Final Ph.D. Progress Report Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo Dissertation Committee: I. Babuska, L. Demkowicz, C. Torres-Verdin, R. Van de Geijn, M. Wheeler. October 31, 2003 Institute for Computational Engineering and Sciences (ICES)
OVERVIEW 1. Overview. 2. Motivation. 3. Maxwell s Equations. 4. hp-adaptivity. 5. The Fully Automatic hp-adaptive Strategy. 6. A Two Grid Solver for SPD Problems. 7. A Two Grid Solver for Electromagnetics. 8. Performance of the Two Grid Solver. 9. Electromagnetic Applications. 10. Conclusions and Future Work.
2. MOTIVATION Radar Cross Section (RCS) Analysis SURFACE WAVES MULTIPLE REFLECTIONS WAVEGUIDE MODES EDGE DIFRACTION Power scattered to receiver per unit solid angle RCS=4π = lim Incident power density r 4πr 2 E s E. i Goal: Determine the RCS of a plane. 2
2. MOTIVATION Waveguide Design Goal: Determine electric field intensity at the ports. 3
2. MOTIVATION Modeling of a Logging While Drilling (LWD) electromagnetic measuring device SIDE VIEW T1 EM FIELD R1 R2 DIFFERENT MATERIAL LAYERS T2 Goal: Determine EM field at the receiver antennas. 4
2. MOTIVATION Modeling of a Logging While Drilling (LWD) electromagnetic measuring device Coil Receiver Formation Simplest case: ONE COIL TRANSMITTER Coil Transmitter Mud Goal: Determine EM field at the receiver antennas. 5
3. MAXWELL S EQUATIONS Time Harmonic Maxwell s Equations: Dirichlet BC at a PEC surface: E = jµωh H = jωɛe + σe + J imp Reduced Wave Equation: ( ) 1 µ E (ω 2 ɛ jωσ)e = jωj imp, Boundary Conditions (BC): n E s = n E inc n E = 0 Neumann continuity BC at a material interface: n 1 µ Es = n 1 µ Einc n 1 µ E = jωjimp S Silver Müller radiation condition at : e r ( E s ) jk 0 E s = O(r 2 ) 6
3. MAXWELL S EQUATIONS Variational formulation The reduced wave equation in Ω, A variational formulation ( ) 1 µ E Ω (ω 2 ɛ jωσ)e = jωj imp, Find E H D (curl; Ω) such that 1 Ω µ ( E) ( F )dx (ω 2 ɛ jωσ)e Fdx = Ω { } jω J imp F dx + F ds J imp S Γ 2 A stabilized variational formulation (using a Lagrange multiplier): for all F H D (curl; Ω). Find E H D (curl; Ω), p HD 1 (Ω) such that 1 Ω µ ( E)( F )dx (ω 2 ɛ jωσ)e F dx (ω 2 ɛ jωσ) p F dx = Ω { Ω } jω J imp F dx + J imp S F ds F H D (curl; Ω) { Ω Γ 2 } (ω 2 ɛ jωσ)e q dx = jω J imp q dx + J imp S qds q HD 1 (Ω). Ω Ω Γ 2 7
3. MAXWELL S EQUATIONS De Rham diagram De Rham diagram is critical to the theory of FE discretizations of Maxwell s equations. IR W Q V id Π L 2 0 Π curl Π div P IR W p Q p V p W p 1 0. This diagram relates two exact sequences of spaces, on both continuous and discrete levels, and corresponding interpolation operators. 8
4. HP-ADAPTIVITY Different refinement strategies for finite elements: y z x Given initial grid y z x y z x y z x h-refined grid p-refined grid hp-refined grid 9
4. HP-ADAPTIVITY Orthotropic heat conduction example k=1 k=2 k=3 k=5 k=4 Equation: (K u) [ = f (k) ] K = K (k) K x (k) 0 = 0 K y (k) K x (k) = (25, 7, 5, 0.2, 0.05) y = (25, 0.8, 0.0001, 0.2, 0.05) K (k) Solution: unknown Boundary Conditions: K (i) u n = g (i) α (i) u 39.00097 error 21.36404 SCALES: nrdof^0.33, log(error) 11.70284 6.41061 3.51162 1.92360 1.05372 0.57721 0.31618 0.17320 0.09488 nrdof 45 278 860 1946 3697 6269 9822 14513 20502 y z Convergence history (tolerance error = 0.1 %) Final hp grid 10
