A Study on the use of multiple surrogate models in valve design

Xue Guan Song

1*, Jeong Ju Choi

2, Joon-Ho Lee

2, Young Chul Park

1

1Department of Mechanical Engineering, Dong-A University, Busan, 604-714, Korea,

2 Technical Center for High-Performance Valves, Dong-A University, Busan, 604-714, Korea, *songxguan@gmail.com

1. Abstract

Nowadays, the use of computational fluid dynamics (CFD) in the design of valves is very common. Despite the

continuing growth of computing capability, the computational cost of complex three-dimensional CFD analysis of

butterfly valve maintains high, therefore, the CFD analysis-based optimization becomes more time-consuming and

computational expensive. In this paper, a comparative study on the use of multiple approximate models including

polynomial response surface, Kriging model, support vector regression and radial basis neural networks, which

have been well used for a variety of engineering optimizations, is performed for the prediction and optimization of

fluid performance of a butterfly valve. Several types of error analysis corresponding to the four surrogate models

are compared to identify the final optimum result and which model is more proper for this case. This study gives a

deep insight into the use of multiple surrogate models for the design and optimization of a butterfly valve.

2. Keywords: Butterfly valve, Optimization, Multiple surrogate models, DOE, Error analysis

3. Introduction

Butterfly valves are one of the oldest practical devices for controlling the flow of fluids. They are widely used due

to their simple and compact construction, light weight and facility to operate from fully open to fully close quickly.

Along with the development of computational method, the internal flow visualizations of butterfly valve and

predictions of their performances with computational approach become feasible and popular.

Up to the present, many researchers have studied the fluid characteristics of butterfly valve by using computational

fluid dynamics (CFD). For example, Huang and Kim [1] investigated the incompressible flow characteristics of

butterfly valve flows in three-dimensions (3-D). The velocity fields, pressure distributions, streamlines, particles

paths and flow separations are observed and/or analyzed. Janusz and Czeslaw [2] investigated a commonly used

valve having a thin-flat-sharp-edge disk and blockage ratio d/D = 0.947 by using both the experimental and

numerical (CFD) methods. Zachary Leutwyler and Charles Dalton [3] investigated the use of CFD methods as

applied to the flow of a compressible fluid past a butterfly valve. The flow coefficient and torque coefficient are

denoted as the valve performance. Having understood the characteristics and performances of butterfly valves with

regard to valve opening and different types of valve shape, the size optimization of butterfly valves is brought into

focus. However, despite the continuing growth of computing power and speed, the CFD analysis of complex

three-dimensional butterfly valve remains high time-consuming and computational expensive, thus the CFD

analysis-based conventional optimizations are severely limited, even with the current supercomputer.

To fulfill CFD-based optimization, an alternative is to construct a surrogate model of complicated CFD analysis.

The surrogate model can construct expression between the objective function and design variables with very

simple equations and short time. There have been a lot of successful researches applying one surrogate model to

solve the engineering optimizations [4-6]. Actually, there are many types of surrogate models. Based on different

concepts, these surrogate models sometimes give the same relation expressions and sometime not, and it‟s not

clear which surrogate model is best for any particular case. Researchers using one surrogate model must combine it

with their past experience. In another word, one surrogate model is not perfect and/or enough. To account for

uncertainties in predictions, a simple way [7, 8] is to use more than one surrogate model simultaneously.

The objective of the present study is (Ι) to compare the multiple surrogate models and identify which one is best for

the optimization of the performance of butterfly valve, which is characterized by the pressure loss coefficient, (Π)

to optimize the butterfly valve and predict its performance.

4. Fluid analysis of butterfly valve

4.1. Flow characteristics of butterfly valve

There are many types of indexes/coefficients to denote the characteristics of butterfly valve. Herein, the pressure

loss coefficient “K” is used to relate the pressure loss of a valve to the discharge of the valve at a given valve

opening angle. In this research, it is used to estimate the characteristics of the butterfly valve. It can be expressed as

Eq.(1):

2 / 2

PK

v g

(1)

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where ΔP refers to the static pressure difference between upstream and downstream of the valve, v is the resultant

average velocity through the valve, and g is the gravity acceleration.

