Meshfree particle methods for analysis of elastic plates and shells

工程中大量的结构构件都可以归为板类。土木工程结构的典型例子是楼板和基础板、闸门、薄挡土墙、桥面和板桥。板材在造船、汽车和航空航天工业中也是不可或缺的。薄板的应力合成可以通过三维弹性力学方程计算。然而，在某些假设下，三维弹性方程的板减少到二维方程。基尔霍夫板模型非常适合于薄板。然而，该板模型的位移控制方程是四阶微分方程。由于构造光滑有限元的复杂性，传统的有限元方法难以应用。此外，中厚板还有其它困难，如边界层问题和剪切锁定问题。 无网格方法（其中使用光滑灵活的近似函数，并且不需要复杂的网格生成）与传统的有限元方法相比具有几个优点。 然而，这些方法有几个主要的局限性，包括在处理本质边界条件，大矩阵条件数，并在构建单位划分的复杂性的效率低下。 最近，PI发明了一种最灵活的封闭形式单位分割，称为广义单位乘积分割。 PI建议引入无网格粒子方法，通过使用广义乘积单位分解，以及新的局部近似函数，可以处理几何边界条件以及各种板模型中出现的力边界条件。此外，PI建议应用无网格粒子方法来获得板和壳的高精度应力分析，用于相关工程结构的设计和维护。 智力优势：本文的研究成果将极大地改善板壳和复合材料层合板的应力分析，使汽车、飞机、船舶和所有与板壳有关的工程结构的设计和维护更加有效和安全。 此外，没有任何困难，所提出的方法是能够外科手术的近似空间丰富的任何类型的奇异函数。 该方法处理奇异性灵活有效，可用于裂纹扩展的精确预测。 更广泛的影响：这项研究的结果可用于设计省油的汽车、更安全的飞机、船舶和更耐故障的新材料。拟议研究 还可用于改善老化客机、桥梁、船舶、建筑物的维护，以及应密切监测结构完整性的许多其他应用。最终，拟议的研究将通过提高日常使用的常见交通方式和结构的效率和安全性，对公共安全和环境产生直接影响。

A large number of structural components in engineering can be classified as plates. Typical examples of civil engineering structures are floor and foundation slabs, lock-gates, thin retaining walls, bridge decks, and slab bridges. Plates are also indispensable in ship building, automobile, and aerospace industries. The stress resultants of a thin plate can be calculated through the 3-dimensional elasticity equations. However, under certain hypotheses, the 3-dimensional elasticity equations for a plate are reduced to the 2-dimensional equations. The Kirchhoff plate model is well suited for thin plates. However, the governing equation for the displacement of this plate model is the fourth order differential equation. The conventional finite element method is difficult to apply because of the complexity of constructing smooth finite elements. Furthermore, moderately thick plates have other difficulties such as boundary layer problems and shear locking problems. Meshless methods (in which smooth flexible approximation functions are used and complicated mesh generation is not necessary) have several advantages over the conventional finite element method. However, these methods have several major limitations including the inefficiency in handling essential boundary conditions, large matrix condition numbers, and complexity in constructing partitions of unity. Most recently, the PI invented one of the most flexible closed form partition of unity, called the Generalized Product Partition of Unity. The PI proposes to introduce Meshfree particle methods by using the Generalized Product Partition of Unity, together with new local approximation functions that can handle geometric boundary conditions as well as force boundary conditions arising in various plate models. Furthermore, the PI proposes to apply Meshfree particle methods to obtain highly accurate stress analysis of plates and shells for design and maintenance of related engineering structures. The Intellectual Merit: The proposed research will greatly improve the stress analysis of plates, shells, and laminated composite plates so that design and maintenance of automobiles, airplanes, ships, and all other engineering structures related to plates and shells may be more effective and safer. Moreover, without any difficulty, the proposed method is able to surgically make the approximation space enriched with any type of singular function. The proposed method is flexible and effective in dealing with singularities and it can be used for accurate prediction of crack propagation. The Broader Impacts: Results from the proposed research can be used to design fuel efficient automobiles, safer airplanes, ships and new materials more resistant to failure. The proposed research can also be applied to improve maintenance of aging airliners, bridges, ships, buildings, and numerous other applications where structural integrity should be closely monitored. Ultimately, the proposed research will have direct impacts on the public safety and the environment by increasing efficiency and safety of common modes of transportation and structures used everyday.