4. HP-ADAPTIVITY Convergence comparison 10 2 Orthotropic heat conduction example h Adaptivity A priori hp adaptivity hp Adaptivity ERROR IN THE RELATIVE ENERGY NORM (%) 10 1 10 0 10 1 10 2 0 1000 8000 27000 64000 125000 216000 343000 512000 NUMBER OF DOF 11
5. THE FULLY AUTOMATIC HP-ADAPTIVE STRATEGY Fully automatic hp-adaptive strategy global hp-refinement y z x y z x global hp-refinement y z x y z x 12
5. THE FULLY AUTOMATIC HP-ADAPTIVE STRATEGY Automatic hp-adaptivity delivers exponential convergence and enables solution of challenging EM problems Coarse Grid hp grid Fine Grid h/2,p+1 grid Next optimal coarse grid NEED FOR A TWO GRID SOLVER Minimization of the projection based interpolation error 13
6. A TWO GRID SOLVER FOR SPD PROBLEMS We seek x such that Ax = b. Consider the following iterative scheme: r (n+1) = [I α (n) AS]r (n) x (n+1) = [I α (n) S]r (n) where S is a matrix, and α (n) is a relaxation parameter. α (n) optimal if: α (n) = arg min x (n+1) x A = (A 1 r (n), Sr (n) ) A (Sr (n), Sr (n) ) A Then, we define our two grid solver as: 1 Iteration with S = S F = A 1 i + 1 Iteration with S = S C = P A 1 C R 14
6. A TWO GRID SOLVER FOR SPD PROBLEMS Error reduction and stopping criteria Let e (n) = x (n) x the error at step n, ẽ (n) = [I S C A]e (n) = [I P C ]e (n). Then: e (n+1) 2 A e (n) 2 A = 1 (ẽ(n), S F Aẽ (n) ) A 2 ẽ (n) 2 A S F Aẽ (n) 2 A = 1 (ẽ(n), (P C + S F A)ẽ (n) ) A 2 ẽ (n) 2 A S F Aẽ (n) 2 A Then: e (n+1) 2 A e (n) 2 A sup e [1 (e, (P C + S F A)e) A 2 e 2 A S F Ae 2 A ] C < 1 (Error Reduction) For our stopping criteria, we want: Iterative Solver Error Discretization Error. That is: e (n+1) A e (0) A 0.01 (Stopping Criteria) 15
A TWO GRID SOLVER FOR ELECTROMAGNETICS We seek x such that Ax = b. Consider the following iterative scheme: r (n+1) = [I α (n) AS]r (n) x (n+1) = [I α (n) S]r (n) where S is a matrix, and α (n) is a relaxation parameter. α (n) optimal if: α (n) = arg min x (n+1) x B = (A 1 r (n), Sr (n) ) B (Sr (n), Sr (n) ) B (NOT COMPUTABLE) Then, we define our two grid solver for Electromagnetics as: 1 Iteration with S = S F = A 1 i + 1 Iteration with S = S = G 1 i + 1 Iteration with S = S C = P A 1 C R 16
A TWO GRID SOLVER FOR ELECTROMAGNETICS A two grid solver for discretization of Maxwell s equations using hp-fe Consider the following two problems: Problem I: E k 2 E = J Matrix form: Au = v Two grid solver V-cycle: T G = (I α 1 S F A)(I α 2 S A)(I S C A C ) Problem II: E + E = J Matrix form: Âu = v Two grid solver V-cycle: T G = (I α 1 Ŝ F Â)(I α 2 Ŝ Â)(I Ŝ C Â C ) Theorem: If h is small enough, then: T Ge (n) T Ge (n) +Ch Notice that C is independent of h and p. 17
A TWO GRID SOLVER FOR ELECTROMAGNETICS A two grid solver for discretization of Maxwell s equations using hp-fe Helmholtz decomposition: H D (curl; Ω) = (Ker(curl)) (Ker(curl)) We define the following subspaces (T =grid, K =element, v =vertex, e =edge): Ω v k,i = int( { K T k : v k,i K}) ; Ω e k,i = int( { K T k : e k,i K}) Domain decomposition M v k,i = {u M k : supp(u) Ω v k,i } ; M e k,i = {u M k : supp(u) Ω e k,i } Nedelec s elements decomposition W v k,i = {u W k : supp(u) Ω v k,i } ; W e k,i = {u W k : supp(u) Ω e k,i } = ø P olynomial spaces decomposition Hiptmair proposed the following decomposition of M k : M k = e M e k,i + v W v k,i Arnold et. al proposed the following decomposition of M k : M k = v M v k,i 18
