微分求积法和微分求积单元法——原理与应用
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摘要
微分求积法和微分求积单元法是在近年里发展起来，并正在进一步发展的一种数
值计算方法，本文对这两种方法的原理和它们在结构力学中的应用进行了较深入系统
地研究。全面介绍和总结了两种方法的原理和性质，证明了微分求积法实际是一种特
殊的混合配点法，并首次证明了微分求积单元法是加权残值法中子域法与特殊混合配
点法相结合的一种方法。探讨了微分求积法应用过程中边界条件处理方法和节点分布
对结果的影响规律，通过推导和应用板的非线性问题边缘约束的通用公式，展示了微
分求积法所具有的部分半解析法的特点。构造了带边界一阶导数的插值公式，研究了
组成该插值公式的基函数的性质。导出了插值函数各阶导数权系数矩阵的矩阵计算式
和显式计算表达式，详细研究了权系数矩阵的性质。首次建立了曲梁单元、截圆锥壳
单元、环形浅球壳单元、压电线球壳单元等微分求积单元。研究了单元数目和节点数
目变化对计算结果的影响，得到了在总自由度数相等的情况下增加微分求积单元节点
数比增加微分求积单元数对提高结果的精度更为有效的结论，据此提出了用微分求积
单元离散结构必须遵循单元数最少的原则。本文对众多板、壳结构的静动特性和稳定
性进行了分析和研究，给出了大量有价值的数值计算结果，这些结果可供工程设计参
考。
本文的研究结果表明微分求积法和微分求积单元法兼有半解析法和数值计算方
法的一些优点，在结构工程领域具有非常广阔的应用前景。
The Differential Quadrature Method (DQM) and the Differential Quadrature Element
Method (DQEM) are numerical methods developed recently and still under developing.
The theory and applications of these two methods in Structural Mechanics are
systematically studied in this dissertation. The principles and properties of these two
methods are described, studied, and summarized in detail. It is shown that DQM is
equivalent to a special kind of mixed collocation method. It is also shown for the first time
that DQEM could be regarded as the combinations of the sub-domain method with the
special mixed collocation method. The effects of the application of boundary conditions
and the selection of grid points on the final results are investigated. The generalized
formulations to apply the boundary constrains are derived for the nonlinear plate bending
problems, and numerical results show that the DQM exhibits behaviors of semi-analytical
methods. The interpolation functions with degree-of-freedom of first derivatives at
boundary points are formulated, the properties of their base functions are also studies, the
explicit formulae to compute the weighting coefficient matrices for various orders of
derivatives are given, and the properties of the weighting coefficient matrices are
investigated in details. Several Differential Quadrature elements are established for the first
time, such as the circular arch element, conical shell element, shallow spherical shell
element, and piezo-electrical shallow spherical shell element. The effects of the number of
elements and the number of grid points on the accuracy of the numerical results are
investigated. Based on the numerical results, it is shown that increasing the number of grid
points in each element yields more accurate results than increasing the number of elements
if the total number of degrees-of-freedom is the same. Thus, one should use as smaller
numbers of element as possible when one establishes the computational model for analysis.
The DQM and DQEM are thus used to obtain static, buckling, and free vibration solutions
for various plate and shell structures, and results may be useful in engineering practice.
Based on the results reported in this dissertation, one may conclude that the DQM
and DQEM are the numerical methods exhibiting behaviors of semi-analytical methods,
thus they will find wide application areas in structural engineering.
值计算方法，本文对这两种方法的原理和它们在结构力学中的应用进行了较深入系统
地研究。全面介绍和总结了两种方法的原理和性质，证明了微分求积法实际是一种特
殊的混合配点法，并首次证明了微分求积单元法是加权残值法中子域法与特殊混合配
点法相结合的一种方法。探讨了微分求积法应用过程中边界条件处理方法和节点分布
对结果的影响规律，通过推导和应用板的非线性问题边缘约束的通用公式，展示了微
分求积法所具有的部分半解析法的特点。构造了带边界一阶导数的插值公式，研究了
组成该插值公式的基函数的性质。导出了插值函数各阶导数权系数矩阵的矩阵计算式
和显式计算表达式，详细研究了权系数矩阵的性质。首次建立了曲梁单元、截圆锥壳
单元、环形浅球壳单元、压电线球壳单元等微分求积单元。研究了单元数目和节点数
目变化对计算结果的影响，得到了在总自由度数相等的情况下增加微分求积单元节点
数比增加微分求积单元数对提高结果的精度更为有效的结论，据此提出了用微分求积
单元离散结构必须遵循单元数最少的原则。本文对众多板、壳结构的静动特性和稳定
性进行了分析和研究，给出了大量有价值的数值计算结果，这些结果可供工程设计参
考。
本文的研究结果表明微分求积法和微分求积单元法兼有半解析法和数值计算方
法的一些优点，在结构工程领域具有非常广阔的应用前景。
The Differential Quadrature Method (DQM) and the Differential Quadrature Element
Method (DQEM) are numerical methods developed recently and still under developing.
