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In this manuscript, we study and examine the time‐fractional modified anomalous sub‐diffusion model of distributed‐order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second‐order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full‐discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the δ$$ \delta $$‐Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table. [ABSTRACT FROM AUTHOR]

In the realm of computational mechanics, the reproducing kernel particle method (RKPM) shines in addressing challenges associated with intricate geometries and substantial deformations. However, in the Galerkin framework, RKPM has a substantial computational workload due to the complexities of numerical integration. To mitigate this limitation, we present a pioneering approach: the finite volume reproducing kernel particle method (FV-RKPM). FV-RKPM amalgamates the strengths of meshfree methods and finite volume methods, offering benefits such as grid independence, conservation properties, and numerical stability. In this paper, we elaborate on the construction methodology for FV-RKPM, alongside its associated integration strategies. We consistently validate the efficacy of this innovative methodology through a series of numerical results. [ABSTRACT FROM AUTHOR]

The reproducing kernel particle method (RKPM) is one of the most universal meshless methods. However, when solving three-dimensional (3D) problems, the computational efficiency is relatively low because of the complexity of the shape function. To overcome this disadvantage, in this study, we introduced the dimension splitting method into the RKPM to present a hybrid reproducing kernel particle method (HRKPM), and the 3D Helmholtz equation is solved. The 3D Helmholtz equation is transformed into a series of related two-dimensional (2D) ones, in which the 2D RKPM shape function is used, and the Galerkin weak form of these 2D problems is applied to obtain the discretized equations. In the dimension-splitting direction, the difference method is used to combine the discretized equations in all 2D domains. Three example problems are given to illustrate the performance of the HRKPM. Moreover, the numerical results show that the HRKPM can improve the computational efficiency of the RKPM significantly. [ABSTRACT FROM AUTHOR]

The unique superelastic properties of the Ni-Ti shape memory alloys (SMAs) have been found extensive practical application in many engineering fields, but the Ni-Ti SMAs exhibits elastic behavior in small load, so the elastic behavior of the Ni-Ti SMAs is studied by using meshless method of reproducing kernel particle method (RKPM) in this paper. The displacement boundary conditions are applied by utilizing the penalty function method, and the impact of penalty factor and parameter of influence domain on errors are discussed, and computational stability is analyzed. At last, the correctness of the RKPM in studying the elastic behavior of the Ni-Ti SMAs is evidenced by using three numerical examples. [ABSTRACT FROM AUTHOR]

This work presents an approach for automating the discretization and approximation procedures in constructing digital representations of composites from micro-CT images featuring intricate microstructures. The proposed method is guided by the Support Vector Machine (SVM) classification, offering an effective approach for discretizing microstructural images. An SVM soft margin training process is introduced as a classification of heterogeneous material points, and image segmentation is accomplished by identifying support vectors through a local regularized optimization problem. In addition, an Interface-Modified Reproducing Kernel Particle Method (IM-RKPM) is proposed for appropriate approximations of weak discontinuities across material interfaces. The proposed method modifies the smooth kernel functions with a regularized Heaviside function concerning the material interfaces to alleviate Gibb's oscillations. This IM-RKPM is formulated without introducing duplicated degrees of freedom associated with the interface nodes commonly needed in the conventional treatments of weak discontinuities in the meshfree methods. Moreover, IM-RKPM can be implemented with various domain integration techniques, such as Stabilized Conforming Nodal Integration (SCNI). The extension of the proposed method to 3-dimension is straightforward, and the effectiveness of the proposed method is validated through the image-based modeling of polymer-ceramic composite microstructures. [ABSTRACT FROM AUTHOR]

Summary: Folded plate roofs are widely used in building roofs due to their light weight and high stiffness. However, traditional folded plate roofs are often limited in application due to insufficient stiffness under the current trend of pursuing larger‐span roofs. Ribbed folded plate roofs can effectively solve this problem because ribs not only increase the stiffness of the roof and reduce stress concentration, but they are also easy to be formed into various shapes to meet different needs for buildings. Therefore, the detailed analysis was carried out on a reinforced concrete ribbed folded plate roof (RCRFR) under the vertical uniform load to investigate its static behavior. The bearing capacity, failure mode, load–displacement relationship, and strain variation were obtained through the test. Additionally, based on the finite element (FE) method, not only a comparison was conducted between RCRFR and reinforced concrete ribless folded plate roof (RCLFR), but the influence of material nonlinearity and geometric nonlinearity on the structure was also investigated. Moreover, the bending characteristics of RCRFR under design load were analyzed based on the reproducing kernel particle method. The experimental results showed that this structure had good mechanical properties within the design load. In the overload stage, the concrete on the underside of the structure was severely damaged. Furthermore, the yielding of the ribbed plate's reinforcing bars caused the increased vertical deformation difference between the ribbed plate and the top chord, and the ribbed plate and the central ridge beam. Eventually, the failure of the anchorage of the ribbed plate's reinforcing bars anchored in the central ridge beam and the top chord led to the loss of structural load‐bearing capacity. The FE analysis results demonstrated that ribs enhance the stiffness of the structure, with material nonlinearity having the primary impact and geometric nonlinearity exerting a secondary effect. The meshfree method analysis was in concordance with the experimental results as well as those of the FE analysis. [ABSTRACT FROM AUTHOR]

