摘要
无网格方法是继有限元法和边界元法等传统数值方法之后兴起的一种很有发展前途的数值方法。与传统的数值分析方法相比,无网格方法具有许多突出的优点,最主要的优点在于克服了对网格的依赖性,彻底或部分消除了网格的划分,因此无网格方法在处理大变形、裂纹扩展和高速冲击等非线性问题时具有明显的优势。近年来,国内外学者在无网格方法的研究方面已经取得了许多开创性的重要成果。无网格局部径向点插值法(LRPIM)是近几年发展起来的一种无网格方法,它不需要借助于任何单元或网格进行积分或插值,是一种真正的无网格方法。而且,其形函数具有Kronecker delta函数性质,可以直接施加本质边界条件。本文将无网格LRPIM应用于求解中厚板的弯曲、动力学以及弹塑性等问题。
本文首先介绍了无网格方法的发展历史和国内外研究现状,按照不同离散方式对各种主要的无网格方法进行了回顾和评价,总结了无网格方法的特点、优越性以及目前无网格方法的难点和存在的问题。概述了无网格方法在板壳问题当中的应用情况。本文的无网格LRPIM采用径向基函数耦合多项式构造形函数,消除了系统矩阵的奇异性,形函数及其导数比较简单,计算效率和精度都比较高。
尽管无网格方法在板壳问题的研究中已经有了一系列的成果,但无网格LRPIM在中厚板问题中的研究却很少有报道。本文的主要工作与创新点是,首次将无网格LRPIM应用于求解中厚板的弯曲问题、动力学问题以及弹塑性问题等。在中厚板的静力平衡方程和动力学方程等基础上,采用局部加权残值法推导出了各种情况下的离散系统方程。在中厚板的弯曲问题中,计算了各种边界条件下和各种外加载荷作用下的中厚板的弯曲变形、内力和应力;讨论了径向基函数的形状参数对计算结果的影响;分析了附加不同阶数多项式的计算效率问题;考虑了积分域和影响域大小对计算结果的影响问题;对剪切自锁所产生的原因以及避免剪切自锁的措施进行了分析,发现无网格方法相对于有限元法等传统数值方法能更好地避免剪切锁死现象;利用无网格LRPIM分析了非均质中厚板的静力弯曲问题。在弹性地基厚板的弯曲问题中,推导了局部径向点插值离散方程,分析了弹性地基上四边简支厚板、四边固支厚板以及建筑筏板基础的挠度和弯矩,计算了挠度和弯矩的相对误差和收敛率。对于中厚板的动力学问题,推导了自由振动和强迫振动的离散系统方程,采用子空间迭代方法求解特征方程,动力学方程则采用Newmark方法进行时域离散,介绍了数值实施方法和主要计算步骤;给出了各种不同边界条件和不同形状中厚板的自由振动和强迫振动数值算例;利用无网格LRPIM分析了非均质中厚板的动力弯曲问题。最后用无网格LRPIM法分析了中厚板的弹塑性弯曲问题。分析了中厚板弹塑性应力应变关系;采用增量Newton-Raphson迭代法来求解中厚板非线性增量形式的离散系统方程。
无论在中厚板的弯曲问题,弹性地基厚板的弯曲问题中,还是在中厚板的动力学问题和弹塑性问题中,所有数值算例结果都表明,本文方法对于中厚板的问题的求解是可行的和有效的,并且所得到的结果具有较好的精度和收敛性。
The meshless method is a new numerical method with a great prospect developed after traditional numerical methods such as Finite Element Method, Boundary Element Method et al. The meshless method possesses many advantages, among these the most outstanding advantage is independent of meshes, and thoroughly or partly eliminates meshing. By using this method, it becomes easy to solve large deformation problems, crack propagation problems and high velocity impact problems et al. A lot of the important pioneering effort has been done on the meshless methods by scholars in a recent decade. The meshless local radial point interpolation method (LRPIM) is a new numerical technique presented in recent years. It doesn’t need any element or mesh for the energy integral or the purpose of interpolation. Therefore it is a truly meshless method. The shape functions have the Kronecker delta function property, and the essential boundary conditions can be easily imposed. Applications of the meshless LRPIM to bending and dynamic problems as well as elasto-plastic problems of moderately thick plates are presented in this dissertation.
At the beginning of the dissertation, recent developments of the meshless method are overviewed. Several typical meshless methods are reviewed and appraised in term of their discretization schemes. Characteristics, advantages and disadvantages of all kinds of meshless methods are pointed out. Applications of the meshless methods to the plate and shell problems are introduced. The shape function of the meshless LRPIM is all constructed by using the radial basis functions with polynomial basis functions, the singularity of the system matrix is overcome. The shape functions and their derivatives are simple, consequently, lower computational cost. The efficient and accurate results can be obtained.
Although a lot of achievements are obtained about meshless methods for the plate and shell problems, the solution of moderately thick plate problem is rarely reported in the use of the meshless LRPIM. In this dissertation, the meshless LRPIM is used to investigate bending and dynamic problems as well as elasto-plastic problems of moderately thick plates. Based on the equilibrium equations and dynamic equations of a moderately thick plate, the various discretized system equations for moderately thick plates are derived using locally weighted residual method. In the analysis of the bending problems for moderately thick plates with various boundary conditions and under various loads, deformations and bending (torsional) moments as well as stresses are calculated. Effects of the shape parameters of the radial basis function on the numerical results are investigated. Computing efficiency is studied when polynomials of the low and high order are used. Effects of sizes of the quadrature sub-domain and the influence domain on the numerical results are investigated. The reason of the shear locking and the measure of avoiding the shear locking are analyzed. It is found that the shear locking is easier avoided in the meshless method than in FEM. The static bending problems of a nonhomogeneous moderately thick plate are analyzed using the meshless LRPIM, too. The discretized system equation for moderately thick plates on the elastic foundation with two parameters is derived using a locally weighted residual method. Bending problems for the raft and moderately thick plates on the elastic foundation with simply supported and clamped boundary conditions are analyzed by the meshless LRPIM. The relative error and convergence rate for deflections and bending moments are studied. For the dynamic analysis of moderately thick plates, the discretized system equations of the free vibration and forced vibration for the moderately thick plate are presented. The subspace iterative method is adopted to solve the eigenvalue equation of the free vibration problem, and the Newmark method is used to discrete the time domain. The approaches of numerical implement are presented, and several numerical examples are presented for the free vibration and forced vibration of moderately thick plates with various boundary conditions. The dynamic bending problems of a nonhomogeneous moderately thick plate are analyzed using the meshless LRPIM, too. In the end, the elasto-plastic bending problems of moderately thick plates are analyzed by the meshless LRPIM. The elasto-plastic stress-strain relation of the moderately thick plate is studied, and an incremental Newton-Raphson iterative algorithm is employed to solve the nonlinear incremental discretized system equation of the moderately thick plate.
Numerical results show that the present method possesses not only feasibility and validity, but also high accuracy and good performance of convergence for moderately thick plate problems including bending and dynamic problems as well as the elasto-plastic problem.
引文
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