Meshless Methods

In this paper, a computational technique is presented based on the natural element method (NEM) for large plastic deformation simulation of the metal forming problems. NEM is a numerical technique in the field of computational mechanics and can be considered as a meshless method. The selected process is backward extrusion for circular shape hollow components from round billets. The punch stroke value is divided into sub-steps, whereas a new set of nodes become active at the end of each sub-step of deformation. Solutions are obtained for different area reductions under friction condition. Hollman-Ludwik law is selected to explain the material behavior after the yielding point. The experiments are carried out with fully annealed commercial aluminum alloy billets at room temperature, using various punch sizes. A set of die and punches are designed and constructed for experimental works. The validity of the proposed method is verified by comparing the results from deformed geometry, contours of equivalent strain and forming loads with those obtained from finite element simulation and experimental measurements. It is concluded that the results obtained by NEM are in good agreement with those from FEM and experiments and therefore, the meshless natural element method is capable of handling large plastic deformation.

The Rajendran-Grove (RG) ceramic damage model is a three-dimensional internal variable based constitutive model for ceramic materials, with the considerations of micro-crack extension and void collapse. In the present paper, the RG ceramic model is implemented into the newly developed computational framework based on the Meshless Local Petrov-Galerkin (MLPG) method, for solving high-speed impact and penetration problems. The ability of the RG model to describe the internal damage evolution and the effective material response is investigated. Several numerical examples are presented, including the rod-on-rod impact, plate-on-plate impact, and ballistic penetration. The computational results are compared with available experiments, as well as those obtained by the popular finite element code (Dyna3D).

A short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost.

The Generalized finite difference method (GFDM) is a meshfree method that can be applied for solving problems defined over irregular clouds of points. The GFDM uses the Taylor series development and the moving least squares approximation to obtain explicit formulae for the partial derivatives. In this paper, this meshfree method is used for solving elliptic and parabolic partial differential equations in 3-D. The influence of the main parameters involved in the approximation and the treatment of the Neumann boundary condition are shown. Parabolic equations have been solved using an explicit method and the criterion for stability has been improved taking into account the irregularity of the cloud of points. The numerical results show the high accuracy obtained.

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points. In this paper the extension of the GFD to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one, two or three space dimensions. The convergence of the method has been studied and the truncation errors over irregular grids are given. Different examples have been solved using the explicit finite difference formulae and the criterion of stability. This has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one, two or three space dimensions. The numerical results show the accuracy obtained over irregular grids. This paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time-dependent equations.

In this paper it is possible to appreciate the great eciency of the generalized ®nite dierence method (GFD), that is to say with an irregular arrangements of nodes, to solve second-order partial dierential equations which represent the behaviour of many physical processes. The method solves any type of second-order dierential equation, in any type of domain and boundary condition (Dirichlet, Neumann and mixed), and immediately obtains the values of derivatives of the nodes through the application of the formulae in dierences obtained. This paper analyses the in¯uences of key parameters of the method, such as the number of nodes of the star, the arrangement of the same, the weight function and the stability parameter in time-dependent problems. This analysis includes solutions obtained for dierent types of problems, represented by dierent dierential equations, including time-dependent equations and under dierent boundary conditions.

The paper reviews recent advances and ongoing technologies on which today's industry work upon in the metal forming processes. This paper has also included some revolutionary researches which later became the base for many processes. Metal forming is advancing field as we can observe different, innovative and easy methods are been introduced for sheet metal process also heavy research been done in forging and extrusion process for its optimization. With growing technologies, we adapt with advance processes. Simulation is one of the way through which the mimic of the process could be done which further helps in analysis of defects, loading profile, stress-strain curve. The analysis is done via Finite Element Analysis or Finite Volume Method in which the mesh grid over the desired surface is made and then simulated. The more refined the grid the better the results. The simulation is provided by the software in which we must give specifications of whatever we process or material we need to simulate. This is the great tool for observing and get a predictable idea of process and thereby obtain desirable result by altering the specification. Also, the integration of CAD/CAM helps further processes to regulate with software and has also been covered in the paper.

