Flow-induced vibration depends on a large number of parameters or features. On the one hand, the number of candidate physical features may be too big to construct an interpretable and transferrable model. On the other hand, failure to account for key dependence among features may oversimplify the model. Feature selection is found to be able to reduce the dimension of the physical problem by identifying the most important features for a certain prediction task. In this paper, a weighted sparse-input neural network (WSPINN) is proposed, where the prior physical knowledge is leveraged to constrain the neural network optimization. The effectiveness of this approach is evaluated when applied to the vortex-induced vibration of a long flexible cylinder with Reynolds number from 104 to 105. The important physical features affecting the flexible cylinders' crossflow vibration amplitude are identified.
Vortex-induced vibration (VIV) is a multi-physics problem
associated with a number of features (or variables) that
characterize either the structure or the flow individually or
their interaction. As flow passes around a cylinder, the
wake becomes unstable. The periodically shed vortices
induce unsteady forces on the cylinder which lead to VIV.
Moreover, the VIV of long cylinders in ocean currents may
vary from single mode dominated, narrow-band random
vibration to multi-mode response, characterized by
broadband random vibration. Different current profiles may
cause structural vibration with standing waves or travelling
wave patterns
Copyright © 2020, for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
To identify the key mechanisms and the governing
dimensionless parameters behind the complicated
fluidstructure interaction process, extensive investigations have
been made through structural response measurement, flow
visualization and various force modeling techniques
Feature selection algorithms are intended to extract the
most important features out of the full set of candidate
features with the goal of keeping prediction accuracy at a
desirable level, but with a reduced set of features. A
preanalysis of a features’ importance can be conducted by
examining the statistical correlations among the features.
However, the statistical analysis often fails to consider the
complicated interactions among the physical input
parameters (features). To solve this problem, the importance of
each feature subset can be assessed according to their
prediction accuracy using a learning machine, such as deep
neural network (DNN). Several learning machine-based
feature selection approaches have already been developed
to iteratively search the optimal feature subset that gives
similar prediction accuracy as the full feature set, but they
can be computationally expensive especially when the
number of input features becomes very large
To efficiently identify important features in a learning
machine, several regularization techniques are introduced
to the machine learning process.
However, for physical problems, some of the system’s
properties may be known in advance or can be obtained
from the governing physical laws and dimensional analysis
In this paper, the sparse-input neural network proposed
by
We consider a fully connected DNN with P input features
x ∈ RP in the input layer and M neurons in the first hidden
layer that predict a certain target output y ∈ R1 . The weight
connecting the pth input feature and mth neuron in the first
hidden layer is denoted as wpm. Figure 1 shows an example
of the DNN with P=3 and M=4. The sparse-input neural
network
M 2
∑ ( wpm ) ≠ 0 ≤ k
m=1
Where w 0 is the l0 norm of weight vector w. |…| is the
cardinality of the weight groups; p and m are the index for
min L ( y, yˆ )
w
subject to
(1)
the input feature and the neuron in the first hidden layer,
respectively. The magnitude of weight group for feature p
is measured by W . Since the l0 norm is non-convex and
p
non-differentiable, l1 norm, which calculates the sum of
absolute values of the vectors, is often used as a convex
proxy
1 N M w N n =1 p∈P m =1 (2) The second term in Equation (2) introduces bias term for prediction. The hyperparameter λ is known as the group lasso penalty, which adjusts the sparsity of the input features versus the prediction accuracy. When λ grows, the neural network will try to minimize the sum of the weight groups, and therefore more weight groups are likely to shrink to near 0. The input features with nonzero weight groups are the remaining features that contribute to the prediction. In this way, the model can be built out of fewer input features, but the prediction accuracy may decrease due to the loss of information.
However, for many physical problems, some of the
features are known to be important in advance, which are
termed as prior knowledge. In this case, the objective is to
select a small number of additional features that will
complement the input features that are considered prior
knowledge and lead to predictions of acceptable accuracy.
Since the conventional sparse-input neural network cannot
tell the difference between prior knowledge and additional
features, the optimization objective in Equation (2) needs
to be modified as follows,
min L ( y, yˆ ) + spλ ∑ ∑M ( wpm )2 + saλ ∑ ∑M ( wpm )2 (3)
w p∈Pp m =1 p∈Pa m =1
Where Pp denotes the feature set representing the prior
knowledge, and Pa denotes the set of all the additional
features. The parameters sp and sa are the weights assigned to
the prior knowledge and additional features, respectively.
