POD方法中的协方差模态截断准则与方差补偿技术
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摘要
本征正交分解(POD)是以能量表述的随机场的最优分解，是一种数据压缩和特征萃取的工具。本文分别采用瑞利商概念及其极值性质和主成分分析方法推导了POD原理。分别对矢跨比为1∶4和1∶6的球壳屋盖采用同步多点压力扫描系统进行了风洞试验，在POD分析中采用方差比和能量比做为模态截断准则来控制前若干阶本征模态重建屋盖风压场，对重建风压包括均值、均方根值、极值在内的统计特性，时程曲线以及功率谱进行了比较，并与实测结果进行比较。在采用方差比做为模态截断准则时，利用重建前后各测点脉动风压系数的方差比值，提出了方差补偿技术，以保证重建前后所有测点脉动风压系数的方差相等。利用方差补偿技术对重建风压时程进行修正，并与修正前的重建结果进行比较。
在响应计算中，采用方差比做为模态截断准则来选取恰当的模态数结合完全二次型方法缩阶计算屋盖结构所有测点的均方根响应，并与采用能量比做为模态截断准则选取的模态数进行缩阶计算的结果进行比较，提高计算精度。最后利用方差补偿技术对屋盖结构所有测点的风致响应进行修正计算，并与修正前的响应结果进行比较，讨论了在保证模态数目不变的情况下，提高CQC响应的计算精度以及在保证计算精度的情况下，提高计算效率的可能性。
The Proper Orthogonal Decomposition (POD) is an optimal decomposition of random fields in terms of an energy representation. It is a tool for data compression and characteristic analysis of extractable. The principle of POD is deduced separately by the concept of Rayleigh quotient and its extreme property, principal component analysis method in this article. The wind tunnel tests of a spherical shell roof when the rise-to-span is 1:4 and 1:6 are conducted by using the synchronous multi-pressure scanning system. Based on the POD technique, the wind pressure field of the roof has been reconstructed by the adoption of variance ratio and energy ratio as the truncation criterion with the control of the first several proper modes. The statistics characteristic of the reconstructed wind pressure, including the mean value, root mean square value, peak value, time-history curves as well as the power spectrum has carried on the comparison, and carried on the comparison with the measured pressures. When we adopt the variance ratio as the truncation criterion, in order to assure the results that the variance ratio of the fluctuating wind-pressure coefficient around all measured point is equal, use the variance ratio on the reconstruction measured points' fluctuating wind-pressure coefficient, and the method for technology of the variance ratio compensation is proposed. The variance ratio compensation technology is utilized to modify the reconstructed wind pressure, and compared with the reconstructed result before revised.
On calculation for responses, the author uses the variance ratio as the truncation criterion to select an appropriate mode number combined with Complete-Quadratic-Combination method, then shrink steps to process the root mean square response to all the measured points of the roof structure. In order to increase the calculation precision, the results are compared with those model numbers by shrinking step through energy ratio as truncation criterion. The variance ratio compensation technology is also utilized to modify the wind-induced response, and compared with the response before revised. When the model number is assured to be invariable, increase the calculation precision of CQC (Complete-Quadratic-Combination) response and make sure of the calculation precision, the possibility of increasing the calculation efficiency is discussed.