8. PERFORMANCE OF THE TWO GRID SOLVER 2002 Importance of the choice of shape functions. Importance of the relaxation parameter. Selection of patches for the block Jacobi smoother. Effect of averaging. Error estimation. Smoothing vs two grid solver. Numerical Studies 2003 Guiding hp-adaptivity with a partially converged fine grid solution for EM problems. Efficiency of the two grid solver. Number of elements per wavelength required by the two grid solver to converge. Control of the dispersion error. Guiding hp-adaptivity with a partially converged fine grid solution. Applications to real world problems. 19
8. PERFORMANCE OF THE TWO GRID SOLVER Orthotropic heat conduction example k=1 k=2 k=3 k=5 k=4 Equation: (K u) [ = f (k) ] K = K (k) K x (k) 0 = 0 K y (k) K x (k) = (25, 7, 5, 0.2, 0.05) y = (25, 0.8, 0.0001, 0.2, 0.05) K (k) Solution: unknown Boundary Conditions: K (i) u n = g (i) α (i) u 39.00097 error 21.36404 SCALES: nrdof^0.33, log(error) 11.70284 6.41061 3.51162 1.92360 1.05372 0.57721 0.31618 0.17320 0.09488 nrdof 45 278 860 1946 3697 6269 9822 14513 20502 y z Convergence history (tolerance error = 0.1 %) Final hp grid 20
8. PERFORMANCE OF THE TWO GRID SOLVER Guiding automatic hp-refinements Orthotropic heat conduction. Guiding hp-refinements with a partially converged solution. RELATIVE ERROR IN % 10 2 10 1 10 0 0.3 0.1 0.01 0.001 Frontal Solver RELATIVE ERROR IN % 10 2 10 1 10 0 10 1 0.3 0.1 0.01 0.001 Frontal Solver NUMBER OF ITERATIONS 150 100 50 0.3 0.1 0.01 0.001 0 125 1000 3375 8000 15625 27000 NUMBER OF DOF Energy error estimate 10 2 0 125 1000 3375 8000 15625 27000 NUMBER OF DOF Discretization error estimate 0 0 5 10 15 NUMBER OF DOF IN THE FINE GRID x 10 4 Number of iterations 21
8. PERFORMANCE OF THE TWO GRID SOLVER Plane Wave incident into a screen (diffraction problem) Screen Geometry y z Second component of electric field 401 665 1026 1499 2098 2838 3736 4805 6061 nrdof 0.07 0.14 0.29 0.62 1.31 2.79 5.91 12.53 26.56 56.31 119.41 error SCALES: nrdof^0.33, log(error) Convergence history (tolerance error = 0.1 %) x y z Final hp-grid 22
Numerical Results Guiding automatic hp-refinements Diffraction problem. Guiding hp-refinements with a partially converged solution. 10 2 0.3 0.1 0.01 Frontal Solver 20 18 0.3 0.1 0.01 16 RELATIVE ERROR IN % 10 1 10 0 NUMBER OF ITERATIONS 14 12 10 8 6 4 10 1 2 512 1000 1728 2744 4096 5832 8000 10648 NUMBER OF DOF Discretization error estimate 0 0 0.5 1 1.5 2 2.5 3 3.5 NUMBER OF DOF IN THE FINE GRID x 10 4 Number of iterations 23
31 October 2003 David Pardo 8. PERFORMANCE OF THE TWO GRID SOLVER Waveguide example y z x Module of Second Component of Magnetic Field 2Dhp90: A Fully automatic hp-a hp-adaptive Finite Element code 14.68error SCALES: nrdof^0.33, log(error) 10.54 7.57 5.43 3.90 2.80 2.01 1.44 1.04 0.74 0.53 552 1045 1766 2761 4074 5748 7827 10356 nrdof 13378 y z x Convergence history (tolerance error = 0.5 %) Final hp-grid 24