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

In this paper the extrusion process of a cross-shaped profile was investigated. In particular, the study was focused on the distortion of extruding profiles when the workpiece and die axis are not aligned. The process was simulated using the finite element method (FEM) and the natural element method (NEM), both implemented in an updated-Lagrangian formulation, in order to avoid the burden associated with the description of free surfaces in ALE or Eulerian formulations. Furthermore, an experimental equipment was developed in order to obtain reliable data in terms of deformed entity, required process load and calculated pressure. At the end, a comparison between the numerical predictions and the experimentally measured data was carried out. The main results are presented

In this paper the extrusion process of a cross-shaped profile was investigated. In particular, the study was focused on the distortion of extruding profiles when the workpiece and die axis are not aligned. The process was simulated using the finite element method (FEM) and the natural element method (NEM), both implemented in an updated-Lagrangian formulation, in order to avoid the burden associated with the description of free surfaces in ALE or Eulerian formulations. Furthermore, an experimental equipment was developed in order to obtain reliable data in terms of deformed entity, required process load and calculated pressure. At the end, a comparison between the numerical predictions and the experimentally measured data was carried out. The main results are presented

Porthole die extrusion is a process typology that can give great advantages in the forming processes. Due to the complexity of the die assembly, experimental analyses are often carried out in order to investigate the parameter influence on the quality of the final parts. Finite Element Analyses, however, have been often used for the cost reducing and for a better local investigation of variables like pressure and effective stress inside the welding chamber. In spite of that, up to now, commercial FE codes present a "structural" limit during the welding phase due to the impossibility to simulate element joining when material reaches the required process conditions. From this point of view, the Natural Element Method (NEM) provides significant advantages; in fact, the meshless characteristic of NEM is "natively" able to simulate joining of free surfaces, as it occurs during porthole die extrusion, simulating the welding line formation inside the welding chamber. In this paper, using experimental tests recognizable in literature, the authors tried to validate the effectiveness of this technique; moreover, even a comparison between NEM and FEM results was carried out. More in detail, different geometries of the welding chamber were analyzed; in some cases, the process conditions were suitable to guarantee material welding while, in other cases, the material came out from the porthole die without joint formation. The variable that was used to verify the process goodness is the maximum pressure inside the welding chamber. Furthermore, to evaluate the effectiveness of 2D analyses, even in a complex shape, a significant section was extrapolated for each die, performing a NEM vs. FEM assessment of the results. A good comparison was obtained between the two different methods that, moreover, were in agreement with the experimental tests.

In this paper, the Natural Element Method (NEM) together with the alpha shapes and some extra numerical procedures are used in the simulation of hollow profiles, emphasizing on the simulation of the welding lines. Numerical results are compared with experimental ones, checking the accuracy of the method.

Porthole die extrusion is a process typology that can give great advantages in the forming processes. Due to the complexity of the die assembly, experimental analyses are often carried out in order to investigate the parameter influence on the quality of the final parts. Finite Element Analyses, however, have been often used for the cost reducing and for a better local investigation of variables like pressure and effective stress inside the welding chamber. In spite of that, up to now, commercial FE codes present a "structural" limit during the welding phase due to the impossibility to simulate element joining when material reaches the required process conditions. From this point of view, the Natural Element Method (NEM) provides significant advantages; in fact, the meshless characteristic of NEM is "natively" able to simulate joining of free surfaces, as it occurs during porthole die extrusion, simulating the welding line formation inside the welding chamber. In this paper, using experimental tests recognizable in literature, the authors tried to validate the effectiveness of this technique; moreover, even a comparison between NEM and FEM results was carried out. More in detail, different geometries of the welding chamber were analyzed; in some cases, the process conditions were suitable to guarantee material welding while, in other cases, the material came out from the porthole die without joint formation. The variable that was used to verify the process goodness is the maximum pressure inside the welding chamber. Furthermore, to evaluate the effectiveness of 2D analyses, even in a complex shape, a significant section was extrapolated for each die, performing a NEM vs. FEM assessment of the results. A good comparison was obtained between the two different methods that, moreover, were in agreement with the experimental tests.

In this paper the extrusion process of a cross-shaped profile was investigated. In particular, the study was focused on the distortion of extruding profiles when the workpiece and die axis are not aligned. The process was simulated using the finite element method (FEM) and the natural element method (NEM), both implemented in an updated-Lagrangian formulation, in order to avoid the burden associated with the description of free surfaces in ALE or Eulerian formulations. Furthermore, an experimental equipment was developed in order to obtain reliable data in terms of deformed entity, required process load and calculated pressure. At the end, a comparison between the numerical predictions and the experimentally measured data was carried out. The main results are presented

