Towards parametric model order reduction for nonlinear PDE systems in networks

Transcription

1 Towards parametric model order reduction for nonlinear PDE systems in networks MoRePas II 2012 Michael Hinze Martin Kunkel Ulrich Matthes Morten Vierling Andreas Steinbrecher Tatjana Stykel Fachbereich Mathematik Universität Hamburg October 4, 2012

2 page 1 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

3 page 2 Motivation: Coupled circuit and semiconductor models Aim Accurate reduced order models for semiconductors in networks Validity over relevant parameter range Accurate physical reduced order model of the coupled system

4 page 3 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

5 Parametric MOR for PDEs in networks page 4 Coupled circuit and semiconductor models [M. Günther 01, C. Tischendorf 03] Kirchhoff s laws (no semiconductors) read +12 V Aj = 0, v = A e R2 R3 C2 A: incidence matrix. C1 Voltage-current relations of components: vin R1 R5 R j C = dq C dt (v C, t), j R = g(v R, t), v L = dφ L (j L, t) dt R4 C3 Modified Nodal Analysis: join all equations to DAE system A C dq C dt ( ) ( ) A C e(t), t + A R g A R e(t), t + A L j L (t) + A V j V (t) = A I i s(t), dφ L dt (j L (t), t) A L e(t) = 0, A V e(t) = v s(t).

6 page 5 Coupled circuit and semiconductor models [M. Günther 01, C. Tischendorf 03] How can semiconductors be introduced? replace semiconductor by a (possibly nonlinear) electrical network, stamp semiconductor network into surrounding network, apply Modified Nodal Analysis. Here: use PDE model for semiconductors DD equations.

7 Parametric MOR for PDEs in networks page 6 Coupled circuit and semiconductor models [M. Günther 01, C. Tischendorf 03] PDE-model (drift-diffusion equations) for semiconductors div (ε ψ) = q(n p C), q tn + div J n = qr(n, p), q tp + div J p = qr(n, p), J n = µ nq( U T n n ψ), J p = µ pq( U T p p ψ), on Ω [0, T ] with Ω R d (d = 1, 2, 3). Dirichlet boundary constraints at Γ O,k : Γ Γ Ο,1 Ο,2 ψ(t, x) = next slide, n(t, x) = ñ(x), p(t, x) = p(x) and Neumann boundary constraints at Γ I : Γ Ι + ψ(t, x) ν(x) = J n ν(x) = J p(t, x) ν(x) = 0 Γ Ο,3 Ω or mixed boundary conditions at MI contacts (MOSFETs).

8 page 7 Couple semiconductor to circuit [M. Günther 01, C. Tischendorf 03]

9 page 8 Couple semiconductor to circuit [M. Günther 01, C. Tischendorf 03] Coupling conditions: Γ Γ Ο,1 Ο,2 j S,k (t) = (J n + J p ε t ψ) ν dσ, Γ O,k Γ Ι + ψ(t, x) = ψ bi (x) + (A S e(t)) k for (t, x) [0, T ] Γ O,k, Γ Ο,3 Ω and add current j S to Kirchhoff s current law: A C dq C dt ( ) ( ) A C e, t + A R g A R e, t + A L j L + A V j V +A S j S = A I i s, dφ L dt (j L, t) A L e = 0, A V e = v s. Add DD-equations + coupling conditions for each semiconductor.

10 page 9 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

11 page 10 Mixed formulation The electric field E = ψ plays dominant role in DD-equations. Mixed formulation [Brezzi et al. 05] Provide additional variable g ψ and equation g ψ = ψ. Scaled DD equations then read: λ div g ψ = n p C, tn + ν n div J n = R(n, p), tp + ν p div J p = R(n, p), g ψ = ψ, J n = n ng ψ, J p = p pg ψ.