Since meshless reproducing kernel particle method (RKPM) is easily affected by different kernel functions in numerical accuracy and stability, the Hermit-type function is introduced to the construction of trial function on boundary nodes. Meanwhile, the radial basis function is applied to establish the trial function in internal nodes, then a meshless Hermit-type radial basis function RKPM (Hermit-type RRKPM) is constructed. The advantage of this proposed method is that the numerical solution is more stable and the accuracy is higher. Furthermore, the meshless Hermit-type RRKPM is used to analyze viscoelastic problems. Finally, to illustrate the accuracy and stability, we compare the results of the proposed method and RRKPM with finite element method (FEM) solutions in solving viscoelastic problems.

The reproducing kernel particle method (RKPM) is one of the most universal meshless methods. However, when solving three-dimensional (3D) problems, the computational efficiency is relatively low because of the complexity of the shape function. To overcome this disadvantage, in this study, we introduced the dimension splitting method into the RKPM to present a hybrid reproducing kernel particle method (HRKPM), and the 3D Helmholtz equation is solved. The 3D Helmholtz equation is transformed into a series of related two-dimensional (2D) ones, in which the 2D RKPM shape function is used, and the Galerkin weak form of these 2D problems is applied to obtain the discretized equations. In the dimension-splitting direction, the difference method is used to combine the discretized equations in all 2D domains. Three example problems are given to illustrate the performance of the HRKPM. Moreover, the numerical results show that the HRKPM can improve the computational efficiency of the RKPM significantly.

Abstract The numerical modelling of natural disasters such as landslides presents several challenges for conventional mesh-based methods such as the finite element method (FEM) due to the presence of numerically challenging phenomena such as severe material deformation and fragmentation. In contrast, meshfree methods such as the reproducing kernel particle method (RKPM) possess unique features conducive to modelling extreme events such as the absence of a structured mesh and the ease of adaptive refinement, among others. While the semi-Lagrangian reproducing kernel (SL-RK) shape functions of RKPM defined in the current configuration have proven to be effective in extreme event modelling, the computational cost for the re-evaluation of the shape functions at every time step is costly. In this work, a deformation-dependent coupling of the Lagrangian reproducing kernel (L-RK) and SL-RK approximations is proposed for the solution of a hydro-mechanical formulation for effective simulations of landslides. The ramp function is constructed based on an equivalent plastic strain as a deformation-dependent transition from L-RK shape functions to SL-RK ones as the deformation progresses. The particular focus of the paper will be on modelling seepage-induced landslides with a mixed $$u$$ u – $$p$$ p formulation to couple the solid and fluid phases. Examples are presented to examine the effectiveness of this coupled Lagrangian/semi-Lagrangian reproducing kernel (L–SL RK) formulation and to highlight its performance in landslide modelling.

The state-of-the-art locking-free meshfree Galerkin formulation for modeling the Reissner–Mindlin plate problems is plagued by the following issues: (1) the requirement of a large enough kernel support size in avoiding kernel instability because of the quadratic basis in meeting the Kirchhoff mode reproducing condition, as well as (2) the tedious construction of the conforming representative domains and the smoothed strain in the stabilized conforming integration scheme. This study introduces an efficient and stabilized approach that circumvents the above-mentioned issues. A quasi-consistent reproducing kernel approximation is first developed to enable a smaller kernel support size to be used without the moment matrix singularity issue under a controllable loss of completeness; thus, the approximation construction is accelerated. Then, a bending consistent nodal integration method is proposed where the bending consistency in Galerkin formulation is achieved via an assumed strain approach without using the conforming cell. A variational multiscale stabilization method from our earlier research is implemented to avoid low energy instability while maintaining the locking-free property. The performance of the present formulation is validated in several benchmark problems. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, the nonlinear vibration behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates is investigated by meshless method. Methods: The efficiency parameters, which clearly affect the mechanical properties of the composite materials, are characterized by polynomial expressions for different CNT contents, and applied as the inputs in the vibration analysis of plate structure. Four distributions of CNTs along the thickness of composite plate are considered, namely, the uniform distribution (UD), FG-O, FG-X and FG-V. Based on the classical plate theory and von Karman strain-displacement relation, the governing equations of motion are derived by the virtual displacement principle. Based on the reproducing kernel particle method (RKPM), the discrete governing equations for the vibration of the FG-CNTRC plates are derived. The linearized updated mode (LUM) method is adopted to solve the iteration process of governing equations. Results: Numerical results are employed to investigate the effects of boundary condition, aspect ratio, volume fraction and arrangement of CNT on the nonlinear vibration characteristics of composite plates. [ABSTRACT FROM AUTHOR]

