Theory And Application Of The Meshless Natural Neighbour Petrov-Galerkin Method

Meshless methods, in which the approximation of solution variables is constructed only by scattered nodes, alleviate the difficulties of traditional finite element method (FEM) in such situations as shape optimization, dynamic crack propagation, extremely large deformation, hypervelocity impact simulation and explosion simulation etc., where degradation of accuracy may occur due to the mesh distortion and the time consuming task of remeshing is inevitable. Meshless methods are developed rapidly and have become more and more attractive in recently years. The current study, application and development trends of meshless methods are summarized and analyzed in this paper. The approximation, discretization, integration scheme and treatments of essential boundary condition used in meshless methods are overviewed. Among meshless methods, the meshless local Petrov-Galerkin method (MLPG), which is constructed based on a local weak form, attracted much attention. As the trial and test functions can be chosen from different functional spaces, the MLPG method is a generalized method to construct different meshless methods as the trial and test functions and the integration schemes are selected appropriately.The natural neighbour interpolation is a multivariable interpolation scheme based on Voronoi diagram of scattered nodes, and any user-defined parameters are not needed. This approximation scheme is used in natural element method and natural neighbour Galerkin method successfully. The non-Sibsonian interpolation is a kind of natural neighbour interpolation developed relatively recently, and proved to be more efficient. As the non-Sibsonian interpolation is interpolant in stead of fitting, meshless methods based on it can impose the essential boundary condition directly. In this paper, the non-Sibsonian interpolation is used to approximate the solution variables. The construction of the non-Sibsonian interpolation is proposed and two modifications are made to improve the efficiency of evaluating the shape function, which includes a local search algorithm to search natural neighbours of any sample point and a simple algorithm to construct the local Voronoi diagram. The prominent features ofnon-Sibsonian interpolation including completeness, local approximation, continuity, compact support, etc., are analyzed and discussed. The Partition of Unity (PU) property of non-Sibsonian shape function is used, and the non-Sibsonian PU method is proposed as a general tool to construct more accurate approximation functional spaces.By using non-Sibsonian interpolation to approximate the trial functions, an efficient and accurate meshless natural neighbour Petrov-Galerkin method (NNPG) is proposed based on the local Petrov-Galerkin discretization scheme. In this method, a local weak form of the equilibrium equation and the boundary conditions are satisfied in each local sub-domain and on its boundary, and the essential boundary condition can be enforced directly by virtue of the non-Sibsonian interpolation used. This method is quite stable, accurate, and the assembly process of global stiffness matrix is avoided. Due to the flexibility of the NNPG method, different meshless methods can be derived from it and truly meshless method can be constructed.Based on the theory of natural neighbour Petrov-Galerkin method, two distinctive meshless methods are derived, which are coined as NNPG-P and NNPG-M respectively. The NNPG-P is quite efficient and fit for the problems where the efficiency is more important and the NNPG-M is a truly meshless method as the interpolation and integration are totally independent of mesh. The construction of local sub-domains and nodal test functions are discussed and the polygonal sub-domains and circular sub-domains are used in these two methods respectively. The local integration schemes used in these methods are also studied. Both of the two methods are stable, accurate and easy to implement by virtue of the theory of natural neighbour Petrov-Galerkin method.Shape optimization is an iterative process, in which the design boundary needs to be modified frequently. Traditional FEM may face difficulties dealing with such problem, while meshless methods are quite suitable. With natural neighbour Petrov-Galerkin method and continuum material derivative concept, a local continuum variational method for shape design sensitivity analysis (DSA-LWF) is proposed. This method eliminates the shortcomings of discrete DSA method and proved to be more flexible than traditional global variational method. The obtained sensitivity is accurate and thus reduces the reanalysis time.Based on the mathematical programming method, a shape optimization method is proposed, and the natural neighbour Petrov-Galerkin method is used to calculate the structure responds and their sensitivities. As the cubic spline function is used to model the design boundary, smoothed design boundary is obtained after the optimization. An update technique for discrete model is also proposed. The optimization process based on the proposed method is totally automatic, efficient and absolutely no mesh is needed, which indicates the natural neighbour Petrov-Galerkin method has advantages for shape optimization problem.