Numerical and symbolic MOR techniques using hierarchical circuit structure

Numerical and symbolic MOR techniques using hierarchical circuit structure

Numerical and symbolic MOR techniques using hierarchical circuit structure COMSON Autumn School Future Developments in Model Order Reduction September 21-25, 2009, Terschelling, The Netherlands Oliver Schmidt

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Overview Symbolic Analysis and Reduction Techniques Hierarchical Modelling, Reduction, and Examples An Algorithm for the Exploitation of the System Structure Example: Operational Amplifier Outlook Slide 4

Nonlinear Symbolic Analysis simplified DAE system original DAE system netlist description behavioral model reference simulation original simplified Slide 5

Symbolic Model Reduction specifications: error control: goal: inputs u, outputs v, numerical analysis A (AC, transient, ) error boundû, error function E find DAE system G with reduced complexity and defined accuracy DAE F R1 R 2 R k DAE G v F = A( F, u) E( v, F vg ) < ε v G = A( G, u) Slide 6

Symbolic Reduction Techniques algebraic manipulations: elimination of variables, removal of independent blocks of equations x 0 = = f ( y) g( x, y) ( f ( y y) 0= g ), term reductions: cancellation of terms, applicable to nested expressions (level concept) F j : N i= 1 t ( x) = 0 i G j N : ti ( x ɶ ) = 0 k i= 1 Slide 7

Symbolic Reduction Algorithm Slide 8

Overview Symbolic Analysis and Reduction Techniques Hierarchical Modelling, Reduction, and Examples An Algorithm for the Exploitation of the System Structure Example: Operational Amplifier Outlook Slide 9

Hierarchical Modelling system is composition of building blocks/subsystems with connection structure subsystems themselves composed of different components ( levels) hierarchical reduction: - reduce subsystems singly - replace subsystems by obtained reduced models - obtain entire reduced system entire system 1 1 1 1 2 2 2 2 f ( x, y, z ) = 0 f ( x, y, z ) = 0 f 3 f ( x 0 3 ( x, y 3 1,, z 3 y 1, x 2, y 2, x, x ) = 0 f 3, y 3 4, y 4 ( x 4 4 ) = 0 subsystem 1 subsystem 2 subsystem 3 subsystem 4, y 4, z 4 ) = 0 netlist-based model behavioral model (DAE) distributed model (PDE) Slide 10

Differential Amplifier Circuit transmission lines between sources and circuit components discretized PDE model with 20 line segments / discretization points 12 V original system: 167 equations 645 terms (level 0) -12 V Slide 11

Non-hierarchical Reduction symbolic non-hierarchical reduction: error bound 2% and 10% error function E( v, v ) = sup v ( t) v ( t) F G F G t I 2% 10% approx. 2h 10min 124 equations, 425 terms (level 0) 44 equations, 284 terms (level 0) Slide 12

A modelling approach is aspired that transmits this information into the set of equations! Intuitive Hierarchy intuitive hierarchy standard graph theoretical methods like MNA / STA lose this information DUT DUT2 Slide 13 Rleitung1

Slide 14 sub sub sub subsystem system system system a b c d test test test test bench bench bench bench sub sub sub subsystem system system system sub sub sub subsystem system system system sub sub sub subsys sys sys sys. sub sub sub subsys sys sys sys. Subsystem Reduction Workflow

Hierarchical Reduction hierarchical reduction: numerical reduction via state space form and Arnoldi s algorithm within seconds: 50 8 resp. 4 equations symbolic reduction within seconds (2% error permitted): 16 9 equations, 59 20 terms symbolic reduction within seconds (2% error permitted): 22 13 equations, 91 50 terms Almost no changes if 10% error is permitted! Slide 15

Results replace subsystems by reduced models: 167 62 equations, 645 252 terms Mostly dependent on the numerical reduction of the transmission lines! usage of more modes for transmission line 1 ( 8 12 equations): 167 66 equations, 645 396 terms Slide 16

Model Check three other inputs V1 Slide 17

Overview Symbolic Analysis and Reduction Techniques Hierarchical Modelling, Reduction, and Examples An Algorithm for the Exploitation of the System Structure Example: Operational Amplifier Outlook Slide 18

