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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. [ABSTRACT FROM AUTHOR]

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples. [ABSTRACT FROM AUTHOR]

In this paper truncated hierarchical B-spline (THB-spline) is coupled with reproducing kernel particle method (RKPM) to blend advantages of the isogeometric analysis and meshfree methods. Since under certain conditions, the isogeometric B-spline and NURBS basis functions are exactly represented by reproducing kernel meshfree shape functions, recursive process of producing isogeometric bases can be omitted. More importantly, a seamless link between meshfree methods and isogeometric analysis can be easily defined which provide an authentic meshfree approach to refine the model locally in isogeometric analysis. This procedure can be accomplished using truncated hierarchical B-splines to construct new bases and adaptively refine them. It is also shown that the THB-RKPM method can provide efficient approximation schemes for numerical simulations and represent a promising performance in adaptive refinement of partial differential equations via isogeometric analysis. The proposed approach for adaptive locally refinement is presented in detail and its effectiveness is investigated through well-known benchmark examples. [ABSTRACT FROM AUTHOR]

Meshfree methods such as the Reproducing Kernel Particle Method and the Element Free Galerkin method have proven to be excellent choices for problems involving complex geometry, evolving topology, and large deformation, owing to their ability to model the problem domain without the constraints imposed on the Finite Element Method (FEM) meshes. However, meshfree methods have an added computational cost over FEM that come from at least two sources: increased cost of shape function evaluation and the determination of adjacency or connectivity. The focus of this paper is to formally address the types of adjacency information that arises in various uses of meshfree methods; a discussion of available techniques for computing the various adjacency graphs; propose a new search algorithm and data structure; and finally compare the memory and run time performance of the methods. [ABSTRACT FROM AUTHOR]

Interaction between conduction and radiation during the phase-change process within multidimensional participating media was numerically investigated. The solutions were based on the reproducing kernel particle method (RKPM). Test cases were analyzed and compared to results from analytical solutions and other studies to examine the performance of RKPM. Comparisons indicate that the RKPM is stable and accurate. The effect of thermo-optics effect on the phase-change process was carried out with the refractive index increases/decreases linearly with the temperature. The solidification process involving radiation was studied in a 2D enclosure. The effects of different parameters, i.e., extinction coefficients, scattering albedos, conduction-radiation parameters, and latent heat, on the temperature profiles were investigated, and the results show the parameters have significant effects on the solidification process. In addition, the solidification process during convection-radiation cooling was analyzed in a cuboid enclosure with the effect of various phase transition temperature ranges. [ABSTRACT FROM AUTHOR]

The conventional numerical approximation of Mindlin plate equations can lead to erroneous solutions for thin plates. The so-called shear-locking problem has been well studied in the context of the finite element method (FEM) whereas the development of numerical formulations for its successful elimination in meshfree methods is still a subject of intensive research. This paper studies the effectiveness of some of the most commonly adopted techniques for the reduction of shear-locking and presents the application of a shear-locking-free formulation based on first-order Mindlin plate theory. In this modified formulation, the shear strains are incorporated as degrees of freedom (DOFs) in lieu of the rotational DOFs in the conventional Mindlin theory formulation. A straightforward transformation technique is presented for the enforcement of boundary conditions and comparisons are made with available analytical and numerical solutions. The generalised reproducing kernel particle method (RKPM) is adopted as the numerical tool and a series of numerical examples are presented to demonstrate the accuracy and performance of the presented method. [ABSTRACT FROM AUTHOR]

When high-order numerical approximation method is applied to model the propagation of shock wave or discontinuity, it usually creates unstable unreal numerical oscillations around the discontinuous regions. In this research, we propose a non-oscillation meshfree scheme based on reproducing kernel particle method (RKPM) which can maintain the accuracy and minimize the oscillation in the modeling of shock wave propagation. In the proposed method, the original influence domain of high-order RK approximation is divided into several subdomains. Then we apply low-order RK approximation within each subdomain. Instead of directly using the discrete particles to build the numerical approximation, we consider that the high-order approximation is constructed by the summation of those low-order approximations multiplied by a local weight function. By adjusting these local weights with the "smoothness indicator", we can determine the "effect" of the corresponding subdomain and the discrete particles inside this subdomain. Therefore, the subdomain containing discontinuity would not participate in the high-order approximation, and the numerical oscillation is automatically suppressed. The proposed method does not need artificial viscosity or numerical damping to stabilize the solution. Several benchmark problems with shock wave propagation are tested. The results show that the proposed method can maintain high-order accuracy without numerical oscillation. [ABSTRACT FROM AUTHOR]

