Meshfree Shape Functions Research Articles

An immersed boundary fast meshfree integration methodology with consistent weight learning

An immersed boundary fast integration methodology featured by a consistent weight learning is proposed to accelerate Galerkin meshfree computation. In the proposed approach, the problem domain is embedded in a rectangular spatial domain discretized by regular distributions of meshfree nodes and integration sampling points with virtual integration cells. A trimming operation of the rectangular spatial domain by the physical problem boundary with nodal discretization yields the discrete model for meshfree computation, where the integration sampling points are grouped into the interior sampling points and the near boundary sampling points which form a training set. For the interior sampling points, a natural alignment between the influence domains of meshfree shape functions and virtual integration cells can be easily realized through employing integer support sizes, which ensures a satisfactory accuracy for the basis degree correspondent normal order Gauss integration. The background cells for the domain integration are completely avoided, which greatly simplifies the preprocessing for numerical integration. Meanwhile, a machine learning module is devised to optimize the weights of near boundary integration sampling points. The key step to construct this machine learning module for weight optimization is the selection of the variational integration consistency condition as the loss function, which guarantees the convergence of Galerkin meshfree formulation. The accuracy and efficiency of the proposed weight learning immersed boundary fast integration methodology is thoroughly validated through numerical results.

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 superconvergent meshfree collocation formulation for laminated composite plates with particular focus on convergence analysis

A superconvergent meshfree collocation formulation is presented for the accurate analysis of laminated composite plates, where the convergence characteristic is particularly investigated. The shear locking is resolved through the employment of a mixed formulation that takes the in-plane displacements, deflection, rotations and shear forces as primary variables. The resulting meshfree collocation equations are attained by introducing the recursive gradients constructed from the first order gradients of meshfree shape functions. Subsequently, the convergence of both the standard meshfree collocation approach and the proposed recursive gradient meshfree collocation formulation is theoretically analyzed for the system of mixed discrete equations corresponding to laminated composite plates. These theoretical results clearly identify the basis degree discrepancy issue for meshfree collocation analysis of laminated composite plates. More importantly, this accuracy deficiency is successfully removed by the recursive gradient meshfree collocation formulation, which essentially achieves a desirable superconvergence for the meshfree laminated plate solutions. The superior accuracy and locking free performance of the present meshfree collocation formulation is thoroughly demonstrated by numerical results.

A consistent projection integration for Galerkin meshfree methods

An efficient and inherently consistent integration method, named projection integration(PI), is proposed for Galerkin meshfree methods with arbitrary order. In contrast to traditional numerical integration methods, PI defines a projection operator based on a weighted norm to systematically approximate meshfree shape functions in an appropriate and consistent function space. Then a substitute function space is established by utilizing projected shape functions as a basis, and Galerkin weak form is approximately calculated by substituting the projected shape functions for the original meshfree shape functions. Owing to the projection consistency of the projection operator, the projected shape functions can still meet consistent conditions. For convenient implementations, the triangular finite element space is selected to approximate meshfree shape functions and the weight function in the weighted norm is carefully chosen to simplify the projection operator in this paper. Arbitrary order integration constraint conditions can be easily met by utilizing the corresponding order triangular finite element shape functions with low order Gauss quadrature rules, which means the optimal convergence rate can be obtained. A series of numerical examples are presented to demonstrate the proposed method’s efficiency and superior convergence performance.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

Vibro-acoustic analysis of laminated composite cylindrical and conical shells using meshfree method

This paper intends to investigate the vibro-acoustic responses of cross-ply laminated composite shell including cylindrical and conical shells subjected to acoustic medium by employing meshfree method. The energy principle, first-order shear deformation theory (FSDT) and the Kirchhoff-Helmholtz integral equation are employed to deduce the theoretical formulations of the structural and acoustic models. The displacement and acoustic pressure of laminated composite shells and acoustic medium are described by employing the meshfree shape function and Fourier series. The interior CHIEF points is introduced for eliminating the non-unique solution with regard to the acoustic integral equations. The reliability and accuracy of presented method are verified for forecasting the vibro-acoustic characteristics of the laminated composite shells by comparing to the vibration and the acoustic response results of published articles. Furthermore, the parametric analysis is performed by investigating the influences of geometric dimensions, material properties, boundary conditions and external force types on the acoustic responses of shell, which is beneficial to the preliminary design of the shell structure immersed in acoustic medium.