12 page 11 Finite Element approximation Finite elements piecewise constant ansatz functions for ψ, n and p. Basis functions: ϕ i, i = 1,..., N, N = T. Raviart-Thomas elements for g ψ, J n and J p. Basis functions: φ j, i = 1,..., M, M = E E N. RT 0 := {y : Ω R d : y T (x) = a T + b T x, a T R d, b T R, [y] E ν E = 0, for all inner edges E}. Galerkin ansatz: ψ h (t, x) = N ψ i (t)ϕ i (x), gψ(t, h x) = i=1 M g ψ,j (t)φ j (x), j=1 and analogously for n, p, J n, and J p.

13 Parametric MOR for PDEs in networks page 12 Full model +12 V C1 R2 R3 C2 A C dq C dt ) ) (A C e(t), t + A R g (A R e(t), t +A L j L (t) + A V j V (t)+a S j S (t) = A I i s(t), vin R1 R4 R5 C3 R dφ L dt (j L (t), t) A L e(t) = 0, A V e(t) = vs(t), j S (t) C 1 J n(t) C 2 J p(t) C 3 ġ ψ (t) = 0, 0 ψ(t) M L ṅ(t) n(t) M L ṗ(t) p(t) + A 0 FEM + F(n h, p h, g g ψ (t) ψ h ) b(a S e(t)) = 0. 0 J n(t) 0 J p(t)

14 page 13 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

15 page 14 Basic test circuit, simulation results e 1(t) j V (t) e 2(t) v s(t) ψ(t, x) n(t, x) p(t, x) R input voltage: v s(t) = 5[V ] sin(2πf t) similar results obtained by MECS [Selva Soto] 0.2 frequency = 1 MHz 0.2 frequency = 1 GHz 0.2 frequency = 5 GHz current j V (t) [ma] current j V (t) [ma] current j V (t) [ma] scaled time scaled time scaled time

16 page 15 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

17 page 16 Snapshot-POD (Proper Orthogonal Decomposition) [L. Sirovich 87] Full simulation yields snapshots (here: y = ψ, n, p,...) { y(ti, ) } i=1,...,m span{ ϕ j } j=1,...,n, with y(t i, x) = N y j (t i )ϕ j (x). j=1 Gather coefficients in matrix Y := ( y(t 1 ),..., y(t m) ) R N m. POD in Hilbert space X as eigenvalue problem: Kv k = σ 2 k v k, with K ij := y(t i, ), y(t j, ) X. Note that K = Y MY with M ij = ϕ i, ϕ j X. Write POD in terms of SVD: Then, the s-dimensional POD basis is { u i := N j=1 ŨΣṼ = L Y, with LL := M. } u j i ϕ j ( ), U := ( u 1,..., u s ) := L Ũ (:,1:s). i=1,...,s

18 page 17 Model Order Reduction Simulate the complete network at one or more reference parameters. Take snapshots of the state of each semiconductor at time points t i. Perform POD component wise on ψ, n, p, g ψ, J n and J p. Use the POD basis functions as (non local) FEM ansatz functions: ψ POD (t, x) = s γ ψ,i (t)uψ(x) i i=1 1D FEM ansatz functions for J n first 5 POD basis functions for J n

19 page 18 Reduced model +12 V C1 R2 R3 C2 A C dq C dt ) ) (A C e(t), t + A R g (A R e(t), t +A L j L (t) + A V j V (t)+a S j S (t) = A I i s(t), vin R1 R4 R5 C3 R dφ L dt (j L (t), t) A L e(t) = 0, A V e(t) = vs(t), j S (t) C 1 U Jn γ Jn (t) C 2 U Jp γ Jp (t) C 3 U gψ γ gψ (t) = 0, 0 γ ψ (t) γ n(t) γ n(t) γ p(t) γ p(t) + A 0 POD + U F(n POD, p POD, g γ gψ (t) ψ POD ) U b(a S e(t)) = 0. 0 γ Jn (t) 0 γ Jp (t)

20 page 19 Computational complexity Computational complexity of reduced model still depends on n FEM : U F(n POD, p POD, gψ POD ) = }{{} U }{{} F ( U n γ }{{} n, U pγ p, U gψ γ gψ ). n POD n FEM n FEM n FEM n POD With matrix-matrix multiplications in Jacobian computation: U }{{} n POD n FEM, block-dense F (...) }{{} n FEM n FEM, sparse }{{} U. n FEM n POD, block-dense

