From a finite element model to
system level simulation by
means of model reduction
MUMPS Users Meeting 2010, Toulouse
Evgenii Rudnyi, CADFEM GmbH
erudnyi@cadfem.de
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
Ambient
P _RE F
Q
Electrical Domain
z_up
z_vp
z_wp
E1
IN_A
IN_B
IN_C
z_um
z_vm
R6
R5
R4
A
A
A
OUT_A
OUT_B
OUT_C
R1
250.00
Induction_Motor_20kW
I_mot
Simplorer4
VM311
500.00
Mechanical
Domain
R2
A
N
B
ROT1
C
ROT2
MASS_ROT1
Y1 [V]
P8
P7
P6
P5
P4
P3
P2
P9
P 12
P 11
P 10
P1
Thermal Domain
0.00
R3
z_wm
-250.00
+
0
V
-500.00
1500.00
RZM
MASS_ROT1.OMEGA [rpm]
SINE1
1495.00
RZM
SINE2
DR1
DR1
1485.00
1475.00
 How to make a model at system level from that at device level?
2
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
4 Compact Models.” Springer, 2006.
Comparison: Electrical vs. Thermal
Electrical flow
cond /
ins
Heat flow
108
Conductor
cond /
ins
102
Insulator
 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
Content
 System level simulation and
compact modeling
 Model order reduction
 Case studies with MOR for
ANSYS
 Experience with MUMPS solver
6
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
7
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.
one-side
Arnoldi process
MOR for ANSYS
8
Low-rank
Grammian,
SVD-Krylov
double-side
Lancsoz algorithm
Model Reduction as Projection
 Projection onto low-
Ex Kx
Bu
dimensional subspace
E
x Vz
V T EVz V T KVz V T Bu
z
x =
V
 How to find
subspace?
 Mode
superposition is
not the best way
to do it.
9
Er
Hankel Singular Values
 Dynamic system in the statespace form:
x
Ax Bu
y
H(s) C(sI A) 1 B
Cx
 Lyapunov equations to
determine controllability and
observability Grammians:
 Hankel singular values (HSV):
AP PAT
BBT
AT Q QA C T C 0
 square root from eingenvalues
for product of Grammians.
i
10
0
i (PQ)
Global Error Estimate
 Infinity norm
H(s) Hˆ (s)
max s abs(H(s) Hˆ (s))
 Global error for a reduced model
of dimension k
H(s) Hˆ (s)
2(
k 1
n)
 Model reduction success
depends on the decay of HSV.
 Log10[HSV(i)] vs. its number.
 From Antoulas review.
11
Implicit Moment Matching
 Padé approximation
 Matching first moments for the
Ex Kx
transfer function
H ( s)
Bu
sE K
1
B
mi ( s s0 )i
H
0
mi ,red ( s s0 )i
H red
0
mi
s0
 Implicit Moment Matching:
 via Krylov Subspace
12
mi,red , i 0, ,r
V
0
span{ (K 1 E,K 1b)}
Second Order Systems
Mx Ex Kx
y Cx
Bu
default
-1
Ignore the damping
matrix during MOR:
Proportional Damping
13
Transform to 1st
order system.
-2
Use Second Order
ARnoldi.
Error Indicator
 Key question :
What is a suitable order of the reduced
system for a desired accuracy?
error estimate:
a way to automate MOR
r=?
User
 “Rule of thumb”: r = 30-50
 Proposed engineering approach:
 comparison of reduced systems of order
r and r + 1
14
r
System of
n ODEs
MOR
Reduced
System of
r << n ODEs
Convergence of Relative Error
True error:
Error indicator:
Er ( s )
Eˆ r ( s )
H (s) H r (s)
H (s)
H r (s) H r 1 (s)
H r (s)
Main result:
E r (s)
Eˆ r (s)
1,5m
15
m
Nonlinear Input
R R(T ) R0 (1
T
T2
)
Nonlinear input
C T
K T
F Q(t , T )
U 2 (t )
F
R(T )
Input function does not
participate in MOR
V T CV z V T KV z V T F Q(t , V * z )
Control temperature is a
single node temperature TN
Q(t,T ) Q(T N (t)) Q(V * z)
16
Uheater=14V
Information on ModelReduction.com
17
Content
 System level simulation and
compact modeling
 Model order reduction
 Case studies with MOR for
ANSYS
 Experience with MUMPS solver
18
MOR for ANSYS: http://ModelReduction.com
Simulink,
Simplorer, VerilogA,
…
ANSYS Model
Small dimensional
matrices
FULL files
Mx Ex Kx
y Cx
Bu Linear Dynamic
System, ODEs
Current version 2.5
19
MOR Algorithm
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.
