Development and Application of Compatible ... - Semantic Scholar

UCRL-PRES-203950

Development and Application of Compatible Discretizations of Maxwell’s Equations Past, Present, and Future Research at LLNL

Daniel White, Joe Koning, Rob Rieben Lawrence Livermore National Laboratory IMA workshop on Compatible Spatial Discretizations for Partial Differential Equations May 11-15, 2004 This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.

Agenda • Review of Maxwell’s Equations • Review of previous methods — FDTD — Mixed Finite Element — Modified Finite Volume, Discrete Surface Integral

• Review of EMSolve — Higher order H(curl) & H(div) basis functions — Symplectic time integration

• Applications — Photonics — Accelerators — Magnetohydrodynamics D. White 5/17/2004- 2

Maxwell’s Equations ε ∂E =∇× H − J

∇•ε E = ρ

D =ε E

µ ∂H = −∇× E

∇• µ H = 0

B = µH

∂t

∂t

E electric field, D electric flux density, H magnetic field, B magnetic flux density J current density, ρ charge density • Electromagnetic waves (similar equation for H) 2 −1∇× E − ∂J ε∂ E = −∇× µ 2

∂t

∂t

• Divergence constraints can be considered initial conditions • Important boundary conditions

n × E = f n ×∇× E = g n ×(α E + β∇× E) = q D. White 5/17/2004- 3

Maxwell’s Equations: Considerations • Boundary conditions across material interfaces Electric Current

Electric Field

ε1

ε2

σ1

tangential continuity • Conservation of energy

stored energy

E •ε E

H •µ H D. White 5/17/2004- 4

σ2

normal continuity

power entering P = E × H (or exiting) through surface

J •E energy gain (or loss)

Finite Difference Time Domain • Dual Cartesian mesh with edge-based variables (Yee, 66) — Electric field is constant on primary mesh edges — Magnetic field is constant on dual mesh edges

• Leapfrog in time — Given E field, update H field

∂H = 1 ∇× E ∂t µ

— Given H field, update E field z y x

n+1

⎛ ⎞ ⎜ y⎟ ⎜ ⎟ ⎝ ⎠ijk

h

n

⎛ ⎞ ⎜ y⎟ ⎜ ⎟ ⎝ ⎠ijk

=h

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎤ n n ⎥ − e + z ijk (i+1) jk ⎥ ⎥ n n ⎥ e − x ijk ij(k +1) ⎥⎥

(ez ) t ∆ − µ∆h (ex )

( ) ( )

And similar equations for other field components D. White 5/17/2004- 5

⎦

FDTD : Dispersion Analysis • The method is 2nd order accurate

— Plane waves of the form exp(iω − k • x) are solutions of free-space Maxwell’s equations, with k 2 = ω 2 / c2

— The solution to the discrete equations satisfies

1 sin2(ω∆t ) = 1 sin 2(k ∆x) + 1 sin 2(k ∆y) + 1 sin 2(k ∆z) z x y 2 c2∆t 2 ∆x2 ∆y2 ∆z 2 — Taylor expansion about ∆t = 0, ∆x = 0, ∆y = 0, ∆z = 0

yields

⎛ ⎞ 2∆y 2 ⎛ ⎞ 2 2 2 ⎜ 2 k 2 2 2 kz ∆z +...⎟⎟ ω ⎜⎜1− ω ∆t +...⎟⎟ = k 2 ⎜1− kx ∆x − y − ⎜ ⎟ 12 12 12 12 ⎟ c2 ⎜ ⎝

D. White 5/17/2004- 6

⎠

⎜⎜ ⎝

⎟⎟ ⎠

FDTD: Stability Analysis • The method is conditionally stable — Assume ∆x=∆y=∆z. The dispersion relation can be expressed as ⎛

⎞

ω = 2 sin−1⎜⎜ c∆t sin(k ∆x) ⎟⎟ ∆t ⎜ ∆x 3 ⎟⎠ ⎝

— We require that ω be real, otherwise we will have exponential growth of the numerical solution. This yields ∆x

∆t