INTER-NOISE 2006
3-6 DECEMBER 2006
HONOLULU, HAWAII, USA
Krylov Subspace Techniques for Low Frequency
Structural Acoustic Analysis and Optimization
R Srinivasan Puria, Denise Morreya
Andrew Bella, John F Durodolaa
a
School of Technology,
Oxford Brookes University,
Gipsy Lane Campus,
Headington, Oxford OX3 0BP,
United Kingdom
Evgenii B Rudnyib and Jan G Korvinkb
b
IMTEK-Department of Microsystems
Engineering, University of Freiburg,
Georges-Koehler-Allee 103,
D-79110, Freiburg,
Germany.
ABSTRACT
A reduced order model is developed for low frequency, fully coupled, undamped and
constantly damped structural acoustic analysis of interior cavities, backed by flexible
structural systems. The reduced order model is obtained by applying a Galerkin projection of
the coupled system matrices, from a higher dimensional subspace to a lower dimensional
subspace, whilst preserving some essential properties of the coupled system. The basis vectors
for projection are computed efficiently using the Arnoldi algorithm, which generates an
orthogonal basis for the Krylov subspace containing moments of the original system. The key
idea of constructing a reduced order model via Krylov subspaces is to remove the
uncontrollable, unobservable and weakly controllable, observable parts without affecting the
transfer function of the coupled system. The reduced order modelling technique is applied to a
frame-panel two-way coupled vibro-acoustic optimization problem, with stacking sequences of
the composite structure as design variables. The optimization is performed via a hybrid search
strategy combining outputs from Latin Hypercube Sampling (LHS) and Mesh Adaptive Direct
Search (MADS) algorithm. It is shown that reduced order modelling technique results in a very
significant reduction in simulation time, while maintaining the desired accuracy of the
optimization variables under investigation.
1
INTRODUCTION
Improving the acoustic behaviour of vehicle and aerospace interiors is an ever increasing
challenge for manufacturers. It is now common practice to evaluate the low frequency noise,
vibration, harshness NVH behaviour of vehicle or aerospace interiors using fully coupled finite
element-finite element (FE/FE) or finite element-boundary element (FE/BE) discretized models
at the design phase of the product development process. Due to the coupling between the fluid
and structural domains in the FE/FE formulation, the resulting mass and stiffness matrices are no
longer symmetrical. Such a multi disciplinary approach requires the solution of these coupled
fluid and structural equations, causing an inevitable increase of computational time and expense
[1]. Since there exists two forms of solution (coupled and uncoupled), it is often left to the
engineer to decide on the approach best suited to the problem under investigation. However, a
‘one-way’ coupled analysis ignores the fluid loading on the structure, which is often the cause of
cavity boom at low frequencies.
___________________
a
s.puri@brookes.ac.uk
b
rudnyi@imtek.uni-freiburg.de
Therefore, a fully coupled analysis is preferred in many vehicle/aerospace applications, but the
computational time required to solve the higher dimensional model restricts its subsequent use.
This is often the case when the structure is spatially damped i.e. different parts of the structure
carry different damping values. Furthermore, this presents a major problem where optimization is
required, especially when a large number of design variables are to be optimized.
The two most popular approaches currently used to reduce the computational time of such coupled
problems are the mode superposition and the component mode synthesis (CMS) method. The
former method uses the dominant natural frequencies and mode shapes, extracted from a normal
modal analysis, and the response is assumed to be a linear combination of the modes. However, the
reduction thus obtained is often not substantial. Further, the CMS method relies on the user to
select interface nodes to enforce coupling conditions, which is a possible source of additional error.
Other approaches to decrease computational time include generation of Ritz vectors, truncated
coupled FE/FE analysis, and the patented Acoustic Transfer Vector (ATV) method, to name a few.
The reader is referred to [1], for a review of some other approaches to reduce computational time.
