应用点基局部光滑点插值法的固有频率上下界计算
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摘要
针对传统有限元方法得到的刚度阵过硬导致所提供的固有频率上限值较为粗糙、点基光滑点插值方法得到的刚度阵过软导致所提供的固有频率下限值较为宽泛且存在时间不稳定性等问题,提出了点基局部光滑点插值法(NLS-PIM)。该方法将有限元和点基光滑点插值法相结合,对背景网格基础上形成的点基光滑域进行局部应变光滑,通过调整二者结合的权重来控制计算模型的整体刚度。研究发现,采用局部梯度光滑方法控制模型刚度而形成的点基局部光滑点插值法,克服了点基光滑点插值方法的时间不稳定性及有限元法的固有缺陷,且能提供更为精细的固有频率上下界区间。所提方法简便、实用、易于实现,可用于复杂问题求解。
The traditional finite element method(FEM)gives over-hard stiffness and hence provides rough upper bounds for natural frequencies,but the node-based smoothed point interpolation method(NS-PIM)gives over-soft stiffness hence provides lower bounds with temporal instability for natural frequencies.A node-based locally smoothed point interpolation method(NLS-PIM)is thus proposed.FEM is combined with NS-PIM by conducting locally gradient smoothing operation on the node-based smoothing domains based on background cells.And the stiffness can be adjusted by changing the parameterα,which is the proportion of the locally smoothed domain over each background cell.It is found that the proposed method can successfully overcome the temporal instability of NS-PIM and the inherent shortcomings of FEM and provide much tighter upper and lower bounds for natural frequencies due to the controlledmodel stiffness by the node-based locally gradient smoothing operation.
The traditional finite element method(FEM)gives over-hard stiffness and hence provides rough upper bounds for natural frequencies,but the node-based smoothed point interpolation method(NS-PIM)gives over-soft stiffness hence provides lower bounds with temporal instability for natural frequencies.A node-based locally smoothed point interpolation method(NLS-PIM)is thus proposed.FEM is combined with NS-PIM by conducting locally gradient smoothing operation on the node-based smoothing domains based on background cells.And the stiffness can be adjusted by changing the parameterα,which is the proportion of the locally smoothed domain over each background cell.It is found that the proposed method can successfully overcome the temporal instability of NS-PIM and the inherent shortcomings of FEM and provide much tighter upper and lower bounds for natural frequencies due to the controlledmodel stiffness by the node-based locally gradient smoothing operation.
引文
[1]ZIENKIEWICZ O C,TAYLOR R L.The finite element method:vol 1[M].5th ed.Oxford,UK:Butterworth/Heinemann,2000:58-60.
[2]胡海昌.放大倍数和本征值的上下限[J].振动与冲击,1983(2):1-8.HU Haichang.Factors of amplification and upper and lower bounds of eigenvalues[J].Journal of Vibration and Shock,1983(2):1-8.
[3]CRANDAL S H.Engineering analysis[M].Columbus,OH,USA:McGraw-Hill,1956:69-70.
[4]BATHE K J,WILSON E L.Numerical methods in finite element analysis[M].New Jersey,USA:Prentice Hall,1976:55-58.
[5]胡海昌.论几种本征值包含定理的内在联系[J].固体力学学报,1983(1):1-10.HU Haichang.On the interconnections among several inclusion theorems for eigenvalues[J].Acta Mechanica Solids Sinica,1983(1):1-10.
[6]DOKUMACI E.On super accurate finite elements and their duals for eigenvalue computation[J].Journal of Sound and Vibration,2006,298(1/2):432-438.
[7]张雄,刘岩.无网格法[M].北京:清华大学出版社,2004:1-2.
[8]刘桂荣,顾元通.无网格法理论及程序设计[M].济南:山东大学出版社,2007:23-24.
[9]LY B L.An upper and lower bound for natural frequencies[J].Journal of Sound and Vibration,1987,112(3):547-549.
[10]PENNY J E,REED J R.An integral equation approach to the fundamental frequency of vibrating beams[J].Journal of Sound and Vibration,1971,19(4):393-400.
[11]SPENCE J P,HORGAN C O.Bounds on natural frequencies of composite circular membranes:integral equation methods[J].Journal of Sound and Vibration,1983,87(1):71-81.
[12]BICKFORD W B,WU S Y.Bounds on natural frequencies of composite circular plates:integral equation methods[J].Journal of Sound and Vibration,1988,126(1):19-35.
[13]FAUCONNEAU G,MARANGONI R D.Natural frequencies and elastic stability of a simply-supported rectangular plate under linearly varying compressive loads[J].International Journal of Solids and Structures,1971,7(5):473-493.
[14]BAZLEY N W,FOX D W.Methods for lower bounds to frequencies of continuous elastic systems[J].Zeitschrift fur Angewandte Mathematik und Physic,1966,17:1-37.
[15]BAZLEY N W,FOX D W,STADTER J T.Upper and lower bounds for the frequencies of rectangular clamped plates[J].Zeitschrift fur Angewandte Mathematik und Mechc,1967,47(3):191-198.
[16]MARANGONI R D,COOK L M,BASAVANHAL-LY N.Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates[J].International Journal of Solids and Structures,1978,14(8):611-623.
[17]LIU G R.A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods[J].International Journal of Computational Methods,2008,5(2):199-236.
[18]LIU G R,ZHANG G Y,Smoothed point interpolation methods:G space and weakened weak forms[M].Singapore:World Scientific,2013:305-306.
