ADD first chapter qmd

01_introduction.qmd

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+---
+title: "Introduction"
+---
+
+## Differential operators
+
+* single derivative in space $\pdv{x}$ or time $\pdv t$
+
+* gradient $\grad=(\pdv x, \pdv y, \pdv z)^T$
+
+* divergence $\div \vb F = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z}$
+
+::::: {.fragment}
+:::: {.columns}
+::: {.column width="70%"}
+Gauss': *what's in (volume) comes out (surface)*
+$$\int_V \div\vb F\ dV = \iint_S \vb F \vdot \vb n\ dS$$
+:::
+::: {.column width="25%"}
+![Gauss's theorem in EM](pics/Divergence_theorem_in_EM.svg)
+:::
+::::
+:::::
+
+## Curl (rotation)
+
+* curl $\curl \vb F = (\pdv{F_z}{y}-\pdv{F_y}{z}, \pdv{F_x}{z}-\pdv{F_z}{x}, \pdv{F_y}{x}-\pdv{F_x}{y})^T$
+
+::::: {.fragment}
+:::: {.columns}
+::: {.column width="70%"}
+Stoke: *what goes around comes around*
+$$\int_S \curl \vb F \vdot \vb dS = \iint_S \vb F \vdot \vb dl$$
+:::
+::: {.column width="25%"}
+![Stokes' theorem in EM](pics/Curl_theorem_in_EM.svg)
+:::
+::::
+:::::
+
+:::{.fragment}
+* curls have no divergence: $\div (\curl \vb F)=0$
+:::
+
+:::{.fragment}
+* potential fields have no curl $\curl (\grad u)=0$ 
+:::
+
+## Numerical simulation
+
+Mostly: solution of partial differential equations (PDEs) for either scalar (potentials) or vectorial (fields) quantities
+
+PDE Types ($u$-function, $f$-source, $a$/$c$-parameter):
+
+* elliptic PDE: $-\nabla^2 u=f$ (Poisson) or $\nabla^2 u + k^2 u = f$ (Helmholtz)
+* parabolic PDE $-\nabla^2 u + a \frac{\partial u}{\partial t}=f$
+* hyperbolic  $-\nabla^2 u + c^2 \frac{\partial^2 u}{\partial t^2}=f$ (plus diffusive term)
+$$\frac{\partial^2\ u}{{\partial x}^2} - c^2\frac{\partial^2 u}{\partial t^2} = 0$$
+* coupled $\nabla\cdot u=f$ & $u = K \nabla p=0$ (Darcy flow)
+* nonlinear $(\nabla u)^2=s^2$ (Eikonal equation)
+
+## Poisson equation
+potential field $u$ generates field $\vec{F}=-\nabla u$
+
+causes some flow $\vec{j}=a \vec{F}$ 
+
+$a$ is some sort of conductivity (electric, hydraulic, thermal)
+
+continuity of flow: divergence of total current $\vb j + \vb j_s$ is zero
+
+$$ \div (a \nabla u) = - \div \vb j_s $$
+
+## Darcy's law
+
+:::: {.columns}
+::: {.column width="50%"}
+
+volumetric flow rate $Q$ caused by gradient of pressure $p$
+
+$$ Q = \frac{k A}{\mu L} \Delta p $$
+
+$$\vb q = -\frac{k}{\mu} \nabla p $$
+
+$$\div\vb q = -\div (k/\mu \grad p) = 0$$
+
+:::
+::: {.column width="45%"}
+![Darcy's law](pics/darcy_law.svg)
+:::
+::::
+
+## The heat equation in 1D
+
+sought: Temperature $T$ as a function of space and time
+
+heat flux density $\vb q = \lambda\grad T$
+
+$q$ in W/m², $\lambda$ - heat conductivity/diffusivity in W/(m.K)
+
+Fourier's law: $\pdv{T}{t} - a \nabla^2 T = s$  ($s$ - heat source)
+
+temperature conduction $a=\frac{\lambda}{\rho c}$ ($\rho$ - density, $c$ - heat capacity)
+
+## Stokes equation
+
+$$ \mu \nabla^2 \vb v - \grad p + f = 0 $$
+
+$$\div\vb v = 0 $$
+
+## Navier-Stokes equations
+(incompressible, uniform viscosity)
+
+$$ \pdv{\vb u}{t} +(\vb u \vdot \grad) \vb u = \nu \nabla^2 \vb u - 1/\rho \grad p + f $$
+
+## Maxwell's equations
+
+* Faraday's law: currents & varying electric fields $\Rightarrow$ magnetic field 
+$$ \curl \vb H = \pdv{\vb D}{t} + \vb j $$ 
+* Ampere's law: time-varying magnetic fields induce electric field
+$$ \curl\vb E = -\pdv{\vb B}{t}$$ 
+* $\div\vb D = \varrho$ (charge $\Rightarrow$), $\div\vb B = 0$ (no magnetic charge)
+* material laws $\vb D = \epsilon \vb E$ and $\vec B = \mu \vb H$
+
+## Helmholtz equations
+
+$$ \nabla^2 u + k^2 u = f $$
+
+results from wavenumber decomposition of diffusion or wave equations
+
+approach: $\vb F = \vb{F_0}e^{\imath\omega t} \quad\Rightarrow\quad \pdv{\vb F}{t}=\imath\omega\vb F \quad\Rightarrow\quad \pdv[2]{\vb F}{t}=-\omega^2\vb F$
+
+$$ \nabla^2 \vb F - a \nabla_t \vb F - c^2 \nabla^2_t \vb F = 0 $$
+
+$$ \Rightarrow \nabla^2\vb F - a\imath\omega\vb F + c^2 \omega^2\vb F = 0 $$
+
+## The eikonal equation
+
+Describes first-arrival times $t$ as a function of velocity ($v$) or slowness ($s$)
+
+$$ |\grad t| = s = 1/v $$
+
+![](pics/tt4.png){.fragment width="458" height="420"}

