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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 Péclet numbers. It is reported that for low Péclet numbers, MQ-RBF provides excellent agreement, while for high Péclet numbers, TPS-RBF is better than MQ-RBF.

This paper is an overview of the main ideas of the Generalized Finite Element Method (GFEM). We present the basic results, experiences with, and potentials of this method. GFEM is a generalization of the classical Finite Element Method — in its h, p, and h-p versions — as well as of the various forms of meshless methods used in engineering.

In this paper, we propose a simple and direct numerical procedure to obtain particular solutions for various types of differential equations. This procedure employs the power series expansion of a differential operator. Chebyshev polynomials are selected as basis functions for the approximation of the inhomogeneous terms of the given partial differential equation. This numerical scheme provides a highly efficient and accurate approximation for the evaluation of a particular solution for a variety of classes of partial differential equations. To demonstrate the effectiveness of the proposed scheme, we couple the method of fundamental solutions to solve a modified Helmholtz equation with irregular boundary configuration. The solutions were observed to have high accuracy.

A simplified meshless methods for brittle fracture and nonlinear material is presented. In this method, the crack is modeled by a set of discrete crack segments crossing the entire domain of influence of the meshless shape functions. The key advantage of this method is its simplicity since no representation of the crack topology is needed. A nonlocal stress tensor around the crack tip is used as fracture criterion. A neo-Hooke material in the bulk material is used and a cohesive zone model is employed once discrete cracks occur. We also present consistent linearization of the cohesive zone model. The method is applied to fracture modeling in concrete that is accompanied by excessive cracking and therefore methods that represent the crack path have major drawbacks. We demonstrate the accuracy of the proposed method for complex problems involving mode-I and mixed mode failure.

Meshless methods are widely investigated and successfully implemented in many applications, including mechanics, fluid-dynamics, and thermo-dynamics. Within this context, this paper introduces a novel particle approach for elasticity, namely the modified finite particle method (MFPM), derived from existing projection particle formulations, however presenting second-order convergence rates when used to solve elastic boundary value problems. The formulation is discussed and some applications to bi-dimensional elastic and elasto-plastic problems are presented. The obtained numerical results confirm the accuracy of the method, both in elasticity and in plasticity applications.

In this work, the natural neighbor radial point interpolation method (NNRPIM) is extended to the numeric analysis of crack propagation problems. Here, the advanced discretization meshless technique is combined with a linear elastic crack growth algorithm. The algorithm simulates the crack propagation by displacing iteratively the crack tip, which consequently requires a local remeshing. In each iteration, it is estimated the stress state in the crack tip and afterwards the direction of the crack propagation is obtained considering the maximum circumferential stress criterion.

The required local remeshing does not represent a numeric difficulty for the NNRPIM. The main advantage of the NNRPIM is its capability to fully discretize the problem domain using only an unstructured nodal distribution. Being a truly meshless method, the NNRPIM is able to define autonomously the nodal connectivity and the background integration mesh.

The classic NNRPIM formulation permits to enforce the nodal connectivity by means of two kind of influence-cells: first degree influence-cells or second degree influence-cells. This work investigates the influence of the nodal connectivity on the simulated crack propagation path. Thus, demanding benchmark crack propagation examples are studied and the obtained results are compared with reference solutions available in the literature.

To reduce the error on the boundary and improve computational accuracy, the normal derivative of radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the Hermit-type reproducing kernel particle method (Hermit-type RKPM) is proposed. The Hermit-type RKPM approximation function is constructed and the governing equation of the elasticity problems is deduced. Then the Hermit-type RKPM is applied to the numerical simulation of the elasticity problems, and the results illustrate that the proposed method is more accurate than the RKPM.

Partial differential equations of hyperbolic type when considered with mixed Dirichlet/Neumann constraints as well as nonlocal conservation conditions model many physical phenomena. The prime motivation of the current work is to apply the recently developed meshfree method to such differential equations. The scheme is built on series expansion of the solution via proper base functions akin to the Galerkin approach. In many cases, the simple polynomials are adequate to convert the hyperbolic partial differential equation and boundary conditions of nonlocal kind into easily treatable algebraic equations concerning the coefficients of the series. If the sought solutions are polynomials of any degree, then the method has the ability of resolving the equations in an exact manner. The validity, applicability, accuracy and performance of the method are illustrated on some well-analyzed hyperbolic equations available in the open literature.

This paper presents a machine learning (ML) surrogate modeling for fast processing in meshless/ meshfree methods. The main idea is to leverage the universal approximation (UA) propriety of supervised ML models (shallow/ deep learning and other regression models) to surrogate the heavy shape function construction in meshless methods. The resulting ML metamodel preserves the same accuracy of the meshless interpolation while avoiding costly matrix inversion operations. The total computation time for solving 3D test simulation problems (using more than 20k nodes) is reduced by a factor of 1k in the case of the Gaussian process (GP) metamodel.