These weights represent the level of confidence on the
feature’ importance for prediction
Figure 2 is a sketch of a tensioned elastic cylinder under a linearly sheared current profile U(z) distributed along axis z, which causes the cylinders’ vibration in both the inline (IL) and crossflow (CF) directions with respect to the incoming current. The vibration of the elastic cylinder can be approximated as a tensioned Euler-Bernoulli beam. The equation of motion in crossflow direction and inline direction can be expressed as, m ( z ) ∂2 y + cs ( z ) ∂∂yt − P ( z, t ) ∂2 y + EI ( z ) ∂4 y =Fcf(4) m ( z ) ∂t 2 ∂2 x ∂t 2 + cs ( z ) ∂x ∂t ∂z2 ∂2 x ∂z2 ∂z4 ∂4 x ∂z4 − P ( z, t ) + EI ( z ) =Fil (5)
Where x and y are the displacement in inline and
crossflow direction. m(z) is the cylinder’s mass per unit length,
P(z,t) is tension of the vibrating cylinder, EI(z) represents
bending stiffness. cs is the structural damping coefficient
per unit length, Fcf and Fil are the vortex induced forces on
the cylinder. The loading transfers energy from fluid to the
structure in a well-defined region with length Lin, which is
the “power-in” region. Outside this region, the vortex
loading dissipates energy by transferring energy from the
structure to the fluid through hydrodynamic damping
coefficient ch(z). The location of the “power-in” region can be
identified from structural vibration measurements in
experiments or simulation
Fcf = f (U ( z ) , x ( z, t ) , y ( z, t ) , ρ ,µ , L, L in , D, m ( z ) , (6) (7) (8) cs,cf ( z ) , ch,cf ( z ) , P ( z, t ) , EI ( z )) Fil = f (U ( z ) , x ( z, t ) , y ( z, t ) , ρ ,µ , L, L in , D, m ( z ) ,
cs,il ( z ) , ch,il ( z ) , P ( z, t ) , EI ( z )) Where ρ , µ , L and D are fluid density and dynamic viscosity, cylinder’s length and diameter, respectively.
If the spatiotemporal root-mean-square (rms) amplitude of crossflow vibration Arms,cf in the power-in region is the target output, then from Equations (4)-(7), the predictive model can be expressed as,
Arms,cf = f (U ( z ) , x ( z, t ) , y ( z, t ) , ρ ,µ , L, L in , D, m ( z ) ,
cs,cf ( z ) , ch,cf ( z ) , P ( z, t ) , EI ( z ))
It can be observed that Equations (6)-(8) involves spatial-temporal distribution of structural response and system properties, which will be further simplified and represented by some global VIV features.
VIV measurements from the 2011 Shell experiments on a
38-m-long cylinder
The measured crossflow displacements in a linearly sheared current are presented in Figure 3. The top figures are the CF response time series at two locations within the “power-in” region, while their corresponding wavelet analysis is shown at the bottom. The vibration is found to be narrow-banded with the dominant frequency drifting in time. Given the dominant vibration frequency and structural properties, the corresponding wavenumber k can be estimated by the dispersion relationship.
Meanwhile, Figure 4 shows the corresponding spatialtemporal distribution of crossflow displacement for the same test condition. The response is nonstationary, with a mixture of standing wave and travelling wave components. To better capture the temporal variation of the vibration signal, a moving window analysis is conducted. The vibration signal is windowed into overlapping time frames over each 3 vibration cycles, with 75% overlap.
Complex proper orthogonal decomposition (POD) is
conducted on the crossflow displacement in each
spatialtemporal window in the “power-in” region to decompose
the displacement in each window into several orthogonal
complex modes
κ cf ,κ il , ρ ,µ , L, L in , D, m, cs , ce,cf , ce,il , P0 , P, EI )
Where Urms and ∆U / L are the spatial root-mean-square
and the shear gradient of the current profile, respectively
within the power-in region. Arms,cf and Arms,il are the
spatiotemporal rms for the crossflow and inline VIV amplitude
in the power-in region. cs and ce are respectively, the
structural damping coefficient and the equivalent rigid cylinder
damping coefficients that will lead to the same power
dissipation as discussed by
Non-dimensionalizing Equation (9) gives,
Ac*f = f (Re,β , Ai*l ,Vrcf ,Vril , Lkcf ,α cf ,α il ,κ cf ,κ il ,
(9)
L / L in , L / D, m*ζ , cc*f , ci*l , P0 L2 EI
, P
( EIkc2f ))
(10)
Where Ac*f = Arms,cf / D , Ai*l = Arms,il / D are the
dimensionless crossflow and inline response amplitude, respectively.
