Scientific Computing

Transcription

1 Lecture on Scientific Computing Dr. Kersten Schmidt Lecture 4 Technische Universität Berlin Institut für Mathematik Wintersemester 2014/2015 Syllabus Linear Regression Fast Fourier transform Modelling by partial differential equations (PDEs) Maxwell Helmholtz Poisson Linear elasticity Navier-Stokes equation boundary value problem eigenvalue problem boundary conditions (Dirichlet Neumann Robin) handling of infinite domains (wave-guide homogeneous exterior: DtN PML) boundary integral equations Computer aided-design (CAD) Mesh generators Space discretisation of PDEs Finite difference method Finite element method Discontinuous Galerkin finite element method Solvers Linear Solvers (direct iterative) preconditioner Nonlinear Solvers (Newton-Raphson iteration) Eigenvalue Solvers Parallelisation SIMP: OpenMP MIMP: MPI VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/2014 2

2 Differential operators ( 1 ) Differential operators = 2 3 ( ) 1 u grad u(x) = ( u)(x) = 2 u 3 u ( ) 2 q 3 3 q 2 curl q(x) = ( q)(x) = 3 q 1 1 q 3 1 q 2 2 q 1 div q(x) = ( q)(x) = 1 q q q 3 Rules curl grad u = 0 div curl q = 0 u q VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Electrostatics Electric charges Q positive and negative charges multiple of the elementar charge: Q = ne e = C (1 C = 1 As) electron: n = 1 proton: n = +1 charge density ρ(x) : Q = ρ(x) dx V point charges q i : Q = i q i = ρ(x) dx with ρ(x) = V i q iδ(x x i ) dx Dirac δ function: δ(x x 0 ) = 0 if x x 0 δ(x x 1 if x 0 V V 0) dx = 0 otherwise Electrostatic force between point charges Coloumb law F 2 1 = q 1q 2 x 1 x 2 x 1 x 2 3 x 1 x 2 2 force measured at x 1 attracting if q 1 q 2 < 0 repulsive if q 1 q 2 > 0 12 As vacuum permittivity ε 0 = Vm VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/2014 4

3 Electrostatics Electrostatic force between point charges Coloumb law F 2 1 = q 1q 2 x 1 x 2 x 1 x 2 3 x 1 x 2 2 force measured at x 1 Electrostatic force of charge density F(x) = q x x i q i x x i q 3 i ρ(x ) x x x x 3 dx Electrostatic force due to charges can be measured everywhere Electric field E(x) = 1 Electric field in electrostatics is a potential ρ(x ) x x x x 3 dx E(x) = grad φ(x) with φ(x) = 1 ρ(x 1 ) x x dx since x x 1 = x x 3 x. x x VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Electrostatics Electrostatic force due to charges can be measured everywhere Electric field E(x) = 1 Electric field in electrostatics is a potential ρ(x ) x x x x 3 dx E(x) = grad φ(x) with φ(x) = 1 ρ(x 1 ) x x dx since x x 1 = x x 3 x moving a charge from x 1 to x 2 in an electric field is independent of the path C U = U... voltage = potential difference electric field in electrostatics is irrotational C E(x) d s = φ(x 2 ) φ(x 1 ) x x. curl E(x) = 0 due to the identity curl grad = 0 VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/2014 6

4 Electrostatics Physical Gauß law div E(x) = φ(x) = 1 ρ(x 1 ) x x x dx = 1 ρ(x) ε 0 since δ(x x ) = 1 4π x Poisson problem (a) Electric field for given charge density (BVP in an infinite domain) { div E(x) = φ = 1 ε 0 ρ(x) in R 3 E(x) decays for x (b) Electric field for given potential / voltage (Dirichlet BVP) ß div E(x) = φ = 0 in R 3 \Ω φ(x) = U(x) on Ω (c) Electric field for given boundary values (Neumann BVP) ß div E(x) = φ = 0 in R 3 \Ω 1 x x φ(x) n = E(x) n on Ω VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Electrostatics Electric field due to charges in vacuum div E(x) = 1 ε 0 ρ(x) curl E(x) = 0 Inside materials we have an huge number of charges (atomic nuclei electrons e.g in volume of 1mm 3 ) average process of the microstructure depend on the local structure. Effect of charges of opposite sign to charges outside the material is reduced. Example: Linear material law ε r 1... relative permittivity D... electric displacement field = auxilliary field (not measured) VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/2014 8

5 Magnetostatics Electric current density = moving charges : charge conservation continuity equation In a volume V the charge is only changed if it is transported from outside Q t = ρ(x) dx = div j(x) dx = j n ds(x) t V V V no local creation of charges no source term in charge balance Electric current = integration over a cross section of a conductor I = j(x) n ds(x) A VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Magnetostatics There exist a force Lorentz force which acts on moving charges for some vector field B the magnetic B-field (dt. magnetische Flußdichte). it can be caused by a permant magnet or by moving charges so currents (electromagnet) (Ampère s law) (Magnetic Gauß law no magnetic charges ) µ 0 = 1 6 Vs = vacuum permeability ε 0 c 2 Am M... magnetization H... auxilliary field for permantent magnet j = 0 M = M(x) linear magnetic materials M(x) = (µ r (x) 1)H(x) µ r 1... permeability of linear materials (not all) µ r = 1 for non-magnetic materials for ferro-magnetic materials M = M(H) (possibly hysteresis memory effect) VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/

6 Magnetostatics Magnetic field in vacuum generated by a coil of stationary current (div j(x) = 0) not flowing out of the conductor (j(x) n = 0 on Ω): Biot-Savart law it can be shown that curl H S = j note in (linear) magnetic materials (µ r 1) div B is not fulfilled i. e. H S has to be corrected (H = H S + H R with curl H R = 0) Ohm s law (in case of Ohmic conductors) linear dependance of electric current inside conductor and electric field note that conductors may have currents but no charge accumulation : ρ = 0 Electric dissipation power (dt. Leistung) P elec = Ω is transformated into heat E(x) j(x) }{{} electric power density dx VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Magnetostatics Summary Magnetostatics Magnetic H field is generated by currents j (Biot-Savart) or magnetization in materials M Currents j in conductors are caused by electric fields E (outside my changing charge distribution) Electric field (in conductors) is caused by voltages (potential differences) Problems in Magnetostatics (a) Magnetic field for given current density no magnetized material (BVP in an infinite domain) 0 = div B(x) = (b) Magnetic field due to magnetized material no current density (BVP in an infinite domain) 0 = div B(x) = VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/

7 Electromagnetics Maxwell s equations = first principles equations curl E(t x) = Faraday s law of induction curl H(t x) = Ampe re s law div D(t x) = Gauß law div B(t x) = Magnetic Gauß law plus the charge conservation ρ (t x) + div j(t x) = 0 t completed by constitutive equations (material laws) e.g. j(t x) = σ(x)e(t x) Ohm s law for conductors D(t x) = εr (x)ε0 E(t x) linear dielectric material B(t x) = µr (x)µ0 H(t x) linear magnetic material. VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/ Electromagnetics Magnetic field (absolute value) for a circular coil (dt. ringfo rmige Spule) Schmidt K. Sterz O. and Hiptmair R. IEEE Trans. Magn. 44: Jun VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/

8 Electromagnetics Shielding effect of two circular coils (dt. kreisförmige Spulen) Schmidt K. and Chernov A. SIAM J. Appl. Math. 73(6): VL Scientific Computing WS 2014/2015 Dr. K. Schmidt 10/23/