More recently, however, model order reduction (MOR) via implicit moment matching, has received
considerable attention among mathematicians and the circuit simulation / Micro ElectroMechanical Systems (MEMS) community [2, 3, 4, 5]. The aim of MOR is to construct a reduced
order model, from the original higher dimensional model, which is a good representation of the
system input/output behavior at certain points in the time or frequency domain. The reduction is
achieved by applying a projection from a higher order to a lower order space using a set of Krylov
subspaces, generated by the Arnoldi algorithm. Additionally, the reduced model preserves certain
essential properties such as maintaining the second order form and stability.
The paper focuses on the application of such Krylov based MOR techniques to undamped and
structurally damped, fully coupled structural acoustic problems. The rest of the paper is laid out as
follows: In Section- 2, the general framework for model order reduction for second order systems is
introduced. In Section- 3 the Arnoldi procedure adapted for model order reduction for the coupled
damped structural acoustic problem is described. In Section- 4 a numerical example from is solved
using the direct approach in ANSYS FE code and the MOR via Arnoldi approach. In Section- 5
MOR is incorporated via the Arnoldi process in the structural-acoustic optimization process to
speed up simulation time, whilst maintaining the desired accuracy of the objective function under
investigation. Section- 6 summarizes the paper with a short discussion of the results.
2
MODEL ORDER REDUCTION FOR SECOND ORDER SYSTEMS
After discretization of a general dynamical model of mechanical system, one obtains a system of
second order ordinary differential equations in matrix form as follows:
[M ] ẍ t[C ] ẋ t [ K ] x t= F u t
y t= LT x t
(1)
where (t) is the time variable, x(t) is the vector of state variables, u(t) is the input force vector, and
y(t) the output measurement vector. The matrices M, C and K are mass, damping and stiffness
matrices, F and L are the input distribution matrix and output measurement matrix at certain
j t
points respectively. A harmonic simulation, assuming {F }=F 0 e
and ignoring damping in (1)
yields:
2
[− [M ][K ]]{x }={F }
T
y = L x
(2)
where, {} denotes the circular frequency, and {x} ,{F } denote complex vectors of state
variables and inputs to the system respectively. The principle of model reduction is to find a lower
dimensional subspace V ∈ℜNxn , and,
x=Vz where, z ∈ ℜn , n≪ N
(3)
such that the time dependent behaviour of the original higher dimensional state vector {x} can be
well approximated by the projection matrix V in relation to a considerably reduced vector {z } of
order n with the exception of a small error ∈ℜN . Once the projection matrix V is found, the
original equation (2) is projected onto it. The projection produces a reduced set of system
equations, as follows:
2
(4)
[− [M r ][ K r ]]{z }={F r }
T
y r =L r z
where the subscript r denotes the reduced matrix and:
M r =V T MV , K r =V T KV , F r=V T F , Lr =V T L
(4-A)
It is worth noting that y r ≈ y . Due to its low dimensionality, the solution to (4) is much
faster than the original higher dimensional model. The input and output vectors are the same
dimension as (2). Several methods exist to choose V. In this work, we choose the projection matrix
V to be a Krylov subspace in order to provide the moment matching properties [2, 3, 6].
