Reproducing Kernel Particle Method Research Articles

Research on Superelasticity of Shape Memory Alloys Based on Reproducing Kernel Particle Method

Shape memory alloys (SMAs), known for their unique superelastic effect, exhibit superior mechanical properties compared to traditional materials. In this study, the reproducing kernel particle method (RKPM) is applied to investigate the superelastic effect of the SMAs. A segmented linearization RKPM model is developed to establish the stress–strain relationship for the superelastic effect, and relevant formulas of superelasticity for the SMAs are derived. Finally, the applicability and accuracy of the RKPM in analyzing the superelastic effect of the SMAs are demonstrated through numerical examples.

Research on elasticity behavior of Ni-Ti shape memory alloys using the reproducing kernel particle method

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.

Data-driven modeling of an unsaturated bentonite buffer model test under high temperatures using an enhanced axisymmetric reproducing kernel particle method

In deep geological repositories for high level nuclear waste with close canister spacings, bentonite buffers can experience temperatures higher than 100 °C. In this range of extreme temperatures, phenomenological constitutive laws face limitations in capturing the thermo-hydraulic behavior of the bentonite, since the pre-defined functional constitutive laws often lack generality and flexibility to capture a wide range of complex coupling phenomena as well as the effects of stress state and path dependency. In this work, a deep neural network (DNN)-based soil–water retention curve (SWRC) of bentonite is introduced and integrated into a Reproducing Kernel Particle Method (RKPM) for conducting thermo-hydraulic simulations of the bentonite buffer. The DNN-SWRC model incorporates temperature as an additional input variable, allowing it to learn the relationship between suction and degree of saturation under the general non-isothermal condition, which is difficult to represent using a phenomenological SWRC. For effective modeling of the tank-scale test, new axisymmetric Reproducing Kernel basis functions enriched with singular Dirichlet enforcement representing heater placement and an effective convective heat transfer coefficient representing thin-layer composite tank construction are developed. The proposed method is demonstrated through the modeling of a tank-scale experiment involving a cylindrical layer of MX-80 bentonite exposed to central heating.

Meshfree digital image correlation using reproducing kernel particle method and its degenerate derivations

Meshfree digital image correlation (MF-DIC) is a recently-proposed advanced DIC technique that deeply integrates meshfree method with DIC. MF-DIC is of potential due to its close relationship with computational mechanics and the excellent balance between spatial resolution and measurement resolution. A new MF-DIC algorithm based on another advanced meshfree method, namely the reproducing kernel particle method (RKPM), is proposed. The RKPM-based MF-DIC is proven to be equivalent to the recently-proposed Element Free Galerkin Method (EFGM) based MF-DIC in certain conditions. This work also briefly introduces two additional MF-DIC methods based on smooth particle hydrodynamics (SPH) and the point interpolation method (PIM). In certain cases, they are the degenerative derivations of the RKPM-based MF-DIC. Benchmark tests on DIC Challenge 2.0 show that these MF-DIC methods also have a superior balance between spatial resolution and measurement resolution compared to state-of-the-art DIC algorithms.

Phase-field modeling of anisotropic crack propagation based on higher-order nonlocal operator theory

This paper presents a novel higher-order nonlocal operator theory for the phase-field modeling of brittle fracture in anisotropic materials. Incorporating higher order nonlocal operators can enhance the accuracy of the phase-field model by effectively capturing long-range interactions that hold significance in numerous materials. The reproducing kernel particle method is employed to derive a nonlocal differential operator to enhance computational stability and accuracy. Moreover, the proposed method eliminates the need for direct computation of derivatives of the modified kernel function, which avoids the calculation of moment matrix derivatives and improves computational efficiency. The phase-field modeling of polycrystalline materials, considering the anisotropic fracture resistance of each grain, is implemented using this numerical framework. The present method is able to capture different scenarios intergranular and transgranular crack propagation patterns in polycrystalline materials. The proposed method involves a detailed representation of the complex process of crack initiation and propagation in 2D and 3D models of polycrystalline materials.

The numerical analysis of viscoelastic problems in signal propagation using a novel meshless Hermit-type RRKPM

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.

