Jasmin Smajic1, Christian Hafner2, Jürg Leuthold2, March 23, 2015

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1 Jasmin Smajic, Christian Hafner 2, Jürg Leuthold 2, March 23, 205 Time Domain Finite Element Method (TD FEM): Continuous and Discontinuous Galerkin (DG-FEM) HSR - University of Applied Sciences of Eastern Switzerland Institute of Energy Technology (IET) Obersstrasse 0, Rapperswil, Switzerland jasmin.smajic@hsr.ch 2 Swiss Federal Institute of Technology (ETH) Institute of Electromagnetic Fields (IEF) Gloriastrasse 35, CH-8092 Zürich, Switzerland christian.hafner@ief..ethz.ch, juerg.leuthold@ief..ethz.ch

2 Contents Introduction Time domain continuous Galerkin FEM (CG-FEM) Wave equation and scattering boundary condition Initial-boundary value problem (IBVP) and its weak form FEM discretization of the vector weak form Implicit scheme for time integration Time domain discontinuous Galerkin FEM (DG-FEM) Local Integration of Maxwell equations DGTD-Central Time marching equations 3-D simple example: parallel plate waveguide Summary

3 Introduction For very long time after its initial applications FEM was used for static and frequency domain electromagnetic simulations for the following reasons: A large sparse system of equations has to be solved in each time step of a time domain analysis. Practical problems are mainly dealing with harmonic sources of known frequency. Then, by using Fourier Transformation, the time is eliminated from the field equations and the fields are complex (frequency domain). A very large class of practical problems are static (such as electrostatic, magnetostatic, stationary current distribution, etc.) where FEM simulations are dominant. Until the development of DG-FEM for electromagnetics, time domain simulations were usually performed by using either Finite Difference Time Domain (FDTD) or Finite Volume Time Domain (FVTD) because these two techniques do not assemble and solve large equation system. They solve instead Maxwell Equations locally (over each cell) and couple them numerically with the adjacent cells.

4 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Wave Equation To obtain the electromagnetic wave equation our analysis starts with the following Maxwell curl-equations: HH = JJ SS + σσee + εε EE EE = μμ HH where EE is the electric field, HH is the magnetic field, JJ SS is the source current density, σσ is the electric conductivity, εε is the electric permittivity, μμ is the magnetic permeability, and tt is time. By applying the curl-operator on the E-field equation the magnetic field can be eliminated from it by using the first H-field equation: μμ EE EE + σσ + εε 2 EE tt 2 = JJ SS

5 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Scattering Boundary Condition Over the boundary ΩΩ RR 2 of the computational domain ΩΩ RR 3 for the E-field can be written the following boundary condition: nn μμ EE + YY SSnn nn EE = UU where nn is the outward unit vector perpendicular to the boundary ΩΩ, YY SS is the surface wave admittance of the boundary ΩΩ, and UU is the known boundary source term. The surface wave admittance is: YY SS = εε μμ

6 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Initial-Boundary Value Problem (IBVP) An electromagnetic wave propagation problem in a 3-D computational domain ΩΩ RR 3 can be described by the following initial-boundary value problem (IBVP): EE iiiiii nn dddd μμ EE EE + σσ + εε 2 EE tt 2 = JJ SS, in ΩΩ RR3 (Ω) nn μμ EE + YY SSnn nn EE = UU, over ΩΩ RR 2 ( )

7 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Weak Form By using the weighted residual method the IBVP (strong form) can be transformed into the so-called weak form: ww ii μμ EE dddd + ΩΩ ΩΩ ww ii σσ EE + εε 2 EE tt 2 + JJ SS dddd = 0 where ww ii is an arbitrary weighting function. The function under the first integral can be transformed further according to the following well known equation of the vector analysis: ww ii μμ EE = μμ EE ww ii + μμ EE ww ii

8 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Weak Form According to the last vector identity the integral form becomes: ΩΩ + ww ii ΩΩ μμ EE ww ii dddd + ΩΩ σσ EE + εε 2 EE tt 2 + JJ SS μμ EE ww iidddd + dddd = 0 The first integral of the above form by using the theorem of Gauss can be transformed into a surface integral: ΩΩ μμ EE ww ii dddd = ΩΩ μμ EE ww ii nn dddd = ww ii nn EE dddd μμ ΩΩ

9 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Weak Form The obtained integral form can be further transformed: ww ii nn μμ EE dddd + ΩΩ ΩΩ μμ EE ww iidddd + ΩΩ ww ii σσ EE + εε 2 EE tt 2 + JJ SS dddd = 0 by using the scattering boundary condition: ΩΩ + μμ EE ww iidddd + ΩΩ YY SS nn ww ii nn EE ΩΩ ww ii σσ EE + εε 2 EE tt 2 + JJ SS dddd + ww ii UU dddd = 0 ΩΩ dddd

10 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Approximation of Vector Function Over a single tetrahedral element the unknown vector E-field can be approximated by using the following so-called vector shape functions: EE rr, tt = jj NN jj rr EE jj tt where NN jj is the vector shape function related to the j-th edge and EE jj is the tangential component of the E-field along the j-th edge

11 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Approximation of Vector Function The 3-D vector shape functions have the following form: NN jj rr = ll jj NN jjjj rr NN jjnn2 rr NN jjnn2 rr NN jjnn rr where j is the j-edge of the e-tetrahedron, n is the first node of the j-edge, n 2 is the second node of the j-edge, and ll jj is the length of the j-edge of the e-tetrahedron.