2.1
Model Order Reduction for coupled structural acoustic systems:
For a mutually coupled structural acoustic case, we start off from Cragg’s pressure formulation[7]:
Ms
0
Mfs Ma
{ }
Cs 0
ü
0 Ca
p̈
{ }
{}
Ks
u̇
0
ṗ
T u
y t= L
p
Kfs
Ka
{ } { }
u
Fs
=
p
0
(5)
where, Ms is the structural mass matrix, Mfs is the coupled mass matrix, Ma is the acoustic mass
matrix, Cs is the structural damping matrix, Ca is the acoustic damping matrix, Ks is the structural
stiffness matrix, Kfs is the coupled stiffness matrix, Ka is the acoustic stiffness matrix, Fs is the
structural force vector, y (t) the output measurement vector and u, p are the displacements and
pressures at nodal co-ordinates respectively. Ignoring damping for the structure and fluid, the
coupled equations in the case of harmonic response analysis become:
[
−
2
Ms
Mfs
0
Ks
Ma
0
Kfs
Ka
]{ } { }
u
Fs
=
p
0
(6)
Constant structural or acoustic damping ratio's can be incorporated into the system matrices of (6)
avoiding a direct participation of [C], as it is frequency independent by the definition. Although
there exists techniques to reduce system matrices with [C], in this paper, we restrict ourselves to
undamped and constant structural damping. A straightforward extension can be made to constant
acoustic damping. The finite element software, ANSYS [11] formulates constant damping via the
command DMPRAT and MP, DMPR which adds imaginary terms to the stiffness matrix according
to the relationship:
2
c =
(6-A)
Where, c is the constant multiplier applied to structural parts of the coupled stiffness matrix, and
is the frequency in rad/s and is the constant damping ratio. This implies that the matrix
[ K ] is complex-valued. In other words, the structural stiffness matrix Ks in Eqn (5), Eqn (6)
becomes K si2 K s
In this case, the approximation becomes:
{up}={x}=Vz
(7)
The transfer function of the system H s=[Y s /U s] using the Laplace transform can be
written as:
T
2
−1
(8)
H s= L s M sa sC saK sa F sa
Ignoring damping, and expanding (8) using the Taylor series about
∞
∞
i =0
i =0
s=0 results in:
i
−1
2i
2i
H s=∑ −1i LT K −1
sa M sa K sa F sa s = ∑ m i s
(9)
−1
−1
Where mi = (− 1) i LT ( K sa M sa ) i K sa Fsa ( for i = 0, ∞ ) are called the moments of H (s) and,
M sa =
Ms
0
Ks
, K sa =
Mfs Ma
0
{ }.
Kfs
Fs
, F sa =
Ka
0
By matching some of these moments of the higher dimensional system about s=0, the reduced
order model can be constructed, as it directly relates the input to the output of the system.
Theoretically, any expansion point within the frequency range of interest can be used, and a real
choice depends on required approximation properties. However, explicitly computing such
moments tends to be numerically unstable [3, 4], and it is therefore preferable to attempt to
implicitly match these moments via the Arnoldi process. Su and Craig [6], showed that if the
projection matrix V is chosen from a Krylov subspace of dimension q,
K q K −1 M , K −1 F =span {K −1 F , K −1 M K −1 F , ...... K −1 M q−1 K −1 F }
(10)
then, the reduced order model matches q+1 moments of the higher dimensional model. The block
vectors K-1F and K-1M can be interpreted as the static deflection due to the force distribution F, and
the static deflection produced by the inertia forces associated with the deflection K-1F respectively.
3
THE ARNOLDI ALGORITHM
To avoid numerical problems while building up the Krylov subspace, an orthogonal basis is
constructed for the given subspace. This is done using the Arnoldi algorithm. Given a Krylov
subspace Kq (A1, g1), the Arnoldi algorithm finds a set of vectors with norm one which are
orthogonal to each other, given by:
T
V V=I
and V T A1 V =H q
(11)
Where H q=ℜqxq is a block upper Hessenberg matrix and I q=ℜqxq is the identity matrix.
Figure: 1 describes the implemented algorithm, which is used to generate the Arnoldi vectors for
the coupled structural acoustic system. For multiple inputs, the block version of the algorithm can
be found in [4]. For the coupled structural acoustic case, we obtain:
−1
−1
Colspan V =K q K sa M sa , K sa F sa
T
−1
V K sa M sa V = H q and V T V = I
(12)
Algorithm: 1:
Input: System Matrices Ksa,Msa, Fsa, L and n (number of vectors), expansion poin s= E B /2
Output: n Arnoldi vectors
0. Set v 1=g 1
1. For
i=1→ n , do:
h i ,i −1=∥v i∥
*
1.2 Normalization: v i =[v i ] /[hi ,i −1 ]
*
1.3 Generation of next vector: v i1= Av 1
1.1 Deflation check:
1.4 Orthogonalization with old vectors: for j=1 to i:
T
1.4.1 h j , 1=v j v i1
*
*
v j 1=v i1−h j ,i v j
2. Discard resulting H q , and project M sa , K sa , Fsa , L onto V to obtain reduced system matrices
M Rsa , K Rsa , FRsa , LRsa where the subscript Rsa represents the reduced structural acoustic matrices.