The results presented here constitute a brief summary of an on-going multi-year effort to investigate hierarchical/wavelet bases for solving PDE's and establish a rigorous foundation for these methods. A new, hierarchical, wavelet-Galerkin solution strategy based upon the Donovan-Geronimo-Hardin-Massopust (DGHM) compactly-supported multi-wavelet is presented for elliptic partial differential equations. This multi-scale wavelet-Galerkin method uses a wavelet transform to yield nearly mesh independent condition numbers for elliptic problems as opposed to the multi-scaling functions that yield condition numbers which increase as the square of the mesh size. In addition, the results of von Neumann analyses for the DGHM multi-wavelet element and the Reproducing Kernel Particle Method (RKPM) are presented for model hyperbolic partial differential equations. RKPM exhibits excellent dispersion characteristics using a consistent mass matrix with the proper choice of refinement parameter and mass matrix formulation. In comparison, the wavelet-Galerkin formulation using the DGHM element delivers a frequency response comparable to a Bubnov-Galerkin formulation with a quadratic element. and on the DGHM multi-wavelet element[2]. . This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recornmend;rtion, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Introduction Wavelet bases promise the capability to compute multi-scale solutions to partial differential equations with potentially higher convergence rates than conventional finite difference and finite element methods, and their built-in adaptive nature offers the possibility of "automatic" adaptivity. Similarly, the Reproducing Kernel Particle Method (RKPM) has many attractive properties that make it ideal for treating a broad class of physical problems. For example, RKPM may be implemented in a "mesh-full" or a "mesh-free" manner and provides the ability to selectively tune the method, via the selection of the refinement parameter and window function, in order to achieve the requisite numerical performance. RKPM also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi-scale problems. Despite the promise of wavelet bases and RKPM, the application of new numerical methods to physical proble...

A meshless Radial Point Interpolation Collocation Method (RPICM) is applied for the modeling of heterogeneous structures consisting of homogeneous materials. Two homogeneous isotropic materials with different material properties are considered. The solution for the entire heterogeneous structure is obtained by enforcing appropriate displacement continuity and traction reciprocity conditions along the interface of homogeneous subdomains. For the approximation of the unknown field variables, the Radial Point Interpolation Method with Polynomial Reproduction (RPIM) is utilized. In the RPIM Gaussian radial basis function with one dimensionless shape parameter c  is applied. The dependance of the accuracy and numerical efficiency on the choice of shape parameter c  is tested. Both standard fully displacement (primal) and mixed approaches in RPICM method are considered and their efficiency is demostrated on a representative numerical example.

A new mixed meshless approach using the interpolation of both stress and displacement has been proposed for the analysis of plate deformation responses. A kinematic of three dimensional solid is adopted and discretization is performed by the nodes located on the upper and lower plate surfaces. The governing equations are derived by employing the local Petrov-Galerkin approach. The approximation of all unknown field variables is carried out by using the B-spline interpolation and the Moving Least Squares functions in the in-plane directions, while linear polynomials are applied in the transversal direction. In order to eliminate the thickness locking effect, the hierarchical quadratic interpolation of the transversal displacement component through the thickness is used. The shear locking effect is efficiently suppressed by the interpolation of the stress field independently from the displacement. The numerical efficiency of the derived algorithm is demonstrated by the numerical examp...

A mixed MLPG collocation method is applied for the modeling of material discontinuity in heterogeneous materials composing of homogeneous domains. Two homogeneous isotropic materials with different linear elastic properties are considered. The solution for the entire domain is obtained by enforcing the corresponding continuity conditions along the interface of homogeneous materials. For the approximation of the unknown field variables MLS functions with interpolatory conditions are applied. The accuracy and numerical efficiency of the mixed approach is compared with a standard primal meshless approach and demonstrated by a representative numerical example.

In this contribution, meshfree methods are applied for the modeling of gradient elasticity and hyperelasticity using higher-order theories based on only one microstructural parameter [1]. Both the Mixed Meshless Local Petrov-Galerkin (MLPG) approach [2] and the Optimal Transportation Method (OTM) [3] are considered. Firstly, the MLPG collocation method based on the staggered solution strategy [4] is utilized for structures undergoing small strain. Thereafter, the OTM [3] based on the weak-form of gradient continuum for finite strain is considered. Both methods are used for the modeling of homogeneous and heterogeneous structures. In the mixed collocation approach the fourth-order equilibrium equations are solved as an uncoupled sequence of two sets of second-order differential equations, while in the OTM method the weak-form of the original fourth-order equilibrium equations is employed. The independent variables in both approaches are approximated using meshless interpolating funct...