A variety of physical problems in science may be expressed using the advection-diffusion-reaction (ADR) equation that covers heat transfer and transport of mass and chemicals into a porous or a nonporous media. In this paper, the meshless generalised reproducing kernel particle method (RKPM) is utilised to numerically solve the time-dependent ADR problem in a general n -dimensional space with variable coefficients and boundary conditions. A time-dependent Robin boundary condition is formulated and precisely enforced in a novel approach. The accuracy and robustness of the meshless solution is verified against finite element simulations and a general one-dimensional analytical solution obtained in this study. [ABSTRACT FROM AUTHOR]

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment-impact processes that commonly exist in extreme events. A new reproducing kernel formulation with 'quasi-linear' reproducing conditions is introduced. In this approach, the first-order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first-order completeness, nearly second-order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi-linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment-impact problems. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This work presents a coupling of the finite strip method (FSM) and the meshless reproducing kernel particle method (RKPM), which is inspired by the complementary features of both methods. A straightforward and effective method is developed to accurately integrate these methods, and the obtained coupled technique is utilised for analysing two-dimensional linear elastic plates weakened by arbitrarily shaped perforations or cracks. The RKPM is used in the regions surrounding the cracks, i.e . parts of the domain with high gradients and irregular boundaries, whereas the FSM is adopted in the rest of the domain. The sub-domains of these methods are conjoined over transition zones by utilising a modified collocation approach which ensures the linear consistency of the approximation field. A series of numerical examples and studies of the obtained convergence rates are presented to demonstrate the accuracy and effectiveness of the developed coupled method. [ABSTRACT FROM AUTHOR]

Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non-convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems. [ABSTRACT FROM AUTHOR]

Free vibration of current-carrying nano-scaled beams immersed in a magnetic field is of huge interest. To bridge this scientific gap, Rayleigh, Timoshenko, and higher-order beam models accounting for the surface energy are employed and their equations of motion are established appropriately. For spatial discretization of the deformations fields, a meshless approach is exploited. The effects of surface and shear deformation, electric current, magnetic field strength, and geometric parameters of the nanobeam on the first ten natural frequencies are examined and discussed. The critical values of slenderness ratio, nanobeam's diameter, electric current, and magnetic field strength, which are corresponding to the dynamically unstable nanobeams, are graphically identified. The obtained results indicate that the discrepancies between the frequencies by considering the surface effect and those evaluated without consideration of the surface energy would increase notably at the above-mentioned critical values. [ABSTRACT FROM AUTHOR]

Many methods have been developed in order to study the impact behaviour of solids and structures. Two common methods are meshfree method and the finite element method. The smoothed finite element method has become a powerful alternative to the finite element method as it promises to model large deformations and high mesh distortion without much loss of accuracy. In this manuscript, we present a comparative study of the finite element method, the smoothed finite element method and nodal stabilised reproducing kernel particle method with respect to their ability to predict impact events. The same constitutive models are applied for all computational methods. Through numerical examples, we show that meshfree methods can predict impact events most accurately but they are also computational more costly. We also show that the smoothed finite element method does not yield in an improvement compared to the finite element method. All results are compared to experimental data. [ABSTRACT FROM AUTHOR]

Convective transport terms in Eulerian conservation laws lead to numerical instability in the solution of Bubnov–Galerkin methods for these non-self-adjoint PDEs. Stabilized Petrov–Galerkin methods overcome this difficulty, however gradient terms are required to construct the test functions, which are typically expensive for meshfree methods. In this work, the implicit gradient reproducing kernel particle method is introduced which avoids explicit differentiation of test functions. Stabilization is accomplished by including gradient terms in the reproducing condition of the reproducing kernel approximation. The proposed method is computationally efficient and simplifies stabilization procedures. It is also shown that the implicit gradient resembles the diffuse derivative originally introduced in the diffuse element method in Nayroles et al. (1992), and maintains the desirable properties of the full derivative. Since careful attention must be paid to efficiency of domain integration in meshfree methods, nodal integration is examined for this class of problems, and a nodal integration method with enhanced accuracy and stability is introduced. Numerical examples are provided to show the effectiveness of the proposed method for both steady and transient problems. [ABSTRACT FROM AUTHOR]

