from a finite element model to system level simulation by means of

From a finite element model to

system level simulation by

means of model reduction

MUMPS Users Meeting 2010, Toulouse

Evgenii Rudnyi, CADFEM GmbH

[email protected]

http://ModelReduction.com

Content

System level simulation and

compact modeling

Model order reduction

Case studies with MOR for

ANSYS

Experience with MUMPS

solver

1

Device and System Level Simulation

How to make a model at system level from that at device level?

2

Thermal Domain

Electrical Domain MechanicalDomain

0

R1 R2 R3

MASS_ROT1

DR1

SINE2

SINE1

A

A

A

IN_A

IN_B

IN_C

OUT_A

OUT_B

OUT_C

I_motSimplorer4

A

B

C

N

ROT1

ROT2

Induction_Motor_20kW

Q

Ambient

P1

P2

P3

P4

P5

P6

P7

P8 P

9

P1

0

P1

1

P1

2

P_

RE

F

z_um z_vm z_wm

z_wpz_vpz_up

+

V

VM311 R6 R5 R4

E1

RZM

DR1

RZM

0.00 2.50 5.00 7.50 10.00 12.50 15.00 17.50 20.00Time [ms]

-500.00

-250.00

0.00

250.00

500.00

Y1 [

V]

Curve Info rms

R1.VTR 230.9405

R4.VTR 313.3812

0.00 2.50 5.00 7.50 10.00 12.50 15.00 17.50 20.00Time [ms]

1475.00

1485.00

1495.00

1500.00

MA

SS

_R

OT

1.O

ME

GA

[rp

m]

0.00

50.00

100.00

150.00

180.00

MA

SS

_R

OT

1.P

HI

[deg]

Curve Info

MASS_ROT1.OMEGATR

MASS_ROT1.PHITR

Compact Modeling: Transistor Compact Model

Figure from J. Lienemann, E. B. Rudnyi and J. G. Korvink. MST MEMS model

order reduction: Requirements and Benchmarks. Linear Algebra and its

Applications, v. 415, N 2-3, p. 469-498, 2006.

Compact Thermal Models

Looks understandable – but how to do it in practice?

Figure from the book “Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic

Compact Models.” Springer, 2006.4

Comparison: Electrical vs. Thermal

Thermal phenomena are much more distributed, it is hard to lump

them.

Figure from the book “Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic

Compact Models.” Springer, 2006.5

Electrical flow

cond / ins 108

Heat flow

cond / ins 102

Conductor

Insulator

Content


compact modeling



ANSYS

Experience with MUMPS solver

6

7

Model Order Reduction

Relatively new technology:

It is not mode superposition;

It is not Guyan reduction;

It is not CMS.

Solid mathematical background:

Approximation of large scale

dynamic systems

Dynamic simulation:

Harmonic or transient simulation

Industry application level:

Linear dynamic systems

8

Linear Model Order Reduction

Linear Dynamic

System, ODEs

Control Theory

BTA, HNA, SPA

O(N3)

Moment Matching

via Krylov subspaces.

Iterative, based on

matrix vector product.

Low-rank

Grammian,

SVD-Krylov

one-side

Arnoldi process

double-side

Lancsoz algorithm

MOR for ANSYS

9

Model Reduction as Projection

Projection onto low-

dimensional subspace

How to find subspace?

Mode superposition is not the best way to do it.

zx V

x V

z

=

uxx BKE

uzz BVKVVEVV TTT

E

Er

Hankel Singular Values

Dynamic system in the state-

space form:

Lyapunov equations to

determine controllability and

observability Grammians:

Hankel singular values (HSV):

square root from eingenvalues

for product of Grammians.

Cxy

BuAxxH(s) C(sI A) 1B

AP PAT BBT 0

AT Q QA CT C 0

i i (PQ)

10

Global Error Estimate

Infinity norm

Global error for a reduced model

of dimension k

Model reduction success

depends on the decay of HSV.

Log10[HSV(i)] vs. its number.

From Antoulas review.

H(s) ˆ H (s)

maxs abs(H(s) ˆ H (s))

H(s) ˆ H (s)

2( k 1 n )

11

12

Implicit Moment Matching

Padé approximation

Matching first moments for the

transfer function

Implicit Moment Matching:

via Krylov Subspace

0

0 )( i

i ssmH

mi mi,red, i 0, ,r

BKsEsH1

)(

0

0, )( i

redired ssmH

s0 0

V span{ (K 1E,K 1b)}

uxx BKE

13

Second Order Systems

Cxy

BuKxxExM

Ignore the damping

matrix during MOR:

Proportional Damping

Transform to 1st

order system.

