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Showing papers on "Meshfree methods published in 2003"


Proceedings ArticleDOI
01 Oct 2003
TL;DR: This work presents a new computational paradigm, the meshfree particle method, where the object representation and the numerical calculation are purely based on the nodal points and do not require the meshing of the analysis domain, and can naturally handle large deformation and domain discontinuity issues.
Abstract: Many of the computer vision algorithms have been posed in various forms of differential equations, derived from minimization of specific energy functionals, and the finite element representation and computation have become the de facto numerical strategies for solving these problems. However, for cases where domain mappings between numerical iterations or image frames involve large geometrical shape changes, such as deformable models for object segmentation and nonrigid motion tracking, these strategies may exhibit considerable loss of accuracy when the mesh elements become extremely skewed or compressed. We present a new computational paradigm, the meshfree particle method, where the object representation and the numerical calculation are purely based on the nodal points and do not require the meshing of the analysis domain. This meshfree strategy can naturally handle large deformation and domain discontinuity issues and achieve desired numerical accuracy through adaptive node and polynomial shape function refinement. We discuss in detail the element-free Galerkin method, including the shape function construction using the moving least square approximation and the Galerkin weak form formulation, and we demonstrate its applications to deformable model based segmentation and mechanically motivated left ventricular motion analysis.

367 citations


Journal ArticleDOI
TL;DR: A survey of meshless methods can be found in this article, where the authors provide a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references.
Abstract: In the past few years meshless methods for numerically solving partial differential equations have come into the focus of interest, especially in the engineering community. This class of methods was essentially stimulated by difficulties related to mesh generation. Mesh generation is delicate in many situations, for instance, when the domain has complicated geometry; when the mesh changes with time, as in crack propagation, and remeshing is required at each time step; when a Lagrangian formulation is employed, especially with nonlinear PDEs. In addition, the need for flexibility in the selection of approximating functions ( e.g. , the flexibility to use non-polynomial approximating functions), has played a significant role in the development of meshless methods. There are many recent papers, and two books, on meshless methods; most of them are of an engineering character, without any mathematical analysis. In this paper we address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations, using variational principles. We give a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references to the current literature. The aim of the paper is to provide a survey of a part of this new field, with emphasis on mathematics. We present proofs of essential theorems because we feel these proofs are essential for the understanding of the mathematical aspects of meshless methods, which has approximation theory as a major ingredient. As always, any new field is stimulated by and related to older ideas. This will be visible in our paper.

350 citations


Journal ArticleDOI
TL;DR: In this article, the simulation of concrete fragmentation under explosive loading by a mesh-free Lagrangian method, the smooth particle hydrodynamics method (SPH), is described, and two improvements regarding the com- pleteness of the SPH-method are examined, first a normalization developed by Johnson and Beissel (NSPH) and second a moving least square (MLS) approach as modied by Scheer (MLSPH).
Abstract: SUMMARY The simulation of concrete fragmentation under explosive loading by a meshfree Lagrangian method, the smooth particle hydrodynamics method (SPH) is described. Two improvements regarding the com- pleteness of the SPH-method are examined, �rst a normalization developed by Johnson and Beissel (NSPH) and second a moving least square (MLS) approach as modied by Scheer (MLSPH). The SPH-Code is implemented in FORTRAN 90 and parallelized with MPI. A macroscopic constitutive law with isotropic damage for fracture and fragmentation for concrete is implemented in the SPH-Code. It is shown that the SPH-method is able to simulate the fracture and fragmentation of concrete slabs under contact detonation. The numerical results from the dierent SPH-methods are compared with the data from tests. The good agreement between calculation and experiment suggests that the SPH-program can predict the correct maximum pressure as well as the damage of the concrete slabs. Finally the fragment distributions of the tests and the numerical calculations are compared. Copyright ? 2003 John Wiley & Sons, Ltd.

