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SPH using the moving least square (MLS) approximation and fixed kernel function is proposed and is applied to large deformation elasto-plastic analysis. SPH is one of the particle methods and is a powerful tool for large-scale hydro dynamics analysis. SPH is, however, not efficient when the kernel functions are calculated at every step and the spatial approximation is less accurate due to only the 0-th order consistency. In this study, Fixed Kernel method in which the kernel is calculated only at the original configuration is utilized in order to reduce numerical cost and MLS approximation is applied for the spatial approximation to improve accuracy. Furthermore, the Poisson’s equation for pressure is solved to take account of incompressibility in plastic state. Numerical analysis of elastic problem and elasto-plastic problem are demonstrated to validate the present method.

The scaled boundary method is an excellent way to model unbounded domains. However, it is limited to linear problems. Many soft-ground geotechnical problems require both non-linear constitutive behaviour for the soil, to capture pre-failure deformations, and the presence of an unbounded domain. Adaptive meshfree methods are ideally suited to such problems. This paper couples a meshless local Petrov–Galerkin method for the near field with a meshless scaled boundary method of similar type for the far field. The method presented is novel as the degrees of freedom of all nodes in the support of the interface nodes are coupled to the stiffness of the unbounded domain, rather than just the nodes on the interface.

Analysis of cerebral contusion using the smoothed particle hydrodynamics (SPH) method is presented. When a human head is subjected to external impacts in cases of crashing with cars in traffic accidents, a person falling down and etc., damage is often observed in the brain. It is called cerebral contusion which means damage of the brain of the human head. Although cerebral contusion which is called coup is usually observed at an impacted side of the head, it may be also observed at an opposite side, which is called contrecoup. There is no reasonable explanation of contrecoup.

The smoothed particle hydrodynamics method, which is one of the meshfree methods, can calculate impact analyses including fracture.

In the present paper, we analyse the damage of the head using the SPH method and obtain the stress distribution of the brain when the head collides with a glass of a car. We obtain the distribution of stress to cause cerebral contusion.

This chapter investigated the irreversible adsorption of colloidal particles from pressure driven flow in a microchannel under the influence of colloidal (DLVO) interactions and external forces such as gravity. A theoretical model was developed on the basis of the stochastic Langevin equation, incorporating the Brownian motion of particles. Brownian dynamics simulation technique was employed to calculate the particle surface coverage. To validate the proposed theoretical model, experiments were carried out using the parallel plate flow cell technique, enabling direct Videomicroscopic observation of the deposition kinetics of polystyrene latex particles in NaCl electrolyte solution. The theoretical predictions were compared with experimental results and a good agreement was found.

Radial basis functions (RBFs) have shown excellent interpolation properties and great promise in meshless methods for solving partial differential equations. However, the accuracy of the RBF meshless method is found to depend on the shape parameters, $c$, of RBFs; too large a value of c leads to severe ill-conditioning. In this paper, the selection of shape parameters of Multiquadrics (MQ) used in the Heaviside weighted MLPG meshless method has been investigated and a relationship between the parameter $c$ and the nodal distance is proposed for solving the stress analysis of two-dimensional solids.

In this paper, an adaptive algorithm using meshfree strong-form collocation methods for nonlinear partial differential equations is proposed. The meshfree method uses polynomial point interpolation method and radial point interpolation method to form shape functions, and applicable for both static and dynamic problems. The adaptivity of the scheme relies on rules of both refinement and coarsening of scattered nodes. An error estimation based on solution interpolation is used for static and time-dependent partial differential equations. The nodal refinement scheme in each adaptive step is performed using Delaunay triangulation and Voronoi diagram. The numerical examples confirm the good performance of the present adaptive meshfree collocation method.

The meshless natural neighbour method (MNNM) is a truly meshless method, which does not need the Delaunay tessellation of the whole domain to construct the Laplace interpolation. At the same time, some difficulties in other meshless methods, such as the imposition of essential boundary conditions, the treatment of material discontinuities and the choice of weight functions are avoided. The governing equations of elasto-plastic for MNNM are obtained to apply the MNNM to the analysis of two-dimensional elasto-plastic problems. The numerical results indicate that the theory and programmes are accurate and effective.

A meshless local Petrov–Galerkin method (MLPG) is presented for solving the elasto-plasticity problem in the paper. It is a truly meshless method using the moving least square (MLS) approximation as a trial function and the MLS weighted function as a test function in the weighted residual method. The incremental tangent stiffness method is applied in computation. Numerical examples show that the local Petrov–Galerkin method is applicable and effective for solving the elasto-plasticity problem.

The Allman’s rotational dofs are often used to develop advanced elements in modeling of shells as an assembly of flat elements in FEM. Yet the technique does not extend to the area of meshfree methods. In this paper we first present an extremely simple expression of the Allman’s rotational dofs of 1984 and then develop a meshfree approximation containing the rotational dofs.

The present study examines two representative viscous diffusion models in combination with the vortex method. It is shown that the MPS Laplacian model is superior to the conventional core spreading model in terms of calculating the decay rate of enstrophy and energy spectra. Computational time is remarkably reduced by using fast multi-pole method while retaining accuracy.