1.4.2
Figure 1: The Arnoldi Process [3] [4]
4
NUMERICAL TEST CASE
The first test case is a frame-panel structure which was built to test new modelling techniques. The
scaled car cabin is modeled as a simple seven sided structure as shown in Figure: 2(a). The
structural model is a frame panel structure coupled with the acoustic cavity. Faces of the acoustic
model, other than that of the roof were assumed to be acoustically rigid. The acoustic model was
modeled using eight noded acoustic brick elements with one pressure degree of freedom at each
node. The thicknesses of the beams and the flat panel are 2.5mm and 1.5mm respectively. The
frequency range of interest for the coupled dynamic analysis is 0-300Hz. A unit harmonic force is
applied to one of the structural nodes of the lower front member to excite the coupled system as
show in Figure: 2(a). The pressure response is computed at two nodes in the fluid domain using
the Direct Method in ANSYS and MOR via the Arnoldi process. The noise transfer function
(Pressure/Force) at nodes representative of driver and passenger ear location is shown in figures
3(a) and 3(b). All computations described in this paper were performed using a Pentium 3GHz,
2GB RAM machine.
Figure 2(a): Structural FE model
Figure 3(a): Noise Transfer Function
Figure 2 (b) Coupled FE Model
Figure 3(b) Noise Transfer Function
For the reduced order model, the computational time is a combination of four steps (a) Running a
partial stationary solution to extract *.FULL file from ANSYS (b) Reading matrices and generating
of Arnoldi vectors (c) Projection to second order form and (d) Simulation of the reduced order
model. The spilt computational times for the test case are given in Table: 1. The reduced order
model is set up and solved in Mathematica/MATLAB environment. A comparison of the solution
times using MOR and the Direct method in ANSYS are given in Table: 2.
Model
TC 1
Writing FULL Read Matrices, Generate Arnoldi Reduced Model
file from ANSYS
Vectors and Project
Simulation
3s
492 s [100 Arnoldi Vectors]
22 s
Total: MOR via
Arnoldi Process
517 s
1
Table: 1: MOR Split Computational Times; TC : Test Case-1
Model Elements
TC 1
6061
Active DOF's ANSYS Direct MOR via Arnoldi Process Time Reduction
9832
7932 s
517 s
1
Table: 2: Computational Times; TC : Test Case-1
93.48%
5
COUPLED VIBRO-ACOUSTIC OPTIMIZATION
Fiber reinforced composites have generated significant interest among automotive and aerospace
manufacturers in the development of structural materials due to their low density, high stiffness and
excellent damping characteristics. Additionally, the orthotropic nature of such fiber reinforced
composite materials implies that the directional stiffness depends on the orientation of fibers. In
this work, the feasibility of reducing interior noise levels through optimal lamination angles of a
laminated composite structure via reduced order modelling is demonstrated.
5.1
Optimization Test Case
The optimization test case is coupled model described in Section:4. In this case, the roof panel is a
symmetric, eight layered Glass fiber reinforced composite structure, with initial lamination angles
of [90/0/90 /0]s . The lamination angles of the fibers, denoted by θ in this study, take the form
of design variables for the optimization problem. The design variables are subject to a lower and
upper bounds 0 ≤ θ ≤ 180 degrees, where 00 represents a unidirectional lay-up of the fibers and
angles beyond 900 represent lay-up in the negative direction. The structural FE model is modeled
using a combination of ANSYS SHELL181 and BEAM4 elements, with material properties for the
uni-directional composite (for SHELL181) as given in Table: 3. A structural damping ratio of 4%
is applied to all elements with composite material properties. In addition to this, a constant overall
structural damping of 2% is specified for the analysis.