213 citations


BookDOI
01 Jan 2003
TL;DR: A Meshfree Method for the Analysis of Planar Flows of Inviscid Fluids and Some Regularized Versions of the Method of Fundamental Solutions.
Abstract: ESPResSo 3.1 - Molecular Dynamics Software for Coarse-Grained Models: A. Arnold, O. Lenz, S. Kesselheim, R. Weeber, F. Fahrenberger, D. Roehm, P. Kosovan, C. Holm.- On the Rate of Convergence of the Hamiltonian Particle-Mesh Method Onno Bokhove, Vladimir Molchanov, Marcel Oliver, Bob Peeters.- Peridynamics: A Nonlocal Continuum Theory.- Etienne Emmrich, Richard B. Lehoucq, Dimitri Puhst.- Immersed Molecular Electrokinetic Finite Element Method for Nano-Devices in Biotechnology and Gene Delivery: Wing Kam Liu, Adrian M. Kopacz, Tae-Rin Lee, Hansung Kim, Paolo Decuzzi.- Corrected Stabilized Non-conforming Nodal Integration in Meshfree Methods: Marcus Ruter, Michael Hillman, Jiun-Shyan Chen.- Multilevel Partition of Unity Method for Elliptic Problems with strongly Discontinuous Coefficients: Marc Alexander Schweitzer.- HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness: Agustin Bompadre, Luigi E. Perotti, Christian J. Cyron, Michael Ortiz.- A Meshfree Splitting Method for Soliton Dynamics in Nonlinear Schrodinger Equations: Marco Caliari, Alexander Ostermann, Stefan Rainer.- A Meshless Discretization Method for Markov State Models Applied to Explicit Water Peptide Folding Simulations: Konstantin Fackeldey, Alexander Bujotzek, Marcus Weber.- Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations: Gregory E. Fasshauer, Qi Ye.- A Meshfree Method for the Analysis of Planar Flows of Inviscid Fluids: Vasily N. Govorukhin.- Some Regularized Versions of the Method of Fundamental Solutions: Csaba Gaspar.- A Characteristic Particle Method for Traffic Flow Simulations on Highway Networks: Yossi Farjoun, Benjamin Seibold.- Meshfree Modeling in Laminated Composites: Daniel C. Simkins, Jr., Nathan Collier, Joseph B. Alford

164 citations


Journal ArticleDOI
TL;DR: In this article, a method for the solution of the incompressible fluid flow equations using a Lagrangian formulation is presented, which simplifies the connections with fixed or moving solid structures, thus providing a very easy way to solve fluid-structure interaction problems.

139 citations


Journal ArticleDOI
TL;DR: A survey of the most relevant advances in natural neighbour Galerkin methods is presented in this article, where the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions.
Abstract: In this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide family of so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy. A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two- and three-dimensions. Application of the NEM to linear and non-linear problems in solid as well as fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of this numerical method.

138 citations


Journal ArticleDOI
TL;DR: In this article, a mesh free weak-strong (MWS) form method is proposed based on a combined formulation of both the strong-form and the local weak-form, which only needs the local numerical integration, is only used for nodes on or near the natural boundaries.
Abstract: A novel meshfree weak–strong (MWS) form method is proposed based on a combined formulation of both the strong-form and the local weak-form. In the MWS method, the problem domain and its boundary is represented by a set of distributed points or nodes. The strong form or the collocation method is used for all nodes whose local quadrature domains do not intersect with natural (Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak-form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The locally supported radial point interpolation method and the moving least squares approximation are used to construct the meshfree shape functions. The final system matrix will be sparse and banded for computational efficiency. Numerical examples of two-dimensional solids are presented to demonstrate the efficiency, stability, accuracy and convergence of the proposed meshfree method.

126 citations


Journal ArticleDOI
TL;DR: In this paper, a pseudo-spectral point collocation mesh-free method is proposed, which is based on the moving least-square reproducing kernel approximations of shape functions.
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.

103 citations


Journal ArticleDOI
TL;DR: The influence of key parameters, as the number of nodes to add in each step or the minimum distance between nodes, is analyzed through the analysis of the obtained solutions for different types of differential equations.

102 citations


Journal ArticleDOI
TL;DR: The element-free Galerkin method is exploited to analyze gradient plasticity theories and it is shown that the regularization properties of the higher-order gradients are necessary, since, similar to finite element methods, a severe discretization sensitivity is encountered otherwise.