A meshfree formulation under the reproducing kernel particle method (RKPM) was introduced for modeling the penetration and perforation of brittle geomaterials such as concrete. RKPM provides a robust framework to effectively model the projectile-target interaction and the material failure and fragmentation behaviors that are critical for this class of problems. A stabilized semi-Lagrangian formulation, in conjunction with a multiscale material damage model for brittle geomaterials and a kernel contact algorithm, were introduced for penetration modeling. In this work, the accuracy of the meshfree impact and penetration formulation was studied using a series of large-caliber projectile concrete slab perforation experiments with impact velocities in the ballistic regime. These experiments were selected due to the challenging nature of concrete perforation, and the results were used to validate the effectiveness of the proposed method to model the penetration processes and the concrete target failure. Simulation results confirm the formulation's accuracy for this type of high-rate ballistic problem and establish a basis for extension to other types of impact problems. The results show the importance of properly formulating the method of domain integration to maintain accuracy in the presence of concrete fragmentation, and also highlight the method's ability to capture the fragmentation response without a need for non-physical treatments commonly used in conventional methods. [ABSTRACT FROM AUTHOR]

The mechanical properties of nano-scaled composites reinforced by carbon nanotubes are enhanced by application of appropriate magnetic fields, however, little is known on the free dynamic response of the magnetically affected stocky single-walled carbon nanotubes (SWCNTs) with elastic supports. Using nonlocal Rayleigh, Timoshenko, and higher-order beam theory, the equations of free transverse vibration of elastically embedded SWCNTs subjected to a longitudinally varying magnetic field are obtained. Since finding an analytical solution to the equations of motion is a very difficult task, an efficient meshless method is proposed. The frequencies of the magnetically affected stocky SWCNTs are evaluated for different boundary conditions. The convergence checks of the proposed numerical models are carried out. In a special case, the obtained results are also compared with those of assumed mode method, and a reasonably good agreement is achieved. Subsequently, the roles of the slenderness ratio of the SWCNT, small-scale parameter, strength of the magnetic field, lateral and rotational interactions of the SWCNT with its surrounding medium on the fundamental frequency are addressed in some detail. The capabilities of the proposed models in capturing the frequencies of the magnetically affected nanostructure are also comprehensively investigated. [ABSTRACT FROM AUTHOR]

This study is devoted to examine load-bearing capacity of a nanosystem composed of two adjacent perpendicular single-walled carbon nanotubes (SWCNTs) which are embedded in an elastic matrix. Accounting for the nonlocality and the intertube van der Waals forces, the governing equations are established based on the nonlocal Euler–Bernoulli, Timoshenko, and higher-order beam theories. These are sets of coupled integro-ordinary differential equations whose analytical solutions are unavailable. Hence, an efficient meshless methodology is proposed and the discrete governing equations are obtained via Galerkin approach. By solving the resulting set of eigenvalue equations, the axial buckling load of the elastically embedded nanosystem is evaluated. The roles of the radius and slenderness ratio of the constitutive SWCNTs, free distance between two tubes, small-scale parameter, aspect ratio, transverse and rotational stiffness of the surrounding matrix on the axial buckling load of the nanosystem are comprehensively addressed. The obtained results can be regarded as a pivotal step for better understanding the mechanism of elastic buckling of more complex systems such as elastically embedded-orthogonal membranes or even forests of SWCNTs. [ABSTRACT FROM AUTHOR]

