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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 561875, 10 pages
Comparison Study on Linear Interpolation and Cubic B-Spline Interpolation Proper Orthogonal Decomposition Methods
National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, China
Received 18 January 2013; Accepted 23 February 2013
Academic Editor: Shuyu Sun
Copyright © 2013 Xiaolong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In general, proper orthogonal decomposition (POD) method is used to deal with single-parameter problems in engineering practice, and the linear interpolation is employed to establish the reduced model. Recently, this method is extended to solve the double-parameter problems with the amplitudes being achieved by cubic B-spline interpolation. In this paper, the accuracy of reduced models, which are established with linear interpolation and cubic B-spline interpolation, respectively, is verified via two typical examples. Both results of the two methods are satisfying, and the results of cubic B-spline interpolation are more accurate than those of linear interpolation. The results are meaningful for guiding the application of the POD interpolation to complex multiparameter problems.
When the complex heat and fluid flow problems are simulated by computers, usually large groups of equations need to be solved at high computational cost. To reduce the computational burdens while maintaining acceptable resolutions for engineering application, the POD method, which was proposed and developed in 1940s, has been used recently in computational fluid dynamics [1–5]. The advantage of the POD method is that we solve the problems by a reduced model; that is, the physical field can be accurately and rapidly reconstructed [6, 7] by a linear combination of orthogonal basis functions (eigenfunctions) and their amplitudes in a low-dimensional way.
The reduced model by POD can be classified into two groups. One is to project the governing equations onto a low-dimensional space spanned by eigenfunctions with the Galerkin method to obtain the equations for amplitudes, while the other is to calculate amplitudes by interpolation. Both methods have their own advantages and disadvantages. The former is suitable for either steady or unsteady problems and is able to convert the partial differential equations of the complex physical process into the ordinary differential ones. However, the deduction of the reduced model is complex, and the method is not applicable to the processes, which cannot be described in the form of partial differential equations. The latter is suitable for the steady problems and accurate even for strong nonlinear situation, which is usually used to deal with single-parameter problems in engineering practice  by using linear interpolation to establish the reduced model. Recently, this method is extended to the double-parameter problems by cubic B-spline interpolation . To the authors’ best knowledge, the POD interpolation method has not been applied in triple-parameter problems. In this paper, the applicability of the reduced model by POD interpolation is tested for multi-parameter problems, and then the accuracy of the reduced models achieved by linear interpolation and cubic B-spline interpolation is examined via two typical examples, that is, Czochralski crystal growth problem  and lid-driven cavity flow problem.
2. A Brief Introduction to POD and SVD
POD is a powerful and elegant method for model reduction aimed at obtaining low-dimensional approximate descriptions of a high-dimensional space . The low-dimensional space is spanned by orthogonal eigenfunctions. The reduced order description of physical process can be reconstructed by linear superposition of eigenfunctions and corresponding amplitudes:
The most striking features of the POD are its optimality of minimizing the average squared distance between the original data and its reduced linear representation with only a few leading eigenfunctions capturing nearly all energy of dynamic physical process . The feature can be described in the following mathematical ways: where the inner product in is presented by ; is the averaging operation; stands for the norm; | |: denotes the modulus.
Introduce a Lagrange multiplier to obtain the eigenfunctions:
This condition can be reduced to the following integral eigenvalue problem : where represents the averaged autocorrelation function.
The energy is defined as the sum of the eigenvalues; that is, . Thus, the energy percentage of the th eigenfunctions is given by .
The POD eigenfunctions and corresponding amplitudes can be obtained by applying the singular-value decomposition (SVD) method to the sample space. Klema and Laub indicated that the SVD was established for real-square matrices in the 1870s by Beltrami and Jordan, for complex square matrices in 1902 by Autonne, and for general rectangular matrices in 1939 by Eckart and Young . SVD is much more general than the eigenvalue decomposition and intimately relates to the matrix rank and reduced-rank least-squares approximation; thus, SVD can be viewed as the extension of the eigenvalue decomposition for the case of nonsquare matrices .
The implementation of POD by using the SVD is described as follows.
Suppose that are sample vectors given out by sampling properly, where (). Assume that the number of samples is large enough, which means . Denote ; thus, and is the auto-correlation matrix. The eigenvalues of in decreasing order are given out as follows: . are defined as the singular values of . Then the eigenvectors of are with their corresponding eigenvalues . Defining with , , and is the rank of matrix . Denote with and . Due to the basis extension theorem of vector space, there exist orthonormal vectors orthogonal to in . Let ; then is an orthonormal matrix. For SVD method, and are defined as left and right singular vectors of corresponding to eigenvalues , respectively .
