Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Suite of finite element algorithms for accurate computation of soft tissue deformation for surgical simulation

Abstract: The ability to predict patient-specific soft tissue deformations is key for computer-integrated surgery systems and the core enabling technology for a new era of personalized medicine. Element-Free Galerkin (EFG) methods are better suited for solving soft tissue deformation problems than the finite element method (FEM) due to their capability of handling large deformation while also eliminating the necessity of creating a complex predefined mesh. Nevertheless, meshless methods based on EFG formulation, exhibit three major limitations: (i) meshless shape functions using higher order basis cannot always be computed for arbitrarily distributed nodes (irregular node placement is crucial for facilitating automated discretization of complex geometries); (ii) imposition of the Essential Boundary Conditions (EBC) is not straightforward; and, (iii) numerical (Gauss) integration in space is not exact as meshless shape functions are not polynomial. This paper presents a suite of Meshless Total Lagrangian Explicit Dynamics (MTLED) algorithms incorporating a Modified Moving Least Squares (MMLS) method for interpolating scattered data both for visualization and for numerical computations of soft tissue deformation, a novel way of imposing EBC for explicit time integration, and an adaptive numerical integration procedure within the Meshless Total Lagrangian Explicit Dynamics algorithm. The appropriateness and effectiveness of the proposed methods is demonstrated using comparisons with the established non-linear procedures from commercial finite element software ABAQUS and experiments with very large deformations. To demonstrate the translational benefits of MTLED we also present a realistic brain-shift computation.

Point collocation methods using the fast moving least-square reproducing kernel approximation

Abstract: A pseudo-spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least-square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.

A meshless Total Lagrangian explicit dynamics algorithm for surgical simulation

Abstract: A method is presented for computing deformation of very soft tissue. The method is motivated by the need for simple, automatic model creation for real-time simulation. The method is meshless in the sense that deformation is calculated at nodes that are not part of an element mesh. Node placement is almost arbitrary. Fully geometrically nonlinear Total Lagrangian formulation is used. Geometric integration is performed over a regular background grid that does not conform to the simulation geometry. Explicit time integration is used via the central difference method. As an example the simple but fully nonlinear Neo-Hookean material model is employed. The results are compared with a finite element simulation to verify the usefulness of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Boundary meshfree methods based on the boundary point interpolation methods

Abstract: A group of meshfree methods based on Boundary Integral Equation have been developed in order to overcome drawbacks in the conversional boundary element method (BEM) that requires boundary elements in constructing shape functions. In this paper, the radial basis point interpolation is firstly used to formulate the boundary radial point interpolation method (BRPIM). Two boundary meshfree methods: the boundary point interpolation method (BPIM) using the polynomial PIM and the BRPIM, are then examined in detail. The numerical implementations of these two methods are studied to address several technical issues, including the size of the compact support domain, the convergence, the performance, and so on. These two boundary-type meshfree methods are also compared with the Boundary Node Method and the conventional BEM in terms of both efficiency and performance. Several numerical examples of 2D elastostatics are analyzed using BPIM and BRPIM. It is found that BPIM and BRPIM are very easy to implement, and very robust for obtaining numerical solutions for problems of computational mechanics with very good accuracy. Key issues related the future development of boundary meshfree methods are also discussed.