四边固定矩形板大挠度问题的摄动最小二乘解

本文将摄动法与加权残值法相结合,提出了求解大挠度问题的摄动加权法,编制了PWS计算程序,计算了四边固定矩形板在均布载荷下的大挠度问题。算例的结果与实验一致。     

不连续Reissner矩形板的自由振动

用分区加权残值法研究Ressner矩形板在几何形状,边界条件等有突变时的自由振动问题,将研究对象按照结构几何形状和边界条件的具体情况划分为若干子域,在每个子域内用不同的试函数代入该域的内的控制方程到内部残值,并代入板的边界条件和各子域的协调条件得到边界残值和连续性残值,然后用最小二乘法消除残值,得到特征方程,文中讨论了该方法的收敛性和计算精度,求解了开孔矩形板的固有频率,并与已有结果进行了比较,结果表明：？该方法收敛性好,精度较高,适用范围广。     

紧支试函数加权残值法

将紧支函数引入加权残值法中，提出了紧支试函数加权残值法，其数值格式具有和有限元相似的窄带系数矩阵，提高了加权残值法的计算效率．在紧支试函数加权残值的基础上，导出了紧支试函数直接配点法、紧支试函数Hermite配点法和紧支试函数最小二乘配点法的具体格式，并且对几个典型算例进行了分析．与配点法相比，这些方法精度高，稳定性好，而与Galerkin法相比，这些方法效率高．     

复合载荷作用下圆板大挠度弯曲问题的伽辽金法

1.引言 关于复合载荷作用下圆板的大挠度弯曲问题,文献[1,2]曾分别选取不同的参数用摄动方法给予求解。本文用一个简便的方法来分析此问题。即先假设一个挠度试函数,使相容方程完全满足,求出薄膜力;然后再用伽辽金加权残数法求解平衡微分方程。 已知均布荷载及中心集中力联合作用下圆板的大挠度方程为     

有效集法在数学规划加权残值法中的应用

本文研究有效集法在数学规划加权残值法求解微分方程问题中的应用。从理论上和实际应用方面 阐明，对于数学规划加权残值法和线性规划问题，有效集法比常用的单纯形法更简便、易程序化、计算速度更快，因此更适用于数学规划加权残值法求解微分方程问题。     

分步迭代加权残值法

本文提出一种分步迭代的加权残值方法,并从理论上证明了分步迭代的最小二乘法用于求解线性问题时的收敛性。文中应用所提出的方法计算了方形固支板的位移,方形杆的扭转刚度及柔性圆板的大挠度位移和应力。结果表明了文中的方法可逐步提高计算精度。     

非线性方程的求解—最优化样条函数康托诺维奇加权残值法

本文是得新提出的一种微分方程的新解法，最优化样条函数康托诺维奇加权残值法。来求解非线性微分方程。该法把优化理论引入微分方程的数值解法，揉最优化算法，加权残值法，样条函数法，康托诺维奇法于一体，具有精度高、收敛快、易于处理各种边界条件的优点，文中有基于原始微分方程的算例，对流体力学中Ｂｕｒｇｅｒｓ方程的成功求解，展示了该法的应用前景。     

轴对称圆薄板大挠度微分方程的数值分析方法

根据轴对称问题的特点,利用级数展开和求极限法则,证明了轴对称大挠度圆薄板在圆心处应满足的边界条件,并以圆薄板轴对称大挠度弯曲变形微分方程为基础,建立了圆心处非奇异的轴对称大挠度圆板弯曲微分方程,从而可以方便地利用现有的常微分方程数值求解方法(如变步长龙格-库塔法)对实心圆板的轴对称问题进行数值求解,又不必像摄动法那样推导复杂的公式。在数值求解轴对称圆板大挠度弯曲变形微分方程时,将非线性微分方程的求解主要归结为迭代求解圆心处三个未知边界条件的问题,即圆心处的径向膜力、圆心处的挠度、圆心处挠度的二阶导数,并提出了相应的求解方法。实例中,对于圆薄板受均布横向荷载的问题,分析了周边固支边界条件下的非线性弯曲问题,给出了中心挠度参数大范围变化时的荷载和部分边界值变化曲线,并与经典摄动解进行了对比。对比结果可见,本文方法和摄动法的解非常接近,在量纲归一化中心挠度不超过4.0时,两种方法解的相对误差均小于5.0%。另外,本文还分析了与挠度有关的液体压力作用下和集中荷载作用下周边固支圆板的非线性弯曲问题。通过算例可见:本文方法可以灵活处理不同的荷载问题;对于不同的问题,计算过程相似,不必推导复杂的计算公式,计算精度容易控制。     

板分析的滑动最小二乘插值函数残值法

利用滑动最小二乘插值数作为加权残值法中试函数，对试函数中的基函数的以及权函数的选取提出了建议；分析了形函数的特性；对试函数拟合原函数的效果进行了分析，进而提出了权函数及相应的影响半径的取值；采用最小二北配点法求解定解问题的近似解；利用该试函数对矩形薄板和L形板的弯曲进行了数值计算，并与理论结果和有限元数值结果进行对比，结果表明，该试函数适用于多种边值问题，且精度高。该法简化了选择试验的过程，尤其适用于工程中的各种数值计算。     