E11=E33 GPa
28
E22 GPa
G12=G21 GPa
G13 GPa
v12=v13
21
1.39
1.40
0.4
Density
Kg /m 3
1480
Table:3: Mechanical Properties for the Glass fiber reinforced composite
5.2
Design Optimization Procedure
The reduced order modelling technique outlined in Section: 2 is incorporated into the optimization
process to speed up simulation time, while maintaining the accuracy of the nodal sound pressure
values. A general framework of optimization via reduced order modelling is shown in Figure: 5.
The reduced order model does not allow us to preserve geometry related information, and after
changes in the original higher dimensional model, the reduced order model must be regenerated
again. Fortunately, the time required to generate a reduced order model is comparable with that for
a single frequency evaluation [9]. The number of vectors required to represent the higher
dimensional system is calculated using the convergence models presented in [8,13]. In this case, 85
Arnoldi generated vectors were sufficient to represent the higher dimensional system for
=1Hz. and =300Hz. The non-linear optimization problem can be stated as:
Find a vector of design variables: =1 ,2 ,3 , ... n
Which minimizes the objective function f
upper
Subject to lower and upper bounds lower
i i
i
Numerous possibilities exist to formulate the objective function, and their effectiveness depends on
the nature of the coupled problem. For the optimization problem stated above, the objective
function is formulated as:
f = F
1
n
obj
{
max
n
1
p − p ref
; F obj =
p i d , where , = i
∫
max − min
0
min
p i p ref
p i p ref
for
for
}
(13)
Where, the function ϑ is a weighting function applied to the nodal sound pressure level (SPL)
value pi . It can be seen that the weighting function depends on reference pressure p ref , determined
as 45dB for the current study. For n=2, this formulation of objective function (13) results in a
frequency averaged root mean square value.
The optimization is carried out using MATLAB GA/PS Toolbox [12] using the Mesh Adaptive
Direct Search (MADS) algorithm. Each iteration of MADS is divided into two steps, SEARCH and
POLL. The SEARCH step allows the evaluation of the objective function at a finite set of points.
Any search strategy can be used, including none. When a SEARCH step fails to improve the
objective function value, a POLL step is invoked. In addition to the mesh size parameter, a poll size
parameter is defined to ensure that the local exploration of the design variable space is not
restricted to a finite set of directions [10]. The set of trial points considered during the POLL step is
called a frame. Depending on the result of the POLL step, i.e. successful or unsuccessful, the mesh
resolution is decreased or increased .A general MADS algorithm is shown in Figure: 4. In this
work, we chose to evaluate initial trial points in the first iteration of MADS using Latin Hypercube
Sampling (LHS).
Algorithm:2:
.
.
INITIALIZATION: Define mesh point, mesh size and poll size parameters, set k 0
SEARCH AND POLL STEP: Perform SEARCH and POLL steps until an improved mesh
point is found on mesh.
1) OPTIONAL SEARCH: Evaluate function on a finite subset of trial points on mesh.
2) LOCAL POLL: Evaluate function on computed frame.
PARAMETER UPDATE: Update mesh size and poll size, set k k 1 and return to
SEARCH and POLL steps.
Figure: 4: A simplified MADS Algorithm [10].
F re q u e n c y V s . S P L
120
START
SPL BEFORE OPTIMIZATION
100
SUGGESTED
CONVENTIONAL
MOMENT MATCHING
OPTIMIZATION
VIA THE
LOOP
(1) DIRECT SOLUTION
(2) MODE SUPERPOSITION
ARNOLDI PROCESS
OBJECTIVE FUNCTION
d B : S o u n d P re s s u re L e v e l
SPL AFTER OPTIMIZATION
DESIGN
PARAMETERIZATION
80
60
40
20
EVALUATION
0
STOP
Figure: 5: Optimization Via Moment Matching
0
50
100
150
F re q u e n c y (H z .)
200
250
300
Figure: 6: Sound pressure levels before and
after optimization.