The theory and applications of these two methods in Structural Mechanics are
systematically studied in this dissertation. The principles and properties of these two
methods are described, studied, and summarized in detail. It is shown that DQM is
equivalent to a special kind of mixed collocation method. It is also shown for the first time
that DQEM could be regarded as the combinations of the sub-domain method with the
special mixed collocation method. The effects of the application of boundary conditions
and the selection of grid points on the final results are investigated. The generalized
formulations to apply the boundary constrains are derived for the nonlinear plate bending
problems, and numerical results show that the DQM exhibits behaviors of semi-analytical
methods. The interpolation functions with degree-of-freedom of first derivatives at
boundary points are formulated, the properties of their base functions are also studies, the
explicit formulae to compute the weighting coefficient matrices for various orders of
derivatives are given, and the properties of the weighting coefficient matrices are
investigated in details. Several Differential Quadrature elements are established for the first
time, such as the circular arch element, conical shell element, shallow spherical shell
element, and piezo-electrical shallow spherical shell element. The effects of the number of
elements and the number of grid points on the accuracy of the numerical results are
investigated. Based on the numerical results, it is shown that increasing the number of grid
points in each element yields more accurate results than increasing the number of elements
if the total number of degrees-of-freedom is the same. Thus, one should use as smaller
numbers of element as possible when one establishes the computational model for analysis.
The DQM and DQEM are thus used to obtain static, buckling, and free vibration solutions
for various plate and shell structures, and results may be useful in engineering practice.
Based on the results reported in this dissertation, one may conclude that the DQM
and DQEM are the numerical methods exhibiting behaviors of semi-analytical methods,
thus they will find wide application areas in structural engineering.
引文
1． Bellman R E, and Casti J. Differential quadrature and long-term integration. J. Math. Anal.Appl., 34, (1971) :235-238
2． Jang S K. Application of differential quadrature to the analysis of structural components. Ph. D. Dissertation, University of Oklahoma, Norman, OK, USA, 1987
3． Bert C W, Jang S K, Striz A J. New methods for analyzing vibtation of structural
components. Proc. AIAA Dynamics Specialist Conference, Monterey, CA, USA, Part 2,(1987) :936-943
4． Sherbourne A N, Pandey M D. Differential quadrature method in the buckling analysis of beams and composite plates. Comp. & Struct, 40,4 (1991) :903-913
5． Bert C W, Wang X, Striz A G New developments in differential quadrature analysis of structural elements. Presented at the 29th Annual Technical Meeting, Society of Engineering Science, La Jolla, CA, USA, 1992 (Sept. 14-16)
6． Wang X. Differential quadrature and applications to analysis of anisotropic plates. Proc. of the 1st Postdoctora Academic Congress, Beijing, China, (1993) : 875-879
7． Bert C W, Wang X, Striz A G Differential quadrature for static and free vibration analysis of anisotropic plates. Int. J. Solids Struct. 30,13 (1993) : 1737-1744
8． Wang X, and Bert C W. A new approach in applying differential quadrature to static and free vibration analyses of beams and plates. J. Sound & Vibration, 162, 3 (1993) : 566-572
9． Wang X, Striz A G, Bert C W. Buckling and vibration analysis of skew plates by the DQ method. AIAA J. 32, 4 (1994) :886-889
10． Bert C W, Wang X, Striz A G Staic and free vibrational analysis of beams and plates by the differential quadrature method. Ada Mechanics, 102, (1994) : 11-24