This study deals with transverse vibrations of two adjacent-parallel-mislocated single-walled carbon nanotubes (SWCNTs) under various end conditions. These tubes interact with each other and their surrounding medium through the intertube van der Waals (vdW) forces, and existing bonds between their atoms and those of the elastic medium. The elastic energy of such forces due to the deflections of nanotubes is appropriately modeled by defining a vdW force density function. In the previous works, vdW forces between two identical tubes were idealized by a uniform form of this function. The newly introduced function enables us to investigate the influences of both intertube free distance and longitudinal mislocation on the natural transverse frequencies of the nanosystem which consists of two dissimilar tubes. Such crucial issues have not been addressed yet, even for simply supported tubes. Using nonlocal Timoshenko and higher-order beam theories as well as Hamilton's principle, the strong form of the equations of motion is established. Seeking for an explicit solution to these integro-partial differential equations is a very problematic task. Thereby, an energy-based method in conjunction with an efficient meshfree method is proposed and the nonlocal frequencies of the elastically embedded nanosystem are determined. For simply supported nanosystems, the predicted first five frequencies of the proposed model are checked with those of assumed mode method, and a reasonably good agreement is achieved. Through various studies, the roles of the tube's length ratio, intertube free space, mislocation, small-scale effect, slenderness ratio, radius of SWCNTs, and elastic constants of the elastic matrix on the natural frequencies of the nanosystem with various end conditions are explained. The limitations of the nonlocal Timoshenko beam theory are also addressed. This work can be considered as a vital step towards better realizing of a more complex system that consists of vertically aligned SWCNTs of various lengths. [ABSTRACT FROM AUTHOR]

In this paper, the complex variable reproducing kernel particle method (CVRKPM) for the bending problem of arbitrary Kirchhoff plates is presented. The advantage of the CVRKPM is that the shape function of a two-dimensional problem is obtained one-dimensional basis function. The CVRKPM is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the CVRKPM for the bending problem of Kirchhoff plates are presented in detail. Several numerical examples of Kirchhoff plates with different geometry and loads are given to demonstrate that the CVRKPM in this paper has higher computational precision and efficiency than the reproducing kernel particle method under the same node distribution. And the influences of the basis function, weight function, scaling factor, node distribution and penalty factor on the computational precision of the CVRKPM in this paper are discussed. [ABSTRACT FROM AUTHOR]

This paper presents vibration and buckling analysis of functionally gradedbeams with different boundary conditions, using reproducing kernel particle method(RKPM). Vibration of simple Euler-Bernoullibeam using RKPM is already developed and reported in the literature. Modeling of FGM beams using theoretical method or finite element technique is not evolved with accurate results for power law form of FGM with large power of "n" value so far. Accuracy of the RKPM results is very good and is not sensitive to n value. System of equations of motion is derived using Lagrange's method under the assumption of Euler-Bernoulli beam theory. Boundary conditions of the beam are taken into account using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction according to the power-law form. RKPM is implemented to obtain the equation of motion and consequently natural frequencies and buckling loads of the FGM beam are evaluated. Results are verified for special cases reported in the literature. Considering the displacement of the neutral axis, buckling loads with respect to length and material distribution are evaluated. For the special case of homogenous beam, RKPM matches theoretical evaluation with less than one percent error. [ABSTRACT FROM AUTHOR]

This paper presents a meshfree smooth contact formulation for application to metal forming problems. The continuum-based contact formulation requires $$\text {C}^{2}$$ continuity in the approximation of contact surface geometry and displacement variables, which is difficult for the conventional $$\text {C}^{0}$$ finite elements. In this work, we introduce a reproducing kernel approximation to achieve arbitrary degree of smoothness for contact surface representation and displacement field approximation. This approach allows the employment of continuum-based contact formulation, leading to a continuous contact force vector and a consistent tangent particularly advantageous in the Newton iteration of contact analysis. The proposed meshfree smooth contact formulation has been applied to the simulation of metal forming processes and is shown to improve the convergence significantly in comparison with the finite element-based $$\text {C}^{0}$$ contact formulation. [ABSTRACT FROM AUTHOR]

Abstract: Using reproducing kernel particle method (RKPM), concentrically and eccentrically functionally graded stiffened plates (FGSPs) are analyzed based on first order shear deformation theory (FSDT). The plates are subjected to uniformly distributed loads with simply supported and clamped boundary conditions. The interactions between the plate and stiffeners are imposed by compatibility equations. Metal-ceramic composition is assumed as the functionally graded material (FGM). Material properties vary through the thickness direction according to the power law of volume fraction. Mori–Tanaka scheme is used to obtain effective material properties. Poisson’s ratios of plates and stiffeners are taken to be constant. Full transformation approach is used to enforce essential boundary conditions. Effects of eccentricity of the stiffeners, dimensionless support domain parameter, dimensionless thickness, boundary conditions and the volume fractions of the constituents on the behavior of the stiffened plates are investigated. [Copyright &y& Elsevier]

