A Time Marching Scheme for Solving Volume Integral Equa9ons on Nonlinear Sca<erers

Transcription

1 A Time Marching Scheme for Solving Volume Integral qua9ons on Nonlinear Sca<erers Hakan Bağcı ivision of Computer, lectrical, and Mathema9cal Sciences and ngineering (CMS) KAUST SRI Center for Uncertainty Quan9fica9on in Computa9onal Science and ngineering King Abdullah University of Science and Technology (KAUST) SRI UQ Annual Mee9ng January 6, 2015 KAUST, Thuwal , Saudi Arabia

2 Outline Mo9va9on: Need for Uncertainty Quan9fica9on (UQ) in lectromagne9c Simula9ons UQ Framework Surrogate Model Assisted Monte Carlo Method Probabilis9c Colloca9on Method Mul9 lement Probabilis9c Colloca9on Method Time omain eterminis9c Simulator for Kerr- nonlinear Sca<erers Time omain Volume Integral qua9on (TVI) Solver TVI Formula9on iscre9za9on xplicit Marching- on- in- Time (MOT) Solu9on Numerical Results

3 Mo9va9on: UQ for lectromagne9c Simula9ons lectromagne9c simulators are needed in a broad range of applica9ons MC/MI characteriza9on Wireless network design in harsh environments evice design at microwave and op9cal frequencies Almost all these systems are replete with uncertain9es Fabrica9on and installa9on ambigui9es Contamina9on of materials Approxima9ons in cons9tu9ve models dispersion models, homogeniza9on techniques, etc. Noise sources State- of- the- art electromagne9c simulators Large scale problems (paralleliza9on, adap9ve meshing and 9me stepping, FMM/FFT accelera9on) Mul9- scale physics (high/low frequency, geometry details, hybridiza9on of different equa9ons, precondi9oners) Objec9ve: fficient UQ techniques in electromagne9c simula9on frameworks to allow for design and characteriza9on of complex electromagne9c and photonic systems

4 UQ Framework Surrogate Model Assisted Monte Carlo Method Probabilis9c Colloca9on Method Mul9 lement Probabilis9c Colloca9on Method with: Abdulkadir Yucel and ric Michielssen (University of Michigan)

5 Uncertainty Quan9fica9on Framework x parameterizes N dof random variables distributed with an assumed/known pdf W (x) and represents a set of M observables cable positions/bend/electrical characteristics, source locations (ext/int. generated MI), angle of arrival (ext. generated MI), x = [x 1,x 2,,x N dof ] = source strengths/signatures (int. generated MI), incident field polarization (ext. generated MI), widths and lengths of interconnects, constitutive parameters of substrates, electronic system positions/parameters, N dof W (x) = w(x i ) i=1 x i i = a i,b i N = = dof i 1 i V (x) = { -field(x), H-field(x), Power(x)} Sta9s9cal moments of M observable V (x) V (x) = V (x)w (x)dx ( ) 2 W (x)dx std V (x) = V 2 (x) V (x) s9ma9ng sta9s9cal informa9on of V (x) via Monte Carlo (MC) method

6 Uncertainty Quan9fica9on Framework irect Monte Carlo Method Random Variables pdf of x Monte Carlo M Binning Method x Solver i x V (x i ) Mean Std. dev pdf of V (x) i = 1,..., N MC irect Monte Carlo (MC) Method Slow convergence is high Very costly since each N MC V (xi ) is a computer simula9on

7 Uncertainty Quan9fica9on Framework Surrogate Model Assisted Monte Carlo Method Random Variables pdf of x x x Monte Carlo Method x i Surrogate Model V (xi ) Binning Mean Std. dev pdf of V (x) i = 1,..., N MC p = 1,..., N SM N SM N MC x k V (x k ) M Solver Surrogate model assisted MC Method ras9cally cheaper if N SM N MC Accurate pdfs if V (x) V (x)

8 Uncertainty Quan9fica9on Framework Probabilis9c Colloca9on Method The probabilis9c colloca9on (PC) generates surrogate model by p th - order generalized polynomial chaos (gpc) expansion N p V (x) V (x) = v m P m (x), N p = ( N dof + p)! ( N dof!p!) 1 P m (x) is the mul9variate orthogonal polynomial basis selected due to W (x) m=0 Legendre polynomials are selected as polynomial bases regardless of W (x) Invert this rela9on to get the coefficients v m = To numerically evaluate mul9dimensional integral Tensor- product (TP) integra9on rule Sparse grid (SG) integra9on rule v m V (x)p m (x)dx = V (x k )P m (x k )α k, m = 0,, N p N SM k=1 Observable Surrogate Model