21 page 20 Discrete Empirical Interpolation Md. (DEIM) [S. Chaturantabut, D. Sorensen 09] DEIM Do POD on snapshots {F(n(t i ), p(t i ), g ψ(t i ))}, obtain basis W R n FEM n DEIM (block diagonal matrix). Ansatz is overdetermined. Select n DEIM useful rows: F(U nγ n(t), U pγ p(t), U gψ γ gψ (t)) Wc(t) P F(...) P Wc(t). If P W is regular: F(...) Wc(t) = W (P W ) 1 P F(...) The regularity of P W can be guaranteed, see [CS09]. Again we apply the method component-wise.

22 page 21 Discrete Empirical Interpolation Md. (DEIM) [S. Chaturantabut, D. Sorensen 09] Reduced model with DEIM: U F(U nγ n, U pγ p, U gψ γ gψ ) (U W (P W ) 1 ) }{{} n POD n DEIM, block-dense P }{{ F } (U nγ n, U pγ p, U gψ γ gψ ) }{{} n DEIM n FEM Results for 1D-diode: n FEM FEM n POD ROM n DEIM ROM + DEIM sec sec sec sec sec sec sec sec sec. order nfem 1.18 nfem 1.10 nfem 0.578

23 page 22 Discrete Empirical Interpolation Md. (DEIM) [S. Chaturantabut, D. Sorensen 09]

24 page 23 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

25 page 24 Reduced model depends on position of diode in network Bridge rectifier with 4 diodes: e 2(t) 6 Input: v s (t) Output: e 3 (t) e 1 (t) S 1 S 2 4 v s(t) e 1(t) R e 3(t) potential [V] 2 0 S 4 S time [sec] x 10 9

26 page 25 Reduced model depends on position of diode in network The distance between the spaces U 1 and U 2 which are spanned, e.g., by the POD-functions Uψ 1 of the diode S 1 and Uψ 2 of the diode S 2 respectively, is measured by d(u 1, U 2 ) := max u U 1 u 2 =1 min u v 2 = 2 2 λ, v U 2 v 2 =1 where λ is the smallest eigenvalue of the positive definite matrix SS with S ij = u 1 ψ,i, u 2 ψ,j 2. d(u 1, U 2 ) d(u 1, U 3 ) Table: Distances between reduced models in the rectifier network.

27 page 26 Modes MOR yields a similar but different model for the diodes S 1 and S 2 : J n POD basis functions (modes) 0 L mode 1 0 L mode 2 0 L mode 3 0 L mode 4

28 page 27 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

29 page 28 Problem setting MOR test problem Basic circuit with frequency f of the voltage source v s(t) = 5[V ] sin(2πf t) as model parameter. e1(t) jv (t) e2(t) vs(t) ψ(t, x) n(t, x) p(t, x) R Lack of information Select number of snapshots so that (s) = m i=s+1 σ2 i mi=1 σ 2 i tol.

30 page 29 Reduced model at a fixed frequency First test: Compare reduced and unreduced system at a fixed frequency. relative error simulation time [sec] reduced simulation full simulation (s) (s)

31 page 30 Reduced model over parameter space Construction of reduced model requires snapshots from full simulations at reference parameters. Is the model valid over a large parameter space? reference parameter: P 1 := {f 1 } := {10 10 [Hz]} parameter space P = [10 8, ] 10 2 error 10 1 error reference frequencies parameter (frequency)

32 page 31 Reduced model over parameter space - sampling Goal Find new sampling parameter f k+1 (reference frequency) without simulating the full, unreduced system. Set P k+1 := P k {f k+1 }. We do not consider the PDE discretization error. Rigorous upper bound for the error not available E(f ; P k ) = y h (f ) y POD (f ; P k )?(s) where y h := (ψ h, n h, p h, g h ψ, J h n, J h p ), y POD := (ψ POD, n POD,...). Rigorous RB methods, Greedy algorithm [see e.g. A. Patera, G. Rozza 07]: a-posteriori error estimates required. Linear ODEs [see e.g. B. Haasdonk, M. Ohlberger 09]: build difference between residual and unreduced equation to derive an ODE for the error.