88 citations


Journal ArticleDOI
TL;DR: In this article, the vorticity-stream function form of N-S equations is taken as the governing equations and the LRPIM method is adopted to simulate the two-dimensional natural convection problems within enclosed domain of different geometries.
Abstract: The LRPIM method is adopted to simulate the two-dimensional natural convection problems within enclosed domain of different geometries. In this paper, the vorticity-stream function form of N-S equations is taken as the governing equations. It was observed that the obtained results agreed very well with others available in the literatures, and with the same nodal density, the accuracy achieved by the LRPIM method is much higher than that of the finite difference (FD) method. The numerical examples show that the present LRPIM method can successfully deal with incompressible flow problems on randomly distributed nodes.

Journal ArticleDOI
TL;DR: Efficient computational methods for scattered point and meshless analysis of electrostatic microelectromechanical systems (MEMS) are presented, showing flexible, efficient, and attractive alternatives compared to conventional finite element/boundary element methods for self-consistent electromechanical analysis.
Abstract: We present efficient computational methods for scattered point and meshless analysis of electrostatic microelectromechanical systems (MEMS). Electrostatic MEM devices are governed by coupled mechanical and electrostatic energy domains. A self-consistent analysis of electrostatic MEMS is implemented by combining a finite cloud method-based interior mechanical analysis with a boundary cloud method (BCM)-based exterior electrostatic analysis. Lagrangian descriptions are used for both mechanical and electrostatic analyses. Meshless finite cloud and BCMs, combined with fast algorithms and Lagrangian descriptions, are flexible, efficient, and attractive alternatives compared to conventional finite element/boundary element methods for self-consistent electromechanical analysis. Numerical results are presented for MEM switches, a micromirror device, a lateral comb drive microactuator, and an electrostatic comb drive device. Simulation results are compared with experimental and previously reported data for many of the examples discussed in this paper and a good agreement is observed.

Journal ArticleDOI
TL;DR: In this paper, a one-dimensional mesh-free particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena, based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity.
Abstract: In this paper, a one-dimensional meshfree particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena. This new formulation is based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity. The new formulation inherits the meshfree Lagrangian and particle nature of SPH, and is a natural extension and improvement on the traditional SPH method and the recently proposed corrective smoothed particle method (CSPM). The formulation is consistent even in the discontinuous regions. The resultant kernel and particle approximations consist of a primary part similar to that in CSPM, and a corrective part derived from the discontinuity. A numerical study is carried out to examine the performance of the formulation. The results show that the new formulation not only remedies the boundary deficiency problem but also simulates the discontinuity well. The formulation is applied to simulate the shock tube problem and a 1-D TNT slab detonation. It is found that the proposed formulation captures the shock wave at comparatively lower particle resolution. These preliminary numerical tests suggest that the new meshfree particle formulation is attractive in simulating hydrodynamic problems with discontinuities such as shocks waves.

Journal ArticleDOI
TL;DR: The natural element method (NEM) has the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fiber orientation equation without the mesh quality requirement characteristics of the standard finite elements method as discussed by the authors.
Abstract: Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics (coupled problem). Some industrial flows involve moving or free boundaries (injection, extrusion, …). Lagrangian descriptions allow an accurate description of the flow front tracking as well as an accurate integration of transport equations along the flow trajectories. However, Lagrangian techniques in the context of finite elements have the important drawback of requiring frequent remeshing in order to avoid large elements distortions. The natural element method (NEM) has the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fiber orientation equation without the mesh quality requirement characteristics of the standard finite elements method.

Journal ArticleDOI
TL;DR: In this article, a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction is presented, which is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions.
Abstract: We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions.

Journal ArticleDOI
TL;DR: In this paper, corrected smooth particle hydrodynamics (CSPH) is used to simulate fluid flow in the high pressure die casting cavity, and the fundamental governing equations are derived based on a variational formulation.
Abstract: Mould filling simulation in high pressure die casting has been an attractive area of research for many years. Several numerical methodologies have been attempted in the past to study the flow behaviour of the molten metal into the die cavities. However, many of these methods require a stationary mesh or grid which limits their ability in simulating highly dynamic and transient flows encountered in high pressure die casting processes. In recent years, the advent of meshfree methods have expanded the capabilities of numerical techniques. Hence, these methods have emerged as an attractive alternative for modelling mould filling simulation in pressure die casting processes. In the present work, a Lagrangian particle method called corrected smooth particle hydrodynamics (CSPH) is used to simulate fluid flow in the high pressure die casting cavity. This paper mainly focuses on deriving the fundamental governing equations based on a variational formulation and presents a number of mould filling examples to demonstrate the capabilities of the CSPH numerical model.