8. PERFORMANCE OF THE TWO GRID SOLVER Guiding automatic hp-refinements Waveguide example. Guiding hp-refinements with a partially converged solution. 10 1 0.3 0.1 0.01 Frontal Solver 25 20 0.3 0.1 0.01 RELATIVE ERROR IN % NUMBER OF ITERATIONS 15 10 10 0 5 512 1000 1728 2744 4096 5832 8000 10648 13824 17576 NUMBER OF DOF Discretization error estimate 0 0 1 2 3 4 5 6 7 8 NUMBER OF DOF IN THE FINE GRID x 10 4 Number of iterations 25
8. PERFORMANCE OF THE TWO GRID SOLVER Efficiency of the two grid solver We studied scalability of the solver with respect h and p. Speed = Coarse grid solve +O(p 9 N) We implemented an efficient solver. Fast integration rules. Fast matrix vector multiplication. Fast assembling. Fast patch inversion. Fast construction of prolongation/restriction operator. 26
8. PERFORMANCE OF THE TWO GRID SOLVER 3D shock like solution example z x y Equation: u = f Geometry: unit cube Solution: u = atan(20 r 3)) r = (x.25) 2 + (y.25) 2 + (z.25) 2 Dirichlet Boundary Conditions 10 1 z x y 10 0 525 1024 1845 3125 5032 7776 11602 Convergence history (tolerance error = 1%) Final hp grid 27
8. PERFORMANCE OF THE TWO GRID SOLVER Performance of the two grid solver 350 300 3D shock like solution example TOTAL TIME PATCH INVERSION MATRIX VECTOR MULTIPLICATION INTEGRATION COARSE GRID SOLVE OTHER 250 TIME IN SECONDS 200 150 100 50 0 2 4 6 8 10 12 14 NUMBER OF DOF x 10 4 In core computations, AMD Athlon 1 Ghz processor. 28
8. PERFORMANCE OF THE TWO GRID SOLVER Performance of the two grid solver 3D shock like solution problem PATCH INVERSION MATRIX VECTOR MULTIPLICATION INTEGRATION COARSE GRID SOLVE ASSEMBLING OTHER 6% 2% 10% 21% 15% 4% 6% 2% 6% 17% 15% 33% 33% 36% 13% 82% Nrdofs 2.15 Million Total time 8 minutes Memory* 1.0 Gb p=2 Nrdofs 0.27 Million Total time 10 minutes Memory* 2.0 Gb p=8 Nrdofs 2.15 Million Total time 50 minutes Memory* 3.5 Gb p=4 *Memory = memory used by nonzero entries of stiffness matrix In core computations, IBM Power4 1.3 Ghz processor. 29
8. PERFORMANCE OF THE TWO GRID SOLVER Convergence history 30 3D shock like solution example. Scales: ERROR VS TIME. RELATIVE ERROR IN % (LOGARITHMIC SCALE) 15 10 5 3 2 1 5 10 19 32 52 80 119 172 243 336 TIME IN SECONDS (ALGEBRAIC SCALE) 30
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example (Baker-Hughes): Electrostatics ( 1, 1) d=10 6 2 degrees d=2*10 6 Dirichlet Boundary Conditions u(boundary)= ln r, r=sqrt (x*x+y*y) 31
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example: final hp-grid, Zoom = 1 x y z 32
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example: final hp-grid, Zoom = 10 13 x y z 33
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example: Comparison between exact and approximate solution at distances 0.01-1 from the singularity 18.385059 16.854529 15.323999 13.793469 12.262939 10.732409 9.201879 7.671349 6.140819 4.610289 3.079759 1.549229 0.018699-1.511831-3.042361-4.572891-6.103421 x -1.007193-1.096212-1.185230-1.274248-1.363267-1.452285-1.541303-1.630322-1.719340 34
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example: Comparison between exact and approximate solution at distances 0.0001-0.01 from the singularity 38.328379 37.080578 35.832777 34.584976 33.337175 32.089374 30.841572 29.593771 28.345970 27.098169 25.850368 24.602567 23.354765 22.106964 20.859163 19.611362 18.363561 x -1.000072-1.000962-1.001852-1.002742-1.003633-1.004523-1.005413-1.006303-1.007193 35