在响应计算中，采用方差比做为模态截断准则来选取恰当的模态数结合完全二次型方法缩阶计算屋盖结构所有测点的均方根响应，并与采用能量比做为模态截断准则选取的模态数进行缩阶计算的结果进行比较，提高计算精度。最后利用方差补偿技术对屋盖结构所有测点的风致响应进行修正计算，并与修正前的响应结果进行比较，讨论了在保证模态数目不变的情况下，提高CQC响应的计算精度以及在保证计算精度的情况下，提高计算效率的可能性。
The Proper Orthogonal Decomposition (POD) is an optimal decomposition of random fields in terms of an energy representation. It is a tool for data compression and characteristic analysis of extractable. The principle of POD is deduced separately by the concept of Rayleigh quotient and its extreme property, principal component analysis method in this article. The wind tunnel tests of a spherical shell roof when the rise-to-span is 1:4 and 1:6 are conducted by using the synchronous multi-pressure scanning system. Based on the POD technique, the wind pressure field of the roof has been reconstructed by the adoption of variance ratio and energy ratio as the truncation criterion with the control of the first several proper modes. The statistics characteristic of the reconstructed wind pressure, including the mean value, root mean square value, peak value, time-history curves as well as the power spectrum has carried on the comparison, and carried on the comparison with the measured pressures. When we adopt the variance ratio as the truncation criterion, in order to assure the results that the variance ratio of the fluctuating wind-pressure coefficient around all measured point is equal, use the variance ratio on the reconstruction measured points' fluctuating wind-pressure coefficient, and the method for technology of the variance ratio compensation is proposed. The variance ratio compensation technology is utilized to modify the reconstructed wind pressure, and compared with the reconstructed result before revised.
On calculation for responses, the author uses the variance ratio as the truncation criterion to select an appropriate mode number combined with Complete-Quadratic-Combination method, then shrink steps to process the root mean square response to all the measured points of the roof structure. In order to increase the calculation precision, the results are compared with those model numbers by shrinking step through energy ratio as truncation criterion. The variance ratio compensation technology is also utilized to modify the wind-induced response, and compared with the response before revised. When the model number is assured to be invariable, increase the calculation precision of CQC (Complete-Quadratic-Combination) response and make sure of the calculation precision, the possibility of increasing the calculation efficiency is discussed.
引文
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[2] Lee, B. E. The effects of turbulence on the surface pressure field of a square prism. Journal of Fluid mechanics (1975), 69,263-282.
[3] Best, R. J., Holmes, J. D. Use of eigenvalues in the covariance integration method for determination of wind load effects. Journal of Wind Engineering and Industrial Aerodunamics (1983) 13,359-370.
[4] Kareem, A., Cermak, J.E. Pressure fluctuations on a square building model in boundary-layer flows. Journal of Wind Engineering and Industrial Aerodunamics (1984) 16, 17-41.
[5] Holmes, J. D. Analysis and synthesis of pressure fluctuations on bluff bodies using eigenvectors. Journal of Wind Engineering and Industrial Aerodunamics (1990) 33, 219-230.
[6] Holmes, J. D. Optimised peak load distributions. Journal of Wind Engineering and Industrial Aerodunamics (1992) 41-44, 267-276.
[7] Holmes, J. D., Sankaran, R., Kwok, K. C. S., Syme, M. J. Eigenvector modes of fluctuating pressures on low-rise building models. Journal of Wind Engineering and Industrial Aerodunamics (1997) 69-71,687-707.
[8] Bienkiewicz, B., Tamura, Y., Ham, H. J., Ueda, H., Hibi, K. Proper orthogonal decomposition and reconstruction of multi-channel roof pressure. Journal of Wind Engineering and Industrial Aerodunamics (1995)54,369-381.
[9] Kikuchi, H., Tamura, Y., Ueda, K. Hibi, K. Dynamic wind pressure acting on a tall building model Proper Orthogonal Decomposition. Journal of Wind Engineering and Industrial Aerodunamics (1997) 69-71,631-646.
[10] Tamura, Y., Ueda, H., Kikuchi, H., Hibi, K., Suganuma, S., Bienkiewicz, B. Proper orthogonal decomposition study of approach wind-building pressure correlation. In Proceedings of the Ninth International Conference on wind Engineering (1995), 2115-2126.
[11] Bienkiewicz, B., Tamura, Y., Ham, H. J., Ueda, H., Hibi, K. Proper orthogonal decomposition and reconstruction of multi-channel roof pressure. Journal of Wind Engineering and Industrial Aerodunamics (1995)54, 369-381.