20
s0
0
V
V
P
A 1E
r
A 1b
span{ (A 1 E, A 1b)}
span{r,Pr,P 2 r,
 Multiple inputs:
 Block Krylov Subspace;
 Block Arnoldi;
 Superposition Arnoldi.
P k 1r}
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:
vi
A 1 Evi
1
ui
1
Aui
A 1ui
1
ui
 Number of vectors x time for
linear solve.
 Direct solver has strong
advantage.
21
The only right hand side is
different.
Can be used to speed it up.
MOR for ANSYS Timing: MOR as Fast Solver
 Reduced model of dimension 30
Dimension,
DoFs
22
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 transistors
The 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
Lamp Currents at turn-on
Ansoft LLC
FET_Light_Bulb_P21W_2
35.00
Curve Info
SL1='1' SL2='0' SL
SL1='1' SL2='1' SL
SL1='1' SL2='1' SL
30.00
25.00
E2.I [A]
20.00
15.00
3 lamps
10.00
2 lamps
5.00
Ansoft LLC
0.00
1 lamp
Junction
10.00
0.00
Temperature
at turn-on
20.00
30.00
Time [ms]
550.00
500.00
FET_Light_Bulb_P21W_2
40.00
50.00
Curve Info
THM1.T
SL1='1' SL2='0' SL3='0'
THM1.T
SL1='1' SL2='1' SL3='0'
3 lamps
THM1.T
SL1='1' SL2='1' SL3='1'
THM1.T [kel]
450.00
400.00
2 lamps
350.00
1 lamp
300.00
250.00
28
0.00
10.00
20.00
Time [ms]
30.00
40.00
50.00
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.
Modelling
Acoustic-structure coupling
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.
MOR for ANSYS in turbine dynamics | CADFEM2009 | Felix Lippold | 2009-11-19 | 30
Modelling
Coupled acoustic-structure equations
Equations of motion
MS
M SA
MA

u

p
ES
EA
u
p
KS
K SA
KA
u
p
bS
bA
q(t )
Formally equivalent to
M x E x K x b q (t )
But:
 non-symmetric matrices
 non-proportional damping matrix
 fluid damping EA only for impedance boundaries
MOR for ANSYS in turbine dynamics | CADFEM2009 | Felix Lippold | 2009-11-19 | 31
Results
No damping
Axial displacement@centre trailing edge – no damping
Amplitude spectrum (Dim=100)
Deviation related to ANSYS results
MOR for ANSYS in turbine dynamics | CADFEM2009 | Felix Lippold | 2009-11-19 | 32
Results
Verification of reduced order method
Pressure distribution on runner for f = 40 Hz
REAL part of pressure
solution
Deviation < 0.2%
IMAG part of pressure
solution
Deviation < 0.2%
Ansys (90 000 DOFs)
Reduced (Dim=100)
MOR for ANSYS in turbine dynamics | CADFEM2009 | Felix Lippold | 2009-11-19 | 33
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
 System level simulation and
compact modeling
 Model order reduction
 Case studies with MOR for
ANSYS
www.msnbc.msn.com/id/12409082/
 Experience with MUMPS solver
39
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
 K
 M
nnz
91627731
31982656
dim
1292663
1292663
 K
 M
nnz
99934728
35774706
dim
1495341
1495341
 27.7 Gb memory
 20.1 Gb memory
 Factorization is 287 s
 Factorization is 243 s
 250 vectors is 1500 s
 6 s pro vector
 200 vectors is 1300 s
 6.5 s pro vector
41
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
matrices
42
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