11． 游仁长,王永亮,王鑫伟.层合板稳定性分析的微分求积法.航空学报,第15卷,第9期 (1994) : 1085-1090
12． He B, Wang X. Error analysis in differential quadrature method. Transaction of NUAA, 11, 2 (1994) : 194-200
13． Bert C W, Wang X, Striz A G Convergence of the DQ method in the analysis of anisotropic plates. J. Sound & Vibration, 170, 1 (1994) : 140-144
14． Striz A G, Chen W L, Bert C W. Static analysis of structures by the quadrature element method (QEM). Int. J. Solids Struct. 31 (1994) : 2807-2818
15. Wang X. Differential quadrature element method and its applications to structural analysis. SES. 32nd Annual Techical Meeting, New Orleans, USA. Oct.29- Nov. 1 (1995)
16．王鑫伟．微分求积法在结构力学中的应用．力学进展．第25卷第2期(1995)：232-241
17．王鑫伟，何柏庆．调和微分求积法权系数矩阵的一种显式计算式．南京航空航天大学学报，第27卷第4期(1995)：496-501
18. Striz A G, Wang X, Bert C W. Harmonic differential quadrature method and applications to structural components, Acta Mechanica, 111, 1-2 (1995): 85-94
19．王永亮．边缘弹性约束圆薄板的大挠度分析．江苏力学，第10辑，(1995)：6-12
20．王鑫伟，何柏庆，陈文．中心对称或反中心对称线性方程组的缩减算法．南京航空航天大学学报，第28卷第5期(1996)：599-606
21．王永亮，王鑫伟．多外载联合作用下圆板的非线性弯曲．南京航空航天大学学报，第28卷第1期(1996)：53-58
22. Bert C W, Malik M. Free vibration analysis of tapered rectangular plates by the differential quadrature method: A semi-analytical approach. J. Sound & Vibration, 190 (1996): 41-63
23. Bert C W, Malik M. Differential quadrature method in computational mechanics: A review. Appl. Mech. Rev. 49, 1(1996): 1-28
24. Wang X, Gu H Z. Static analysis of frame structures by the differential quadrature element method. Int. J. Numer. Methods Engng. 40(1997): 759-772
25. Gu H Z, Wang X. On the free vibration analysis of circular plates with stepped thickness over a concentric region by the differential quadrature element method. J. Sound & Vibration, 202, 3 (1997): 452-459
26. Striz A G, Chen W L, Bert C W. Free vibration of high-accuracy plates by the quadrature element method. J. Sound & Vibration, 202 (1997): 689-702
27． Chen W L, Striz A G, Bert C W. A new approach to the differential quadrature method for forth-order equations. Int. J. Numer. Meth. Engng. 40 (1997) : 1941-1956
28． Malik M, Bert C W. Differential quadrature: a powerful new technique for analysis of composite structures. Compose. Struct. 39 (1997) : 179-189
29． 王永亮,刘人怀,王鑫伟.弹性支承环形板的非线性弯曲,固体力学学报,第19卷 (1998) :107-110
30． Wang X, Wang Y L, Chen R B. Static and free vibration analysis of rectangular plates by the differential quadrature element method. Commun. Numer. Meth. Engng. 14 (1998) : 1133-1141
31． Malik M, Bert C W. Three-dimensional elasticity solutions for free vibration of thick rectangular plates by the differential quadrature method. Int. J. Solids Struct. 35 (1998) : 299-318
32． Mierfakhraei P, Redekop D. Buckling of circular cylindrical shells by the differential quadrature method. Int. J. Pressure Vessels Piping, 75, 4(1998) : 347-353
33． Moradi S, Taheri F. Differential quadrature approach for delarnination buckling analysis of composites with shear deformation. AIAA J.36,10(1998) : 1869-1873
34． Liu F L, Liew K M. Static analysis of Reissner-Mindlin plates by the differential quadrature element method. J. Applied Mechanics, 65 (1998) : 705-710
35． Moradi S, Taheri F. Application of differential quadrature to delamination buckling analysis of composites plates. Comput. Struct. 70, 6(1999) : 615-623
36． Liu F L, Liew K M. Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method. J. Sound & Vibration, 225, 5 (1999) : 915-934
37． Wang Y L, Liu R H, Wang X. Free vibration analysis of truncated conical shells by the differential quadrature method. J. Sound & Vibration, 224,2(1999) : 387-394
38． Shu C, Chen W. On optimal selection of interior points for applying discretized
boundary conditions in DQ vibration analysis of bearms and plates. J. Sound & Vibration, 222, 2(1999) : 239-257
39． Han J B, Liew K M. Static analysis of Mindlin plates: the differential quadrature element method(DQEM). Computer Methods in Applied Mechanics & Engineering, 177, 1-2(1999) :51-75
