NumGy_LV10.qmd

---
title: "Numerical Simulation Methods in Geophysics, Part 10: 2D Helmholtz equation"
subtitle: "1. MGPY+MGIN"
author: "thomas.guenther@geophysik.tu-freiberg.de"
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--- 
# Recap

* Maxwell equations in time domain
* harmonic approach (or decomposition)<br>
  $\Rightarrow$ (complex-valued) Helmholtz equation for $E$ and $B$
* solve 1D Helmholtz equation complex-values
  * compare with analytic solution

## Next lectures and exercises

* LV10: 15.01. exercise on 16.01.
* LV11: 22.01. exercise on 23.01.
* LV12: 29.01. exercise on 30.01.
* VL13: 05.02., exercise on 06.02.
* report on 2D Helmholtz equations

## Todo

* solve 2D Helmholtz equation 
  * use secondary field approach
* use wide range of frequencies
  * combine E and H to yield MT sounding curves
* excurse on 3D vectorial Maxwell solvers
* overview on equation solvers and high-performance computing
* outlook to computational fluid dynamics

## Electromagnetic fields in the Earth

Maxwell equations lead to diffusion equation

$$\pdv[2]{\vb B}{z}=\mu_0\sigma\pdv{\vb B}{t}$$

A periodic excitation ($B_0 e^{\imath\omega t}$) leads to (cf. temperature problem)

$$B=B_0 e^{-z/d}\cos(\omega t-z/d)$$

with the skin depth $d=\sqrt{2/(\mu_0 \sigma\omega)}\approx 503\sqrt{\rho/f}$

## Electromagnetic fields in the Earth

Assume E field in x direction

$$E_x=-\frac{1}{\mu_0\sigma}\pdv{B_y}{z}=\frac{B_0}{\mu_0\sigma d}e^{-z/d}\sqrt{2}\cos(\omega t-z/d+\pi/4)$$

$\Rightarrow$ phase shift of 45° ($\pi/4$) between $E_x$ und $B_y$

$$\frac{|E_x|}{|B_y|}=\frac{\sqrt{2}}{\mu_0\sigma d} \Rightarrow \rho=\frac{\mu_0}{\omega}\left|\frac{E_x}{B_y}\right|^2$$


## The magnetotelluric (MT) method
::::{.columns}
:::{.column width="40%"}
![](pics/MT_Prinzip.png){fig-align="center"}
:::
:::{.column width="60%"}
* inductive electromagnetics with $f$=0.001-100\,Hz ($T$=0.01-1000s)
* source in ionosphere (natural source) or on ground (controlled source)
* measure magnetic and electric fields
* analyse (complex) ratio in frequency domain
* depth sounding ($\rho_a$ & $\phi$) with $T$
* MT course of Anna Marti (U Barcelona)
:::
::::

## Magnetotelluric depth sounding

![](pics/mt1d.png){fig-align="center"}

## 2D/3D Magnetotellurics

![Imaging of a subduction zone (Worszewski et al., 2011)](pics/mt2d.png){fig-align="center"}

## Helmholtz equations

$$ \curl \mu^{-1} \curl \vb E + \imath\omega\sigma\vb E - \omega^2\epsilon \vb E = - \imath\omega \vb j_s$$ 

$$ \curl \sigma^{-1} \curl \vb H + \imath\omega\mu\vb H - \omega^2\epsilon\mu/\sigma \vb H=\curl \sigma^{-1}\vb j_s$$ 

PDEs identical $\vb E$ and $\vb H$ through exchanging $\mu$ and $\sigma$

component perpendicular to modelling frame (E/H polarization)

$$ \curl a \curl = -\div a \grad $$


## Finite element discretization

* weak formulation (for E)
$$ \int_\Omega \mu^{-1}\grad v_i \cdot \grad v_j \dd\Omega + \imath\omega \int_\Omega \sigma v_i v_j \dd\Omega = \int_\Omega v_i f \dd\Omega $$
* stiffness = second derivative $\div\vb v_i$, expressed by 2 gradients
$$ \vb A_{i,j} = \int_\Omega \mu^{-1} \grad v_i \cdot \grad v_j \dd\Omega$$

## Finite element discretization

* weak formulation (for E)
$$ \int_\Omega \mu^{-1}\grad v_i \cdot \grad v_j \dd\Omega + \imath\omega \int_\Omega \sigma v_i v_j \dd\Omega = \int_\Omega v_i f \dd\Omega $$
* mass matrix resembles functions $\vb v_i$
$$ \vb M_{i,j} = \int_\Omega \sigma v_i \cdot v_j \dd\Omega $$

## Complex or real-valued?

Either discretize the complex system 

$$(\vb A+\imath\omega\vb M) (\vb u_r + \imath \vb u_i) = \vb b_r + \imath \vb b_i$$

by complex shape functions OR transfer into real

$$\vb A\vb u_r + \imath\vb A \vb u_i + \imath\omega\vb M \vb u_i - \omega\vb M \vb u_i = \vb b_r + \imath \vb b_i$$

$$\mqty(\vb A & -\omega \vb M\\ \omega \vb M & \vb A) \mqty(\vb u_r\\ \vb u_i) = \mqty(\vb b_r\\ \vb b_i) $$


## Secondary field approach

Consider the field to consist of a primary (background) and an secondary (anomalous) field $F=F_0+F_a$  (or $F_p+F_s$)

solution for $F_0$ known, e.g. analytically or 1D (semi-analytically)

$\Rightarrow$ form equations for $F_a$, because

* $F_a$ is weaker or smoother (e.g. $F_0\propto 1/r^n$ at sources)
* boundary conditions easier to set (e.g. homogeneous Dirichlet)