[19]ZHANG G Y,LIU G R,WANG Y Y,et al.A linearly conforming point interpolation method(LC-PIM)for three-dimensional elasticity problems[J].International Journal for Numerical Methods in Engineering,2007,72(13):1524-1543.
[20]ZHANG G Y,LIU G R,NGUYEN T T,et al.The upper bound property for solid mechanics of the linearly conforming radial point interpolation method(LC-RPIM)[J].International Journal of Computational Methods,2007,4(3):521-541.
[21]LIU G R,ZHANG G Y.Upper bound solution to elasticity problems:a unique property of the linearly conforming point interpolation method(LC-PIM)[J].International Journal for Numerical Methods in Engineering,2008,74(7):1128-1161.
[22]ZHANG Z Q,LIU G R.Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems[J].Computational Mechanics,2010,46(2):229-246.
[23]杜超凡,章定国.光滑节点插值法:计算固有频率下界值的新方法[J].力学学报,2015,47(5):839-847.DU Chaofan,ZHANG Dingguo.Node-based smoothed point interpolation method:a new method for computing lower bound of natural frequency[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):839-847.
[24]彭金晓.光滑点插值无网格法及在工程力学中的应用[D].长春:吉林大学,2011:19-21.
[2]胡海昌.放大倍数和本征值的上下限[J].振动与冲击,1983(2):1-8.HU Haichang.Factors of amplification and upper and lower bounds of eigenvalues[J].Journal of Vibration and Shock,1983(2):1-8.
[3]CRANDAL S H.Engineering analysis[M].Columbus,OH,USA:McGraw-Hill,1956:69-70.
[4]BATHE K J,WILSON E L.Numerical methods in finite element analysis[M].New Jersey,USA:Prentice Hall,1976:55-58.
[5]胡海昌.论几种本征值包含定理的内在联系[J].固体力学学报,1983(1):1-10.HU Haichang.On the interconnections among several inclusion theorems for eigenvalues[J].Acta Mechanica Solids Sinica,1983(1):1-10.
[6]DOKUMACI E.On super accurate finite elements and their duals for eigenvalue computation[J].Journal of Sound and Vibration,2006,298(1/2):432-438.
[7]张雄,刘岩.无网格法[M].北京:清华大学出版社,2004:1-2.
[8]刘桂荣,顾元通.无网格法理论及程序设计[M].济南:山东大学出版社,2007:23-24.
[9]LY B L.An upper and lower bound for natural frequencies[J].Journal of Sound and Vibration,1987,112(3):547-549.
[10]PENNY J E,REED J R.An integral equation approach to the fundamental frequency of vibrating beams[J].Journal of Sound and Vibration,1971,19(4):393-400.
[11]SPENCE J P,HORGAN C O.Bounds on natural frequencies of composite circular membranes:integral equation methods[J].Journal of Sound and Vibration,1983,87(1):71-81.
[12]BICKFORD W B,WU S Y.Bounds on natural frequencies of composite circular plates:integral equation methods[J].Journal of Sound and Vibration,1988,126(1):19-35.
[13]FAUCONNEAU G,MARANGONI R D.Natural frequencies and elastic stability of a simply-supported rectangular plate under linearly varying compressive loads[J].International Journal of Solids and Structures,1971,7(5):473-493.
[14]BAZLEY N W,FOX D W.Methods for lower bounds to frequencies of continuous elastic systems[J].Zeitschrift fur Angewandte Mathematik und Physic,1966,17:1-37.
[15]BAZLEY N W,FOX D W,STADTER J T.Upper and lower bounds for the frequencies of rectangular clamped plates[J].Zeitschrift fur Angewandte Mathematik und Mechc,1967,47(3):191-198.
[16]MARANGONI R D,COOK L M,BASAVANHAL-LY N.Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates[J].International Journal of Solids and Structures,1978,14(8):611-623.
[17]LIU G R.A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods[J].International Journal of Computational Methods,2008,5(2):199-236.
[18]LIU G R,ZHANG G Y,Smoothed point interpolation methods:G space and weakened weak forms[M].Singapore:World Scientific,2013:305-306.
[19]ZHANG G Y,LIU G R,WANG Y Y,et al.A linearly conforming point interpolation method(LC-PIM)for three-dimensional elasticity problems[J].International Journal for Numerical Methods in Engineering,2007,72(13):1524-1543.
[20]ZHANG G Y,LIU G R,NGUYEN T T,et al.The upper bound property for solid mechanics of the linearly conforming radial point interpolation method(LC-RPIM)[J].International Journal of Computational Methods,2007,4(3):521-541.
[21]LIU G R,ZHANG G Y.Upper bound solution to elasticity problems:a unique property of the linearly conforming point interpolation method(LC-PIM)[J].International Journal for Numerical Methods in Engineering,2008,74(7):1128-1161.
[22]ZHANG Z Q,LIU G R.Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems[J].Computational Mechanics,2010,46(2):229-246.
[23]杜超凡,章定国.光滑节点插值法:计算固有频率下界值的新方法[J].力学学报,2015,47(5):839-847.DU Chaofan,ZHANG Dingguo.Node-based smoothed point interpolation method:a new method for computing lower bound of natural frequency[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):839-847.
[24]彭金晓.光滑点插值无网格法及在工程力学中的应用[D].长春:吉林大学,2011:19-21.