40． Liu F L, Liew K M. Differential quadrature element method static analysis of Reissner-Mindlin pollar plates. Int. J. Solids Struct. 36,33 (1999) : 5101-5123
41． Liu F L, Liew K M. Vibration analysis of discontinuous Mindlin plates by differential quadrature element method. J. Vibration & Acoustic, 121,2 (1999) : 204-208
42． Liu F L. Rectangular plates on Winkler foundation: differential quadrature element solution. Int. J. Solids Struct. 37(2000) : 1743-1763
43． Liu F L. Static analysis of thick rectangular laminated plates: three dimensional elasticity solutions via differential quadrature element method. Int. J. Solids Struct 37 (2000) : 7671-7688
44． Chen W L, Striz A G, Bert C W. High-accuracy plane stress and plate elements in the quadrature element method. Int J. Solids Struct 37 (2000) : 627-647
45． Malik M, Bert C W. Vibration analysis with curvilinear quadrilateral planforms by DQM using blending functions. J. Sound A Vibration, 230,4(2000) : 949-954
46． Wang Y L, Liu R H, Wang X. On free vibration analysis of nonlinear piezoelectric circular shallow spherical shells by the differential quadrature element method. J. Sound & Vibration, (in press), 2001
2． Jang S K. Application of differential quadrature to the analysis of structural components. Ph. D. Dissertation, University of Oklahoma, Norman, OK, USA, 1987
3． Bert C W, Jang S K, Striz A J. New methods for analyzing vibtation of structural
components. Proc. AIAA Dynamics Specialist Conference, Monterey, CA, USA, Part 2,(1987) :936-943
4． Sherbourne A N, Pandey M D. Differential quadrature method in the buckling analysis of beams and composite plates. Comp. & Struct, 40,4 (1991) :903-913
5． Bert C W, Wang X, Striz A G New developments in differential quadrature analysis of structural elements. Presented at the 29th Annual Technical Meeting, Society of Engineering Science, La Jolla, CA, USA, 1992 (Sept. 14-16)
6． Wang X. Differential quadrature and applications to analysis of anisotropic plates. Proc. of the 1st Postdoctora Academic Congress, Beijing, China, (1993) : 875-879
7． Bert C W, Wang X, Striz A G Differential quadrature for static and free vibration analysis of anisotropic plates. Int. J. Solids Struct. 30,13 (1993) : 1737-1744
8． Wang X, and Bert C W. A new approach in applying differential quadrature to static and free vibration analyses of beams and plates. J. Sound & Vibration, 162, 3 (1993) : 566-572
9． Wang X, Striz A G, Bert C W. Buckling and vibration analysis of skew plates by the DQ method. AIAA J. 32, 4 (1994) :886-889
10． Bert C W, Wang X, Striz A G Staic and free vibrational analysis of beams and plates by the differential quadrature method. Ada Mechanics, 102, (1994) : 11-24
11． 游仁长,王永亮,王鑫伟.层合板稳定性分析的微分求积法.航空学报,第15卷,第9期 (1994) : 1085-1090
12． He B, Wang X. Error analysis in differential quadrature method. Transaction of NUAA, 11, 2 (1994) : 194-200
13． Bert C W, Wang X, Striz A G Convergence of the DQ method in the analysis of anisotropic plates. J. Sound & Vibration, 170, 1 (1994) : 140-144
14． Striz A G, Chen W L, Bert C W. Static analysis of structures by the quadrature element method (QEM). Int. J. Solids Struct. 31 (1994) : 2807-2818
15. Wang X. Differential quadrature element method and its applications to structural analysis. SES. 32nd Annual Techical Meeting, New Orleans, USA. Oct.29- Nov. 1 (1995)
16．王鑫伟．微分求积法在结构力学中的应用．力学进展．第25卷第2期(1995)：232-241
17．王鑫伟，何柏庆．调和微分求积法权系数矩阵的一种显式计算式．南京航空航天大学学报，第27卷第4期(1995)：496-501
18. Striz A G, Wang X, Bert C W. Harmonic differential quadrature method and applications to structural components, Acta Mechanica, 111, 1-2 (1995): 85-94
19．王永亮．边缘弹性约束圆薄板的大挠度分析．江苏力学，第10辑，(1995)：6-12
20．王鑫伟，何柏庆，陈文．中心对称或反中心对称线性方程组的缩减算法．南京航空航天大学学报，第28卷第5期(1996)：599-606
21．王永亮，王鑫伟．多外载联合作用下圆板的非线性弯曲．南京航空航天大学学报，第28卷第1期(1996)：53-58
22. Bert C W, Malik M. Free vibration analysis of tapered rectangular plates by the differential quadrature method: A semi-analytical approach. J. Sound & Vibration, 190 (1996): 41-63