9. ELECTROMAGNETIC APPLICATIONS Edge diffraction example: Comparison between exact and approximate solution at distances 0.000001-0.0001 from the singularity 58.199630 56.956577 55.713524 54.470471 53.227418 51.984366 50.741313 49.498260 48.255207 47.012154 45.769102 44.526049 43.282996 42.039943 40.796891 39.553838 38.310785 x -1.000001-1.000010-1.000019-1.000027-1.000036-1.000045-1.000054-1.000063-1.000072 36
9. ELECTROMAGNETIC APPLICATIONS Time Harmonic Maxwell s Equations E = jµωh H = jωɛe + σe 3 4 Reduced Wave Equation: ( ) 1 µ E (ω 2 ɛ jωσ)e = jωj imp 2 1 d=0.01 m d=4 m Boundary Conditions (BC): Dirichlet BC at a PEC surface: 2 n E = 0 on Γ 2 Γ 4 d=0.01 m 4 Neumann BC s: 3 n 1 µ E = jω on Γ 1 z 4 d=2 m r n 1 µ E = 0 on Γ 3 37
9. ELECTROMAGNETIC APPLICATIONS Battery example: Convergence history 27 183 579 1325 2534 4317 6786 10053 14229 nrdof 9.38 11.76 14.75 18.50 23.20 29.09 36.47 45.74 57.35 71.92 90.19 error SCALES: nrdof^0.33, log(error) 38
9. ELECTROMAGNETIC APPLICATIONS Battery example: final hp-grid, Zoom = 1 x y z 39
9. ELECTROMAGNETIC APPLICATIONS Why the optimal grid is so bad? Optimization is based on minimization of the ENERGY NORM of the error, given by: error 2 = error 2 + error 2 Interpretation of results: The grid is optimal for the selected refinement criteria, but our refinement criteria is inadequate for our pourposes. 40
9. ELECTROMAGNETIC APPLICATIONS Waveguide example with five iris Geometry of a cross section of the rectangular waveguide 0 S 11 (db) 5 10 15 20 25 30 35 H-plane five resonant iris filter. Dominant mode (source): T E 10 mode. Dimensions 20 2 1 cm. Operating Frequency 8.8 9.6 Ghz Cutoff frequency 6.56 Ghz 40 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 Frequency (Ghz) Return loss of the waveguide structure 41
H x 9. ELECTROMAGNETIC APPLICATIONS FEM solution for frequency = 8.72 Ghz H y Hx 2 + H y 2 0 5 10 S 11 (db) 15 20 25 30 35 40 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 Frequency (Ghz) 42
H x 9. ELECTROMAGNETIC APPLICATIONS FEM solution for frequency = 8.82 Ghz H y Hx 2 + H y 2 0 5 10 S 11 (db) 15 20 25 30 35 40 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 Frequency (Ghz) 43
H x 9. ELECTROMAGNETIC APPLICATIONS FEM solution for frequency = 9.58 Ghz H y Hx 2 + H y 2 0 5 10 S 11 (db) 15 20 25 30 35 40 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 Frequency (Ghz) 44
H x 9. ELECTROMAGNETIC APPLICATIONS FEM solution for frequency = 9.71 Ghz H y Hx 2 + H y 2 0 5 10 S 11 (db) 15 20 25 30 35 40 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 Frequency (Ghz) 45
9. ELECTROMAGNETIC APPLICATIONS Griding Techniques for the Waveguide Problem Our refinement technology incorporates: An hp-adaptive algorithm Low dispersion error Small h is not enough Large p required Waveguide example: p 3 A two grid solver Convergence of iterative solver Insensitive to p-enrichment (1 p 4) Coarse grid sufficiently fine Waveguide example: λ/h 9 Limitations of the hp-strategy for wave propagation problems: We need large p and small h. 46
