中文摘要：

详细论述了近年来迅速发展的无网格法的理论基础及其在各个领域内的应用．无网格法网格依赖性弱，避免了传统的有限元、边界元等基于网格的数值方法中可能出现的网格畸变和扭曲，在一些有限元、边界元等方法难以较好处理的领域体现出独特的优势．以加权余量法为主线归纳了已有的30多种无网格法，各类无网格法的主要区别在于使用了不同的加权余量法和近似函数．详尽介绍了各种无网格近似方案（包括移动最小二乘近似、核近似和重构核近似、单位分解近似、径向基函数近似、点插值近似、自然邻接点插值近似等）和无网格法中常用的各类加权余量法（伽辽金格式、配点格式、局部弱形式、加权最小二乘格式和边界积分格式等），并讨论了数值积分方法和边界条件的处理等问题．在此基础上较系统地总结了无网格法在冲击爆炸、裂纹传播、超大变形、结构优化、流固耦合、生物力学和微纳米力学等领域的应用，展示了无网格法相对于传统数值方法的优势．

英文摘要：

Meshfree methods, both their theoretical foundation and their applications to a variety of fields, are reviewed in detail. There is much less mesh-dependency in meshfree methods, which can eliminate possible mesh distortion and entanglement in mesh-based numerical methods, such as the finite element method or boundary element method. Meshfree methods show particular advantages in some fields where the finite element or boundary element method encounter difficulties. More than 30 kinds of meshfree methods are reviewed in this paper in the light of weighted residual method, and different meshfree methods can be viewed as different forms of weighted residual method and/or with different approximation functions. Various kinds of meshfree approximate schemes are presented in detail, including moving least square approximation, kernel and reproducing kernel approximation, partition of unity approximation, radial basis approximation, radial point interpolation and natural neighbor interpolation. Different forms of weighted residual method, such as Galerkin form, collocation form, local weak form, weighted least-square form, boundary integral form are also described, and corresponding numerical quadrature algorithms and implementation of boundary conditions are discussed. Furthermore, applications of meshfree methods to the fields of impact and explosion, crack propagation, hyper large deformation, structural optimization, fluid-solid interaction, biomechanics and micro- and nano-mechanics are reviewed, and their advantages over conventional numerical methods are demonstrated.