02_finite-differences.qmd

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+---
+title: "Finite Differences"
+jupyter: python3
+---
+
+
+
+
+
+## Helmholtz equations
+
+e.g. from Fourier assumption $u=u_0 e^{\imath\omega t}$
+
+$$ \div (a\grad u) + k^2 u = f $$
+
+* Poisson operator assembled in stiffness matrix $\vb A$
+* additional terms with $u_i$ $\Rightarrow$ mass matrix $\vb M$
+
+$$ \Rightarrow \vb A + \vb M = \vb b $$
+
+## Hyberbolic equations 
+
+Acoustic wave equation in 1D
+$$ \pdv[2]{u}{t} - c^2\pdv[2]{u}{x} = 0 $$ 
+
+$u$..pressure/velocity/displacement, $c$..velocity
+
+Damped (mixed parabolic-hyperbolic) wave equation
+$$ \pdv[2]{u}{t} - a\pdv{u}{t} - c^2\pdv[2]{u}{x} = 0 $$ 
+
+
+## Discretization
+
+:::: {.columns}
+::: {.column}
+$$ \pdv[2]{u}{t}^{n} \approx \frac{u^{n+1}-u^n}{\Delta t} - \frac{u^{n}-u^{n-1}}{\Delta t}
+$$
+
+$$ = \frac{u^{n+1}+u^{n-1}-2 u^n}{\Delta t^2} = c^2\pdv[2]{u}{x}^n $$
+
+
+$$ u^{n+1} = c^2 \Delta t^2 \pdv[2]{u}{x}^{n} + 2 u^n - u^{n-1} $$
+
+<!-- \frac{1}{\Delta t}  -->
+
+$$ \vb M \vb u^{n+1} = (\vb A + 2\vb M) \vb u^n - \vb M \vb u^{n-1} $$ 
+:::
+::: {.column}
+![Second derivative](pics/dtsecond.svg)
+:::
+::::
+
+## Example: velocity distribution
+
+```{python}
+#| output-location: column-fragment
+#| eval: true
+#| echo: true
+import numpy as np
+import matplotlib.pyplot as plt
+x=np.arange(0, 600.01, 0.5)
+c = 1.0*np.ones_like(x) # velocity in m/s
+c[100:300] = 1 + np.arange(0,0.5,0.0025) 
+c[900:1100] = 0.5  # low velocity zone
+
+# Plot velocity distribution.
+plt.plot(x,c,'k')
+plt.xlabel('x [m]')
+plt.ylabel('c [m/s]')
+plt.grid()
+```
+
+
+## Initial displacement
+
+Derivative of Gaussian (Ricker wavelet)
+```{python}
+#| output-location: column-fragment
+#| eval: true
+#| echo: true
+l=5.0
+
+# Initial displacement field [m].
+u=(x-300.0)*np.exp(-(x-300.0)**2/l**2) 
+# Plot initial displacement field.
+plt.plot(x,u,'k')
+plt.xlabel('x [m]')
+plt.ylabel('u [m]')
+plt.title('initial displacement field')
+plt.show()
+```
+
+## Time propagation
+
+```{python}
+#| output-location: column-fragment
+#| eval: true
+#| echo: true
+u_last=u
+dt = 0.5
+ddu = np.zeros_like(u)
+dx = np.diff(x)
+for i in range(100):
+    dudx = np.diff(u)/dx
+    ddu[1:-1] = np.diff(dudx)/dx[:-1]
+    u_next = 2*u-u_last+ddu*c**2 * dt**2 
+    u_last = u
+    u = u_next
+  
+plt.plot(x,u,'k')
+```
+
+<video autoplay="true" loop="true">
+<source src="notebooks/wave.mp4"> 
+</video>
+<!-- <video data-autoplay src="notebooks/wave.mp4"></video> -->
+<!-- {{< video notebooks/wave.mp4 video="true">}} -->
+