9 Uncertainty Quan9fica9on Framework Probabilis9c Colloca9on Method The probabilis9c colloca9on (PC) generates surrogate model by p th - order generalized polynomial chaos (gpc) expansion N p V (x) V (x) = v m P m (x), N p = ( N dof + p)! ( N dof!p!) 1 P m (x) is the mul9variate orthogonal polynomial basis selected due to W (x) Legendre polynomials are selected as polynomial bases regardless of Invert this rela9on to get the coefficients To numerically evaluate mul9dimensional integral v m = Tensor- product (TP) integra9on rule Sparse grid (SG) integra9on rule m=0 v m V (x)p m (x)dx = V (x k )P m (x k )α k, m = 0,, N p N SM k=1 W (x) Observable Surrogate Model

10 Uncertainty Quan9fica9on Framework The Mul9- lement Probabilis9c Colloca9on Method The M- PC method* for efficiently handling func9ons with discon9nui9es, rapid, and slow varia9ons together Adap9vely refines (sub- ) domains based on the decay rates of observables local variances τ ( ) 1 γ = γ β j (varp, j var p 1, j) var p, j; j J Refinement is performed along dimensions that sa9sfy α j i i i τ 2 max α i=1,,ndof j ; α j = v j i (var p, j var p 1, j ) * X. Wan and G.. Karniadakis, An adap9ve mul9- element generalized polynomial chaos method for stochas9c differen9al equa9ons. J. Comput. Phys., 209 (2005), pp

11 eterminis9c Simulator- How to compute V(x i )? Circuit Solver MNA Linear elements Nonlinear elements Wave Propaga9on in Tunnels Field Solver 3 MoM FMM- FFT accelerated Parallel computa9on 2012 iffrac9on Gra9ngs Periodic Field Solver 3 T- G- FM FFT- accelerated ACs fficient 9me integra9on MC/MI Characteriza9on Cable Solver 1 MoM FFT accelerated Coaxial cables Mul9conductor High- contrast Sca<erers Transient Field Solver 3 TVI Band- limited interpola9on Stable extrapola9on Field Solver 3 MoM FFT accelerated Parallel computa9on Kerr- nonlinear Sca<erers (3) material Gold H Pre Transient Field Solver 3 TVI Nonlinear cons9tu9ve rela9on fficient explicit 9me marching ε(r,t) = ε 0 ( χ L + χ NL (r,t) ) 2

12 Time omain eterminis9c Simulator for Kerr- nonlinear Sca<erers Time omain Volume Integral qua9on (TVI) Solver TVI Formula9on iscre9za9on xplicit Marching- on- in- Time (MOT) Solu9on Numerical Results with: Sadeed B. Sayed and H. Arda Ulku (KAUST)

13 Introduc9on Time omain Volume Integral qua9on (TVI) Solvers are an alterna9ve to finite difference 9me domain (FT) and classical FM schemes Fields Spa9o- temporal convolu9ons of unknown current and Green func9on of the background medium Boundary/field condi9ons TVI TVIs are classically solved using Marching- on- in- 9me (MOT) schemes Local spa9al and temporal basis func9ons Galerkin tes9ng in space and point tes9ng in 9me Causal temporal basis func9ons Implicit 9me marching Advantages No phase dispersion (Green func9on is analy9cal) No grid trunca9on (exact radia9on condi9on) Time step size is not necessarily constrained by spa9al discre9za9on (no CFL condi9on) Only volume of the object is discre9zed Accurate representa9on of the geometry (higher- order spa9al discre9za9on)

14 Introduc9on Challenges Interac9ons are global in space High computa9onal cost Reduced using previously developed Plane Wave Time omain (PWT) scheme (9me domain version of FMM) Blocked FFT scheme (9me domain version of CG- FFT) Retarded 9me func9ons, f (r,t r r c 0 ), are integrated and tested in space Numerical stability calls for high accuracy xact or quasi exact integra9on methods Band- limited temporal interpola9on with con9nuous deriva9ves and stable extrapola9on schemes Applicability of the TVI solvers should be expanded Applica9on to sca<erers with nonlinear cons9tu9ve rela9on (i.e., material proper9es depend on field values) has been limited MOT schemes are implicit: ifficult to incorporate nonlineari9es xplicit MOT schemes are needed

15 Formula9on: TVI inc (r,t) Cons9tu9ve rela9on (r,t) = ε(r,t)(r,t) J(r,t) V ielectric Object For ε(r,t) = f ((r,t)) cons9tu9ve rela9on becomes nonlinear Volumetric sca<er with and residing in free space with V ε(r) ε 0 and µ 0 µ 0 Total volume: xcita9on: inc (r,t) band- limited to Current induced in : V J(r,t) f max Kerr nonlinearity ( ) ε(r,t) = ε 0 χ L + χ NL (r,t) 2 χ, χ are the first and third order L NL rela9ve permipvity terms