数学规划加权残值法在非线性微分方程中的应用

本文研究数学规划加权残值法在非线性微分方程求解中的应用,利用数学规划加权残值法和ＬＰ模理论,把非线性微分方程边值问题转化为一个可微分的无约束非线性优化问题,从而运用成熟稳定的寻优方法求得问题的解。文中数字计算例子表明本文方法可以快速有效地求解非线性微分方程。     

Large deflection of circular plate under compound load

By means of the perturbation method, this paper presents an approximate solution for large deflection of clamped circular plate under uniform pressure together with a concentrated load at the center. The special case of vanishing central deflection is also discussed. In this paper, a load distribution function is introduced so as to make the compound loads depend on a single load parameter, and average angular deflection is used as the single displacement perturbation parameter.     

The perturbation parameter in the problem of large deflection of clamped circular plates

In the problems of large deflection of clamped circular plates under uniformly distributed loads, various perturbation parameters relating to load, deflection, slope of deflection, membrane force, etc, are studied. For a general perturbation parameter, the variational principle is used for the solution of such a problem. The applicable range of these perturbation parameters are studied in detail. In the case of uniformly loaded plate, perturbation parameter relating to central deflection seems to be the best among all others. The method of determination of perturbation solution by means of variational principle can be used to treat a variety of problems, including the large deflection problems under combine loads. This paper is completed under the guidance of Prof. Chien Wei-zang.     

ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE

In this paper,we reexamine the method of successive approximation presented byProf.Chien Wei-zang for solving the problem of large deflection of a circular plate,and findthat the method could be regarded as the method of strained parameters in the singularperturbation theory.In terms of the parameter representing the ratio of the centerdeflection to the thickness of the plate,we make the asymptotic expansions of thedeflection,membrane stress and the parameter of load as in Ref.[1],and then give theorthogonality conditions(i.e.the solvability conditions)for the resulting equations,bywhich the stiffness characteristics of the plate could be determined.It is pointed out thatwith the solutions for the small deflection problem of the circular plate and theorthogonality conditions,we can derive the third order approximate relations between theparameter of load and the center deflection and the first-term approximation of membranestresses at the center and edge of the plate without solving the differential equ     

轴对称弹性圆锥扁壳非线性自由振动直接摄动解法

本文采用弹性圆锥扁壳中心无量纲振幅和壳体母线的倾角为参数，将挠度、应力函数的导数以及自由振动频率展开为双参量的幂级数形式．用直接摄动法获得各级递推线性偏微分方程．应用变分法求得各级递推方程的近似解答．从而给出弹性圆锥扁壳非线性自由振动频率的基本公式。     

粘弹性大挠度圆板的轴对称弯曲

本文探讨粘弹性大挠度圆板的轴对称弯曲的基本方程和求解方法.用半逆解和摄动法分析挠度与膜力,对标准线性固体进行数例计算,并与小挠度理论相比较.全部方程与解答可退化得相应的弹性大挠度板的结果.     

周边嵌入弹性圆薄板非线性自由振动直接摄动解法

采用弹性圆薄板中心无量纲振幅和板厚与半径的比值为参数。将挠度、应力函数对半径的导数以及自由振动频率展开为双参数的幂级数。用直接摄动法获得各级递推线性偏微分方程。应用变分法求得各级递推方程的近似解，从而给出弹性圆薄板非线性自由振动频率的基本公式。     

The solution of large deflection problem of thin circular plate by the method of composite expansion

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky’s membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

Large deflection analysis of unsymmetrically laminated composite plates: analytical-numerical type approach

Large deflection analysis of laminated composite plates is considered. The Galerkin method along with Newton-Raphson method is applied to large deflection analysis of laminated composite plates with various edge conditions. The von Kármán plate theory is utilized and the governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacement functions. The solutions are compared to that of Dynamic Relaxation and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems.     

均变载荷下固定边圆薄板大挠度问题

关于均佈或中心集中载荷下圆薄板的大撓度平衡問題,已經有好几位学者用不同方法处理过。但用攝动法來处理时,不但計算比较簡單,并且得到的結果合乎实用。文蓀的攝动法选擇載荷为参数,解答的适用范圍小;钱偉長教授的攝动法选擇中心撓度为参数,得到的解答适用范圍大,并且和实驗的結果相符合。錢偉長教授的攝动法要比文蓀的攝动法來得高明。近几年來,錢偉长教授,胡海昌同志,叶开沅同志都采用撓皮为参数的攝动法解决了各种圓板的大撓度平衡問題。     

复合材料层合板的三维研究

本文提出了固支复合材料各向异性层合圆板受均布横向载荷作用下的满足三维弹性力学基本微分方程和边界条件的解析解答。文中采用一种发展的摄动方法进行求解，板中的每个应力和位移都展开为无量纲厚度参数ε的摄动级数，并采用二维板理论解答作为其相应三维摄动解答的一个基本解的形式，通过摄动方法逐级求解而获得完整的三维解答。文中以解析形式和数值形式给出了高精确度的三维应力和位移结果，结果表明，本文求解三维问题的解析方法是合理有效的。