Journal ArticleDOI
TL;DR: The extended Delaunay tessellation (EDT) is presented in this paper as the unique partition of a node set into polyhedral regions defined by nodes lying on the nearby Voronoi spheres, suitable for Lagrangian problems and meshless methods in which only the connectivity information is used and there is no need for any expensive smoothing process.
Abstract: The extended Delaunay tessellation (EDT) is presented in this paper as the unique partition of a node set into polyhedral regions defined by nodes lying on the nearby Voronoi spheres. Until recently, all the FEM mesh generators were limited to the generation of tetrahedral or hexahedral elements (or triangular and quadrangular in 2D problems). The reason for this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements. Nowadays, there are several acceptable shape functions for a very large class of polyhedra. These new shape functions, together with the EDT, gives an optimal combination and a powerful tool to solve a large variety of physical problems by numerical methods. The domain partition into polyhedra presented here does not introduce any new node nor change any node position. This makes this process suitable for Lagrangian problems and meshless methods in which only the connectivity information is used and there is no need for any expensive smoothing process.

Journal ArticleDOI
TL;DR: In this article, the authors present a preprocessor for the generation of nodal points on two-dimensional computational domains and a specialized version of the method of finite spheres using point collocation and moving least squares approximation functions and singular weight functions.
Abstract: In this paper we report some recent advances regarding applications using the method of finite spheres; a truly meshfree numerical technique developed for the solution of boundary value problems on geometrically complex domains First, we present the development of a preprocessor for the generation of nodal points on two-dimensional computational domains Then, the development of a specialized version of the method of finite spheres using point collocation and moving least squares approximation functions and singular weight functions is reported for rapid computations in virtual environments involving multi-sensory (visual and touch) interactions

Journal ArticleDOI
TL;DR: In this article, mesh-free variational methods are used for the solution of incompressible fluid dynamics problems using the R-function method (RFM), which constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non-conforming spatial grid.
Abstract: We show that meshfree variational methods may be used for the solution of incompressible fluid dynamics problems using the R-function method (RFM). The proposed approach constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non-conforming spatial grid. We give detailed implementation of the method for two common formulations of the incompressible fluid dynamics problem: first using scalar stream function formulation and then using vector formulation of the Navier-Stokes problem with artificial compressibility approach. Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back-facing step channel problem reveal strengths and weaknesses of the proposed meshfree method.

Book ChapterDOI
01 Jan 2003
TL;DR: The adaptive meshfree advection scheme for numerically solving linear transport equations is extended to nonlinear transport equations, and an artificial viscosity term is added to the scheme in order to be able to model shock propagation.
Abstract: In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. In this paper, the method is extended to nonlinear transport equations. To this end, in order to be able to model shock propagation, an artificial viscosity term is added to the scheme. Moreover, the local interpolation method and the node adaption rules are modified accordingly. The good performance of the resulting method is finally shown in the numerical examples by using two specific nonlinear model problems: Burgers equation and the Buckley-Leverett equation, the latter describing a two-phase fluid flow in a porous medium.

Journal ArticleDOI
TL;DR: In this paper, a posteriori error estimates and an adaptive refinement scheme of first-order least-squares meshfree method (LSMFM) are presented and the error indicators are readily computed from the residual.
Abstract: A posteriori error estimates and an adaptive refinement scheme of first-order least-squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin–Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently reflect the actual error. In the proposed refinement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive first-order LSMFM, which combines the proposed error indicators and nodal refinement scheme, is effectively applied to the localized problems such as the shock formation in fluid dynamics. Copyright © 2003 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated.
Abstract: Recently, considerable effort has been devoted to the development of the so-called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the 'a posteriori error' should be approximated. An 'a posteriori error' approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors.