16 Formula9on: TVI inc (r,t) J(r,t) V ielectric Object Volumetric sca<er with and residing in free space with V ε(r) ε 0 and µ 0 µ 0 Total volume: xcita9on: inc (r,t) band- limited to Current induced in : f max Fields sa9sfy (TVI) t inc (r,t) = t (r,t) + L{(r,t)} V J(r,t) R = r r ε 0 1 L{(r,t)} Sca<ered electric field operator L{X(r,t)}= µ ε 0 0 4π + 1 4π 1 R V V 1 R 3 X( r,τ )d r t X( r,τ )d r t τ = t R c 0

17 Formula9on: MOT Solu9on To numerically solve the TVI Volume V is divided into tetrahedrons is expanded as (r,t) (r,t) = Unknowns: I n,i Temporal basis func9ons: T i (t) Spa9al basis func9ons: f f n (r) f n f (r) = N N t f i=0 n=1 ± A k ± 3V (r r ± ± ), r V k k k 0, elsewhere I n,i T i (t)f f n (r) ρ n + ρn

18 Formula9on: MOT Solu9on To numerically solve the TVI Volume V is divided into tetrahedrons is expanded as (r,t) (r,t) = Unknowns: I n,i Temporal basis func9ons: T i (t) Spa9al basis func9ons: f h n (r) f n h (r) = N N t h i=0 n=1 A k (r r 3V k ), r V k k 0, elsewhere I n,i T i (t)f h n (r) ρ n

19 Formula9on: MOT Solu9on Tes9ng TVI with MOT matrices Sparse Gram matrix f n h (r) at 9mes yields G t = V j inc Z 0 +ε 0 1 Z 0 P t + ε 0 1 j 1 i=0 t = jδt Z j i I i j 1 i=0 {Z j i } m,n = f m h (r) L{f m h (r)t i (t)} V m Z j i P t I i t= jδt dr MOT matrix system {G} m,n = Vm f m h (r) f n h(r)dr Tested electric field {V j }= f m h(r) inc (r,t)dr V m t= jδt

20 Formula9on: MOT Solu9on Tes9ng TVI with f h n (r) at 9mes yields t = jδt G t = V j inc Z 0 +ε 0 1 Z 0 P t + ε 0 1 j 1 i=0 Z j i I i j 1 i=0 Z j i P t I i MOT matrix system Unknown coefficients {I j } n = I n, j { t I j } n = t I n, j {I j } n = I n, j P Sparse matrix maps f h n (r) space to f f n (r) space

21 Formula9on: MOT Solu9on Tes9ng cons9tu9ve rela9on with f f n (r) at 9mes t = jδt yields PGP t = P G j Auxiliary equa9on Sparse Gram matrices {G} m,n = h f m Vm (r) f n h(r)dr { G h j } m,n = ε(r, jδt)f m (r) f n h(r)dr Vm Unknown coefficients {I j } n = I n, j {I j } n = I n, j iscre9zed permipvity ε(r, jδt) ε 0 χ L + χ N NL I n, j f h n (r) n=1 2

22 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m

23 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part V j fixed = V j inc j 1 i=0 Z j i I i + ε 0 1 j 1 i=0 Z j i P t I i

24 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict,0 = using predictor coefficients k l=1 {p} l 1+l k p +{p} k+l t 1+l k

25 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients ε(r, jδt) P G j Step 2 : Update and and solve for PGP t,0 = P G j,0 p

26 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving P G j p G t,0 = V j fixed Z 0,0 + ε 0 1 Z 0 P t,0

27 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving P G j Step 4 : Iterate un9l convergence (ν 1) p

28 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct,ν = k l=1 P G j (ν 1) p using corrector coefficients {c} l 1+l k +{c} k+l t 1+l k c +{c} 2k+1 t,ν 1

29 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct P G j using corrector coefficients ε(r, jδt) (ν 1) P G j Step 4.2 : Update and and solve for p PGP t,ν = P G j,ν c

30 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct using corrector coefficients Step 4.2 : Update and and solve for Step 4.3 : valuate ε(r, jδt) t P G j (ν 1) by solving P G j p G t,ν = V j fixed Z 0,ν + ε 0 1 Z 0 P t,ν c

31 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct using corrector coefficients Step 4.2 : Update and and solve for Step 4.3 : valuate ε(r, jδt) t P G j (ν 1) by solving P G j Step 4.4 : Apply successive over relaxa9on to p,ν = α,ν + (1 α ),ν 1 c

32 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct using corrector coefficients Step 4.2 : Update and and solve for Step 4.3 : valuate ε(r, jδt) t P G j (ν 1) by solving P G j Step 4.4 : Apply successive over relaxa9on to Step 4.5 : Check for convergence p c