Interpolating meshless methods can directly impose boundary conditions because of the interpolation property which shows advantages in dealing with problems with boundary conditions. The interpolating element-free Galerkin method (IEFGM), the improved interpolating element-free Galerkin method (IIEFGM), and the radial point interpolation method (RPIM) are applied in this paper to solve the two-dimensional and three-dimensional elastic problems. IEFGM and IIEFGM are two different ways to change the status that the traditional element-free Galerkin method (EFG) does not have the interpolation property. IEFGM uses an improved interpolating moving least-squares (IMLS) method that employed singular weight functions while IIEFGM takes the improved interpolating moving least-squares method based on non-singular weight function. RPIM, one of the most widely used interpolating meshless methods, is compared with IEFGM and IIEFGM in this paper. The numerical results of two-dimensional and three-dimensional elastic problems show that the three types of interpolating meshless methods obtain high precision displacement solutions and stress solutions.

This paper aims to extend the meshless smoothed point interpolation methods (SPIMs) to the analysis of the Timoshenko beam problem. These methods are based on the concepts of smoothing domains and weakened-weak (W²) form; their use is made possible by the extension of the weakened-weak form that they are based on to the case of the Timoshenko beam. The provided numerical simulations emphasize that, by changing the number of nodes used to build the shape functions at each interest point, it is possible to obtain a lower-bound or an upper-bound approximation to the analytical solution of the beam problem. This property is further exploited with the concept of $\alpha$PIM shape function that, blending together the different bounds to the analytical solution, allows to improve the convergence. The proposed formulation is naturally locking-free, i.e., no additional treatment is necessary to avoid the spurious stiffer behavior that commonly occurs in FEM simulations of shear-deformable beams.

Adhesive joints are an increasingly important joining method, in part due to usually being lighter than the alternatives, which is very important in the search for more energy efficient transportation. However, the amount of studies focused on the free-vibration behavior of adhesive joints is currently very limited. Since this knowledge is important, to ensure that a joint is working outside its natural frequencies, this work sets out to perform a parametric study of a Double Lap Joint (DLJ) using the Radial Point Interpolation Method (RPIM), a meshless method. Using the RPIM in the free vibration of adhesive joints is the next step for this numerical method, after using it in the static analysis of adhesive joints. Considering that this is one of the first uses of this method in this type of problem it is also necessary to validate it. This task was performed in this work by comparing it with the Finite Element Method (FEM), which is the standard numerical method for this type of problem. The validation was successful, showing very small differences between the two numerical methods. The parametric study should aid future adhesive joint designers to develop safer joint. It showed that changes to the adhesive produce diminutive changes to the natural frequencies of the joints. On the other hand, changes to the adherents change the natural frequencies significantly. The overlap length had different effects on the different modes.

The meshless local Petrov–Galerkin (MLPG) and the local boundary integral equation (LBIE) methods has been introduced approximately three decades ago. These methods are based on writing the local weak form of the governing equation and performing subsequent numerical integration and interpolations in the local subdomains. A key step is the choice of the test function in the local weak form, which has historically led to several different formulations regarding the final form of the local integrals. Considering the application of the methods to acoustics, four different test functions have been employed so far in the literature; all of these approaches resulted in formulations which contain domain integrals. In this paper, we present a new test function to be used in meshless methods, which yields a simple form of the local integral equation without domain integrals and provides significant improvement in terms of the computational time and CPU requirements. The efficiency and the accuracy of the new method are presented and compared with the previous methods.

While finite elements has been considered during the last decades as the universal tool to perform simulations in biomechanics, a recently developed wide family of methods, globally coined as meshless methods, has emerged as an attractive choice for an increasing variety of engineering problems. They present some key advantages such as the absence of a mesh in the traditional sense, particularly important in domains of very complex geometry, a less sensitivity to the nodal distribution and therefore to the implicit mesh distorsion what is especially interesting to handle problems under finite strains and large displacements in a Lagrangian framework. Here, we analyze the convenience and possible advantages of using meshless methods in numerical simulations of soft biological tissues. Biological tissues are usually nonlinear, anisotropic, inhomogeneous, viscoelastic, and undergo large deformations, so these methods seem to be an appealing possibility for this type of applications. In particular, we discuss the use of one of these methods, the so-called natural element method that has specific and important features as interpolatory character, easy handling of geometry, and essential boundary conditions via the so-called alpha-NEM extension, well-defined mathematical properties and a simple computer implementation. Different examples are solved using this approach including the human cornea, the temporo-mandibular joint, knee ligaments, and the passive behavior of the heart.