Re = ρUrms D / µ is Reynolds number; β =D/ ( Urms )( ∆U / L)
is known as the shear parameter; Vrcf = 2π U / (ωcf D ) ,
Vril = 2π U / (ωil D ) are the crossflow and inline reduced
velocities, respectively; m ζ = 4mζ / (πρ D2 ) is known as
*
the mass damping parameter in the VIV literature, which
historically has been thought to be important in controlling
rigid cylinder’s VIV amplitude. cc*f = 2ce,cf ωcf / ρU r2ms and
ci*l = 2ce,ilωil / ρU r2ms are the dimensionless forms of the
equivalent damping parameter in the crossflow and inline
directions, respectively
Although Equation (10) suggests that crossflow response prediction in the “power-in” region may require considering the effect of all the 17 dimensionless variables, it is likely that the dimension of the input features can be further reduced due to redundancy or correlation between features or irrelevance to the prediction target. We are interested in finding a smaller and more manageable subset of parameters that are ultimately the most important out of the full set when it comes to determining the CF response amplitude. The motivation behind this is our interest to understand what causes the CF response variability that is observed in the temporal domain. At the very least, we would like to start associating changes to certain parameters with that variability that is often observed but is too complicated to understand.
The dataset is from a set of experiments conducted by
Shell Oil Co. in 2011 at Marintek. The vibration of
38meter-long cylinders under various current profiles were
measured. The test matrix included two pipes with
different diameters (30-mm and 80-mm) but of the same
bending stiffness. The cylinders were tested in uniform and
linearly sheared current profiles with the maximum flow
speed, Umax, ranging from 0.5 m/s to 2.5 m/s. This resulted
in the Reynolds number Re ranging from 1.0 ×104 –
20×104. The dataset also included cases where the 80-mm
pipe was covered with strakes over 50% of its length. The
pipe tests were conducted in uniform flows with Umax
varying from 0.5 m/s to 1.5 m/s. The strakes dissipated
vibration energy and limited the power-in region to Lin = 0.5L ,
50% of the cylinder’s length. Detailed descriptions of the
experiments can be found in
The structural damping ratio ζ in the experiment was
around 0.5%
The deep neural network was constructed using two hidden
layers. Each hidden layer had twenty neurons using a
sigmoid activation function. The total number of data points
was around 6000. 70% of the experimental data were used
as the training data, while the rest was used as the test data.
The input variables x were standardized to keep the
features at the same scale, while the output variables y were
normalized to values between 0 to 1. The mean absolute
percentage error (MAPE) was chosen as the loss function
between prediction and measurement L ( y, yˆ ) . The neural
network optimization was conducted via FTRL algorithm
Experimental studies on small spring-mounted rigid
cylinders show that the response amplitude increases with
increasing Reynolds number in the range 103 to 104 and
decreases as the dimensionless damping increases
Similarly, studies on long flexible cylinders have shown
that the Reynolds number and the dimensionless damping
continue to play important roles on the VIV response
amplitude
Because it is known that Reynolds number Re and dimensionless damping parameter cc*f are important, these two parameters are designated to be used as prior knowledge. The shear parameter β which is ideally suited to differentiating between uniform or sheared flows was the third parameter that was chosen as prior knowledge before starting the feature selections process.
Feature selection knowledge on top of prior physical The feature selection procedure was conducted by increasing the hyperparameter λ from 0.01 until all the input features except the prior knowledge shrank to 0. Figure 5 demonstrates how varying the value of λ determines the number of features chosen by the proposed algorithm. In the figure the retained features are indicated by the presence of a black bar at each λ value tested.
