NumGy_LV09.qmd

---
title: "Numerical Simulation Methods in Geophysics, Part 9: 2D Helmholtz equation"
subtitle: "1. MGPY+MGIN"
author: "thomas.guenther@geophysik.tu-freiberg.de"
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--- 
# Recap

Done:

* finite difference method for Poisson (1D) and heat transfer
* wave equation modelling in 1D
* finite element method for Poisson equation
* weak form decreases order of derivatives
* Maxwell equations lead to Helmholtz equation 

## The next lectures and exercises

* today (LV09): 08.01.25, no exercise on 09.01. (dies academicus)
* LV10: 15.01.25, exercise on 16.01.
* LV11: 22.01.25, exercise on 23.01.
* LV12: 29.01.25, exercise on 30.01.
* last VL13: 05.02.25, exercise on 06.02.
* report on 2D Helmholtz equations

<!-- ## Recap  -->

<!-- Todo:

* solving Helmholtz equation for 1D and 2D
* complex-valued problem
* secondary field approach -->

## Helmholtz equation for E
Maxwell equation in frequency domain (see also [Theory EM](https://ruboerner.github.io/ThEM/mtfields.html))
$$ \curl \vb H = \imath\omega\epsilon\vb E + \vb j \qquad (\vb j = \sigma\vb E + \vb j_s) $$
$$ \curl\vb E = -\imath\omega\mu\vb H \Rightarrow \vb H = \imath \frac{1}{\mu\omega} \curl\vb E $$ 

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

$$ \curl\mu^{-1}\curl\vb E + (\imath\sigma\omega-\omega^2\epsilon)\vb E =-\imath\omega\vb j_s $$
Quasistatic approximation ($\omega\epsilon\ll\sigma$): neglect $\epsilon$ term

## Helmholtz equation for H
Maxwell equation in frequency domain (see also [Theory EM](https://ruboerner.github.io/ThEM/mtfields.html))
$$ \curl\vb E = -\imath\omega\mu\vb H $$ 

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

$$ \curl\sigma^{-1}\curl\vb H + \imath\mu\omega\vb H = \curl\sigma^{-1}\vb j_s $$
Quasistatic approximation ($\omega\epsilon\ll\sigma$)
neglect $\epsilon$ term

## 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 $\vb v_i$
$$ \vb M_{i,j} = \int_\Omega \sigma v_i \cdot v_j \dd\Omega $$

## Next steps

* solve 1D Helmholtz equation complex-values
  * compare with analytic solution
* 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

## 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

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

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


## 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)

## 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)
```