1  Introduction

1.1 Differential operators

The physical fields are governed by partial differential equations with differential operators in space and time There are single derivative in space \(\pdv{x}\) or time \(\pdv{t}\). All derivatives into all spatial dimensions are subsumed in the nabla operator \(\grad=(\pdv x, \pdv y, \pdv z)^T\). The (negative) gradient of a potential is the corresponding vector field with directions perpendicular to the isosurfaces into the direction of decreasing potential.

The divergence applies the nabla operator to a vectorial field by the dot product: \[\div \vb F = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z}\] The divergence of a field describes the source strength using Gauss’ law.

Gauss’ law:

what’s in (volume) comes out (surface) \[\int_V \div\vb F\ dV = \iint_S \vb F \vdot \vb n\ dS\]

Gauss’s theorem in EM

The curl (rotation) operator occurs particularly in electromagnetic problems, where magnetic fields are caused by (conduction or displacement) currents and electric fields infer changes in the magnetic field. It is described by the cross-product between nabla operator and the field \[\curl \vb F = (\pdv{F_z}{y}-\pdv{F_y}{z}, \pdv{F_x}{z}-\pdv{F_z}{x}, \pdv{F_y}{x}-\pdv{F_x}{y})^T\]

Stokes law:

what goes around comes around \[\int_S \curl \vb F \vdot \vb dS = \iint_S \vb F \vdot \vb dl\]

Stokes’ theorem in EM

There are two laws concerning curl fields:

  1. curls have no divergence: \[\div (\curl \vb F)=0 \tag{1.1}\]
  2. potential fields have no curl \[\curl (\grad u)=0 \tag{1.2}\]

1.2 Partial differential equations (PDEs)

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Mostly: solution of partial differential equations (PDEs) for either scalar (potentials) or vectorial (fields) quantities.

1.2.1 PDE Types

(\(u\)-function, \(f\)-source, \(a\)/\(c\)-parameter):

  • elliptic (potential) PDE of Poisson type \[-\nabla^2 u=f\] or Helmholtz type \[\nabla^2 u + k^2 u = f\]
  • parabolic (diffusion) PDE \[-\nabla^2 u + a \pdv{u}{t} \frac{\partial u}{\partial t}=f\]
  • hyperbolic (wave) PDE \[-\nabla^2 u + c^2 \frac{\partial^2 u}{\partial t^2}=f\]
  • coupled \(\nabla\cdot u=f\) & \(u = K \nabla p=0\) (Darcy flow)
  • nonlinear \((\nabla u)^2=s^2\) (Eikonal equation)

1.2.2 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

\[ \div (a \nabla u) = - \div \vb j_s \]

1.2.3 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)

1.3 Flow and transport

1.3.1 Darcy’s law

volumetric flow rate \(Q\) caused by gradient of pressure \(p\)

\[ Q = \frac{k A}{\mu L} \Delta p \]

\[\vb q = -\frac{k}{\mu} \nabla p \]

\[\div\vb q = -\div (k/\mu \grad p) = 0\]

Darcy’s law

1.3.2 Stokes equation

\[ \mu \nabla^2 \vb v - \grad p + f = 0 \]

\[\div\vb v = 0 \]

1.4 Electromagnetics

1.4.1 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 \(\vec B = \mu \vb H\)

1.5 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 \(\vb M\)

\[ \Rightarrow \vb A + \vb M = \vb b \]

1.6 Helmholtz equations

\[ \nabla^2 u + k^2 u = f \]

results from wavenumber decomposition of diffusion or wave equations

approach: \(\vb F = \vb{F_0}e^{\imath\omega t} \quad\Rightarrow\quad \pdv{\vb F}{t}=\imath\omega\vb F \quad\Rightarrow\quad \pdv[2]{\vb F}{t}=-\omega^2\vb F\)

\[ \nabla^2 \vb F - a \nabla_t \vb F - c^2 \nabla^2_t \vb F = 0 \]

\[ \Rightarrow \nabla^2\vb F - a\imath\omega\vb F + c^2 \omega^2\vb F = 0 \]

2 Wave equations

2.1 The eikonal equation

Describes first-arrival times \(t\) as a function of velocity (\(v\)) or slowness (\(s\))

\[ |\grad t| = s = 1/v \]

2.2 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 \[ \pdv[2]{u}{t} - a\pdv{u}{t} - c^2\pdv[2]{u}{x} = 0 \]

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import pygimli