## Example DC resistivity

$$-\div(\sigma\grad u) = \div j_s = I\delta(\vb r-\vb r_s)$$

Problem: point source leads to infinite potential 

$$u(\vb r)=\frac{I}{2\pi\sigma}\frac{1}{|\vb r - \vb r_s|}$$

Approach: $u=u_p(\sigma_0) + u_s(\sigma-\sigma_0)$:

$$-\div(\sigma\grad u_s) = \div((\sigma-\sigma_0)\grad u_p)$$

## Example DC resistivity

$$-\div(\sigma\grad u_s) = \div((\sigma-\sigma_0)\grad u_p)$$

discrete form using unit conductivity $\sigma_1=1$S/m

$$\vb A^\sigma \vb u_s = \vb A^{\delta\sigma}\vb u_p = \vb A^\sigma \vb u_p - \sigma_0 \vb A^1 \vb u_p$$


## Secondary field Helmholtz equation

The equation $-\grad^2 F - k^2 F = 0$ is solved by the primary field for $k_0$:

$-\grad^2 F_0 - k_0^2 F_0=0$ and the total field for $k_0+\delta k$:

$$ -\grad^2 (F_0+F_a) -(k_0^2+\delta k^2) (F_0+F_a) = 0 $$

$$ -\grad^2 F_a - k^2 F_a = \delta k^2 F_0 $$

::: {.callout-note}
Same operator, source terms at anomalies, weighted by the primary field.
:::

## Secondary field for EM

Maxwells equations $k^2=-\imath\omega\mu\sigma$

$$-\grad^2 \vb E_0 +\imath\omega\mu\sigma \vb E_0=0$$

leads to

$$ -\grad^2 \vb E_a + \imath\omega\mu\sigma \vb E_a = -\imath\omega\mu\delta\sigma \vb E_0 $$

::: {.callout-note}
Source terms only arise at anomalous conductivities and increase with primary field
:::

## Secondary field for EM

$$ -\grad^2 \vb E_a + \imath\omega\mu\sigma \vb E_a = -\imath\omega\mu\delta\sigma \vb E_0 $$

leads to the discretized form ($\vb A$-stiffness, $\vb M$-mass)

$$ \vb A \vb E_a + \imath\omega\vb M_\sigma E_a = -\imath\omega\vb M_{\delta\sigma} \vb E_0 $$

```{python}
#| echo: true
#| eval: false
A = stiffnessMatrix1DFE(x=z)
M = massMatrix1DFE(x=z, a=w*mu*sigma)
dM = massMatrix1DFE(x=z, a=w*mu*(sigma-sigma0))
u = uAna + solve(A+M*w*1j, dM@uAna * w*1j)
```

# 2D problems
Make use of pyGIMLi

See documentation on [pyGIMLi.org](https://pygimli.org)

## The meshtools module
```{python}
#| echo: true
#| eval: true
import pygimli as pg
import pygimli.meshtools as mt
world = mt.createWorld(start=[-10000, -10000], end=[10000, 0])
pg.show(world)
```

## The meshtools module
```{python}
#| echo: true
#| eval: true
mesh = mt.createMesh(world, quality=34, area=1e5)
pg.show(mesh)
```

## Creating a 2D geometry
```{python}
#| echo: true
#| eval: true
anomaly = mt.createRectangle(start=[0, -8000], end=[1000, -1000], marker=2)
pg.show(world+anomaly)
```

## Creating a 2D mesh
```{python}
#| echo: true
#| eval: true
mesh = mt.createMesh(world+anomaly, quality=34, smooth=True, area=1e5)
pg.show(mesh, markers=True, showMesh=True);
```


## Creating a 2D conductivity model
```{python}
#| echo: true
#| eval: true
sigma0 = 1 / 100  # 100 Ohmm
sigma = mesh.populate("sigma", {1: sigma0, 2: sigma0*10})
pg.show(mesh, "sigma", showMesh=True);
```

## The solver module
```{python}
#| echo: false
#| eval: true
import matplotlib.pyplot as plt
import numpy as np
```

```{python}
#| echo: true
#| eval: true
import pygimli.solver as ps
mesh["my"] = 4 * np.pi * 1e-7
A = ps.createStiffnessMatrix(mesh, a=1/mesh["my"])
M = ps.createMassMatrix(mesh, mesh["sigma"])
fig, ax = plt.subplots(ncols=2)
ax[0].spy(pg.utils.toCSR(A), markersize=1)
ax[1].spy(pg.utils.toCSR(M).todense(), markersize=1)
```

## The complex problem matrix

$$\vb B = \mqty(\vb A & -\omega \vb M\\ \omega \vb M & \vb A)$$

```{python}
#| echo: true
#| eval: true
#| output-location: column
w = 0.1
nd = mesh.nodeCount()
B = pg.BlockMatrix()
B.Aid = B.addMatrix(A)
B.Mid = B.addMatrix(M)
B.addMatrixEntry(B.Aid, 0, 0)
B.addMatrixEntry(B.Aid, nd, nd)
B.addMatrixEntry(B.Mid, 0, nd, scale=-w)
B.addMatrixEntry(B.Mid, nd, 0, scale=w)
pg.show(B)
```

## Sparse matrices

Up to now: regular (dense) array: save every element including 0

Save only non-zero components (e.g. using `scipy.sparse`)

* COO - coordinate format
* CSC/CRS - compressed sparse column/row
* BSR - block sparse row format, ...

## Solve systems of equations 

* Gauss elimination (expensive and dense)
* Cholesky (or ILU) decomposition
* Iterative solvers (conjugate gradients)
* incomplete factorizations or factorization of submatrices

## Reordering

![original (left) & reordered (right) matrix (Rücker et al. 2006)](pics/reordering.png)