[12] Jeong, S. H., Bienkiewicz, B., Ham, J. H. Proper Orthogonal Decomposition of building wind pressure specified at non-uniformly distributed pressure Taps. Journal of Wind Engineering and Industrial Aerodunamics (2000) 87, 1-14.
[13] Uematsu, Y., Kuribara, O., Yamada, M., Sasaki, A., Hongo, T. Wind-induced dynamic behavior and its load estimation of a single-layer latticed dome with a long span. Journal of Wind Engineering and Industrial Aerodynamics (2001)89, 1671-1687.
[14] Chen, Y., Kopp, G. A., Surry, D. Spatial extrapolation of pressure time series on low building using proper orthogonal decomposition. Journal of Wind and Structure (2004) 7(6), 373-392.
[15] 李方慧，倪振华，谢壮宁．POD方法在重建双坡屋盖风压场中的应用．工程力学(2005)22增刊，177-182
[16] 周福，倪振华，谢壮宁．利用POD对双坡屋盖风压场预测．力学季刊(2005)26(2)，248-256
[17] Holmes, J. D. Effective static load distributions in wind engineering. Journal of Wind Engineering and Industrial Aerodynamics(2002)90, 91-109.
[18] 武岳，陈波，沈世钊．大跨度屋盖结构等效静风荷载研究．建筑科学与工程学报(2005)22(4)，28-39．
[19] 江卓荣 POD技术以及在屋盖风压场中的应用．中国优秀硕士学位论文(2006)．
[20] 倪振华，李方慧．屋盖风振分析中的模态加速度法．第十二届全国结构风工程学术会议论文集(2005)．
[21] 肖云茹．概率统计计算方法．南开大学出版社(1994)295-317．
[22] 希缪E，斯坎伦R H著．刘尚培，项海帆，谢霁明译．风对结构的作用—风工程导论．上海同济大学出版社．
[23] 中华人民共和国国家标准—建筑结构荷载规范(GB50009—2001)．北京：中国建筑工业出版社，2002．
[24] Chen, X. Z., Kareem, A. Equivalent static wind loans for buffeting response of bridges
[25] 黄本才．结构抗风分析原理及应用[M]，同济大学出版社，2001．
[26] Loeve, M. Probability Theory [M], 4th Edition, Chapter Ⅺ: Second Order Properties, 37.5 Orthogonal decompositions, Springer-Verlag New York Inc. 1978.
[27] Tamura, Y., Suganuma, S., Kikuchi, H., Hibi, K. Proper Orthogonal Decomposition of random wind pressure field. Journal of Fluids and Structures(1999)13, 1069-1095.
[28] Clarence, R. W., Jerrold, M. E. Reconstruction equations and the Karhunen-Loeve expansion for systems with symmetry. Physica D (2000) 142, 1-19.
[29] 倪振华，振动力学，西安交通大学出版社，1989．
[2] Lee, B. E. The effects of turbulence on the surface pressure field of a square prism. Journal of Fluid mechanics (1975), 69,263-282.
[3] Best, R. J., Holmes, J. D. Use of eigenvalues in the covariance integration method for determination of wind load effects. Journal of Wind Engineering and Industrial Aerodunamics (1983) 13,359-370.
[4] Kareem, A., Cermak, J.E. Pressure fluctuations on a square building model in boundary-layer flows. Journal of Wind Engineering and Industrial Aerodunamics (1984) 16, 17-41.
[5] Holmes, J. D. Analysis and synthesis of pressure fluctuations on bluff bodies using eigenvectors. Journal of Wind Engineering and Industrial Aerodunamics (1990) 33, 219-230.
[6] Holmes, J. D. Optimised peak load distributions. Journal of Wind Engineering and Industrial Aerodunamics (1992) 41-44, 267-276.
[7] Holmes, J. D., Sankaran, R., Kwok, K. C. S., Syme, M. J. Eigenvector modes of fluctuating pressures on low-rise building models. Journal of Wind Engineering and Industrial Aerodunamics (1997) 69-71,687-707.