Simulation of the microphone membrane determines whether highest yield and sensitivity is attainable when it comes to the field of microelectromechanical system (MEMS) capacitive microphone design. Consequently, it is significantly critical to predict and understand the behavior of the membrane in the air. The reproducing kernel particle method and element-free Galerkin, RKPM and EFG respectively, are introduced to differentiate from the traditional finite element method (FEM) since RKPM and EFG models are meshless to greatly improve the problems of FEM with large size aspect ratio. The result from a numerical axisymmetry model of 1 mm radius and 10 μm thickness membrane with fixed boundary condition upon 1 mm thickness viscothermal air is identical to that from the theoretical model. Finally, a MEMS axisymmetry model of a 180 μm radius and 10 μm thickness membrane upon 10 μm thickness viscothermal air is simulated in this paper. [ABSTRACT FROM AUTHOR]

Abstract: Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes. [Copyright &y& Elsevier]

Numerical simulation of sheet metal forming process using the meshless method called reproducing kernel particle method (RKPM) is studied in this paper. Owing to limitations of meshless approximation under parametric coordinate system, a constrained reproducing kernel (RK) approximation under Cartesian coordinate system is employed to describe the sheet metal deformation. Several technologies are used to solve the crucial problems involved in the numerical procedure. RKPM shape function is modified to impose essential boundary conditions; stabilized conforming nodal integration method is adopted to implement numerical integration of weak formulation; and an algorithm based on moment method is used to treat contact detection during sheet metal forming process. A program based on constrained RK approximation is developed to simulate two sheet metal forming examples: square-cup deep drawing and hemispherical part drawing. Numerical results are presented to demonstrate the effectiveness of the RKPM in numerical simulation of sheet metal forming. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples. [ABSTRACT FROM AUTHOR]

Microelectromechanical systems (MEMS) have found applications in defence as well as in civilian sectors. Analysis of MEMS devices require complex 3-D meshes due to the presence of mechanical and electrostatic energy domains. On the other hand, meshless methods do not require the generation of mesh and perform the computational analysis by just sprinkling the points covering the domain. In this paper, meshless analysis of MEMS switches based on reproducing kernel particle method is reported. Numerical results for the static analysis of the switch are compared with the simulated results obtained using INTELLISUITE MEMS CAD tool. [ABSTRACT FROM AUTHOR]

Conventional finite element (FE) analysis of bulk metal forming processes often breaks down due to severe mesh distortion. In recent years, meshless methods have been considerably developed for structural applications. The main feature of these methods is that the problem domain is represented by a set of nodes, and a finite element mesh is unnecessary. This new generation of computational methods can avoid time-consuming meshing and remeshing. A meshless method based on reproducing kernel particle method (RKPM) is applied to bulk metal forming analysis. The displacement shape functions are developed from a reproducing kernel (RK) approximation that satisfies consistency conditions. The shape function is modified to impose essential boundary conditions accurately and expediently. A material kernel function that deforms with the material is introduced to assure the stability of the RKPM shape function during large deformations. A program based on RKPM is developed to simulate two examples of bulk metal forming process such as ring compression and cold upsetting, and numerical results demonstrate the performance of the meshless method in bulk metal forming analysis. [ABSTRACT FROM AUTHOR]

We develop an immersed meshfree method under a variational multiscale framework for modeling fluid–structure interactive systems involving shock waves. The proposed method enables flexible non-body-fitted discretization, approximations, and quadrature rules for solid and fluid subdomains. The interfacial compatibility conditions are imposed by a volumetric constraint, which avoids the tedious contour integral and interface tracking. The reproducing kernel particle method (RKPM) is employed for both solid and fluid sub-systems, which allows arbitrary control of the orders of continuity and approximation, as well as flexibility in discretization, making it particularly advantageous for modeling fluid–structure interaction (FSI). In the proposed approach, the fictitious fluid is combined with the foreground solid, forming an "effective solid problem" solved on a moving foreground domain, while the background fluid problem is solved with prescribed solid velocity in the overlapping domain to reduce the leaking instability and mesh sensitivity. The variational multiscale immersed method (VMIM) is employed to enhance accuracy and stability in FS coupling, which leads to a residual-based stabilization. The MUSCL-SCNI shock algorithm provides a natural way of introducing the Riemann solution in the shock algorithm via the SCNI contour integral for desirable accuracy. The employment of SCNI in the proposed framework also provides computational efficiency, accuracy, and stability. Using a larger RKPM support size in the fluid domain can effectively suppress the leaking instability. The effectiveness of the proposed methods is verified in solving several FSI problems with shock waves, and the enhanced stability and accuracy of the proposed methods compared to the classical immersed approach have also been demonstrated. • A variational multiscale immersed method (VMIM) is proposed for FSI with shocks. • The improved immersed FSI formulation avoids the leaking instability. • The VMIM results in an immersed residual-based stabilization for FSI-coupling. • The RKPM is advantageous in constructing the high-order derivatives in VMIM. • The VMIM with MUSCL-SCNI enhances stability and accuracy in shock-capturing. [ABSTRACT FROM AUTHOR]