The present study is related to the utilization of the mixed Meshless Local Petrov-Galerkin (MLPG) methods for solving problems in gradient elasticity, which are governed by fourth-order differential equations. Here, three different numerical MLPG methods are presented, where the continuity requirements for the approximation functions are lowered by applying different mixed procedures to improve the numerical accuracy and efficiency. The first one is based on the direct solution of the problem, where the primary variable (displacement) and its independently chosen higher-order variables are approximated separately. The global discretized system of equations consists of appropriate equilibrium and compatibility equations written for each node and the solution vector contains all unknown independent nodal variables. Such approach demands only the first-order continuity of meshless approximation functions. The second and third procedures are both based on the displacement-based operator-split approach, where the original gradient elasticity problem is solved as two uncoupled problems governed by the second-order differential equations. Herein, in both uncoupled problems only primary variable (displacement) and its first derivative (strain) are approximated independently. In these procedures the original problem is solved by a staggered approach, where the solution of the first uncoupled equation is utilized as an input in the second equation. The main difference in the second and third procedure is that the one is based on the solution of the local weak forms of the governing equations, while the other is based on solution of the strong forms of the same equations. The accuracy of the presented computational methods is compared to analytical solutions and demonstrated on a one-dimensional benchmark problem of axial bar in gradient elasticity.

Discontinuous coefficients in the Poisson equation lead to the weak discontinuity in the solution, e.g. the gradient in the field quantity exhibits a rapid change across an interface. In the real world, discontinuities are frequently found (cracks, material interfaces, voids, phase-change phenomena) and their mathematical model can be represented by Poisson type equation. In this study, the extended finite element method (XFEM) is used to solve the formulated discontinuous problem. The XFEM solution introduce the discontinuity through nodal enrichment function, and controls it by additional degrees of freedom. This allows one to make the finite element mesh independent of discontinuity location. The quality of the solution depends mainly on the assumed enrichment basis functions. In the paper, a new set of enrichments are proposed in the solution of the Poisson equation with discontinuous coefficients. The global and local error estimates are used in order to assess the quality of t...

In a liquid, an ultrasonic field can carry along small bubbles or can produce cavitation bubbles, whose movements determine drastic effects as: erosion, unpassivation and emulsification, chemical reactions, sonoluminescence, pressure variation, that have as a effect oscillations whose frequencies differ from that of the incident ultrasound wave. We found out that the frequency of these electrical signals, generated by the cavitation bubbles, at their exterior, corresponds to the frequency of the mechanical waves, generated by the collapse of the cavitation bubbles. We present the mathematical models for the voltage induced by the cavitation bubbles in diesel.

By using the trigonometric uniform splines of order 3 with a real tension factor, a numericalmethod is developed for solving a linear second order boundary value problems (2VBP) withDirichlet, Neumann and Cauchy types boundary conditions. The moment at the knots isapproximated by central finite-difference method. The order of convergence of the methodand the theory is illustrated by solving test examples. Experimental results demonstrate thatour method is more effective for the problems where the exact solution is trigonometric orhyperbolic.

In the present paper, we use efficient and simple algorithms of the fractional power series and Adomain polynomial methods that provide effective tools for solving such linear and nonlinear fractional differential equations in the sense of conformable derivative. The first method succeeded in finding the fractional power series solution for the Laguerre linear fractional differential equation with some important special cases. The second method is applied to predict and construct the fractional Adomain approximation solution of the general conformable Lane-Emden fractional differential equation with some interesting linear and nonlinear special cases with their graphs. Finally, the results show that the Adomain approximation solutions of those problems are stable at different values of a and assure that this way can be applied successfully for solving many linear and nonlinear fractional differential equations in conformable sense.

In this paper we consider the heat equation with memory in a bounded region Ω ⊂ R d , d ≥ 1, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class C 1 . We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of Ω. We prove approximate controllability of the system and, in contrast with this, we prove the existence of initial conditions which cannot be steered to hit the target 0 in a certain time T , of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the heat equation.

Radial basis function (RBF) networks are the new, recently developed, meshless explicit, piecewise geometry description methods. Among many useful properties the RBFs have, they belong to Reproducing Kernel Hilbert Spaces and have the best approximating property, and they therefore might be suitable for describing complex ship geometry. Moreover, they are the solution of scattered data interpolation problem and can achieve high accuracy. Various types of radial basis functions and their parameters will be investigated and their applicability in 2D ship geometry description studied in this paper.