In this study, a variational multiscale stabilized locking-free meshfree formulation is introduced for modeling Reissner–Mindlin plate problems under arbitrary plate thickness. Under this framework, the plate quantities are decoupled into coarse-scale and fine-scale components in the variational equations, where the fine-scale solution represents a correction to the residual of the coarse-scale equations that can be solved by an effective collocation-type approach with an approximation method meeting locking free conditions. The substitution of fine-scale solutions in the coarse-scale system leads to a residual-based Galerkin formulation. In the proposed framework, the reproducing kernel approximation, as well as the smoothed gradient and divergence, are adopted to ensure the bending exactness in the Galerkin formulation. The multiscale approach is also beneficial for problems exhibiting localized phenomena. The effectiveness of the proposed method is tested by solving a series of numerical examples and compared with classical methods. [ABSTRACT FROM AUTHOR]

Recently, several types of nonlocal discrete differential operators have emerged either from meshfree particle methods or from nonlocal continuum mechanics, such as peridynamics. In this article, we discuss the mathematical formulation as well as construction of the nonlocal discrete differential operators. Based on a least-square minimization procedure and the associated Moore-Penrose inverse, we have found a general form of the shape tensor and a unified expression for the first type nonlocal differential operators. We then conduct a convergence study, which provides the interpolation error estimate for the first type discrete nonlocal different operators. We have shown that as the radius of the horizon approaches to zero, the first type nonlocal differential operators will converge to the local differential operators. Moreover, we have demonstrated the computational performance of the first type nonlocal differential operators in several numerical examples. [ABSTRACT FROM AUTHOR]

Panel cracking for stiffened panels is investigated based on linear elastic fracture mechanics. Meshfree flat shell formulation and fracture modeling are employed. After investigating stress distribution of the intact stiffened panel, the mode-I and mixed-mode crackings are studied using the stress resultant intensity factors (SRIFs). When the crack tip approaches to the stiffener, deformation around the crack is suppressed due to presence of the stiffeners. The mechanical behaviors are carefully examined by changing crack length, loading conditions and thickness of the main panel and stiffeners. Quantitative evaluation of SRIFs is addressed by comparing with the reference solutions. An in-depth discussion is provided for the meshfree modeling of the panel cracking in the stiffened panels. This series studies are useful for examining crack arrest effect of the stiffened panel due to presence of the stiffeners. • Mode-I and mixed-mode panel cracking is studied in stiffened panel. • RKPM-based meshfree flat shell and fracture modeling is employed. • Stress distribution and SRIFs are carefully studied based on linear elastic fracture mechanics. • Fracture in stiffened panels are investigated compared with existing reference solutions and experimental results. [ABSTRACT FROM AUTHOR]

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kernel particle method (SL-RKPM) is introduced into the volumetric strain energy to accurately embed the key shock physics in the absence of a mesh or grid, which is shown to also ensure the conservation of linear momentum. For kernel stability, a deformation-dependent anisotropic kernel support update algorithm is proposed, which is shown to capture excessive plastic flow and material separation. A quasi-conforming nodal integration is adopted to avoid the need of updating conforming cells which is tedious in extreme deformations. It is shown that the proposed formulation effectively captures shocks, jet formation, and smooth-to-wavy interface morphology transition with good agreement with experimental results. [ABSTRACT FROM AUTHOR]