A meshfree approach, the radial basis reproducing kernel particle method (RRKPM), is proposed in this study, which is based on the radial basis functions (RBFs) and the reproducing kernel particle method (RKPM). The presented approach eliminates the negative effects of different kernel functions on numerical accuracy, which has the advantages of greater accuracy and convergence. Furthermore, the presented approach is adopted to solve the elastoplastic problem of functionally graded materials (FGMs). Using Galerkin weak form of elastoplastic problem, the meshfree RRKPM for elastoplastic problem of FGMs is established, and the penalty method is employed to impose the essential boundary conditions, then the corresponding formulas are obtained. The effects of the scaling factor, loading steps, number of nodes and node distributions on computational results of numerical accuracy are discussed in detail, and the influences of different functional gradient indexes on displacements are studied. To validate the applicability and reliability of the presented meshfree RRKPM, several elastoplastic examples of FGMs are performed and compared to the RKPM and the finite element method (FEM) solutions. • The RRKPM for elastoplastic problem of FGMs is proposed. • Combining with the Galerkin weak form, the corresponding formulae are derived. • The different functional gradient indexes are analyzed. • The scaling factor, the loading steps and the node distributions are discussed. • The RRKPM has excellent accuracy and convergence. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the reproducing kernel particle method (RKPM) is used for finding the solution of a two-dimensional parabolic inverse problem with a source control parameter, and the corresponding discrete equations are obtained. Comparing with the numerical methods based on mesh, the reproducing kernel particle method only needs the scattered nodes instead of meshing the domain of the problem. The reproducing kernel particle method is an efficient mesh free technique for the numerical solution of partial differential equations. The result of numerical example is presented. [Copyright &y& Elsevier]

Abstract: Computational modeling is an effective approach for simulating and predicting the development and remodeling of biological systems. Here, we propose to integrate a probabilistic mathematical modeling approach into an emerging bioengineering area known as molecular regenerative engineering. This experimental–computational framework can be used to simulate and predict the efficacy of molecular engineering in controlling and enhancing the regenerative processes of pathologically disordered cells, tissues and organs. We demonstrate the significance and principles of this approach by using a special case as an example: enhancement of endothelial cell adhesion to arterial constructs by siRNA-mediated modulation of an adhesion-inhibiting factor known as Src homology 2 domain-containing protein tyrosine phosphatase (SHP)-1. A continuum approach is used to model endothelial cell adhesion at the cellular level, while the Kramers reaction-rate theory is used to model cell adhesion at the subcellular and molecular levels. We show that the proposed computational model can potentially be used to simulate the processes and alterations of endothelial cell adhesion at the molecular and cellular levels under shear flow and to predict the effectiveness of siRNA-mediated SHP-1 knockdown in enhancing endothelial cell adhesion in arterial constructs. These preliminary investigations suggest that this integrated experimental–computational approach may be used for the simulation and prediction of regenerative processes in response to molecular engineering. [Copyright &y& Elsevier]

Abstract: Mesh-free methods continue to have problems in enforcing the essential boundary conditions because of the absence of the Kronecker delta property in mesh-free shape functions, constructed from moving least squares method, reproducing kernel particle method, and others. In this paper, this problem in dealing with the essential boundary conditions is examined. Boundary locking, induced by the penalty enforcement of the essential boundary conditions in mesh-free methods, is first explored, and the reason for obtaining an over-constrained solution is explained theoretically. To justify these observations, various numerical examples are also presented. Additionally, based on further theoretical investigations, a remedy to circumvent this problem of the over-constraint phenomena, caused by the penalty enforcement of essential boundary conditions, is proposed. [Copyright &y& Elsevier]