5.3
Results
In this paper, symmetric laminates are considered for the optimization study due to structural and
manufacturing considerations. From an initial lay up of [90/0/90 /0]s the lamination angles
move towards a lay up of [127 /125 /35 /2]s . Figure: 6 compares the sound pressure level (SPL)
before and after optimization. The face sheets of the outer layer of the composite material tend to
be moving towards a more even (-53/-55) while the inner lamination angles moves towards an
cross-ply orientation (35/2). Sound pressure levels near frequencies ~50Hz, ~85Hz, and ~138Hz
and ~245Hz. have decreased by 5.5dB, 2.6dB and 8dB and 9.2dB respectively. In the frequency
range of 250-290Hz., peak SPL value has increased from 69.31dB to 74.94dB – an increase of
5.62dB. This is primarily because (a) The optimization is considered over the entire frequency
range of 0-300Hz., and an overall decrease in the root mean square SPL value is considered as a
successful iteration by the optimizer and (b) the value of p ref =45dB. is applied to (13). From a
computational viewpoint, the use of reduced order modelling significantly decreases the
computational time requirements for the optimization process. The time taken by MOR via Arnoldi
is shown in Table: 4. For the optimization test case, the cost of reduction via Arnoldi is
approximately 96% smaller than solving the original higher dimensional ANSYS model. Such
smaller cost enables effective application of hybrid search strategies, to search the multi-modal
space which usually requires more number of function evaluations. If the optimization had been
carried out by the direct method in ANSYS, ~366 hours of computational time would have been
required for 166 function evaluations.
Model Function
Initial Stacking
Evaluations
Sequence
OPT
6
Final Stacking
Sequence
Time
MOR
Time
Time
ANSYS* Reduction
[90/0/90 /0]s
[127 /125 /35 /2]s
166
49539 s
1316691 s
96.24%
Table: 4: Optimization Results: OPT: Optimization Test Case.*Estimated time.
SUMMARY
An efficient method to perform coupled structural analysis and optimization via reduced order
modelling has been outlined. The basis vectors for matching the coupled system moments are
computed by applying the Arnoldi algorithm, which computes the projection vectors spanning the
Krylov subspace, to match the maximum number of moments of the system. The moments in the
test cases shown are matched at approximately half of the analysis range s= E B / 2 . If a
Taylor series expansion is considered around a higher frequency, a reduced order model could be
obtained with better approximation properties around that frequency range. Figure: 3(a), 3(b)
indicates that good approximation properties can be obtained by projecting the higher dimensional
system to a lower dimension and matching some of the low frequency moments of the system.
In this work, the explicit participation of [C] is avoided by using a complex stiffness approach
K s 1i2 . For a higher dimensional model with [C], a reduced order model via the Second
Order Arnoldi (SOAR) process, involving computing orthogonal vectors belonging to the second
order Krylov subspaces [14] or transforming the Equation (6) to first order form and matching
moments via Arnoldi is possible, but a comparison of accuracy and efficiency of such methods is
beyond the scope of this current paper. The number of vectors needed to accurately represent the
system was 85 for both test case and the optimization test case respectively. Comparing the
computational times of the test case and the optimization, it can be seen that the time reduction in
the optimization test case is 2.76% higher. In the first test case described in this paper, 100 vectors
were generated initially to check for error convergence. However, it is also worth noting that the
process of computing the minimum number of required vectors can be completely automated by a
user defined error parameter. Lastly, the reduced order modelling framework via the Arnoldi
process was incorporated in the structural-acoustic optimization process. The lamination angles of
the composite structure took the form of design variables for the optimization problem. As stated
earlier, any change in material properties required generation of reduced order model from the
higher dimensional model. A general decrease in SPL over the entire frequency range of
optimization is evident. Both mode shifting and peak splitting phenomenon’s resulting from change
in lamination angles were accurately captured by the reduced order model. Finally, it should be
noted that MOR via Arnoldi [3,4] would be appropriate only in the low frequency range for
vehicle/aerospace structural acoustic applications, and other techniques exists e.g. Energy Finite
Element Method (EFEA), Statistical Energy Analysis (SEA), to deal with higher frequencies,
where modal density is often high, and the acoustic response is very sensitive to minor structural
modifications.
7
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