[8] Bienkiewicz, B., Tamura, Y., Ham, H. J., Ueda, H., Hibi, K. Proper orthogonal decomposition and reconstruction of multi-channel roof pressure. Journal of Wind Engineering and Industrial Aerodunamics (1995)54,369-381.
[9] Kikuchi, H., Tamura, Y., Ueda, K. Hibi, K. Dynamic wind pressure acting on a tall building model Proper Orthogonal Decomposition. Journal of Wind Engineering and Industrial Aerodunamics (1997) 69-71,631-646.
[10] Tamura, Y., Ueda, H., Kikuchi, H., Hibi, K., Suganuma, S., Bienkiewicz, B. Proper orthogonal decomposition study of approach wind-building pressure correlation. In Proceedings of the Ninth International Conference on wind Engineering (1995), 2115-2126.
[11] Bienkiewicz, B., Tamura, Y., Ham, H. J., Ueda, H., Hibi, K. Proper orthogonal decomposition and reconstruction of multi-channel roof pressure. Journal of Wind Engineering and Industrial Aerodunamics (1995)54, 369-381.
[12] Jeong, S. H., Bienkiewicz, B., Ham, J. H. Proper Orthogonal Decomposition of building wind pressure specified at non-uniformly distributed pressure Taps. Journal of Wind Engineering and Industrial Aerodunamics (2000) 87, 1-14.
[13] Uematsu, Y., Kuribara, O., Yamada, M., Sasaki, A., Hongo, T. Wind-induced dynamic behavior and its load estimation of a single-layer latticed dome with a long span. Journal of Wind Engineering and Industrial Aerodynamics (2001)89, 1671-1687.
[14] Chen, Y., Kopp, G. A., Surry, D. Spatial extrapolation of pressure time series on low building using proper orthogonal decomposition. Journal of Wind and Structure (2004) 7(6), 373-392.
[15] 李方慧，倪振华，谢壮宁．POD方法在重建双坡屋盖风压场中的应用．工程力学(2005)22增刊，177-182
[16] 周福，倪振华，谢壮宁．利用POD对双坡屋盖风压场预测．力学季刊(2005)26(2)，248-256
[17] Holmes, J. D. Effective static load distributions in wind engineering. Journal of Wind Engineering and Industrial Aerodynamics(2002)90, 91-109.
[18] 武岳，陈波，沈世钊．大跨度屋盖结构等效静风荷载研究．建筑科学与工程学报(2005)22(4)，28-39．
[19] 江卓荣 POD技术以及在屋盖风压场中的应用．中国优秀硕士学位论文(2006)．
[20] 倪振华，李方慧．屋盖风振分析中的模态加速度法．第十二届全国结构风工程学术会议论文集(2005)．
[21] 肖云茹．概率统计计算方法．南开大学出版社(1994)295-317．
[22] 希缪E，斯坎伦R H著．刘尚培，项海帆，谢霁明译．风对结构的作用—风工程导论．上海同济大学出版社．
[23] 中华人民共和国国家标准—建筑结构荷载规范(GB50009—2001)．北京：中国建筑工业出版社，2002．
[24] Chen, X. Z., Kareem, A. Equivalent static wind loans for buffeting response of bridges
[25] 黄本才．结构抗风分析原理及应用[M]，同济大学出版社，2001．
[26] Loeve, M. Probability Theory [M], 4th Edition, Chapter Ⅺ: Second Order Properties, 37.5 Orthogonal decompositions, Springer-Verlag New York Inc. 1978.
[27] Tamura, Y., Suganuma, S., Kikuchi, H., Hibi, K. Proper Orthogonal Decomposition of random wind pressure field. Journal of Fluids and Structures(1999)13, 1069-1095.
[28] Clarence, R. W., Jerrold, M. E. Reconstruction equations and the Karhunen-Loeve expansion for systems with symmetry. Physica D (2000) 142, 1-19.
[29] 倪振华，振动力学，西安交通大学出版社，1989．