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM. [ABSTRACT FROM AUTHOR]

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The one-dimensional and the two-dimensional numerical results that support the theory are resented. [ABSTRACT FROM AUTHOR]

In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu ( Z– Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z– Z error estimator are converging to the ideal value of unity as the meshes are refined. [ABSTRACT FROM AUTHOR]

The reproducing kernel particle method based on the irreducible flow formulation is utilised to perform the numerical simulation of bulk metal forming processes. Emphasis is given on analysing the influence of employing triangular or quadrilateral background cells on the predictions of material flow, forming load and distribution of strain. A new proposal to smooth the distribution of average stresses during stress computations in the background cells is also included. The effectiveness of the proposed method is discussed by comparing its numerical predictions with a benchmark test case, finite element calculations and experimental data. The benchmark test case is included with the objective of illustrating the influence of several theoretical and numerical subjects such as; order of the basis correction functions, dimension of the compact support and computation of the volume associated to each nodal point. Experimental data was acquired from metal forming controlled laboratory-based tests that were designed so that the proposed method could be tested on its ability to efficiently handle large plastic deformations. It is shown that adaptive arbitrary triangular background cells are capable of efficiently handling large plastic deformations without remeshing. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a detailed derivation of the numerical method, Immersed Finite Element Method (IFEM), for the solution of fluid-structure interaction problems. This method is developed based on the Immersed Boundary (IB) method that was initiated by Peskin, with additional capabilities in handling nonuniform and independent meshes and applying arbitrary boundary conditions on both fluid and solid domains. A higher order interpolation function is adopted from one of the mesh-free methods, the Reproducing Kernel Particle Method (RKPM), which relieves the uniformity constraint of fluid meshes. Two 2-D example problems are presented to illustrate the capabilities of the algorithm. The accuracy in the numerical analysis demonstrates that the IFEM algorithm is a reliable and robust numerical approach to solve fluid and deformable solid interactions. [Copyright &y& Elsevier]

Abstract: A parallel computational implementation of modern meshless system is presented for explicit for 3D bulk forming simulation problems. The system is implemented by reproducing kernel particle method. Aspects of a coarse grain parallel paradigm—domain decompose method—are detailed for a Lagrangian formulation using model partitioning. Integration cells are uniquely assigned on each process element and particles are overlap in boundary zones. Partitioning scheme multilevel recursive spectrum bisection approach is applied. The parallel contact search algorithm is also presented. Explicit message passing interface statements are used for all communication among partitions on different processors. The parallel 3D system is developed and implemented into 3D bulk metal forming problems, and the simulation results demonstrated the efficiency of the developed parallel reproducing kernel particle method system. [Copyright &y& Elsevier]

Abstract: A new mesh free approach is put forwards for the numerical simulation of bulk metal forming processes. The approach is based on the utilization of the reproducing kernel particle method in conjunction with the flow formulation for incompressible and slightly compressible rigid-plastic materials. Special emphasis is placed on the utilization of adaptive cell procedures that are capable of generating a new set of background cells at the end of each increment of deformation. Results show that the proposed methodology is capable of efficiently handling large plastic deformations without remeshing and providing results that are in close agreement with both finite element predictions and experimental measurements. [Copyright &y& Elsevier]

Abstract: This paper summarizes the newly developed immersed finite element method (IFEM) and its applications to the modeling of biological systems. This work was inspired by the pioneering work of Professor T.J.R. Hughes in solving fluid–structure interaction problems. In IFEM, a Lagrangian solid mesh moves on top of a background Eulerian fluid mesh which spans the entire computational domain. Hence, mesh generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element method and the continuity between the fluid and solid sub-domains is enforced via the interpolation of the velocities and the distribution of the forces with the reproducing Kernel particle method (RKPM) delta function. The proposed method is used to study the fluid–structure interaction problems encountered in human cardiovascular systems. Currently, the heart modeling is being constructed and the deployment process of an angioplasty stent has been simulated. Some preliminary results on monocyte and platelet deposition are presented. Blood rheology, in particular, the shear-rate dependent de-aggregation of red blood cell (RBC) clusters and the transport of deformable cells, are modeled. Furthermore, IFEM is combined with electrokinetics to study the mechanisms of nano/bio filament assembly for the understanding of cell motility. [Copyright &y& Elsevier]