A coupling approach of the isogeometric–meshfree method and peridynamics for static and dynamic crack propagation

A coupling approach of the isogeometric–meshfree method and the peridynamic method is developed for static and dynamic crack propagation. The coupling approach exhibits advantages in the flexibility of modeling cracks and the exactness of geometry representation. The isogeometric–meshfree method, which adopts the moving least-squares approximations to establish the equivalence between meshfree shape functions and isogeometric basis functions, is capable of obtaining the exact geometry. The isogeometric–meshfree nodes located on the geometry can be directly coupled with peridynamic points by the balanced force principle, which is straightforward and effective. With this approach, the boundary conditions can be enforced directly using the isogeometric–meshfree method, and the surface effects of peridynamics are effectively eliminated. Moreover, the present approach is flexible in switching the isogeometric–meshfree nodes to peridynamic points and achieving adaptive coupling with the same convergence rate as the one for the isogeometric–meshfree method, which is higher than the one for the original peridynamic method. The dual-horizon peridynamic method is adopted for non-uniform discretization. Based on the exact geometry representation, the coupling approach is further extended to crack problems with contact loading. Several representative examples are presented to validate the effectiveness of the present approach in solving static/dynamic crack problems and studying fractures caused by contact.

An investigation on the stochastic thermal vibration behaviors for laminated combined composite plate systems

The present work aims to study the stationary and nonstationary stochastic thermal vibration properties of the laminated combined plate structures using meshless method. The combined system structure consists of several rigidly connected laminated rectangular plates, where the thermal effect of each subplate is considered according to the thermo-elastic theory. The Hamilton’s principle combined with the first-order shear deformation theory (FSDT) is adopted for generating the vibration formulas of the divided plates. The boundary and rigid compatibility conditions of the combined plates are imposed by massless springs with variable stiffness. The displacement components of the subplate system are approximated by 2D Chebyshev meshfree shape functions and the stochastic acceleration excitation is taken into account by introducing the pseudo excitation method (PEM). The modal behaviors and the stochastic dynamic solutions of laminated combined plates subjected to thermal loads are then successfully obtained. A considerable number of computational examples on the free vibration and random responses of the combined laminated plates are carried out to verify the convergence, accuracy and efficiency of the presented meshless model. Finally, systematic parametric studies are conducted to investigate the effects of thermal factor, acceleration excitations and boundary constraints on the free vibration and random dynamic characteristics of the laminated combined plates.

Full-Vector Mode Solver for Nonlinear Optical Waveguides Using Meshfree Finite Cloud Method

A full-vector mode solver based on the meshfree (MF) finite cloud method (FCM) is developed for analysis of nonlinear optical waveguide. Self-consistent solutions are approximated through a simple iterative scheme after the nonlinear eigenvalue matrix equation derived using the quadratic basis function as well as the MF shape function. The proposed mode solver can significantly reduce the computational requirements since major parts of formulations are tackled analytically. Results for two numerical examples are presented and compared with those from the well-known finite element method (FEM) and FEM-based imaginary-distance beam propagation method (FEM-IDBPM), respectively, showing the validity and utility of the proposed method.

A meshfree method for free vibration analysis of ply drop-off laminated rectangular plates

The purpose of this paper is to develop a more effective meshfree shape function for free vibration analysis of ply drop-off laminated rectangular plates. The proposed meshfree Tchebychev-radial point interpolation method (TRIPM) shape function uses the Tchebychev polynomials and multi-quadrics (MQ) radial functions as the basis. The first order shear deformation theory (FSDT) is applied to the formulation for free vibration analysis of ply drop-off laminated rectangular plate, and the field functions are approximated by TRIPM shape function. The governing equations and boundary conditions for the substructures of ply drop-off laminated plate are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The accuracy and reliability of the proposed method are verified by comparing the results of the literature and the finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of ply drop-off laminated rectangular plates with various geometrical dimensions and boundary conditions are presented through numerical examples.

