move geometry to geometry2d and add new geometry module (3d)
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<title>pylib.numerical.ode module — pylib 2019.12.21 documentation</title>
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<div class="section" id="module-pylib.numerical.ode">
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<span id="pylib-numerical-ode-module"></span><h1>pylib.numerical.ode module<a class="headerlink" href="#module-pylib.numerical.ode" title="Permalink to this headline">¶</a></h1>
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<p>Numerical solver of ordinary differential equations.</p>
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<p>Solves the initial value problem for systems of first order
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ordinary differential equations.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Date</dt>
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<dd class="field-odd"><p>2015-09-21</p>
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</dd>
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</dl>
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<span class="target" id="module-ode"></span><dl class="function">
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<dt id="pylib.numerical.ode.e1">
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<code class="sig-name descname">e1</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e1" title="Permalink to this definition">¶</a></dt>
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<dd><p>Explicit first-order method /
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(standard, or forward) Euler method /
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Runge-Kutta 1st order method.</p>
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<p>de:
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Euler’sche Polygonzugverfahren / explizite Euler-Verfahren /
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Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
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<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
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<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
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<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
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…)</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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</dl>
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<p>Approximate the solution of the initial value problem</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x} &= f(t,x) \\
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x(t_0) &= x_0\end{split}\]</div>
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<p>Choose a value h for the size of every step and set</p>
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<div class="math notranslate nohighlight">
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\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]</div>
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<p>The derivative of the solution is approximated as the forward
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difference equation</p>
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<div class="math notranslate nohighlight">
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\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]</div>
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<p>Therefore one step <span class="math notranslate nohighlight">\(h\)</span> of the Euler method from
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<span class="math notranslate nohighlight">\(t_i\)</span> to <span class="math notranslate nohighlight">\(t_{i+1}\)</span> is</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
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x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]</div>
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<p>Example 1:</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\
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\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]</div>
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<p>with</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
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x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
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<p>becomes</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x}_1 &= x_2 \\
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\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]</div>
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<p>or</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x} &= f(t,x) \\
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\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
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\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1)
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\end{bmatrix} \\
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&=
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\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
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\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
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\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
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<p>Example 2:</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\
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\ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]</div>
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<p>with</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
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x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
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<p>becomes</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x}_1 &= x_2 \\
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\dot{x}_2 &=
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m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]</div>
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<p>or</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x} &= f(t,x) \\
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\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
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\begin{bmatrix}
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x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1)
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\end{bmatrix} \\
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&=
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\begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} +
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\begin{bmatrix}
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0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2)
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\end{bmatrix}
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\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
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<p>The Euler method is a first-order method, which means that the
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local error (error per step) is proportional to the square of
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the step size, and the global error (error at a given time) is
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proportional to the step size.</p>
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</dd></dl>
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<dl class="function">
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<dt id="pylib.numerical.ode.e2">
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<code class="sig-name descname">e2</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e2" title="Permalink to this definition">¶</a></dt>
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<dd><p>Explicit second-order method / Runge-Kutta 2nd order method.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
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<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
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<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
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<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
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…)</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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</dl>
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</dd></dl>
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<dl class="function">
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<dt id="pylib.numerical.ode.e4">
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<code class="sig-name descname">e4</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e4"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e4" title="Permalink to this definition">¶</a></dt>
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<dd><p>Explicit fourth-order method / Runge-Kutta 4th order method.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
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<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
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<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
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<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
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…)</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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</dl>
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</dd></dl>
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<dl class="function">
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<dt id="pylib.numerical.ode.fpi">
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<code class="sig-name descname">fpi</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">xi</em>, <em class="sig-param">ti</em>, <em class="sig-param">ti1</em>, <em class="sig-param">*p</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#fpi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.fpi" title="Permalink to this definition">¶</a></dt>
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<dd><p>Fixed-point iteration.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em>) – the function to iterate <span class="math notranslate nohighlight">\(f = \dot{x}(x,t)\)</span></p></li>
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<li><p><strong>xi</strong> (<em>list</em>) – initial condition <span class="math notranslate nohighlight">\(x_i\)</span></p></li>
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<li><p><strong>ti</strong> (<em>float</em>) – time <span class="math notranslate nohighlight">\(t_i\)</span></p></li>
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<li><p><strong>ti1</strong> (<em>float</em>) – time <span class="math notranslate nohighlight">\(t_{i+1}\)</span></p></li>
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<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
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…)</p></li>
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<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
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<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum <span class="math notranslate nohighlight">\(\varepsilon\)</span>
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(default = 1e-9)</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p><span class="math notranslate nohighlight">\(x_{i}\)</span></p>
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</dd>
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</dl>
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<div class="math notranslate nohighlight">
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\[x_{i,j=0} = x_{i}\]</div>
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<div class="math notranslate nohighlight">
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\[x_{i,j+1} = x_i + \dot{x}(x_{i,j}, t_{i+1})\cdot(t_{i+1}-t_i)\]</div>
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<div class="math notranslate nohighlight">
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\[\text{residuum} = \frac{\lVert x_{i,j+1}-x_{i,j}\rVert}
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{\lVert x_{i,j+1} \rVert} < \varepsilon\]</div>
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<div class="math notranslate nohighlight">
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\[x_{i} = x_{i,j=\text{end}}\]</div>
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</dd></dl>
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<dl class="function">
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<dt id="pylib.numerical.ode.i1">
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<code class="sig-name descname">i1</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#i1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.i1" title="Permalink to this definition">¶</a></dt>
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<dd><p>Implicite first-order method / backward Euler method.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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||||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
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…)</p></li>
|
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<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
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<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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</dl>
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<p>The backward Euler method has order one and is A-stable.</p>
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</dd></dl>
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<dl class="function">
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<dt id="pylib.numerical.ode.newmark_newtonraphson">
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<code class="sig-name descname">newmark_newtonraphson</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">xp0</em>, <em class="sig-param">xpp0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">gamma=0.5</em>, <em class="sig-param">beta=0.25</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#newmark_newtonraphson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.newmark_newtonraphson" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Newmark method.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||||
…)</p></li>
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||||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
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<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
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||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
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||||
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<dl class="function">
|
||||
<dt id="pylib.numerical.ode.newmark_newtonraphson_rdk">
|
||||
<code class="sig-name descname">newmark_newtonraphson_rdk</code><span class="sig-paren">(</span><em class="sig-param">fnm</em>, <em class="sig-param">x0</em>, <em class="sig-param">xp0</em>, <em class="sig-param">xpp0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">gamma=0.5</em>, <em class="sig-param">beta=0.25</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#newmark_newtonraphson_rdk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.newmark_newtonraphson_rdk" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Newmark method.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||||
…)</p></li>
|
||||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
|
||||
<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
</div>
|
||||
|
||||
|
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<h3>Navigation</h3>
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<li>Previous: <a href="pylib.numerical.integration.html" title="previous chapter">pylib.numerical.integration module</a></li>
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<li>Next: <a href="pylib.numerical.ode_model.html" title="next chapter">pylib.numerical.ode_model module</a></li>
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