03_finite-elements.qmd

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+---
+title: "Finite Elements"
+jupyter: python3
+---
+
+## History and background
+
+* [1943] Courant: Variational Method
+* [1956] Turner, Clough, Martin, Topp: Stiffness
+* [1960] Clough: Finite Elements for static elasticity
+* [1970-80] extension to structural, thermic and fluid dynamics
+* [1990] computational improvements 
+* now main method for almost all PDE types
+
+Geophysics: Poisson equation in 1970s, revival in 1990s and predominant in 2000s up to now
+
+## Variational formulation of Poisson equation
+
+$$ -\div a \grad u = f $$
+
+::: {.fragment}
+Multiplication with test function $w$ and integration $\Rightarrow$ weak form
+
+$$ -\int_\Omega w \div a \grad u \dd\Omega = \int_\Omega w f \dd\Omega $$
+:::
+
+::: {.fragment}
+$$ \div(b\vb c) = b\div \vb c + \grad b \cdot \vb c $$
+:::
+
+::: {.fragment}
+$$ \int_\Omega \grad w a \grad u \dd\Omega - \int_\Omega \div(w a \grad u) \dd\Omega = \int_\Omega w f \dd\Omega $$
+:::
+
+## Variational formulation of Poisson equation
+
+$$ \int_\Omega \grad w a \grad u \dd\Omega - \int_\Omega \div(w a \grad u) \dd\Omega = \int_\Omega w f \dd\Omega $$
+
+::: {.fragment}
+use Gauss' law $\int_\Omega \div \vb A = \int_\Gamma \vb A \cdot \vb n$
+
+$$ \int_\Omega a \grad w \grad u \dd\Omega - \int_\Gamma a w \grad u \dd\Gamma = \int_\Omega fw \dd\Omega $$
+:::
+
+::: {.fragment}
+Let $u$ be constructed by shape functions $v$: $u=\sum_i u_i v_i$
+
+$$ \int_\Omega a \grad w \grad v_i \dd\Omega - \int_\Gamma a w \grad v_i \dd\Gamma = \int_\Omega fw \dd\Omega $$
+:::
+
+## Galerkins method
+
+$$ \int_\Omega a \grad w \grad v_i \dd\Omega - \int_\Gamma a w \grad v_i \dd\Gamma = \int_\Omega fw \dd\Omega $$
+
+::: {.fragment}
+Test functions equal shape (trial) functions $w_i=v_i$
+
+$$ \int_\Omega a \grad v_i \grad v_j \dd\Omega - \int_\Gamma a w \grad v_j \dd\Gamma = \int_\Omega f v_j \dd\Omega $$
+:::
+
+::: {.fragment}
+* choose $v_i$ so that $\grad v_i$ is simple and $\grad v_i \grad v_j$ mostly 0
+* divide subsurface in sub-volumes $\Omega_i$ with constant $a_i$ and $\grad v_j$
+:::
+
+## Hat functions 
+
+![](pics/shapefunctions1d.svg){width="100%"}
+
+## Gradients for hat functions
+Every element is surrounded by two nodes "carrying" a hat. 
+
+The gradients are piece-wise constant $\pm 1/\Delta x_i$
+
+$$ \Rightarrow \int_\Omega a \grad v_i \grad v_{i+1} \dd\Omega = -\frac{a_i}{\Delta x_i^2} \cdot \Delta x_i = -\frac{a_i}{\Delta x_i}$$ 
+
+$$ -\int_\Omega a \grad v_i \grad v_i \dd\Omega = \frac{a_{i-1}}{\Delta x_{i-1}^2}\Delta x_{i-1} + \frac{a_i}{\Delta x_i^2}\Delta x_i = \frac{a_{i-1}}{\Delta x_{i-1}} + \frac{a_i}{\Delta x_i} $$
+
+$$ -\int_\Omega a \grad v_i \grad v_{i+2} \dd\Omega = 0 $$
+
+# System (stiffness) matrix
+
+Matrix integrating gradient of base functions for neighbors
+$A$ 
+$$ \vb A_{i, i+1} = -\frac{a_i}{\Delta x_i^2} \cdot \Delta x_i $$
+
+$$ A_{i,i} = \int_\Omega a \grad v_i \grad v_i \dd\Omega = -A_{i,i+1} + A_{i+1,i}  $$
+
+$\Rightarrow$ matrix-vector equation $\vb A \vb u = \vb b$
+$\Rightarrow$
+
+## Boundary conditions
+
+second term
+$$ - \int_\Gamma a v_i \grad v_j \dd\Gamma $$
+
+
+
+
+## Right-hand side vector
+
+The right-hand-side vector $b=\int v_i f \dd\Omega$ also scales with $\Delta x$
+
+e.g. $f=\div j_s$ $\Rightarrow$ $b=\int v_i \div j_s \dd\Omega = \int_\Gamma $
+
+system identical to FD for $\Delta x=$const
+
+:::{.callout-tip icon="false" title="Difference of FE to FD"} 
+Any source function $f(x)$ can be integrated on the whole space!
+:::
+
+## Solution
+
+$\vb v$ holds the coefficient $u_i$ creating $u=\sum u_i v_i(x)$
+
+:::{.callout-tip icon="false" title="Difference of FE to FD"} 
+$u$ is described on the whole space and approximates the solution, not the PDE!
+:::
+
+Hat functions: $u_i$ potentials on nodes, $u$ piece-wise linear
+
+:::{.callout-tip icon="false" title="Generality of FE"} 
+Arbitrary base functions $v_i$ can be used to describe $u$
+:::