A smooth Crack-Band Model for anisotropic materials: Continuum theory and computations with the RKPM meshfree method

A smooth Crack Band Model (sCBM) is developed for anisotropic materials with both high and low magnitude of anisotropy. sCBM is primarily used as a regularization mechanism to enable capturing the correct size and shape of the fracture process zone (FPZ) as well as the smooth distribution of strain inside the zone. The versatility of the proposed sCBM formulation is demonstrated by its adaptability to any material law of choice. A Reproducing Kernel Particle Method (RKPM) is employed to discretize such formulation, the benefit of which is its ability to compute the Hessian of the displacement field (i.e., the sprain tensor), a key quantity in the sCBM theory, using only the nodal displacements as the problem unknowns. This makes the resulting formulation accurate and efficient, which is shown by the validations against size-effect and radial crushing experiments, illustrating the power of the proposed methodology for practical applications.

Static behavior of ribbed folded plate roof under vertical uniform load—Experimental study and meshfree method analysis

SummaryFolded 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.

A neural network-based enrichment of reproducing kernel approximation for modeling brittle fracture

Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent computational methods to predict the evolution of localizations. In this work, an improved version of the neural network-enhanced Reproducing Kernel Particle Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed method, a background reproducing kernel (RK) approximation defined on a coarse and uniform discretization is enriched by a neural network (NN) approximation under a Partition of Unity framework. In the NN approximation, the deep neural network automatically locates and inserts regularized discontinuities in the function space. The NN-based enrichment functions are then patched together with RK approximation functions using RK as a Partition of Unity patching function. The optimum NN parameters defining the location, orientation, and displacement distribution across location together with RK approximation coefficients are obtained via the energy-based loss function minimization. To regularize the NN-RK approximation, a constraint on the spatial gradient of the parametric coordinates is imposed in the loss function. Analysis of the convergence properties shows that the solution convergence of the proposed method is guaranteed. The NN enrichment allows the modeling of evolving cracks by a fixed coarse RK discretization without adaptive refinement for enhanced computational efficiency. The effectiveness of the proposed method is demonstrated by a series of numerical examples involving damage propagation and branching.

Support vector machine guided reproducing kernel particle method for image-based modeling of microstructures

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.

Three dimensional meshfree analysis for time-Caputo and space-Laplacian fractional diffusion equation

Numerical exploration of spatial fractional differential equations in three dimensions is not trivial due to their complexity, especially for complicated problem domains. In this work, the time-Caputo and space-Laplacian fractional diffusion equations in three dimensions are analyzed using a meshfree technique. In the temporal dimension, the classical L1 finite difference scheme is used to approach the Caputo fractional derivative. The spatial discretization is realized by a three dimensional reproducing kernel particle method (RKPM), which can eliminate the dependence of shape functions on certain meshes. Therefore RKPM is very suitable to approximate the field variable in complicated three dimensional domains compared with other mesh-dependent methods. For the purpose of increasing the computational efficiency, the stabilized conforming nodal integration (SCNI) and lumped mass matrix techniques are adopted in the Galerkin meshfree formulation. In the proposed method, the tedious derivatives computing for meshfree shape functions, numerical integration for Galerkin weak form and time-consuming inverse calculating for the large size mass matrix are all realized by more efficient approaches comparing with the conventional Galerkin RKPM. Several numerical examples in various domains with structured and unstructured discretization are studied to demonstrate the proposed methodology, and the results show very favorable performance of the proposed method regarding the accuracy and effectiveness.

A Hybrid Reproducing Kernel Particle Method for Three-Dimensional Elasticity Problems

This study presents a fast meshless method called the hybrid reproducing kernel particle method (HRKPM) for the solution of three-dimensional (3D) elasticity problems. The equilibrium equations of 3D elasticity are divided into three groups of equations, and two equilibrium equations are contained in each group. By coupling the discrete equations for solving two arbitrary groups of equations, the complete solution of 3D elasticity can be obtained. For an arbitrary group of equations, the 3D elasticity problem is transformed into a series of associated two-dimensional (2D) ones, which is solved by the RKPM to derive the discrete formulae. The discrete equations of 2D problems are combined using the difference method in dimension splitting direction. Then, arbitrarily choosing another group of equilibrium equations, the discrete equation of another group of 2D problems can be obtained similarly. By combining the discrete equations for these two groups of 2D problems, the solution to an original 3D problem will be reached. The numerical results show that the HRKPM performs better than RKPM in solution efficiency.