23. Bert C W, Malik M. Differential quadrature method in computational mechanics: A review. Appl. Mech. Rev. 49, 1(1996): 1-28
24. Wang X, Gu H Z. Static analysis of frame structures by the differential quadrature element method. Int. J. Numer. Methods Engng. 40(1997): 759-772
25. Gu H Z, Wang X. On the free vibration analysis of circular plates with stepped thickness over a concentric region by the differential quadrature element method. J. Sound & Vibration, 202, 3 (1997): 452-459
26. Striz A G, Chen W L, Bert C W. Free vibration of high-accuracy plates by the quadrature element method. J. Sound & Vibration, 202 (1997): 689-702
27． Chen W L, Striz A G, Bert C W. A new approach to the differential quadrature method for forth-order equations. Int. J. Numer. Meth. Engng. 40 (1997) : 1941-1956
28． Malik M, Bert C W. Differential quadrature: a powerful new technique for analysis of composite structures. Compose. Struct. 39 (1997) : 179-189
29． 王永亮,刘人怀,王鑫伟.弹性支承环形板的非线性弯曲,固体力学学报,第19卷 (1998) :107-110
30． Wang X, Wang Y L, Chen R B. Static and free vibration analysis of rectangular plates by the differential quadrature element method. Commun. Numer. Meth. Engng. 14 (1998) : 1133-1141
31． Malik M, Bert C W. Three-dimensional elasticity solutions for free vibration of thick rectangular plates by the differential quadrature method. Int. J. Solids Struct. 35 (1998) : 299-318
32． Mierfakhraei P, Redekop D. Buckling of circular cylindrical shells by the differential quadrature method. Int. J. Pressure Vessels Piping, 75, 4(1998) : 347-353
33． Moradi S, Taheri F. Differential quadrature approach for delarnination buckling analysis of composites with shear deformation. AIAA J.36,10(1998) : 1869-1873
34． Liu F L, Liew K M. Static analysis of Reissner-Mindlin plates by the differential quadrature element method. J. Applied Mechanics, 65 (1998) : 705-710
35． Moradi S, Taheri F. Application of differential quadrature to delamination buckling analysis of composites plates. Comput. Struct. 70, 6(1999) : 615-623
36． Liu F L, Liew K M. Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method. J. Sound & Vibration, 225, 5 (1999) : 915-934
37． Wang Y L, Liu R H, Wang X. Free vibration analysis of truncated conical shells by the differential quadrature method. J. Sound & Vibration, 224,2(1999) : 387-394
38． Shu C, Chen W. On optimal selection of interior points for applying discretized
boundary conditions in DQ vibration analysis of bearms and plates. J. Sound & Vibration, 222, 2(1999) : 239-257
39． Han J B, Liew K M. Static analysis of Mindlin plates: the differential quadrature element method(DQEM). Computer Methods in Applied Mechanics & Engineering, 177, 1-2(1999) :51-75
40． Liu F L, Liew K M. Differential quadrature element method static analysis of Reissner-Mindlin pollar plates. Int. J. Solids Struct. 36,33 (1999) : 5101-5123
41． Liu F L, Liew K M. Vibration analysis of discontinuous Mindlin plates by differential quadrature element method. J. Vibration & Acoustic, 121,2 (1999) : 204-208
42． Liu F L. Rectangular plates on Winkler foundation: differential quadrature element solution. Int. J. Solids Struct. 37(2000) : 1743-1763
43． Liu F L. Static analysis of thick rectangular laminated plates: three dimensional elasticity solutions via differential quadrature element method. Int. J. Solids Struct 37 (2000) : 7671-7688
44． Chen W L, Striz A G, Bert C W. High-accuracy plane stress and plate elements in the quadrature element method. Int J. Solids Struct 37 (2000) : 627-647
45． Malik M, Bert C W. Vibration analysis with curvilinear quadrilateral planforms by DQM using blending functions. J. Sound A Vibration, 230,4(2000) : 949-954
46． Wang Y L, Liu R H, Wang X. On free vibration analysis of nonlinear piezoelectric circular shallow spherical shells by the differential quadrature element method. J. Sound & Vibration, (in press), 2001