33 page 32 Residual based sampling Define residual R(z POD (f ; P k )): insert z POD (f ; P k ) into unreduced equation, 0 ψ POD (t) M L ṅ POD (t) n POD (t) M L ṗ POD (t) p POD (t) R := + A 0 FEM gψ POD(t) + F(n POD, p POD, g ψ POD ) b(epod (t)). 0 Jn POD (t) 0 Jp POD (t) Residual admits different scales. Scale with block diagonal matrix-valued function D(f ) := diag( d ψ(f )I, d n(f )I, d p(f )I, d gψ (f )I, d Jn (f )I, d Jp (f )I ) and choose d ψ(f ) according to d ψ(f j ) R ψ(y POD (f j ; P k )) = ψh (f j ) ψ POD (f j ; P k ), f ψ h j P k. (f j )

34 page 33 Residual based sampling Algorithm: sampling 1. Select f 1 P, P test P, tol > 0, and set k := 1, P 1 := {f 1 }. 2. Simulate the unreduced model at f 1 and calculate the reduced model with POD basis functions U Calculate weight functions d ( ) (f ) > 0 for all f P k. 4. Calculate the scaled residual D(f )R(z POD (f, P k )) for all f P test. 5. Check termination conditions, e.g. max f Ptest D(f )R(z POD (f, P k )) < tol, no progress in weighted residual. 6. Calculate f k+1 := arg max f Ptest D(f )R(z POD (f, P k )). 7. Simulate the unreduced model at f k+1 and create a new reduced model with POD basis U k+1 using also the already available information at f 1,..., f k. 8. Set P k+1 := P k {f k+1 }, k := k + 1 and goto 3.

35 page 34 Numerical example - sampling step 1 Let f 1 := [Hz], P 1 := {10 10 [Hz]}, P = [10 8, ] sampling step error residual reference frequencies parameter (frequency) f 2 = arg max f Ptest D(f )R(z POD (f, P 1 )) = 10 8 [Hz] P 2 = {10 8 [Hz], [Hz]}

36 page 35 Numerical example - sampling step 2 P 2 = {10 8 [Hz], [Hz]} 10 2 sampling step error residual reference frequencies parameter (frequency) f 3 = arg max f Ptest D(f )R(z POD (f, P 2 )) = [Hz] P 3 = {10 8 [Hz], [Hz], [Hz]}

37 page 36 Numerical example - sampling step 3 P 3 = {10 8 [Hz], [Hz], [Hz]} 10 2 sampling step error residual reference frequencies parameter (frequency) Terminate with no progress in residual.

38 page 37 Outline Motivation PDAE-model Finite Element Method Simulation results Construction of the reduced model Location dependence of reduced model Residual based parameter sampling PABTEC and POD, joint work with A. Steinbrecher & Tatjana Stykel Next steps

39 Parametric MOR for PDEs in networks page 38 Combination of PABTEC (Reis & Stykel 2010) and POD; joint work with [A. Steinbrecher, T. Stykel] +12 V subproject V R3 C2 R3 C2 R2 R2 C1 C1 R5 R R5 R vin R1 vin R1 R4 C3 R4 C3 subproject V subproject V vin vin

40 page 39 Combination of PABTEC and POD; Int. J. Numer. Model. 2012

41 page 40 Substitution of nonlinear components for PABTEC and recoupling A. Steinbrecher, T. Stykel (Int. J. Circuits Theory Appl., 2012): Nonlinear inductor current source Nonlinear capacitor voltage source Nonlinear resistor linear circuit with 2 serial resistors and one voltage source parallel to one of the resistors

42 page 41 Combination of PABTEC and POD; Int. J. Numer. Model input in V / output in A time net full / diode full net red / diode full net full / diode red net red / diode red input u V (t)

43 page 42 Next steps Include QDD models. Include EM effects. Generalize approach to other equation networks containing simple and complex components. Thank you for attending! The work reported in this talk is supported by the German Federal Ministry of Education and Research (BMBF), grants 03HIPAE5 & 03MS613D.