The prediction error varies with the retained features in the prediction model, which is presented in the bottom part of Figure 5. At each number of features, a brute force approach that searches all the possible combinations of the additional features is also carried out. The error obtained from the WSPINN is compared with hundreds runs of DNN predictions using combinatorically searched features in addition to the 3 features representing prior knowledge. The comparison suggests that the WSPINN is able to find the feature subsets that gives smallest prediction error among all the feature combinations. Besides, it can be observed that there could be multiple combinations of features that give similar prediction accuracy. For example, both the additional features Ai*l ,κ cf and Ai*l ,κ il gives prediction error around 13%. This suggests the correlations and interactions among some of the VIV features.
After balancing the prediction accuracy with the sparsity of input features, we find that the feature subset containing 5 features: Re, β , cc*f , Ai*l ,κ cf gives 13% MAPE, which is close to 10.6% MAPE using all 17 features.
We have also applied the WSPINN algorithm to other VIV related problems, such as the prediction for rigid cylinder’s VIV amplitude and flexible cylinders’ VIV amplitude at higher harmonics etc. Because of space limitations we cannot demonstrate this here. Moreover, since this paper only studied the important parameters for VIV sheared and uniform current profiles, the importance of the features may be different for more complicated current profiles.
The importance of the identified features for flexible cylinder VIV can be examined by systematically varying the ranges of input features to the constructed neural network models. Figure 6 and Figure 7 show the effect of varying Re, cc*f while constraining the other variables in the prediction model to characteristic values most often observed in the Shell experiments. The black dots are the experimental measurements within 20% from the referenced values and are included to demonstrate that the prediction model (contours) did in fact have data in that vicinity.
The results demonstrate that increasing Reynolds number tends to increase the spatiotemporal CF RMS amplitude. This Reynolds number effect is obvious in the uniform flow data which typically has small dimensionless damping values ( cc*f <0.3-0.4). While the Reynolds number effect is virtually non-existent when looking at the sheared flow cases with damping parameters ( cc*f >0.4).
Figure 8 shows the effect of varying Ai*l and κ cf while
constraining the other variables. It can be found that the
crossflow response tends to increase with inline response.
Such a relationship has also been observed in
springmounted rigid cylinder’s VIV experiments
1. Direct and learning task dependent dimension reduction in the original feature space while retaining the prior information in the model.
The WSPINN is one of the dimension reduction approaches. However, different from widely used PCA or auto-encoders, WSPINN seeks to reduce the dimension directly in the original input feature space. Moreover, through machine learning prediction, WSPINN is able to identify most important input feature with respect to the target output. The much smaller constraints placed on the prior knowledge also allows the prior knowledge to retain in the prediction model to improve prediction and also identify additional important features. 2. High prediction accuracy due to the universal approximation property of the DNN (Hornik, 1993)
The WSPINN is a feature selection approach embedded in DNN, which is able to predict nonlinear inputoutput relationships accurately. For instance, for the crossflow VIV amplitude prediction, the prediction accuracy from the DNN and linear regression given Re, β , cc*f , A* ,κ cf il are 13% and 25%, respectively. However, training DNN with WSPINN requires several rounds of iterations to optimize the weights in each layer, hence it was found to be more computationally expensive than most of the other machine learning methods We consider the computational cost acceptable since our intention is not to create a fast predictive tool but rather to use machine learning to reduce the dimensionality of the problem as we try to understand the importance of each of the many governing parameters.
In this paper, we modify and propose changes to a sparseinput neural network so it can efficiently select additional features which can complement a subset of features known to be important in advance (i.e. prior knowledge). The algorithm was applied to the experimental results from vortex-induced vibration of flexible cylinders. The complicated spatiotemporal response measurements of the continuous system are reduced to an equivalent 2 Degree of Freedom system. The proposed algorithm is then used to investigate the role of Reynolds number, damping parameter, and shear parameter (3 parameters for which we have prior knowledge), as well as 14 other parameters that the dimensional analysis indicated might be important. The algorithm was able to reduce the 14 additional parameters to just 2 additional parameters on top of the prior knowledge. We found that this feature selection technique is much more efficient than a brute force combinatorial search.
This research has been sponsored by the members of the SHEAR7 Joint Industry Project: BP, Chevron, ExxonMobil, Petrobras, SBM Offshore, Shell International Exploration and Production, Equinor, & Technip USA.