Thus, the samples can be presented in the following form:
From (5), it follows that
Define the matrix as .
From (6), it follows that
Thus, the sampled vectors of POD have been obtained.
3. POD Interpolation
The reduced model by the POD interpolation can be achieved by the three steps: first, we should get sufficient data to construct a sample matrix. Then, eigenfunctions and the corresponding amplitudes can be calculated by SVD. Finally, the amplitudes out of sampling parameters are obtained by interpolation. Since the interpolation affects the accuracy of the reduced model and few studies have been made to clarify the effects of different interpolations, we make the accuracy comparison between linear interpolation and cubic B-spline interpolation in this paper.
POD interpolation method is usually used to deal with one-parameter problems in engineering practice  by using linear interpolation to establish the reduced model. Recently, this method is extended to the double-parameter problems by the cubic B-spline interpolation in .
For a linear interpolation problem with parameters (), the nonsample amplitudes can be calculated by where the superscript of parameter is used to differentiate various parameters, while the subscripts and stand for the upside and downside values of interpolation.
The cubic B-spline interpolation is widely used in numerical approximation, the calculation of ordinary differential equations, and computation in science and technology, because it can give out a smooth interpolation curve. For brevity, the expression of a cubic B-spline interpolation is hence omitted. The detailed information about it can be found in the help documentation MATH. PDF in function library IMSL of FORTRAN language.
4. Computational Cases
Accuracy comparisons are carried out for POD reduced models established by linear interpolation and cubic B-spline interpolation, respectively, for two steady flow problems, that is, Czochralski crystal growth problem and lid-driven cavity flow problem sketched in Figure 1. For the first problem, the temperature fields are reconstructed while the streamfunctions are reconstructed for the second problem. The lid-driven cavity flow problem is a very famous one in the field of computational fluid dynamics and the governing equations and boundaries can be easily found in the literature. Here we only present the governing equations and boundary conditions for the Czochralski crystal growth problem as follows:
The general variables, generalized diffusion coefficient, and generalized source term are summarized in Table 1. In Table 1, the definitions of the dimensionless variables are where and are the Reynolds numbers based on the angular velocity of crystal and crucible, respectively. In this paper, , , and are changeable while , , and are fixed in the calculation. The boundary conditions are given in Table 2.
A finite volume method is used for the discretization of the governing equations. 80 × 80 uniform grids are employed for both problems. The convective terms and diffusion terms are discretized by QUICK scheme and a central difference scheme, respectively. A staggered grid system is used to avoid unphysical zigzag pressure field. The SIMPLE algorithm is adopted to couple the velocity field and pressure. The discretized algebraic equations are solved by TDMA method. The details on the above numerical techniques are referred to in .
5. Results and Discussion
The reduced models by means of POD interpolations of single, double, and triple parameters for the Czochralski crystal growth problem as well as double parameters for the lid-driven flow problem are established. To establish the reduced models, first of all, sufficient data are required to construct a sample matrix to obtain the eigenfunctions of a temperature field for the Czochralski crystal growth problem and a flow field for the lid-driven flow problem at different conditions. The matrices can be obtained through finite volume method (FVM), stated in Section 4. The samplings of these two problems are as follows. As for the Czochralski crystal growth problem with one parameter, Gr is selected as a variable, and its 8 values varying in the range of 105 to 106 are listed in column A of Table 3; and for double parameters case, Gr and Re are selected as the two variables, and their values are listed in columns A and B of Table 3, respectively. 48 calculation cases are obtained by the combination of the two parameter values. As for triple parameters case, another variable (the ratio of liquid height to radius of crucible) is added, besides Gr and Re, whose value is in column C of Table 3. The combination of triple parameters increases to 288 calculation cases. As for the lid-driven flow problem, the two parameters are Re and (the ratio of height to width), whose values are, respectively, listed in columns D and E of Table 3 and the number of combinations is 25.
Then the SVD method [15, 16] is applied to obtain the eigenfunctions and corresponding amplitudes. The nonsampling amplitudes can be calculated by linear interpolation and cubic B-spline interpolation, respectively. The test examples are selected in Table 4. For the Czochralski crystal growth problem and lid-driven flow problem, the numbers of test examples by combination are 7, 35, 175, and 16, respectively. Around 280 s and 130 s are taken by FVM to get solutions using a personal computer that contained an I7-860 CPU and 4 GB DDR3 memory for a Czochralski crystal growth case and a lid-driven flow case, respectively, and comparatively only around 0.03 s is taken by POD interpolation to get a solution. Apparently, the POD method much outweighs the FVM method in terms of computational cost.