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

A novel node collocation approach for the application of radial basis function meshless methods in neutron diffusion equations is presented in this paper. By introducing ghost nodes, the number and position of external nodes can be flexibly controlled to investigate their impact on the ill-conditioning of the coefficient matrix and the solution accuracy. The steady-state single-group neutron diffusion equation is solved using the improved method, and the discretization method and solution process are provided. The boundary scale factor, which measures the distance of external nodes from the computational domain, shows a negative correlation with the error once it stabilizes. An optimal boundary scale factor exists regarding its influence on the matrix condition number; as the shape factor of the radial basis function increases, the optimal value also increases. An increase in the number of layers of external nodes slightly enhances solution accuracy but significantly raises the condition number of the matrix. For a shape factor of 0.01, the computational error decreases by 4.01 % when the number of layers is increased from 1 to 4. When the outer number factor is small, fluctuations in the solution error are observed, with stability achieved once it exceeds 0.5. Although increasing the outer number factor raises the matrix condition number, its impact is less pronounced than that of increasing the number of layers.

Regarding "Taylor Series Based Domain Collocation Meshless Method for Problems with Multiple Boundary Conditions including Point Boundary Conditions" This paper deals with handling boundary conditions for the so-called Taylor Meshfree Method (TMM, not to be confounded with Transport Meshfree method, a kernel particle method) and proposes a method to enhance TMM with handling particular multiple boundary conditions. This paper then illustrates numerically the proposed method. It discusses technical choices to tune the model to Helmholtz and Poisson equations on some domains (rectangles, circles, and an

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method for accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. A Moving Least Square (MLS) method is used to construct the shape function making the approach a fully least squares based approach. An error estimate and adaptive refinement strategy is proposed in this paper to increase the efficiency of the DLSM method. For this, a residual based error estimator is introduced and used to discover the region of higher errors. The proposed error estimator has the advantages of being available at the end of each analysis contributing to the efficiency of the proposed method. An enrichment method is then used by adding more nodes to the area of higher errors as indicated by the error estimator. A Voronoi diagram is used to locate the position of the nodes to be added to the current nodal configuration. Efficiency and effectiveness of the proposed procedure is examined by adaptively solving two benchmark problems. The results show the ability of the proposed strategy for accurate simulation of elasticity problems.

A numerical inverse Laplace transform method is established using Bernoulli polynomials operational matrix of integration. The efficiency of the method is demonstrated through some standard nonlinear differential equations: Duffing equation, Van der Pol equation, Blasius equation and jerk equation. The solution approach is to adopt Laplace Adomian decomposition method for solving nonlinear differential equations and then at each step employ the numerical inverse Laplace transform using the developed method based on Bernoulli polynomials operational matrix of integration. The numerical results exemplify that the estimated solutions are in good agreement with exact or numerical methods available in literature wherever the exact solutions are not known.

This paper describes a new topology optimization (TO) technique based on meshless method to evolve two-dimensional truss structures. The meshless method has been considered as a very attractive computational technique since it does not need any mesh generation process during the analysis. It has been gradually and widely used in many engineering disciplines and a particular weak point such as the re-generation of mesh information inherit in other numerical analysis techniques have been naturally solved. However, there have been a few applications of meshless method into structural design optimization so that we here try to apply the meshless method into structural topology optimization problem. We adopt the radial point interpolation method (RPIM) which uses radial basis function (RBF) since it is stable and robust for arbitrary nodal distributions. Then, the hard kill method based on fully stressed design scheme is consistently combined with the adopted meshless method. In order to demonstrate the accuracy of the proposed topology optimization technique, several benchmark tests are tackled to investigate the accuracy and capability of the present TO technique. From numerical results, it is found to be that the proposed TO technique is very simple and easy to produce the optimum topologies of plane structures.

This study proposes a structual topology optimization technique using meshless method. We adopt the radial point interpolation method (RPIM) which uses the radial basis function (RBF). So far, there is a few application of new meshless method into the structural design optimization. Therefore, we here apply the meshless method into structural topology optimization problem. The hard kill method based on fully stressed design technique is consistently combined with RPIM meshless method. In order to demonstrate the accuracy of the proposed topology optimization technique, a benchmark test is tackled to investigate the performance of new topology optimization process.