Use Second Order

ARnoldi.

default-1 -2

14

Error Indicator

Key question :

What is a suitable order of the reduced

system for a desired accuracy?

“Rule of thumb”: r = 30-50

Proposed engineering approach:

comparison of reduced systems of order

r and r + 1

System of

n ODEs

Reduced

System of

r << n ODEsMOR

r

User

r = ?

error estimate:

a way to automate MOR

15

Convergence of Relative Error

)(

)()()(

sH

sHsHsE r

r

)(

)()()(ˆ 1

sH

sHsHsE

r

rr

r

True error:

Error indicator:

E r(s)ˆ E r (s)

1,5mm

Main result:

16

R R(T ) R0 (1 T T 2 )

)(

)(),(

2

TR

tUFTtQFTKTC

Input function does not

participate in MOR

Q(t,T ) Q(TN (t)) Q(V * z)

Nonlinear input

Control temperature is a

single node temperature TN

Uheater=14V

),( * zVtQFVzKVVzCVV TTT

Nonlinear Input

Information on ModelReduction.com

17

Content


compact modeling



ANSYS


18

19

MOR for ANSYS: http://ModelReduction.com

FULL files

Linear Dynamic

System, ODEsCxy

BuKxxExM MOR Algorithm

Small dimensional

matrices

Current version 2.5

ANSYS Model

Simulink,

Simplorer, VerilogA,

…

http://modelreduction.com/

Implicit Moment Matching

Take matrix and vector

corresponding to a given

expansion point.

Compute the orthogonal basis by

the Arnoldi process.

Project the original system on this

basis.

One can prove that this way the

reduced model matches k

moments.

Multiple inputs:

Block Krylov Subspace;

Block Arnoldi;

Superposition Arnoldi.

s0 0 P A 1E r A 1b

V span{ (A 1E,A 1b)}

V span{r,Pr,P2r, Pk 1r}

20

Treating Matrix Inverse

The Arnoldi process requires

matrix vector product.

Yet, we have a matrix inverse.

Instead solve a system of linear

equations.

This is the biggest computational

cost:

Number of vectors x time for

linear solve.

Direct solver has strong

advantage.

vi 1 A 1Evi

ui 1 A 1ui

Aui 1 ui

The only right hand side is

different.

Can be used to speed it up.

21

MOR for ANSYS Timing: MOR as Fast Solver

Reduced model of dimension 30

22

Dimension,

DoFs

nnz MOR Time

/ANSYS

static

4 267 20 861 1.4

11 445 93 781 1.8

20 360 265 113 1.7

79 171 2 215 638 1.5

152 943 5 887 290 2.2

180 597 7 004 750 1.9

375 801 15 039 875 1.7

Chip and its Model in Workbench

Two power MOSFET transistorsThe example is from T. Hauck, I. Schmadlak, L. Voss, E. B. Rudnyi. Electro-Thermal Simulation of Multi-channel Power

Devices: From Workbench to Simplorer by means of Model Reduction. NAFEMS, Seminar: Multi-Disciplinary

Simulations - The Future of Virtual Product Development, 2009

23

Import Reduced Model in Simplorer

Simplorer supports state space model

24

Thermal Impedance and Comparison with ANSYS

ANSYS: about 300 000 DoFs, Reduced model: 30 DoFs

The difference is less than 1%

Timing: 60 timesteps is about 30 min in ANSYS

25

Thermal Runaway

Transistor is considered as temperature dependent RDSon

The VHDL-AMS model

26

Transient Turn-on of an Automotive Light-Bulb

27

Transient Turn-on of an Automotive Light-Bulb

28

0.00 10.00 20.00 30.00 40.00 50.00Time [ms]

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

E2

.I [A

]

Curve Info

SL1='1' SL2='0' SL3='0'

SL1='1' SL2='1' SL3='0'

SL1='1' SL2='1' SL3='1'

Ansoft LLC FET_Light_Bulb_P21W_2Lamp Currents at turn-on

1 lamp

2 lamps

3 lamps

0.00 10.00 20.00 30.00 40.00 50.00Time [ms]

250.00

300.00

350.00

400.00

450.00

500.00

550.00

TH

M1

.T [ke

l]

Curve Info

THM1.TSL1='1' SL2='0' SL3='0'

THM1.TSL1='1' SL2='1' SL3='0'

THM1.TSL1='1' SL2='1' SL3='1'

Ansoft LLC FET_Light_Bulb_P21W_2Junction Temperature at turn-on

3 lamps

2 lamps

1 lamp

MOR for ANSYS in turbine dynamics

Felix Lippold / Dr. Björn Hübner

Voith Hydro Holding

ANSYS Conference & 27. CADFEM Users Meeting, 18 - 20

November 2009, Congress Center Leipzig.