In order to quantitatively validate the accuracy of reduced model given by POD interpolation, we define the relative average error as in the following expression: where is the FVM result, while is the one achieved by POD interpolation. is the grid numbers. The maximum relative error and average error are shown in Tables 5, 6, 7, and 8 under various conditions and various numbers of eigenfunctions. In these tables, and stand for the maximum relative error of linear interpolation and cubic B-spline interpolation, respectively, while and represent the average error by using linear interpolation and cubic B-spline interpolation, respectively.
From Tables 5–8, we can see that when the number of eigenfunctions applied in the reduced model is small (first 1 or 2 groups of eigenfunctions), the errors for both of the POD interpolations are great with the maximum error being 46%. When more eigenfunctions are used, the results are predicted more accurately for the reduced model. On the other hand, if the number of eigenfunctions exceeds 5, the computational accuracy is improved a little. For example, when we choose eight eigenfunctions to describe Czochralski crystal growth problem, the maximum error is 0.365%/0.169%, 0.464%/0.249%, and 1.782%/0.761% for different parameter numbers, that is, single variable, double variables, and triple variables, respectively, while the maximum error for the lid-driven flow problem is 6.167%/2.558%.
When the number of eigenfunctions is less than 3, the numerical accuracy of the two interpolation methods is almost the same. When more eigenfunctions are employed, higher accuracy is achieved by cubic B-spline interpolation. Figure 2 illustrates that the deviation of cubic B-spline interpolation is less than its counterpart, that is, linear interpolation, among all the test examples when the first 8 eigenfunctions are applied. The maximum and minimum errors of the POD interpolation are shown in Figures 3, 4, 5, and 6. In these figures, the solid line stands for the results of FVM method, while the dash line and dotted line are representative of the results of cubic B-spline interpolation and linear interpolation, respectively. Obviously, the results of POD interpolation agree well with those of FVM. In a word, POD interpolation method can simulate the physical process precisely for multivariable; in addition, the cubic B-spline POD interpolation presents higher accuracy.
The POD linear interpolation and cubic B-spline interpolation are employed to work out the nonsample amplitudes, and their accuracy for multivariable problems is verified via two typical heat and fluid flow problems. Through a large number of examples, we can conclude that POD interpolation method outweighs the FVM method in terms of computational cost, and the cubic B-spline POD interpolation has higher accuracy than the linear one.
|:||The amplitude of POD corresponding to the th eigenfunction|
|:||The maximum relative error of linear interpolation|
|:||The maximum relative error of cubic B-spline interpolation|
|:||The average error of linear interpolation|
|:||The average error of cubic B-spline interpolation|
|:||The Grashof number|
|:||The height of crucible|
|:||The point achieved by interpolation|
|, :||Interval circumscribed interpolation points|
|:||The Prandtl number|
|:||The radius of crystal in Czochralski crystal growth problem|
|:||The radius of crucible in Czochralski crystal growth problem|
|:||The Reynolds numbers in lid-driven flow problem|
|:||The Reynolds number due to the rotation of crystal in Czochralski crystal growth problem|
|:||The Reynolds number due to the rotation of crucible in Czochralski crystal growth problem|
|, , :||The spatial cylindrical coordinates in Czochralski crystal growth problem|
|, , :||Thevelocity components in the cylindrical coordinate|
|:||Left singular vectors of sample matrix|
|:||Right singular vectors of sample matrix|
|:||The sample matrix|
|:||The ratio of height to radius in Czochralski crystal growth problem|
|:||The radius ratio of crystal to crucible in Czochralski crystal growth problem|
|:||The ratio of height to width in lid-driven flow problem|
|:||The total energy of the POD eigenfunctions|
|:||The th eigenfunction of POD|
|:||The eigenvalue of the th eigenfunction|
|:||The physical field|
|:||The th singular value of samples|
|:||The angular velocity of crystal in Czochralski crystal growth problem|
|:||The angular velocity of crucible in Czochralski crystal growth problem.|
The study is supported by the National Science Foundation of China (nos. 51206186, 51134006, 51176204) and Science Foundation of China University of Petroleum-Beijing (nos. 2462011LLYJ33, 2462011LLYJ55, 2462012KYJJ0403, 2462012KYJJ0404).
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