We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

• Static, stability and free vibration analysis of FG plate are carried out. • Free vibration of plates with inner point, line and area supports is studied. • The finite strip method with longitudinal RKPM functions based on FSDT is used. • A decent way is used to impose boundary conditions considering meshfree functions. • Elimination of shear-locking, as an undesirable phenomenon, is elaborated. This paper is focused on the application of a novel reproducing kernel particle finite strip method (RKP-FSM) for the static, mechanical and thermal buckling, and free vibration analyses of rectangular functionally graded (FG) plates with different boundary conditions and internal supports. Conventional reproducing kernel particle method (RKPM) is incorporated into the finite strip method (FSM), and the governing equations are derived based on the first-order shear deformation theory (FSDT). As a convenient approach, boundary conditions are imposed using a simple method based on the RKPM correction function and the essence of the FSDT displacement field. Several examples are studied using this method, and the results are compared to those obtained from other numerical methods. The proposed method is also extended to consider different support geometries and complex inner conditions, and the results are validated using COMSOL finite element software. The shear-locking error, which occurs when considering C0 plate theories, is noticeably reduced by considering the RKPM parameters in the finite strip formulation. [ABSTRACT FROM AUTHOR]

The computational accuracy of the traditional reproducing kernel particle method (RKPM) is susceptible to different kernel functions. To eliminate the adverse effects of the different kernel functions on the computational accuracy of the RKPM, the radial basis function (RBF) is introduced and the radial basis reproducing kernel particle method (RRKPM) is proposed. Compared with the RKPM, the proposed method has the advantages of greater computational accuracy and faster convergence speed. Furthermore, the proposed method is applied to analyze the elastic deformation process of materials with various defects. Utilizing the Galerkin weak form of elastic mechanical problem, the RRKPM of elastic mechanics is established and the corresponding formulae are derived. Combined with the numerical examples, the shaped parameter of the RBF, the scaling parameter and the penalty factor are discussed. Finally, several different examples are adopted to demonstrate the applicability and correctness of the RRKPM.

In this paper, the method of radial basis function (RBF) is employed to construct the approximating function of the reproducing kernel particle method (RKPM), which can reduce the adverse effect of different kernel functions on computational accuracy and improve stability in the problem domain, and the radial basis reproducing kernel particle method (RRKPM) is presented. Compared with the RKPM, the RRKPM has higher computational accuracy and better stability. Then RRKPM is applied to wave propagation, and the discretized system equation can be obtained using the integral weak form. The penalty method is applied to imposing the essential boundary condition, and the two-point difference method is selected to discretize the time. The accuracy and stability of the RRKPM for wave propagation problem are illustrated by the numerical examples.

In physically based-based animation, pure particle methods are popular due to their simple data structure, easy implementation, and convenient parallelization. As a pure particle-based method and using Galerkin discretization, the Moving Least Square Reproducing Kernel Method (MLSRK) was developed in engineering computation as a general numerical tool for solving PDEs. The basic idea of Moving Least Square (MLS) has also been used in computer graphics to estimate deformation gradient for deformable solids. Based on these previous studies, we propose a multiphase MLSRK framework that animates complex and coupled fluids and solids in a unified manner. Specifically, we use the Cauchy momentum equation and phase field model to uniformly capture the momentum balance and phase evolution/interaction in a multiphase system, and systematically formulate the MLSRK discretization to support general multiphase constitutive models. A series of animation examples are presented to demonstrate the performance of our new multiphase MLSRK framework, including hyperelastic, elastoplastic, viscous, fracturing and multiphase coupling behaviours etc. [ABSTRACT FROM AUTHOR]

• The radial basis reproducing kernel particle method for the geometrically nonlinear problem of the FGM is proposed. • Combined with the weak form of Galerkin integration and the T-L formulation, the corresponding formulae are derived. • The penalty factor, shaped parameter, influence domain radius, loading step number and node distribution are discussed. • The different functional gradient exponents and functions of the FGM are analyzed. In this paper, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed for solving geometrically nonlinear problem of functionally graded materials (FGM). Compared with the RKPM, the advantages of the proposed method are that it can eliminate the negative effect of different kernel functions on the computational accuracy, and has higher computational accuracy and stability. Using the Total Lagrange (T.L.) formulation and the weak form of Galerkin integration, the corresponding formulae for geometrically nonlinear problem of FGM are derived. The penalty factor, shaped parameter of the RBF, the control parameter of influence domain radius, loading step number and node distribution are discussed. Furthermore, the effects of different gradient functions and exponents on displacement and stress are analyzed. Newton-Raphson (N-R) iterative method is utilized for numerical solution. The proposed method is correct and effective for solving geometrically nonlinear problem of FGM, which can be demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]