Free vibration analysis of elastically connected composite laminated double-plate system with arbitrary boundary conditions by using meshfree method

In this paper, the meshfree method is adopted for the first time to study the free vibration of a composite laminated double-plate system (CLDPS). The first order shear deformation theory is used to analyze the free vibration of the CLDPS, and the artificial elastic spring technique is utilized to generalize its boundary conditions. All of the displacement functions including the boundary conditions are approximated by a meshfree shape function. The reliability and accuracy of the proposed technique are verified through the comparison with the results in the previous literature and by the finite element method. The effects of various boundary conditions, material properties, and geometries of the CLDPS on its natural frequencies are considered in detail.

An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Presently, it has been widely recognized that numerical integration plays a crucial role on the solution accuracy of Galerkin meshfree methods due to the rational nature of meshfree shape functions and the misalignment of shape function supports with integration cells. Nonetheless, the corresponding underlying cause is still not quite clear. To address this issue, an accuracy analysis of Galerkin meshfree methods is presented in this study with a special focus on the contribution of numerical integration to the error estimates. One important fact is that the integration difficulty of meshfree methods leads to a loss of the Galerkin orthogonality condition which usually serves as a basis to develop error bounds. In order to enable the accuracy analysis, an error measure is proposed to evaluate the loss of Galerkin orthogonality condition, which is actually related to the order of integration consistency for Galerkin meshfree formulation. With the aid of this orthogonality error measure, both H1 and L2 error estimates are established for Galerkin meshfree methods. It turns out that these errors essentially consist of two parts, one comes from the standard interpolation error, and the other one attributes to the numerical integration error. In accordance with the proposed error estimates, it is evident that for the conventional Gauss integration schemes violating the integration consistency, the solution errors eventually are controlled by the integration error and optimal convergence cannot be achieved even with high order quadrature rules. On the other hand, for the consistent integration approaches, e.g., the stabilized conforming nodal integration and the more recent reproducing kernel gradient smoothing integration, the integration and interpolation errors have the same accuracy order and thus an optimal convergence of Galerkin meshfree methods is ensured. The proposed theoretical error estimates for Galerkin meshfree methods are well demonstrated by numerical results.

Phase‐field modeling of brittle fracture in a 3D polycrystalline material via an adaptive isogeometric‐meshfreeapproach

SummaryPhase‐field modeling, which introduces the regularized representation of sharp crack topologies, provides a convenient strategy for tackling 3D fracture problems. In this work, an adaptive isogeometric‐meshfree approach is developed for the phase‐field modeling of brittle fracture in a 3D polycrystalline material. The isogeometric‐meshfree approach uses moving least‐squares approximations to construct the equivalence between isogeometric basis functions and meshfree shape functions, thus inheriting the flexible local mesh refinement scheme from a meshfree method. This refinement scheme is improved by introducing an error estimator that includes both the phase field and its gradient. With the present approach, numerical implementations of the adaptive phase‐field modeling that introduces the anisotropy of fracture resistance in polycrystals are proposed. In this way, propagating cracks can be dynamically tracked, and the mesh near cracks is refined in a meshfree manner without requiring a priori knowledge of crack paths. Furthermore, the intergranular and transgranular crack propagation patterns in polycrystalline materials can be simulated by the present approach. A series of numerical examples that deal with the isotropic and anisotropic fracture are investigated to demonstrate the robustness and effectiveness of the proposed approach.

A local gradient smoothing method for solving strong form governing equation

The gradient smoothing method(GSM) is used to approximate the derivatives of the meshfree shape function and it usually generates the smoothing domain by connecting the midpoints of sides and centroids of elements around the interested node. In order to simplify the generating process and geometric computation of smoothing domain, a local gradient smoothing method (LGSM) has been proposed in which the local smoothing domains corresponding to the nodes are independent of each other. The proposed method is very flexible to generate the smoothing domain, which can have a simple and regular shape. The derivatives of the meshfree shape function approximated by the proposed method was applied to the governing strong from of system equations at all nodes of problem domain. The convergence and accuracy of the proposed method according to the size of smoothing domain are examined. Numerical examples to some typical benchmark problems illustrated the efficiency of the proposed method.

A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations

Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann–Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov–Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen–Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results.