Subsystem Sensitivity entire system 1. provide k error bounds 2. choose appropriate reduction methods for all subsystems S 3. reduce subsystem S k- times allowing all of the k error bounds 4. each time check errors on the entire system sensitivity Sε 1 S { ε ε } 1,..., k... Sε k entire system 5. repeat this for all subsystems Slide 19

Algorithm exploitation of the system s hierarchy Slide 20

Overview Symbolic Analysis and Reduction Techniques Hierarchical Modelling, Reduction, and Examples An Algorithm for the Exploitation of the System Structure Example: Operational Amplifier Outlook Slide 21

Example operational amplifier op741 Slide 22

Example operational amplifier op741 CM1 CM3 DP LS CM4 CM2 DAR PP Slide 23

OpAmp741 transient input Vid: sine wave, 0.8V maximum amplitude, 1kHz frequency original system: non-hierarchical reduction: 215 equations, 1050 terms (level 0) 61 equations, 495 terms (level 0) Slide 24 error error: error 10% 10% 10% 10% not not not not always always always always feasible feasible feasible feasible! > > > > 12h 12h 12h 12h!!!!!!!!!!!! perm. error perm perm perm

Error Functions f g : = max min f ( x ) g( x) i x U ε ( x ) i i f g f g f g : = max f ( x ) g( x ) i i i ( ) Slide 25

Subsystem Sensitivities error sweep: reduction method: sw = { 1, 2,5,10, 20,30, 40,50, 60, 70,80,90,100} [%] term cancellation (level 0), algebraic manipulations eqns. eqns. eqns. 5 6 8 4 3 5 4 3 7 6 5 4 2 2 3 1 1 5 10 20 30 40 50 60 70 80 90 100 error 1 1 5 10 20 30 40 50 60 70 80 90 100 error 2 1 1 5 10 20 30 40 50 60 70 80 90 100 error terms terms terms 14 12 10 8 6 4 2 CM1 18 16 14 12 10 8 6 4 2 CM2 30 27 24 21 18 15 12 9 6 3 CM3 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error Slide 26

Subsystem Sensitivities error sweep: reduction method: sw = { 1, 2,5,10, 20,30, 40,50, 60, 70,80,90,100} [%] term cancellation (level 0), algebraic manipulations eqns. 18 16 14 12 10 8 6 4 2 eqns. 40 36 32 28 24 20 16 12 8 4 eqns. 21 18 15 12 9 6 3 eqns. 24 21 18 15 12 9 6 3 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error terms 120 100 80 DAR terms 300 250 200 DP terms 120 100 LS terms 160 140 120 PP 60 150 80 60 100 80 40 20 100 50 40 20 60 40 20 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error 1 5 10 20 30 40 50 60 70 80 90 100 error Slide 27

Results subsystem perm. error # equations CM1 5 50% 20 5 original system: 215 equations, 1050 terms (level 0) CM2 CM3 DAR 1 90% 50 60% 20 90% 21 6 30 8 33 8 DP 10 40% 72 22 (23) hierarchical reduction: 132 equations, LS PP 10 60% 40 90% 39 10 42 13 approx. 1h 48min 336-356 terms (level 0) further non-hierarchical reduction: approx. 17min perm. error: 10% 34 equations, 93 terms (level 0) Slide 28

Model Check three other inputs Vid Slide 29

Overview Symbolic Analysis and Reduction Techniques Hierarchical Modelling, Reduction, and Examples An Algorithm for the Exploitation of the System Structure Example: Operational Amplifier Outlook Slide 30

Outlook subsystem ranking 1. not clear how to spread the entire system s error bound over the subsystems 2. if error check fails: not totally clear which subsystem s error bound has to be altered error function system segmentation ranking / sensitivity analysis results depend on the choice of the error function how to get / what is a good segmentation of the entire system into subsystems? Slide 31

Thank you for your attention. The work presented in this talk has been carried out within the project SyreNe (Systemreduktion für IC Design in der Nanoelektronik, grant no. 03LAPAE6) which is supported by the German Federal Ministry of Education and Research (BMBF). Slide 32