A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

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.

Coarse-graining algorithms for the Eulerian-Lagrangian simulation of particle-laden flows

In the present article, novel Coarse-Graining (CG) algorithms for the Eulerian-Lagrangian (EL) simulation of particle-laden flows are proposed. These include different variants of Reproducing Kernel Particle Methods (RKPM) and an extended Diffusion Two-Step Method (DTSM) for highly polydisperse flows. Owing to the dynamic nature of the kernel function in RKPMs, CG algorithms with high-order consistency properties are constructed and the extra physics of the fluid-particle interaction torque effect on the two-way coupling force distributed to the fluid is taken into consideration. To increase the robustness of RKPMs, they are hybridized with other simple CG algorithms in such a way that each model is activated in the range of its validity. The performance of the new CG algorithms is carefully assessed in comparison to other widely-used algorithms by devising four benchmarks and several grid-independence tests. Based on the analyses, the first-order RKPM demonstrates the best performance in terms of 8 attributes, including conservativity, grid-independency, smoothness, etc., among all algorithms and is recommended when the accuracy is of prime importance. However, due to its higher computational cost compared to the extended DTSM, the latter model would be an affordable alternative for large-scale discrete element method problems when the computational cost is critical.

A comparison of linear consistent correction methods for first-order SPH derivatives

A well-known issue with the widely used Smoothed Particle Hydrodynamics (SPH) method is the neighborhood deficiency. Near the surface, the SPH interpolant fails to accurately capture the underlying fields due to a lack of neighboring particles. These errors may introduce ghost forces or other visual artifacts into the simulation. In this work we investigate three different popular methods to correct the first-order spatial derivative SPH operators up to linear accuracy, namely the Kernel Gradient Correction (KGC), Moving Least Squares (MLS) and Reproducing Kernel Particle Method (RKPM). We provide a thorough, theoretical comparison in which we remark strong resemblance between the aforementioned methods. We support this by an analysis using synthetic test scenarios. Additionally, we apply the correction methods in simulations with boundary handling, viscosity, surface tension, vorticity and elastic solids to showcase the reduction or elimination of common numerical artifacts like ghost forces. Lastly, we show that incorporating the correction algorithms in a state-of-the-art SPH solver only incurs a negligible reduction in computational performance.

The elastoplastic large deformation analysis based on meshless radial basis reproducing kernel particle method

In this paper, the radial basis reproducing kernel particle method (RB-RKPM) is extended to solve the elastoplastic large deformation problem. The Galerkin weak form of the elastoplastic large deformation problem based on the full Lagrange scheme is used to establish the solution equation, which is then solved using the Newton–Raphson iterative method. The RB-RKPM is established for the elastoplastic large deformation problem. The effect of the penalty factor, the scale parameter, and the shape parameter on the calculation accuracy are introduced, and the optimal parameters are obtained. Several numerical examples were used in the simulation. The RB-RKPM solution is compared with the reproducing kernel particle method (RKPM) solution and the finite element method (FEM) solution, the effectiveness of the RB-RKPM in solving elastoplastic large deformation problem is verified, and the RB-RKPM has high computational accuracy.

Numerical Investigation of Kink Strengthening Mechanism due to Kink Band in Long-Period Stacking Ordered Magnesium Alloy

A numerical investigation of the kink strengthening mechanism in long-period stacking ordered magnesium alloy is presented. A higher-order gradient crystal plasticity model is introduced, and the reproducing kernel particle method is adopted for the numerical procedure. The specimen, including a kink band with several kink angles, which is defined as the angle between the inside and outside of the kink band, is considered. A simple shear analysis is conducted to evaluate the kink strengthening due to the kink band. The numerical results suggest two main origins of kink strengthening, namely, orientation and defect strengthening. The former is caused by the spatial distribution of the slip direction due to kink, and the latter is the strengthening induced by crystal defects around the kink boundary. The amplitude of kink strengthening depends on the kink angle, and an optimal kink angle, which maximizes the kink strengthening, might exist. The kink strengthening exhibits a Hall–Petch-like behavior, i.e., the correlation between the inverse of the square root of the kink band width and flow stress is almost linear.