12 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Equation System A combination of the weak form with the tetrahedral mesh, approximation of the unknown vector field, and the Galerkin choice of the weighting function (ww ii = NN ii ) yields the following: EE jj tt jj ii,jj EE jj + jj ii ΩΩ ΩΩ μμ NN ii rr NN jj rr dddd + YY SS nn NN ii or in the following matrix form: nn NN jj dddd = ii jj σσ EE jj + εε 2 EE jj tt 2 ΩΩ ii,jj NN ii JJ SS dddd ii ΩΩ ΩΩ NN ii NN jj dddd + NN ii UU dddd TT dd2 ddtt 2 EE + RR dd ddtt EE + SS EE = bb

13 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Equation System Evidently, the FEM discretization yielded the following ordinary differential equation of second order with respect to time: TT dd2 ddtt 2 EE + RR dd dddd EE + SS EE = bb The matrices of the equation have the following form: TT iiii = ii,jj ΩΩ εε NN ii NN jj dddd, bb ii = ii ΩΩ NN ii JJ SS dddd ii ΩΩ NN ii UU dddd RR iiii = ii,jj SS iiii = ii,jj ΩΩ ΩΩ σσ NN ii NN jj dddd + ii ΩΩ μμ NN ii rr NN jj rr dddd YY SS nn NN ii nn NN jj dddd

14 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Time Integration The obtained differential equation is numerically integrated in time by using the socalled backward difference representation: dd dddd EE EE nn EE nn tt dd2 ddtt 2 EE EE nn 2 EE nn + EE nn 2 tt 2 A combination of the above approximation and the obtained differential equation yields: TT EE nn+ 2 EE nn + EE nn tt 2 + RR dd dddd EE nn+ EE nn tt + SS EE nn+ = bb nn+ or in a more suitable form for the iterative procedure in time: TT tt 2 + RR tt + SS EE nn+ = 2 TT tt 2 + RR tt EE nn TT tt 2 EE nn + bb nn+

15 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Time Integration The obtained scheme is called an implicit scheme because it requires solving a large linear system of equations including the matrix [S]: TT tt 2 + RR tt + SS EE nn+ = 2 TT tt 2 + RR tt EE nn TT tt 2 EE nn + bb nn+ The above implicit scheme is unconditionally stable! The other variants are also possible: The forward difference representation yields a time marching procedure which is numerically unstable. The central difference representation yields an explicit scheme because the linear system to be solved in each time step does not involve the ill-conditioned matrix [S]. The explicit scheme is however stable only if the time step tt is sufficiently small.

16 Time Domain Continuous Galerkin FEM (CG-FEM) EM Waves in 3-D: Summary The obtained scheme is called an implicit scheme because it requires solving a large linear system of equations including the matrix [S]: TT tt 2 + RR tt + SS EE nn+ = 2 TT tt 2 + RR tt EE nn TT tt 2 EE nn + bb nn+ Evidently, the main drawback of the above scheme is the nd to solve a large illconditioned system of linear equations in each time step of the integration. The reason for this drawback is in the continuous character of the approximation of the unknown vector function.

17 Time Domain Discontinuous Galerkin FEM (DG-FEM) Local Integration of Maxwell Equations To present the main ideas of the time domain DG-FEM we will start again with the following Maxwell curl-equations (this time, for the reason of simplicity, in a loss- and current-fr region): HH = εε EE EE = μμ HH The weighted residual method is applied here locally over a single 3-D element: EE ww ii HH εε ΩΩ HH ww ii EE + μμ ΩΩ dddd = 0 dddd = 0

18 Time Domain Discontinuous Galerkin FEM (DG-FEM) Local Integration of Maxwell Equations The weighted residual method is applied here locally over a single 3-D element: EE ww ii εε ΩΩ HH ww ii μμ ΩΩ HH dddd = 0 + EE dddd = 0 By applying the vector identity aa bb = bb aa aa bb and the divergence theorem the following is obtained: εε ww ii ΩΩ μμww ii ΩΩ EE dddd ΩΩ HH ww ii dddd = ΩΩ HH dddd + ΩΩ EE ww ii dddd = ww ii nn HH ddss ww ii EE nn dddd

19 Time Domain Discontinuous Galerkin FEM (DG-FEM) DGTD-Central In the last obtained equation we will use the surface integrals to couple the neighboring elements: εε ww ii ΩΩ μμww ii ΩΩ EE dddd ΩΩ HH ww ii dddd = ΩΩ HH dddd + ΩΩ EE ww ii dddd = ww ii nn HH ddss ww ii EE nn dddd By introducing the following averaging of the fields: ww ii nn HH dddd 2 ww ii nn HH iiiiii + nn HH dddd ww ii EE nn dddd 2 ww ii EE iiiiii nn + EE nn dddd