9. ELECTROMAGNETIC APPLICATIONS Griding Techniques for the Waveguide Problem Does convergence (or not) of the two grid solver depends upon h and/or p? How? Convergence of two grid solver p = 1 p = 2 p = 3 p = 4 Nr. of elements per λ = 7, 13 YES YES YES YES Nr. of elements per λ = 7, 11 NO NO NO YES Nr. of elements per λ = 6, 13 NO NO NO NO Convergence (or not) of the two grid solver is (almost) insensitive to p-enrichment. 47
9. ELECTROMAGNETIC APPLICATIONS Griding Techniques for the Waveguide Problem Estimate of the relative error (%) 10 3 10 2 10 1 10 0 Convergence history for different initial grids p=1, 1620 Initial grid elements p=2, 1620 Initial grid elements p=3, 1620 Initial grid elements p=1, 27 Initial grid elements p=2, 27 Initial grid elements p=3, 27 Initial grid elements p=4, 27 Initial grid elements 10 1 125 1000 3375 8000 15625 27000 42875 64000 91125 Number of unknowns (algebraic scale) Conclusion : We need to control the dispersion error. 48
9. ELECTROMAGNETIC APPLICATIONS Griding Techniques for the Waveguide Problem Estimate of the relative error (%) 10 3 10 2 10 1 10 0 Convergence history for different initial grids p=1, 1620 Initial grid elements p=2, 1620 Initial grid elements p=3, 1620 Initial grid elements p=1, 27 Initial grid elements p=2, 27 Initial grid elements p=3, 27 Initial grid elements 10 1 125 1000 3375 8000 15625 27000 42875 64000 91125 Number of unknowns (algebraic scale) Conclusion : Do we need to control the dispersion error? 49
10. CONCLUSIONS AND FUTURE WORK Exponential convergence is achieved for real world problems by using a fully automatic hp-adaptive strategy. Multigrid for highly nonuniform hp-adaptive grids is an efficient iterative solver. It is possible to guide hp-adaptivity with partially converged solutions. There is a compromise between large p and small h on the design of the initial grid. This numerical method can be applied to a variety of real world EM problems.
10. CONCLUSIONS AND FUTURE WORK Completed tasks Designed and implemented a 2D and 3D version of the two grid solver for elliptic problems. Studied numerically the 2D and 3D versions of the two grid solver. Designed, studied and implemented a two grid solver for 2D Maxwell s equations. Future Tasks Solve the 3D Fickera problem using hpadaptivity and the two grid solver. Completion date NOV 2003 Studied and designed an error estimator for a two grid solver for Maxwell s equations. Studied performance of different smoothers (in context of the two grid solver) for Maxwell s equations. Designed, studied, and implemented a flexible CG/GMRES method that is suitable to accelerate the two grid solver for Maxwell s equations. Developed a convergence theory for all algorithms mentioned above. Applied the hp-adaptive strategy combined with the two grid solver in order to solve a number of problems related to waveguide filters design, and modeling of LWD electromagnetic measuring devices. Implement and study a two grid solver for 3D Maxwell s equations. Utilize this technology to solve a 3D model problem related to Radar Cross Section (RCS) analysis. Write and defend dissertation. DEC 2003 JAN 2004 MAR 2004 51