Meshfree methods and their applications

Meshfree methods, boththeir theoretical foundation and their applications to a varietyof fields, are reviewed in detail. There is much lessmesh-dependency in meshfree methods, which can eliminate possiblemesh 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 wherethe finite element or boundary element method encounterdifficulties. More than 30 kinds of meshfree methods are reviewedin this paper in the light of weighted residual method, anddifferent meshfree methods can be viewed as different forms ofweighted residual method and/or with different approximationfunctions. Various kinds of meshfree approximate schemes arepresented in detail, including moving least square approximation,kernel and reproducing kernel approximation, partition of unityapproximation, radial basis approximation, radial pointinterpolation and natural neighbor interpolation. Different formsof weighted residual method, such as Galerkin form, collocationform, local weak form, weighted least-square form, boundaryintegral form are also described, and corresponding numericalquadrature algorithms and implementation of boundary conditionsare discussed. Furthermore, applications of meshfree methods tothe fields of impact and explosion, crack propagation, hyper largedeformation, structural optimization, fluid-solid interaction,biomechanics and micro- and nano-mechanics are reviewed, and theiradvantages over conventional numerical methods are demonstrated.