A meshless method based on the radial point interpolation method(RPIM) is used to analyze cantilever beam. Meshless methods have been considered is a very attractive as new computational method since it does not need mesh generation in analysis procedure. Especially, RPIM, using radial basis function(RBF), has an advantage that do not consider the essential boundary condition. In order to demonstrate the accuracy of the adopted meshless method, numerical examples are tackled with multiple elastic moduli and the different number of background cell. Numerical results are compared with the reference solution produced by finite element analysis program FEAP.

En este trabajo se presenta una variacion del Metodo de Volumenes Finitos (FVM) para la solucion de problemas de valores en la frontera, consistente en la implementacion de un esquema de interpolacion Hermitica utilizando Funciones de Base Radial (RBFs), en lo que se conoce como el metodo CV-RBF (Control Volume- Radial Basis Function). El esquema de interpolacion es una aplicacion local del Metodo Simetrico utilizado en los metodos Sin Malla, en el cual se realiza una aproximacion de la funcion empleando informacion de la ecuacion diferencial parcial (EDP) gobernante y las condiciones de frontera que definen el problema global. Se utiliza la RBF multicuadrica con un parametro de forma hallado experimentalmente. De acuerdo a la naturaleza sin malla del esquema de interpolacion empleado, se llega a una estrategia de discretizacion espacial sumamente versatil en cuanto a geometrias, mallas y condiciones de frontera. El esquema CV-RBF es acoplado con el metodo de Newton-Rapshon y el esqu...

A family of cell-centered genuinely multidimensional upwind schemes for structured meshes is developed. Two di erent approaches for the numerical ux formulation are applied, based on respectively characteristic variable extrapolation and ux extrapolation. The latter is developed within the context of the uni ed central/dissipation formulation of classical central and upwind schemes. The numerical ux is based on a 4-wave decomposition model of the 2D Euler equations using two a rbitrary characteristic directions. A g e neral theoretical analysis of linear convection schemes is the framework for t h e development of rst-and second-order multidimensional upwind schemes. A monotone second-order zero cross-di usion algorithm is developed introducing classical limiters with multidimensional ratios assuring the monotonicity property. Several combinations of algorithms with di erent characteristic directions are tested using a multigrid solver. Results near discontinuities are showing a sharper resolution than grid aligned methods. A signi cant improvement is obtained with the central/dissipation approach concerning robustness and exibility f o r implementation of multidimensional methods in standard codes.

In this article, the transient dynamic analysis of decagonal quasicrystal (QC) is carried out using the meshless generalized finite difference (GFD) method. The transient behaviors of phonon and phason displacements in these types of QCs, when they are subjected to mechanical shock loading are studied. Also, the meshless GFD method is developed to solve the dynamic problems of QCs, considering coupling between phonon and phason displacements. The governing equations of the problem are obtained in the coupled system of PDEs by applying the coupling coefficient in the governing equations to simulate the interaction between phonon and phason displacements. A 2D rectangular domain made from decagonal (Al-Ni-Co) QCs is assumed for the problem to show the transient behaviors of phonon and phason displacements, when one side of 2D domain is excited by phonon displacement shock loading. Based on the coupling between phonon and phason displacements, any disturbance in phonon field causes some variations in phason field. To study the dynamic behaviors of phonon and phason fields, the coupled governing equations are transferred to Laplace domain. After analysis the problem by GFD method, the obtained results in Laplace domain are transferred to time domain using the Laplace inversion technique. The interactions between phonon and phason fields are studied in details. Also, the effects of coupling parameter and the phason friction coefficient on the dynamic and transient behaviors of phonon and phason displacement fields are obtained and discussed in details.

The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, twodimensional (2D) incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-Cartesian node arrangements and large sizes, i.e., in the order of 10 5 nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favorably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains.

In this paper, the symmetric smoothed particle hydrodynamics (SSPH) as a meshless method is explained in detail. Free vibration analysis of bilayer graphenes with interlayer shear effect is modeled. The bilayer graphene is modeled as a sandwich beam with free-clamp end condition. To obtain the governing equations, each graphene layer is modeled based on the Euler-Bernoulli theory and in-plane displacements are also considered in addition to the transverse displacement. It is also assumed that the graphene layers do not have relative displacement during vibration. The results obtained by the sandwich beam model solved by SSPH method, include the first two natural frequencies of the bilayer graphenes. These results are validated by the molecular dynamic and compared with the GDQM results reported in the literature.

A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.

We employ a nonlinear anisotropic di usion operator like the ones used as a means of ltering and edge enhancement in image processing, in numerical methods for conservation laws. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular, we stabilize the well-known Lax-Wendro formula by means of a nonlinear di usion term.