4.2. Boundary conditions and grid generation

Fluid analysis is firstly conducted to observe the flow characteristics of this butterfly valve at two valve opening

positions. As shown in Figure 1, a half symmetrical CFD model of the butterfly valve is created with a scale of 1:1

to get a better result and to save the computation time. The bolts and nuts for fixation of disk are ignored here. An

upstream pipe with a length of four times the diameter and downstream pipe with a length of eight times the

diameter which have been verified long enough for the analysis, are created to provide a static fluid domain [9]. In

addition, the steady state analysis is done without considering the heat transfer, and water at normal temperature of

25oC (incompressible) is used as a working fluid. The standard k-ε model is adopted for the prediction of turbulent

flow because of its robustness, economy and reasonable accuracy. A uniform velocity of 3m/s is imposed at the

inlet boundary, while the “opening” boundary is utilized as the outlet of the downstream pipe.

Usually, the grid structure has a strong influence on the quality of the numerical results and computation time

required. It has been found that solutions by using structured grid are generally faster and more accurate than those

using unstructured grid, but unstructured grid is very suitable for structures with complex shape. Thus, the hybrid

grid containing both structured grid and unstructured grids are developed in this study. Hexahedron elements

(structured grid) are used in the upstream and downstream pipes away from the section of the grid in close

proximity to the disk. Tetrahedron elements (unstructured grid) are used near the disk, except for the cell elements

(also called as prism layer in ANSYS ICEM-CFDTM

) on the disk face and pipe walls as seen in Figure 1. And to

ensure the CFD analyses at different opening have the same accuracies as much as possible, the same upstream and

downstream pipes are used, only the middle section including disk was modified in each case. The total number of

nodes and elements are approximately 209,831 and 571215, respectively.

4.3. Result of CFD analysis

After creation of the CFD model and definition of the boundary conditions, the commercial software ANSYS CFX

Solver is used to solve it. Despite the little difference of elements/nodes number in each model, each simulation

takes approximately 26 min on a single Intel Xeon processor (2.33GHz, 8GB of RAM). Figure 2 illustrates the

velocity and pressure distributions at 30o and 8

o opening positions. It can be found the valve opening has a great

effect on the velocity and fluid pressure through the valve.

inlet

outlet

Symmetry plane

Butterflyvalve

Figure 1: Hybrid grids and boundary conditions for CFD analysis

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Figure 2: Velocity and pressure distribution of middle plane

5. Approximation for butterfly valve

5.1. Surrogate models

The surrogate models are suitable to reduce computational cost of optimization. For this study, four surrogate

models including 4th order polynomial response surface approximation (PRS) [10, 11], kriging (KRG) [12, 13],

support vector regression (SVR) [14, 15], and radial basis neural networks (RBNN) [16] are comparatively used to

reduce the computational expense involved in optimization. There have been many researches regarding the

principle characteristics of these surrogate models, and they will not be explained here.

5.2. Optimization formulation and DOE

As explained above, low pressure loss coefficient is appreciated for a butterfly valve, since it results in a low

pressure loss under the same velocity condition. Therefore, minimizing pressure loss coefficient K is considered as

the design objective. Since the major dimensions of this valve are not allowed to change though they may have

great effect on the pressure loss coefficient, only two dimensions „L‟ and „S‟ are taken into account as the design

variables, L is the diameter of the fringe and S is the diameter of middle thicker part. These two dimensions specify

the core region of valve disk as indicated as the red region in the front view of Figure 3. Another parameter

„opening α‟ is also considered to study the flow characteristic of the butterfly valve at different openings. The

range of L and S are from 100mm to 140mm and 40mm to 50mm, respectively. Because the CFD analysis of valve

at positions from 75o to fully close position is very inaccurate [9], the opening is set in the range from zero (fully

open) to 75o. Hence, the optimization for the butterfly valve can be formulated as follows:

Minimize ( , , ) (2)

100 140 (3)

40 50 (4)

0 75 (5)

K f L S

Subject to mm L mm

mm S mm

30

o

Velo

city (m

/s)

30

o

Pressu

re (Pa)

8o

Velo

city (m

/s)

8o

Pressu

re (Pa)

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After definition of the variable space, next step is to sample data in the design space, it‟s also called as design of

experiment (DOE). Among many types of DOE methods, Latin hypercube sampling (LHS) and face central cubic

designs (FCCD) are very classical and effective on reducing computation times. To get better results, a

combination of FCCD and ordinary LHS is generated to sample the data. As illustrated in Figure 4, total 45

sampling points are generated, which are adequate to evaluate 35 coefficients of a fourth-order PRS, then they are

scaled to fit the design space defined by the bounds on L, S and opening α.

α (op

enin

g)

L

S

FLOW

Front view

Figure 3: Design variables for butterfly valve

Figure 4: Thirty LHS points (red) and fifteen FCCD points (green)

5.3. Error analysis of surrogate model

To access these four surrogate models, the goodness of fit obtained from “training” data is not sufficient.