Using the total Lagrange formulation, the reproducing kernel particle method (RKPM) for the geometrically nonlinear problem of the functionally graded materials (FGM) is proposed, and the corresponding formulae are derived. The displacement boundary condition is applied by the penalty method, and the numerical solution is solved by Newton–Raphson (N–R) iterative method. Furthermore, penalty factor, the control parameter of influence domain radius, loading step number and node distribution are discussed. Finally, the numerical examples illustrate that the RKPM for the geometrically nonlinear problem of the FGM is correct and effective. [ABSTRACT FROM AUTHOR]

This paper presents a level set topology optimization method in combination with the reproducing kernel particle method (RKPM) for the design of structures subjected to design-dependent pressure loads. RKPM allows for arbitrary particle placement in discretization and approximation of unknowns. This attractive property in combination with the implicit boundary representation given by the level set method provides an effective framework to handle the design-dependent loads by moving the particles on the pressure boundary without the need of remeshing or special numerical treatments. Moreover, the reproducing kernel (RK) smooth approximation allows for the Young's modulus to be interpolated using the RK shape functions. This is another advantage of the proposed method as it leads to a smooth Young's modulus distribution for smooth boundary sensitivity calculation which yields a better convergence. Numerical results show good agreement with those in the literature. [ABSTRACT FROM AUTHOR]

Analysis of stochastic dynamic system is still an open research issue. Recently a family of generalized probability density evolution equation, which provides an available way for general nonlinear systems, is put forward. In this paper, a numerical method based on reproducing kernel particle method (RKPM) for the solution of generalized probability density evolution equation, named the refined algorithm based on RKPM, is developed. Besides, the corresponding implementation procedure is elaborated. In this method, the time dependent probability distributions of the responses of interest can be obtained with less computational efforts. In addition, the mesh sensitivity problem in traditional probability density evolution method is settled well. Some details of parameter analysis are also discussed. To verify both the efficiency and accuracy of the method, a single-degree-of-freedom example and a 10-story frame structure are investigated. The refined algorithm based on RKPM can be applied to uni-variable and multi-variable, one-dimensional and multi-dimensional systems. [ABSTRACT FROM AUTHOR]

Summary In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three‐dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two‐dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element‐free Galerkin method for 3D potential problems. [ABSTRACT FROM AUTHOR]

• The Hermit-type RRKPM for viscoelastic problems is proposed. • The penalty method is applied to imposing the essential boundary condition. • Two-point difference method is selected for the time discretization. • The validity of the Hermit-type RRKPM are illustrated. Since meshless reproducing kernel particle method (RKPM) is easily affected by different kernel functions in numerical accuracy and stability, the Hermit-type function is introduced to the construction of trial function on boundary nodes. Meanwhile, the radial basis function is applied to establish the trial function in internal nodes, then a meshless Hermit-type radial basis function RKPM (Hermit-type RRKPM) is constructed. The advantage of this proposed method is that the numerical solution is more stable and the accuracy is higher. Furthermore, the meshless Hermit-type RRKPM is used to analyze viscoelastic problems. Finally, to illustrate the accuracy and stability, we compare the results of the proposed method and RRKPM with finite element method (FEM) solutions in solving viscoelastic problems. [ABSTRACT FROM AUTHOR]

Summary: Terminal ballistics of concrete is of extreme importance to the military and civil communities. Over the past few decades, ultra‐high performance concrete (UHPC) has been developed for various applications in the design of protective structures because UHPC has an enhanced ballistic resistance over conventional strength concrete. Developing predictive numerical models of UHPC subjected to penetration is critical in understanding the material's enhanced performance. This study employs the advanced fundamental concrete (AFC) model, and it will run inside the reproducing kernel particle method (RKPM)‐based code known as the nonlinear meshfree analysis program (NMAP). NMAP is advantageous for modeling impact and penetration problems that exhibit extreme deformation and material fragmentation. A comprehensive experimental study was conducted to characterize the UHPC. The investigation consisted of fracture toughness testing, the utilization of nondestructive microcomputed tomography analysis, and lastly projectile penetration shots on the UHPC targets. To improve the accuracy of the model, a new scaled damage evolution law (SDEL) is employed within the microcrack informed damage model. During the homogenized macroscopic calculation, the corresponding microscopic cell needs to be dimensionally equivalent to the mesh dimension when the partial differential equation becomes ill posed and strain softening ensues. To ensure arbitrary mesh geometry for which the homogenized stress‐strain curves are derived, a size scaling law is incorporated into the homogenized tensile damage evolution law. This ensures energy‐bridging equivalence of the microscopic cell to the homogenized medium irrespective of arbitrary mesh geometry. Results of numerical investigations will be compared with results of penetration experiments. [ABSTRACT FROM AUTHOR]

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]