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Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the boundary conditions are treated as a constraint to the discrete equations obtained from the Galerkin methods.
Abstract: One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary Γu, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov–Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd.
36 citations
TL;DR: In this article, a superconvergent point interpolation method (SC-PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming PIM using triangular mesh.
Abstract: A superconvergent point interpolation method (SC-PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming point interpolation method (LC-PIM) using triangular mesh. In the SC-PIM, point interpolation methods (PIM) are used for shape functions construction; and a strain field with a parameter α is assumed to be a linear combination of compatible stains and smoothed strains from LC-PIM. We prove theoretically that SC-PIM has a very nice bound property: the strain energy obtained from the SC-PIM solution lies in between those from the compatible FEM solution and the LC-PIM solution when the same mesh is used. We further provide a criterion for SC-PIM to obtain upper and lower bound solutions. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained using the present SC-PIM; (2) there exists an αexact∈(0, 1) at which the SC-PIM can produce the exact solution in the energy norm; (3) for any α∈(0, 1) the SC-PIM solution is of superconvergence, and α=0 is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms; (4) a procedure is devised to find a αprefer∈(0, 1) that produces a solution very close to the exact solution. Copyright © 2008 John Wiley & Sons, Ltd.
36 citations
TL;DR: A meshless method of lines (MMOL) is proposed for the numerical solution of nonlinear Burgers’-type equations that does not require a mesh in the problem domain, and only a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions.
36 citations
TL;DR: In this paper, the mesh-free shape functions are derived from a moving least-squares interpolation scheme and the incremental displacement-based formulation for large deformation is developed by employing the mesh free shape functions and the continuum tangent stiffness tensor in the weak form of the equilibrium equations.
36 citations
TL;DR: In this paper, a block Sakurai-Sugiura method is used to solve the eigenvalue problems governed by the Helmholtz equation, and formulated through two meshless methods.
Abstract: In this paper, a contour integral method (especially the block Sakurai-Sugiura method) is used to solve the eigenvalue problems governed by the Helmholtz equation, and formulated through two meshless methods. Singular value decomposition is employed to filter out the irrelevant eigenvalues. The accuracy and the ease of use of the proposed approach is illustrated with some numerical examples, and the choice of the contour integral method parameters is discussed. In particular, an application of the method on a sphere with realistic impedance boundary condition is performed and validated by comparison with results issued from a finite element method software.
36 citations