A coupled RKPM and dynamic infinite element approach for solving static and transient heat conduction problems

A new accurate and efficient coupled method RKPM-DIEM is proposed. This is a stable and efficient meshfree nodally-integrated reproducing kernel particle method (RKPM) coupled with a dynamic infinite element method (DIEM) for solving half-space problems. The half-space domain is defined as the near field (bounded) and the far field (unbounded) analyzed by the RKPM and DIEM, respectively. Unlike the element-based methods, RKPM is constructed using only nodal data in the global Cartesian coordinates directly to avoid mesh issues such as mesh distortion and entanglement. Also, it provides flexible control of the local smoothness and order of basis, as well as easy construction for a higher-order gradient by changing the kernel function directly. DIEM is first used to show that this approach could solve not only dynamic but also static problems by setting the wave number and the decay coefficient properly. Furthermore, various meshfree integration methods, such as the Gaussian integration, the direct nodal integration, and the natural stabilized nodal integration, are tested to show accuracy and stability. Several benchmark problems are investigated to verify the effectiveness of the proposed method. It has been found that numerical results can achieve high accuracy and stability.

Meshless numerical solution for nonlocal integral differentiation equation with application in peridynamics

The mid-point quadrature rule is often used to solve the nonlocal integral differentiation equations, in which a central material point is used to calculate the integral of the subdomain in the meshless geometry. However, it can only achieve the linear approximation and the first order convergence rate, and it is only accurate enough when the entire cell overlaps with the neighborhood of a material point. In this paper, to achieve the higher precision and the higher rate of convergence speed, a novel coupling method of reproducing kernel particle method and Gaussian-Legendre quadrature scheme is proposed to solve the nonlocal integral differentiation equations in the meshless geometry, in which the mid-point quadrature scheme is replaced with Gauss quadrature scheme to calculate the integral of the subdomain in the meshless geometry, the reproducing kernel particle method is employed to construct the shape functions for material points in the nonlocal model, and to approximate the value of the Gaussian quadrature node. Moreover, the meshless solution for the peridynamic equation is derived detailedly based on the proposed method. Compared with the traditional meshless solution for the nonlocal equations, the proposed solution has higher precision and higher rate of convergence speed in different problems. The numerical results demonstrate improved results in both mathematics and mechanical problems using the proposed method.

A 3D meshless radial basis reproducing kernel particle model for elastoplastic analysis of temperature-dependent FG plates resting on Winkler/Pasternak foundation

• 3D thermo-elastoplastic bending response of temperature-dependent FG plates on Winkler/Pasternak foundation is presented. • A truly meshless approach based on the radial basis reproducing kernel particle method (RRKPM) is developed. • Detailed parametric studies are performed on the scaling factor, shape parameter and node distribution. • The stability and accuracy of the results are confirmed by analyzing several numerical examples. • Different material gradients, elastic foundation parameters, thickness ratios, and boundary conditions are considered. For the first time, a three-dimensional (3D) meshless approach based on the radial basis reproducing kernel particle method (RRKPM) is developed to scrutinize the nonlinear thermo-elastoplastic bending response of temperature-dependent functionally graded (FG) plates resting on Winkler/Pasternak foundation under a combination of mechanical and thermal loads. The modified rule of mixtures is employed to locally evaluate the effective temperature-dependent parameters of the functionally graded material (FGM). To describe the plastic behavior of the FG plate, the von Mises yield criterion, isotropic strain hardening, and the Prandtl-Reuss flow rule are adapted. To demonstrate the accuracy and efficiency of the present method, the outcomes are compared with other existing numerical and analytical results, which shows an excellent agreement. Detailed studies have been conducted on the scaling factor, shape parameter and nodal distribution for finding their desired values. Furthermore, the effect of significant parameters such as material gradient, elastic foundation parameters, thickness ratio, and boundary conditions on the bending response of FG plates has been investigated.