Especially, since KRG interpolates the sample data, many estimators based on the original “training” points

become meaningless except cross-validation. For this reason, additional 15 validation samples are used to verify

the accuracy of models from the viewpoint of newly predicted points, and four types of estimators are used: the

maximum absolute percent error, the average absolute percent error, the root mean square error and the

cross-validation. The equations for them are given in Eq.(6) to Eq.(9), respectively.

ˆ[ ) / ] 1,2,...,i i i vMAPE max Y Y Y i n (6)

1

ˆ[ ( / )] /vn

i i i v

i

AAPE Y Y Y n

(7)

2

1

ˆ( ) /vn

i i v

i

RMSE Y Y n

(8)

2

1

ˆ( ) /sn

i i s

i

CV Y Y n

(9)

where nv is the number of new created points for validation, and ns is the number of original sampling points. nv is

15 and ns 45.

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6. Results

6.1. Error and graphical comparison of surrogate models

To compare the performance of these surrogate models obviously, multiple bar-charts are provided in Figure 5. It

can be observed that no matter what kind of error is estimated (the fitness of the original sampling points or the

prediction of new points), KRG gets the best performance. RBNN and PRS are a little worse than KRG, and SVR

performs worst. In Figure 6, three-dimensional contour plots of pressure loss coefficient are given. In each plot the

opening α is set to be zero. It‟s revealed that the contours from PRS and KRG are very similar, and those from SVR

and RBNN indicate opposite trends. Figure 7 illustrates the pressure loss coefficient at different valve opening

with different L and S. It‟s obvious that the predicted curves of SVR deviate from the overall trends and the

sampling data seriously. This is also evidenced by the high errors displayed in Figure 5. In short, KRG is shown to

be the best model for the overall fitness.

Figure 5：Error comparison of four surrogate models

Figure 6: Error comparison of four surrogate models

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Figure 7: Pressure loss coefficient vs opening in four surrogate models

6.2. Optimization and prediction of K curve

As a final comparison of the accuracy of the four models and optimization of the valve‟s performance, the genetic

algorithm (GA) is used to solve Eq.(2) ~ (4). Each surrogate model with the equations is solved five times to get

optima. The final results of four optimizations are summarized in Table 1 and illustrated in Figure 8. It implies

KRG finds the real optimum pressure loss coefficient at L=140mm, S=50mm and α=0o, on the contrast, PRS yields

the worst optimum result. Combining the results of error analysis it can be summarized that KRG performs best in

both the overall fitness and finding the optimum point. But other three models are good at either the overall fitness

or the prediction of optima.

Table 1: Optimization results using four surrogate models

L (mm) S (mm) α (o) Prediction Real value Error (%)

PRS 100 50 0 0.1022 0.2097 51.26

KRG 140 50 0 0.1624 0.2 18.8

SVR 121.3 44.9 0 0.1848 0.2010 8.06

RBNN 101.4 48.3 0 0.1575 0.2073 24.02

Figure 8: Error of optimization results

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Figure 9: Pressure loss coefficient curves

At last, in terms of the comparison results, the best model KRG is used to predict the pressure loss coefficient of

valve (L=140mm and S=50mm) for different valve opening. Figure 9 indicates the curve of pressure loss

coefficient K, it will be very useful for the valve users to properly use this type of butterfly valve in service.

7. Conclusions

In this paper, the comparative use on four surrogate models (PRS, KRG, SVR and RBNN) for the optimization and

prediction of fluid performance of a butterfly valve is demonstrated. Firstly, the CFD analysis of butterfly valve is

conducted to observe the flow characteristics of butterfly valve. Secondly, the combination of Latin hypercube

sampling and face central cubic design are adopted as the design of experiment to generate the sampling points.

Then multiple surrogate models are constructed and the optimum point for each surrogate model is found with the

help of genetic algorithm. Based on the error comparisons, it‟s confirmed that the Kriging model appears to yield

the best results in both the overall fitness of the real response and prediction of optimum point among the four

models. Had only one surrogate model been conducted and used for the optimization, it might be quite possible to

get a bad result. Thus, it can be said comparatively using multiple surrogate models is a very effective and

economical approach for the design and optimization of butterfly valve. In the near future, this approach will be

used to optimize and predict the other performance of this butterfly valve such as hydraulic toque coefficient, drag

coefficient and the maximum stress on the valve disk under severe work condition.

8. Acknowledgements

This work was supported by Technical Center for High-Performance Valves from the Regional Innovation Center

(RIC) Program of the Ministry of Knowledge Economy (MKE).

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