "editable": true }, "source": [ "and the surface is defined by" ] }, { "cell_type": "markdown", "id": "9f68a0da", "metadata": { "editable": true }, "source": [ "$$\n", "\\mathbf{r} = u \\cos v \\mathbf{i} + u \\sin v \\mathbf{j} + v \\mathbf{k}\n", "$$" ] }, { "cell_type": "markdown", "id": "1fd1537c", "metadata": { "editable": true }, "source": [ "with $0 \\le u \\le 2, 0 \\le v \\le 2\\pi$.\n", "\n", "The implementation is only a few lines, and we end by comparing\n", "to the exact solution $14 \\pi /3$" ] }, { "cell_type": "code", "execution_count": 16, "id": "40e3a695", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:01.094913Z", "iopub.status.busy": "2024-02-19T13:48:01.094689Z", "iopub.status.idle": "2024-02-19T13:48:01.442399Z", "shell.execute_reply": "2024-02-19T13:48:01.441680Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error = 1.7763568394002505e-15\n" ] } ], "source": [ "u, v = psi =sp.symbols('x,y', real=True, positive=True)\n", "rv = (u*sp.cos(v), u*sp.sin(v), v)\n", "B0 = FunctionSpace(0, 'C', domain=(0, 2))\n", "B1 = FunctionSpace(0, 'C', domain=(0, np.pi))\n", "T = TensorProductSpace(comm, (B0, B1), coordinates=(psi, rv))\n", "f = Array(T, buffer=sp.sqrt(1+rv[0]**2+rv[1]**2))\n", "print('Error =', abs(inner(1, f)-14*np.pi/3))" ] }, { "cell_type": "markdown", "id": "80153aef", "metadata": { "editable": true }, "source": [ "In this case the integral measure is" ] }, { "cell_type": "code", "execution_count": 17, "id": "dc0793f7_1", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:01.445135Z", "iopub.status.busy": "2024-02-19T13:48:01.444907Z", "iopub.status.idle": "2024-02-19T13:48:01.448313Z", "shell.execute_reply": "2024-02-19T13:48:01.447785Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sqrt(x**2 + 1)\n" ] } ], "source": [ "print(T.coors.sg)" ] }, { "cell_type": "markdown", "id": "ed8752ba", "metadata": { "editable": true }, "source": [ "## Example 3\n", "\n", "In this third example we use a surface that\n", "looks like a seashell. Again, the example is taken from\n", "[chebfun](http://www.chebfun.org/examples/approx3/SurfaceIntegral3D.html).\n", "\n", "The surface of the seashell is parametrized with position\n", "vector" ] }, { "cell_type": "markdown", "id": "9d475c23", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", "\\mathbf{r} &= \\left(\\left(\\frac{5}{4}-\\frac{5 v}{8 \\pi}\\right) \\cos 2v(1+\\cos u) + \\cos 2v \\right) \\mathbf{i} \\\\ \n", " &+\\left(\\left(\\frac{5}{4}-\\frac{5 v}{8 \\pi}\\right) \\sin 2v (1+\\cos u) + \\sin 2v \\right) \\mathbf{j},\\\\ \n", " &+\\left(\\frac{10 v}{2 \\pi} + \\left(\\frac{5}{4}-\\frac{5 v}{8 \\pi}\\right) \\sin u + 15\\right) \\mathbf{k}\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "d845d419", "metadata": { "editable": true }, "source": [ "for $0 \\le u \\le 2 \\pi, -2 \\pi \\le v \\le 2 \\pi$.\n", "\n", "The function $f$ is now defined as" ] }, { "cell_type": "markdown", "id": "2735adb0", "metadata": { "editable": true }, "source": [ "$$\n", "f(x,y,z) = x+y+z\n", "$$" ] }, { "cell_type": "markdown", "id": "ceb7c81d", "metadata": { "editable": true }, "source": [ "The implementation is" ] }, { "cell_type": "code", "execution_count": 18, "id": "89c0e257", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:01.451297Z", "iopub.status.busy": "2024-02-19T13:48:01.451070Z", "iopub.status.idle": "2024-02-19T13:48:07.239536Z", "shell.execute_reply": "2024-02-19T13:48:07.239022Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-6030.7881470861\n" ] } ], "source": [ "rv = (5*(1-v/(2*sp.pi))*sp.cos(2*v)*(1+sp.cos(u))/4 + sp.cos(2*v),\n", " 5*(1-v/(2*sp.pi))*sp.sin(2*v)*(1+sp.cos(u))/4 + sp.sin(2*v),\n", " 10*v/(2*sp.pi) + 5*(1-v/(2*sp.pi))*sp.sin(u)/4 + 15)\n", "\n", "B0 = FunctionSpace(100, 'C', domain=(0, 2*np.pi))\n", "B1 = FunctionSpace(100, 'C', domain=(-2*np.pi, 2*np.pi))\n", "T = TensorProductSpace(comm, (B0, B1), coordinates=(psi, rv, sp.Q.positive(v-2*sp.pi)))\n", "\n", "f = rv[0]+rv[1]+rv[2]\n", "fb = Array(T, buffer=f)\n", "I = inner(1, fb)\n", "print(I)" ] }, { "cell_type": "markdown", "id": "737091ab", "metadata": { "editable": true }, "source": [ "which agrees very well with chebfun's result. The basis vectors\n", "for the surface of the seashell are" ] }, { "cell_type": "code", "execution_count": 19, "id": "60cc5783", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:07.242781Z", "iopub.status.busy": "2024-02-19T13:48:07.242525Z", "iopub.status.idle": "2024-02-19T13:48:07.253833Z", "shell.execute_reply": "2024-02-19T13:48:07.253128Z" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\mathbf{b}_{u} =\\frac{5 \\left(v - 2 \\pi\\right) \\sin{\\left(u \\right)} \\cos{\\left(2 v \\right)}}{8 \\pi}\\,\\mathbf{i}+\\frac{5 \\left(v - 2 \\pi\\right) \\sin{\\left(u \\right)} \\sin{\\left(2 v \\right)}}{8 \\pi}\\,\\mathbf{j}+\\frac{5 \\left(- v + 2 \\pi\\right) \\cos{\\left(u \\right)}}{8 \\pi}\\,\\mathbf{k} \\\\ \\mathbf{b}_{v} =\\frac{\\left(10 v - 20 \\pi\\right) \\left(\\cos{\\left(u \\right)} + 1\\right) \\sin{\\left(2 v \\right)} - 5 \\left(\\cos{\\left(u \\right)} + 1\\right) \\cos{\\left(2 v \\right)} - 16 \\pi \\sin{\\left(2 v \\right)}}{8 \\pi}\\,\\mathbf{i}+\\frac{\\left(- 10 v + 20 \\pi\\right) \\left(\\cos{\\left(u \\right)} + 1\\right) \\cos{\\left(2 v \\right)} - 5 \\left(\\cos{\\left(u \\right)} + 1\\right) \\sin{\\left(2 v \\right)} + 16 \\pi \\cos{\\left(2 v \\right)}}{8 \\pi}\\,\\mathbf{j}+\\frac{5 \\cdot \\left(8 - \\sin{\\left(u \\right)}\\right)}{8 \\pi}\\,\\mathbf{k} \\\\ $" ], "text/plain": [ "" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Math(T.coors.latex_basis_vectors(symbol_names={u: 'u', v: 'v'}))" ] }, { "cell_type": "markdown", "id": "feec2b32", "metadata": { "editable": true }, "source": [ "which, if nothing else, shows the power of symbolic\n", "computing in Sympy.\n", "\n", "We can plot the\n", "seashell using either plotly or mayavi. Here we choose\n", "plotly since it integrates well with the executable\n", "jupyter book." ] }, { "cell_type": "code", "execution_count": 20, "id": "0aceb331", "metadata": { "collapsed": false, "editable": true, "execution": { "iopub.execute_input": "2024-02-19T13:48:07.256251Z", "iopub.status.busy": "2024-02-19T13:48:07.256068Z", "iopub.status.idle": "2024-02-19T13:48:07.655645Z", "shell.execute_reply": "2024-02-19T13:48:07.655015Z" } }, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "colorscale": [ [ 0.0, "rgb(0,0,131)" ], [ 0.2, "rgb(0,60,170)" ], [ 0.4, "rgb(5,255,255)" ], [ 0.6, "rgb(255,255,0)" ], [ 0.8, "rgb(250,0,0)" ], [ 1.0, "rgb(128,0,0)" ] ], "surfacecolor": [ [ 25.997522722304275, 25.977587445979395, 25.937095803351077, 25.874821915396836, 25.788971936660033, 25.67724966646497, 25.53696283619131, 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