33 Formula9on: MOT Solu9on Coupled matrix system is integrated in 9me using P(C) m At 9me step j th Step 0 : Compute the fixed part Step 1 : Predict using predictor coefficients Step 2 : Update and and solve for Step 3 : valuate ε(r, jδt) t by solving Step 4 : Iterate un9l convergence Step 4.1 : Correct using corrector coefficients Step 4.2 : Update and and solve for Step 4.3 : valuate ε(r, jδt) t P G j (ν 1) by solving P G j Step 4.4 : Apply successive over relaxa9on to Step 4.5 : Check for convergence Step 5 : Store results and march on p c

34 Formula9on: MOT Solu9on Highlights First formula<on and implementa<on of an MOT- TVI solver for nonlinear sca<erers Proposed MOT scheme is quasi- explicit Gram matrices are sparse and well- condi9oned regardless of 9me step size and frequency Matrix inversion is carried out very efficiently using an itera9ve solver More efficient than implicit solver at low frequencies Previously developed PWT and blocked FFT schemes can s9ll be used A Newton- type nonlinear solver is not required Nonlinearity is accounted for as only a func9on evalua9on

35 Numerical xamples: Four wave mixing Sca<erer: ielectric sphere of unit radius ε(r,t) = ε 0 (χ L + χ NL (r,t) 2 ) χ L = 1.5, χ NL = xcita9on: inc (r,t) = ŷg(t r ẑ c 0, f 0 ) G(t, f ) = cos 2π f (t t p ) exp[ (t t p )2 / 2σ 2 ] f 0 = 12.5MHz, f bw = 2.5MHz, σ = 3/ (2π f bw ), t p = 6σ 2 m inc (r,t) ẑ ŷ f 0 3 f0

36 Numerical xamples: Four wave mixing Sca<erer: ielectric sphere of unit radius ε(r,t) = ε 0 (χ L + χ NL (r,t) 2 ) χ L = 1.5, χ NL = xcita9on: inc (r,t) = ŷ 0.75G(t r ẑ c 0, f 1 ) +0.5G(t r ẑ c 0, f 2 ) f 1 = 12.5MHz, f 2 = 8MHz, f bw = 2.5MHz 2 m inc (r,t) ẑ ŷ f 2 f 1 2 f 2 f 1 2 f 1 f 2

37 On- Going and Future Work More work on the MOT scheme for nonlinear sca<erers Time- step size selec9on Valida9on of the accuracy Computa9onal efficiency Applying the MOT- TVI solver to more realis9c nonlinear structures MOT scheme for 9me domain surface integral equa9ons + dispersive cons9tu9ve rela9on for plasmonic interac9ons (Poster session) MOT scheme for 9me domain impedance boundary condi9ons + dispersive cons9tu9ve rela9on for graphene (Poster session) xplicit solver for volume magne9c field integral equa9on (Poster session) UQ framework with mul9- level Monte Carlo method (Poster session)

38 CM Research Group at KAUST People and Facili9es Ph Students Abdulla esmal (Jan May 2015): Sparse reconstruc9on techniques for nonlinear electromagne9c imaging Ismail Uysal (Sep May 2015): Quantum- corrected 9me domain surface integral equa9on solvers for plasmonics Sadeed B. Sayed (Sep May 2016): Highly stable 9me domain volume integral equa9ons for dielectric sca<erers Ali Imran Sandhu (Sep May 2017): Time domain surface integral equa9ons solvers for analyzing graphene structures Noha Alhar9 (June ec. 2017): A parallel FMM- accelerated Nystrom solver for electromagne9cs Post- octoral Researchers Arda Ulku (Aug Jan. 2015) Yifei Shi (May May 2015) Mohamed Farhat (Sep Sep. 2015) Ping Li (Sep Sep 2015) Alumni Muhammad Amin (Ph degree, Sep May 2014): Assistant Professor, lectrical ngineering, Sarhad University of Science and Informa9on Technology (SUIT), Peshawar, Pakistan Abdulla esmal (MS degree, Sep ec. 2010): Ph student at KAUST Umair Khalid (MS degree, ec. 2010): MBA Student, Strategy and Finance, mory University, Atlanta, GA. Kostyantyn Sirenko (Post- oc) (February August 2014) Mohamed Salem, (Post- oc) (Mar Mar. 2013): Post- doctoral Researcher, Poly- Grames Research Center, École Polytechnique de Montréal, Montréal, Canada Ahmed Al- Jarro, (Post- oc) (Nov Nov. 2012): Research Scien9st, lectronic and lectrical ngineering, University College London, London, United Kingdom Meilin Liu, (Post- oc) (Jan Jan. 2013): Satellite Communica9ons Researcher, Shanghai Ins9tute of Satellite ngineering, Shanghai City, China