1  Introduction

1.1 Differential operators

  • single derivative in space \(\pdv{x}\) or time \(\pdv t\)

  • gradient \(\grad=(\pdv x, \pdv y, \pdv z)^T\)

  • divergence \(\div \vb F = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z}\)

Gauss’: 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

1.1.1 Curl (rotation)

  • curl \(\curl \vb F = (\pdv{F_z}{y}-\pdv{F_y}{z}, \pdv{F_x}{z}-\pdv{F_z}{x}, \pdv{F_y}{x}-\pdv{F_x}{y})^T\)

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

Stokes’ theorem in EM
  • curls have no divergence: \(\div (\curl \vb F)=0\)
  • potential fields have no curl \(\curl (\grad u)=0\)

1.2 Partial differential equations (PDEs)

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 PDE: \(-\nabla^2 u=f\) (Poisson) or \(\nabla^2 u + k^2 u = f\) (Helmholtz)
  • parabolic PDE \(-\nabla^2 u + a \pdv{u}{t} \frac{\partial u}{\partial t}=f\)
  • 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)

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