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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: The meshless local Petrov-Galerkin (MLPG) approach as discussed by the authors was proposed to solve nonlinear boundary value problems, as no mesh connectivity is needed for interpolating the solution variables and for integrating the weak form.
Abstract: The meshless local Petrov-Galerkin approach based on a regular local boundary integral equation is successfully extended to solve nonlinear boundary value problems. The present method is truly meshless, as no mesh connectivity is needed for interpolating the solution variables and for integrating the weak form. Compared to the original MLPG method, the present method does not need the derivatives of the shape functions in constructing the stiffness matrix for those nodes with no displacement specified on their local boundaries. Numerical examples show that high rates of convergence with mesh refinement are achievable, and the computational results are quite accurate.
1 citations
DOI•
01 Jan 2006
TL;DR: Lee et al. as mentioned in this paper developed a meshless method for solving PDEs using Radial Basis Functions, which is easy to understand and even easier to implement than the traditional mesh-based methods.
Abstract: Developing Meshless Methods for Partial Differential Equations by Arthur Jonathan Lee Dr. Jichun Li, Examination Committee Chair Assistant Professor o f Mathematics University o f Nevada, Las Vegas In the past, the world o f numerical solutions for Partial Differential Equations has been dominated by Finite Element Method, Finite Difference Method, and Boundary Element Method. These three methods all revolve around using a mesh or grid to solve their problems. This complicates problems with irregular boundaries and domains. In this thesis, we develop methods for solving partial differential equations using Radial Basis Functions. This method is meshless, easy to understand, and even easier to implement.
1 citations
01 Jan 2007
TL;DR: In this paper, a hybrid approximation scheme for the shallow-water equations on the sphere is proposed which utilizes spectral-element approximation coupled with regional meshless collocation, and the issue of satisfying continuity conditions across spectral element-to-collocation interfaces for this domain decomposition method is discussed.
Abstract: A hybrid approximation scheme for the shallow-water equations on the sphere is proposed which utilizes spectral-element approximation coupled with regional meshless collocation. The issue of satisfying continuity conditions across spectral-element to meshless collocation interfaces for this domain decomposition method is discussed and gives an example of a meshless collocation framework which can be successfully coupled with spectral-element approximation. We conclude the paper with numerical examples using the proposed hybrid scheme on two well-known standardized test problems for the rotational shallow-water equations on the sphere.
1 citations
Journal Article•
TL;DR: The smoothed particle hydrodynamics (SPH) method is one of the most important branches of the meshless methods which are being developed and applied extensively, recently as mentioned in this paper.
Abstract: The smoothed particle hydrodynamics(SPH) method is one of the most important branches of the meshless methods which are being developed and applied extensively,recently.It is a new and pure Lagrangian method.In this paper,the fundamental theory of SPH,the conservation equation of continuum mechanics,and the improvement of stability are introduced.Then boundary condition is emphasized.The example using SPH method is presented.Finally,the latest advances of the SPH is described and outlined.
1 citations
01 Jan 2014
TL;DR: In this paper, the Moving Least Square approximation-based meshless collocation method for simulating deformable objects and a verification technique that is based on the Hertzian theory of non-adhesive elastic contact is presented.
Abstract: Mesh-based techniques are well studied and established methods for solving continuum biomechanics problems. When the problem at hand involves extreme deformations or artificial discontinuities, meshless methods provide several advantages over the mesh-based methods. This work discusses the Moving Least Square approximation-based meshless collocation method for simulating deformable objects and presents a verification technique that is based on the Hertzian theory of non-adhesive elastic contact. The effectiveness of the Hertzian contact theory as a means for verification was first tested and proven through a well-established FEM code, FEBio. The meshless method was implemented as a reusable component for the SOFA framework, an open source software library for real-time simulations. Through experimentation, the Hertzian theory has been tested against SOFA hexahedral FEM and the meshless models within the SOFA framework. Convergence studies and L2 error curves are provided for both models. Experimental results demonstrated the effectiveness of the implementation of the meshless method.
1 citations