A semi-analytical mesh-free method for 3D free vibration analysis of bi-directional FGP circular structures subjected to temperature variation

In this present paper, a semi-analytical mesh-free method is employed for the three-dimensional free vibration analysis of a bi-directional functionally graded piezoelectric circular structure. The dependent variables have been expanded by Fourier series with respect to the circumferential direction and have been discretized through radial and axial directions based on the mesh-free shape function. The current approach has a distinct advantage. The nonlinear Green-Lagrange strain is employed as the relationship between strain and displacement fields to observe thermal impacts in stiffness matrices. Nevertheless, high order terms have been neglected at the final steps of equations driving. The material properties are assumed to vary continuously in both radial and axial directions simultaneously in accordance with a power law distribution. The convergence and validation studies are conducted by comparing our proposed solution with available published results to investigate the accuracy and efficiency of our approach. After the validation study, a parametric study is undertaken to investigate the temperature effects, different types of polarization, mechanical and electric boundary conditions and geometry parameters of structures on the natural frequencies of functionally graded piezoelectric circular structures.

Arbitrary order recursive formulation of meshfree gradients with application to superconvergent collocation analysis of Kirchhoff plates

A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an nth order recursive meshfree gradient is formulated as an interpolation of the (n − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a pth degree basis function not only meet the conventional pth order consistency conditions for standard gradients, but also satisfy (p + 1)th to (p + n − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.

A quasi-consistent integration method for efficient meshfree analysis of Helmholtz problems with plane wave basis functions

A quasi-consistent integration method is presented for the efficient meshfree analysis of Helmholtz problems. The plane wave basis functions are employed for the reproducing kernel meshfree approximation to accurately represent the acoustic field resulting from Helmholtz problems. In order to improve the computational efficiency of Galerkin meshfree analysis of Helmholtz problems, a reproducing kernel gradient smoothing approach is introduced into the meshfree formulation with plane wave basis functions. In the proposed method, the smoothed gradients of meshfree shape functions with plane wave basis functions are built upon a reproducing kernel gradient representation and the integration consistency of Galerkin meshfree formulation is implicitly ensured. Furthermore, a quasi-consistent integration scheme is proposed to compute the smoothed gradients, which aims to balance the efficiency and accuracy for meshfree analysis of Helmholtz problems. The proposed integration method leads to fully consistent integration when one wave direction is considered, and nearly consistent integration if two wave directions are taken into account, where the boundary sample points of integration cells are particularly preferred since they are simultaneously used by neighboring integration cells with efficiency gain. Numerical results demonstrate that the proposed methodology is much more efficient and accurate for Galerkin meshfree analysis of Helmholtz problems, in comparison with the standard meshfree formulations using high order Gauss quadrature rules.

Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving least-squares approach

This paper reports an isogeometric-meshfree moving least-squares approach for the adaptive analysis of crack propagation in thin-shell structures within the context of linear elastic fracture mechanics. The present approach is developed based on the equivalence of the moving least-squares meshfree shape functions and the isogeometric basis functions, which provides an effective strategy of adaptive mesh refinement for isogeometric analysis (IGA) in a straightforward meshfree manner. The adaptivity of the mesh refinement is achieved by utilizing a gradient-based error estimator to identify the meshes that need to be refined by adding linear reproducing points. The Kirchhoff–Love theory is further applied in the isogeometric-meshfree moving least-squares formulation to simplify the modeling of cracked thin-shell structures by neglecting the rotational degrees of freedom. In this way, the singularity of stress fields near the crack tip and the discontinuity of displacement fields around the crack surface can be efficiently captured by the adaptive mesh refinement to generate accurate results. A series of two-dimensional static and quasi-static crack propagation problems of thin-shell structures are investigated. It is found that the adaptive refinement strategy makes the present approach achieve higher convergence rate and computational efficiency than IGA and the meshfree method. The predicted propagation paths obtained by the present approach are in good agreement with the previously reported results.

Analysis of laminated composite plates based on THB-RKPM method using the higher order shear deformation plate theory

In the present investigation, static, free vibration and buckling response of laminated composite plates based on the coupling of truncated hierarchical B-splines (THB-splines) and reproducing kernel particle method (RKPM) within higher order shear deformation plate theory are presented. The coupled THB-RKPM method blends the 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 shown that THB-RKPM method is ideally appropriate for local refinement of laminated composite plates in the framework of isogeometric analysis. The flexibility of the proposed method for refining basis functions leads to decrease the computational cost without losing the accuracy of the solution. Numerical examples considering different boundary conditions, various aspect ratios, stiffness ratios and fiber orientations demonstrate validity and versatility of the proposed method.