20 Time Domain Discontinuous Galerkin FEM (DG-FEM) DGTD-Central The obtained two integral forms: εε ww ii ΩΩ EE dddd ΩΩ HH ww ii dddd = 2 ww ii nn HH iiiiii + nn HH dddd μμww ii ΩΩ HH dddd + ΩΩ EE ww ii dddd = 2 ww ii EE iiiiii nn + EE nn dddd Can be discretized by using the presented vector FEM theory: TT EE dd dddd EE SS HH = ff EE RR EE EE + bb EE TT HH dd dddd HH + SS EE = ff HH RR HH HH + bb HH

21 Time Domain Discontinuous Galerkin FEM (DG-FEM) DGTD-Central: Time Marching Equations The obtained set of ordinary differential equations of first order with respect to time: TT EE dd dddd EE SS HH = ff EE RR EE EE + bb EE TT HH dd dddd HH + SS EE = ff HH RR HH HH + bb HH can be transformed into the following time marching equations by using the central difference method: TT EE tt + 2 RR EE EE nn+ = TT EE tt 2 RR EE EE nn + SS nn+ HH 2 + ff nn+ 2 EE + bb nn+ 2 EE TT HH tt + 2 RR HH nn+ HH 3 2 = TT HH tt 2 RR HH nn+ HH 2 SS EE nn+ + ff nn+ EE + bb nn+ HH This scheme is conditionally stable with the known stability condition. According to this scheme the E- and H-field can be computed element by element and step by step in a leapfrog fashion: EE nn HH nn+ 2 EE nn+ HH nn+ 3 2.

22 3-D Example Parallel Plate Waveguide A parallel plate waveguide for comparison of different explicit schemes is chosen because of the existence of the known analytical solution (easy to test the accuracy): PEC: nn EE = 0 m WG port boundary m PMC: nn HH = 0 PMC: nn HH = 0 m WG port boundary PEC: nn EE = 0 X. Li, J. Jin, A Comparative Study of Thr Finite Element-Based Explicit Numerical Schemes for Solving Maxwell s Equations, IEEE Transactions on Antenna and Propagation, Vol. 60, No. 3, pp , March 202.

23 3-D Example Parallel Plate Waveguide A parallel plate waveguide for comparison of different explicit schemes is chosen because of the existence of the known analytical solution (easy to test the accuracy): FETD Finite-element time-domain DFDD dual-field domain decomposition DGTD-Upwind DGTD-Central (presented version) λλ mmmmmm is the shortest wavelength of interest h pp = h pp is the ratio betwn the average element edge length (h) and the order of basis functions (p) X. Li, J. Jin, A Comparative Study of Thr Finite Element-Based Explicit Numerical Schemes for Solving Maxwell s Equations, IEEE Transactions on Antenna and Propagation, Vol. 60, No. 3, pp , March 202.

24 3-D Example Parallel Plate Waveguide A parallel plate waveguide for comparison of different explicit schemes is chosen because of the existence of the known analytical solution (easy to test the accuracy): FETD Finite-element time-domain DFDD dual-field domain decomposition DGTD-Upwind DGTD-Central (presented version) λλ mmmmmm is the shortest wavelength of interest h pp = h pp is the ratio betwn the average element edge length (h) and the order of basis functions (p) X. Li, J. Jin, A Comparative Study of Thr Finite Element-Based Explicit Numerical Schemes for Solving Maxwell s Equations, IEEE Transactions on Antenna and Propagation, Vol. 60, No. 3, pp , March 202.

25 3-D Example Parallel Plate Waveguide A parallel plate waveguide for comparison of different explicit schemes is chosen because of the existence of the known analytical solution (easy to test the accuracy): FETD Finite-element time-domain DFDD dual-field domain decomposition DGTD-Upwind DGTD-Central (presented version) λλ mmmmmm is the shortest wavelength of interest h pp = h pp is the ratio betwn the average element edge length (h) and the order of basis functions (p) X. Li, J. Jin, A Comparative Study of Thr Finite Element-Based Explicit Numerical Schemes for Solving Maxwell s Equations, IEEE Transactions on Antenna and Propagation, Vol. 60, No. 3, pp , March 202.

26 Summary Some electromagnetic simulations such as those involving non-harmonic arbitrary sources and nonlinear materials have to be performed in time domain. Standard Continuous Galerkin (CG) FEM does require solving a large linear equation system in each time step. This is a considerable drawback compared to FDTD or FVTD which do not operate with equation systems. To overcome the problems of FDTD (regular grid) and of FVTD (low order approximation) the discontinuous Galerkin (DG) FEM was suggested. DG-FEM integrate Maxwell equations locally over each element and couple those local solutions through fluxes at the interface betwn different elements. DG-FEM does require inverting small elemental matrices which is very fast and efficient.

27 Literature Wolfram MathWorld, the web s most extensive mathematics resource, J. D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, New York, 998. N. Ida, Enginring Electromagnetics, Second Edition, Springer-Verlag, New York, J. Jin, The Finite Element Method in Electromagnetics, Second Edition, John Wiley & Sons, New York, 2002.

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