scispace - formally typeset
Search or ask a question
Topic

Meshfree methods

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


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of two-dimensional cubic nonlinear Schrodinger equations, and the convergence order of the time discrete scheme is O ( δ t ).
Abstract: In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of two-dimensional cubic nonlinear Schrodinger equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O ( δ t ) . The aim of this paper is to show that the SMRPI method is suitable for the treatment of the nonlinear Schrodinger equations. Also, the SMRPI has less computational complexity than the other methods that have already solved this problem. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

10 citations

Journal ArticleDOI
TL;DR: In this paper, the numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions.
Abstract: The numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions. The results of radial basis functions are compared with the mesh-dependent boundary element and finite difference methods as well as the analytical solution for high Peclet numbers. It is reported that for low Peclet numbers, MQ-RBF provides excellent agreement, while for high Peclet numbers, TPS-RBF is better than MQ-RBF.

10 citations

Journal ArticleDOI
TL;DR: A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area.
Abstract: Red blood cells (RBCs) are the most common type of cells in human blood and they exhibit different types of motions and deformed shapes in capillary flows. The behaviour of the RBCs should be studied in order to explain the RBC motion and deformation mechanism. This article presents a numerical simulation method for RBC deformation in microvessels. A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area. The forces acting on the RBC membrane are obtained from the principle of virtual work. The whole fluid domain is discretized into a finite number of particles using smoothed particle hydrodynamics concepts and the motions of all the particles are solved using Navier--Stokes equations. Minimum energy concepts are used to simulate the deformed shape of the RBC model. To verify the model, the motion of a single RBC is simulated in a Poiseuille flow and the characteristic parachute shape of the RBC is observed. Further simulations reveal that the RBC shows a tank treading motion when it flows in a linear shear flow. References D. A. Fedosov, B. Caswell, and G. E. Karniadakis. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. , 98(10):2215–2225, 2010. doi:10.1016/j.bpj.2010.02.002 T. M. Fischer, M. Stohr-Lissen, and H. Schmid-Schonbein. The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow. Science , 202(4370):894–896, 1978. doi:10.1126/science.715448 R. A. Frcitas. Exploratory design in medical nanotechnology: a mechanical artificial red cell. Artif. Cell. Blood. Sub. , 26(4):411–430, 1998. doi:10.3109/10731199809117682 H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' deformation using SPH method. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' motion : a review. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. Y. T. Gu. Meshfree methods and their comparisons. Int. J. Comput. Meth. , 2(04):477–515, 2005. doi:10.1142/S0219876205000673 D. V. Le, J. White, J. Peraire, K. M. Lim, and B. C. Khoo. An implicit immersed boundary method for three-dimensional fluid–membrane interactions. J. Comput. Phys. , 228(22):8427–8445, 2009. doi:10.1016/j.jcp.2009.08.018 G. R. Liu and Y. T. Gu. An introduction to meshfree methods and their programming . Springer, 2005. G. R. Liu and M. B. Liu. Smoothed particle hydrodynamics: a meshfree particle method . World Scientific, 2003. doi:10.1142/5340 T. W. Pan and T. Wang. Dynamical simulation of red blood cell rheology in microvessels. Int. J. Numer. Anal. Mod. , 6:455–473, 2009. L. Shi, T. W. Pan, and R. Glowinski. Deformation of a single red blood cell in bounded Poiseuille flows. Phys. Rev. E , 85(1):016307, 2012. doi:10.1103/PhysRevE.85.016307 C. Sun and L. L. Munn. Particulate nature of blood determines macroscopic rheology: a 2-D lattice Boltzmann analysis. Biophys. J. , 88(3):1635–1645, 2005. doi:10.1529/biophysj.104.051151 K. I. Tsubota, S. Wada, and T. Yamaguchi. Particle method for computer simulation of red blood cell motion in blood flow. Comput. Meth. Prog. Bio. , 83(2):139–146, 2006. doi:10.1016/j.cmpb.2006.06.005 K. I. Tsubota, S. Wada, and T. Yamaguchi. Simulation study on effects of hematocrit on blood flow properties using particle method. J. Biomech. Sci. Eng. , 1(1):159–170, 2006. doi:10.1299/jbse.1.159 A. Vadapalli, D. Goldman, and A. S. Popel. Calculations of oxygen transport by red blood cells and hemoglobin solutions in capillaries. Artif. Cell. Blood. Sub. , 30(3):157–188, 2002. doi:10.1081/BIO-120004338

10 citations

Journal ArticleDOI
TL;DR: In this article, both of direct and inverse Stokes problems are stably and accurately analyzed by the method of fundamental solutions (MFS) and the Laplacian decomposition.
Abstract: In this study, both of direct and inverse Stokes problems are stably and accurately analyzed by the method of fundamental solutions (MFS) and the Laplacian decomposition. In order to accurately resolve the Stokes problem, the Laplacian decomposition is adopted to convert the Stokes equations into three Laplace equations, which will be solved by the MFS, with an augmented boundary condition. To enforce the satisfactions of continuity equation along whole boundary as an augmented boundary condition will guarantee the satisfactions of mass conservation inside the computational domain. The MFS is one of the most promising boundary-type meshless methods, since the time-consuming tasks of mesh generation and numerical quadrature can be avoided as well as only boundary nodes are needed for numerical implementations. The numerical solutions of the MFS are expressed as linear combinations of fundamental solutions of Laplace equation and the sources are located out of the computational domain to avoid numerical sin...

10 citations


Network Information
Related Topics (5)
Finite element method
178.6K papers, 3M citations
89% related
Numerical analysis
52.2K papers, 1.2M citations
86% related
Discretization
53K papers, 1M citations
86% related
Boundary value problem
145.3K papers, 2.7M citations
82% related
Partial differential equation
70.8K papers, 1.6M citations
81% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202355
2022112
2021102
202092
201996
201897