Journal ArticleDOI
TL;DR: In this paper, a mesh-free method based on the first-order least-squares formulation for linear elasticity is presented, where both primal and dual variables can be approximated by the same function space, leading to higher rate of convergence for dual variables than Galerkin formulation.
Abstract: A meshfree method based on the first-order least-squares formulation for linear elasticity is presented. In the authors' previous work, the least-squares meshfree method has been shown to be highly robust to integration errors with the numerical examples of Poisson equation. In the present work, conventional formulation and compatibility-imposed formulation for linear elastic problems are studied on the convergence behavior of the solution and the robustness to the inaccurate integration using simply constructed background cells. In the least-squares formulation, both primal and dual variables can be approximated by the same function space. This can lead to higher rate of convergence for dual variables than Galerkin formulation. In general, the incompressible locking can be alleviated using mixed formulations. However, in meshfree framework these approaches involve an additional use of background grids to implement lower approximation space for dual variables. This difficulty is avoided in the present method and numerical examples show the uniform convergence performance in the incompressible limit. Therefore the present method has little burden of the requirement of background cells for the purposes of integration and alleviating the incompressible locking.

Journal ArticleDOI
TL;DR: The implementation of a 3-D parallel CFD code using the meshless method and the Total Arbitrary Lagrangian Eulerian formulations using Finite Element Method are developed and implemented in the parallel code.
Abstract: In this paper, the implementation of a 3-D parallel CFD code using the meshless method. Reproducing Kernel Particle Method (RKPM) is described. A novel procedure for implementing the essential boundary condition using the hierarchical enrichment method is presented. The Total Arbitrary Lagrangian Eulerian (ALE) formulations using Finite Element Method are developed and implemented in the parallel code. The flow past a cylinder problem served as examples throughout the paper. Both methods have shown promising results compared with analytical solution. Copyright © 2003 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, both Sibson and non-Sibson interpolants ability to exactly reproduce essential boundary conditions is investigated and a new analytical condition ensuring linear precision along explicitly described (i.e. CAD) boundaries in both two and three dimensions is presented.
Abstract: In this paper issues related to the imposition of essential boundary conditions in Natural Neighbour Galerkin methods are addressed. Both Sibson and non-Sibson interpolants ability to exactly reproduce essential boundary conditions is investigated and a new analytical condition ensuring linear precision along explicitly described (i.e. CAD) boundaries in both two and three dimensions is presented. The paper is completed with some benchmark numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: This paper presents a coupling of the meshless finite cloud method (FCM) and the standard (mesh-based) boundary element method (BEM), which is motivated by the complementary properties of both methods.

Journal ArticleDOI
TL;DR: In this article, the authors compared three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries.
Abstract: The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C 1 extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy.

Journal ArticleDOI
TL;DR: A geometry-based automatic pre-processing environment for the method of finite spheres is reported; a truly meshfree numerical technique developed for the solution of boundary value problems on geometrically complex domains.

Book ChapterDOI
20 Jul 2003
TL;DR: In this paper, a general representation and computation framework based on sampling nodal points is presented for medical image analysis issues where the domain mappings between images involve large geometrical shape changes, such as the cases of nonrigid motion recovery and interobject image registration.
Abstract: For medical image analysis issues where the domain mappings between images involve large geometrical shape changes, such as the cases of nonrigid motion recovery and inter-object image registration, the finite element methods exhibit considerable loss of accuracy when the elements in the mesh become extremely skewed or compressed. Therefore, algorithmically difficult and computationally expensive remeshing procedures must be performed in order to alleviate the problem. We present a general representation and computation framework which is purely based on the sampling nodal points and does not require the construction of mesh structure of the analysis domain. This meshfree strategy can more naturally handle very large object deformation and domain discontinuity problems. Because of its intrinsic h-p adaptivity, the meshfree framework can achieve desired numerical accuracy through adaptive node and polynomial shape function refinement with minimum extra computational expense. We focus on one of the more robust meshfree efforts, the element free Galerkin method, through the moving least square approximation and the Galerkin weak form formulation, and demonstrate its relevancy to medical image analysis problems. Specifically, we show the results of applying this strategy to physically motivated multi-frame motion analysis, using synthetic data for accuracy assessment and for comparison to finite element results, and using canine magnetic resonance tagging and phase contrast images for cardiac kinematics recovery.

Journal ArticleDOI
TL;DR: In this paper, a new hybrid numerical method, the hM-DOR method, which is based on an order-reduction technique for partial differential equations, combines the true-meshless collocation technique with a fixed reproducing kernel approximation.