MOR for ANSYS in turbine dynamics | CADFEM2009 | Felix Lippold | 2009-11-19 | 30

Finite-Element model used in harmonic analysis

Linear-elastic SOLID elements for runner structure.

Acoustic FLUID elements for surrounding water.

Complex load vectors for rotating pressure fields.

ModellingAcoustic-structure coupling

//euro1/HDH/HYDRO/home/10/bjoern.huebner/Praesentationen/MOR4ANSYS/GMA_Workshop_2007/FT_runner_press.avi


ModellingCoupled acoustic-structure equations

Equations of motion

)(A

S

A

SAS

A

S

ASA

Stq

b

b

p

u

K

KK

p

u

E

E

p

u

MM

M

Formally equivalent to

)(tqbxKxExM

But:

non-symmetric matrices

non-proportional damping matrix

fluid damping EA only for impedance boundaries


Axial displacement@centre trailing edge – no damping

Amplitude spectrum (Dim=100) Deviation related to ANSYS results

ResultsNo damping


Pressure distribution on runner for f = 40 Hz

Ansys (90 000 DOFs) Reduced (Dim=100)

REAL part of pressure

solution

Deviation < 0.2%

IMAG part of pressure

solution

Deviation < 0.2%

ResultsVerification of reduced order method

Hard Disk Drive Actuator/Suspension System

Michael R Hatch, Vibration Simulation Using MATLAB and ANSYS

34

Model Reduction

3352 elements

7344 nodes

21227 equation

400 frequencies takes about 12

min

MOR takes only 3 s

Comparison for head

ANSYS

MOR 30

MOR 80

35

Comparison

36

Relative difference (blue modal80 – green MOR80)

37

Information on ModelReduction.com

38

Content

CADFEM


compact modeling



ANSYS


39

www.msnbc.msn.com/id/12409082/

History of solvers in MOR for ANSYS

MUMPS

Fortran 90 – Activation barrier

TAUCS – symmetric matrices

UMFPACK – unsymmetric

matrices

2006 – HPC-EUROPA

SGI Origin 3800 (teras)

Micro-FEM bone models

Bert van Rietbergen

2007 – MUMPS in MOR for

ANSYS

40

Voith Hydro Timing

2 x quad-core Intel X5560 (Nehalem) with 48 GB under Windows

Shared memory

41

nnz dim

K 91627731 1292663

M 31982656 1292663

27.7 Gb memory

Factorization is 287 s

250 vectors is 1500 s

6 s pro vector

nnz dim

K 99934728 1495341

M 35774706 1495341

20.1 Gb memory

Factorization is 243 s

200 vectors is 1300 s

6.5 s pro vector

How MUMPS is used in MOR for ANSYS

Shared memory (serial)

No parallel

Linux and Windows versions (Visual C + Intel Fortran on Windows)

Used to have an Altix version

Intel MKL BLAS

Used to work with ATLAS

Two versions:

32-bit integers

64-bit integers

Mostly in-core, sometimes out-of-core

Symmetric positive definite, symmetric indefinite, unsymmetric

matrices42

Sharing Experience

Oberwolfach Model Reduction Benchmark Collection

http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark

Trabecular Bone Micro-Finite Element Models

http://MatrixProgramming.com

Compiling MUMPS under Microsoft Visual Studio and Intel Fortran with

GNU Make

Sample code in C++ to run MUMPS

Working with Microsoft Visual Studio

Distribution the code with manifests

-02 does not turn safe iterators off (_SECURE_SCL=0)

32-bit code has some problems with memory fragmentation

Using a tree to assemble a matrix

43

Conclusion

Model Order Reduction is a nice extension for a FEM application

Model Order Reduction needs a direct solver

MUMPS plays an important role in MOR

44

from a finite element model to system level simulation by means of

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