Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures

A dispersion analysis is carried out to study the dynamic behavior of the Hermite reproducing kernel (HRK) Galerkin meshfree formulation for thin beam and plate problems. The HRK approximation utilizes both the nodal deflectional and rotational variables to construct the meshfree approximation of the deflection field within the reproducing kernel framework. The discrete Galerkin formulation is fulfilled with the method of sub-domain stabilized conforming integration. In the dispersion analysis following the HRK Galerkin meshfree semi-discretization, both the deflectional and rotational nodal variables are expressed by harmonic functions and then substituted into the semi-discretized equation to yield the characteristic equation. Subsequently the numerical frequency and phase speed can be obtained. The transient analysis with full-discretization is performed by using the central difference time integration scheme. The results of dispersion analysis of thin beams and plates show that compared to the conventional Gauss integration-based meshfree formulation, the proposed method has more favorable dispersion performance. Thereafter the superior performance of the present method is also further demonstrated by several transient analysis examples.     

A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates

A Hermite reproducing kernel Galerkin meshfree approach is proposed for buckling analysis of thin plates. This approach employs the Hermite reproducing kernel meshfree approximation that incorporates both the deflectional and rotational nodal variables into the approximation of the plate deflection and the C ¹ continuous approximation requirement for the Galerkin analysis of thin plates can be easily achieved herein. The strain smoothing operation is consistently introduced to construct the smoothed rotation and curvature fields which appear in the weak form governing the thin plate buckling. The domain integration of the weak form is carried out by the method of sub-domain stabilized conforming integration with the smoothed measures of rotation and curvature, as leads to an efficient discrete meshfree formulation for the eigenvalue problem of thin plate buckling. A series of benchmark buckling problems are presented to assess the proposed algorithm and the results uniformly demonstrate the present approach is very effective and it performs superiorly compared to the conventional Galerkin meshfree formulations whose domain integration are performed by Gauss quadrature rules.     

Quadratically consistent nodal integration for second order meshfree Galerkin methods

Robust and efficient integration of the Galerkin weak form only at the approximation nodes for second order meshfree Galerkin methods is proposed. The starting point of the method is the Hu-Washizu variational principle. The orthogonality condition between stress and strain difference is satisfied by correcting nodal derivatives. The corrected nodal derivatives are essentially linear functions which can exactly reproduce linear strain fields. With the known area moments, the stiffness matrix resulting from these corrected nodal derivatives can be exactly evaluated using only the nodes as quadrature points. The proposed method can exactly pass the quadratic patch test and therefore is named as quadratically consistent nodal integration. In contrast, the stabilized conforming nodal integration (SCNI) which prevails in the nodal integrations for meshfree Galerkin methods fails to pass the quadratic patch test. Better accuracy, convergence, efficiency and stability than SCNI are demonstrated by several elastostatic and elastodynamic examples.     

Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration

A Hermite reproducing kernel (HRK) Galerkin meshfree formulation is presented for free vibration analysis of thin plates. In the HRK approximation the plate deflection is approximated by the deflection as well as slope nodal variables. The nth order reproducing conditions are imposed simultaneously on both the deflectional and rotational degrees of freedom. The resulting meshfree shape function turns out to have a much smaller necessary support size than its standard reproducing kernel counterpart. Obviously this reduction of minimum support size will accelerate the computation of meshfree shape function. To meet the bending exactness in the static sense and to remain the spatial stability the domain integration for stiffness as well as mass matrix is consistently carried out by using the sub-domain stabilized conforming integration (SSCI). Subsequently the proposed formulation is applied to study the free vibration of various benchmark thin plate problems. Numerical results uniformly reveal that the present method produces favorable solutions compared to those given by the high order Gauss integration (GI)-based Galerkin meshfree formulation. Moreover the effect of sub-domain refinement for the domain integration is also investigated.     

A Hermite reproducing kernel approximation for thin‐plate analysis with sub‐domain stabilized conforming integration

A Hermite reproducing kernel (RK) approximation and a sub‐domain stabilized conforming integration (SSCI) are proposed for solving thin‐plate problems in which second‐order differentiation is involved in the weak form. Although the standard RK approximation can be constructed with an arbitrary order of continuity, the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems. By imposing the Kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously, it is demonstrated that the minimum normalized support size (coverage) of kernel functions can be significantly reduced. With this proposed approximation, the Galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived. Subsequently, an SSCI method is developed to achieve the exact pure bending solution as well as to maintain spatial stability. Numerical examples demonstrate that the proposed formulation offers superior convergence rates, accuracy and efficiency, compared with those based on higher‐order Gauss quadrature rule. Copyright © 2007 John Wiley & Sons, Ltd.     

A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition

In this article, we present a novel nodal integration scheme for meshfree Galerkin methods, which draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small-strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell-based Gauss integrated meshfree method.     

Locking-free curved beam element based on curvature

The formulation of a curved beam element with 3 nodes for curvature to eliminate the shear/membrane locking phenomenon is presented. The element is based on curvature so that it may represent the bending energy fully, and the shear/membrane strain energy is incorporated into the formulation by the equilibrium equations. To deal with general boundary conditions, a transformation matrix between nodal curvature and nodal displacement vector is introduced. Several examples are presented in order to verify the element formulation and its analytical capability. The solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently, showing no locking phenomena, and that it is also applicable to the analysis of both thin and thick curved beams.     

A pure bending exact nodal-averaged shear strain method for finite element plate analysis

An averaged shear strain method, based on a nodal integration approach, is presented for the finite element analysis of Reissner–Mindlin plates. In this work, we combine the shear interpolation method from the MITC4 plate element with an area-weighted averaging technique for the nodal integration of shear energy to relieve shear locking in the thin plate analysis as well as to pass the pure bending patch test. In order to resolve the numerical instability caused by the direct nodal integration, the bending strain field is computed by a sub-domain nodal integration approach based on the Sub-domain Stabilized Conforming Integration and a modified curvature smoothing scheme. The resulting nodally integrated smoothed strain formulation is shown to contain only the primitive variables and thus can be easily implemented in the existing displacement-based finite element plate formulation. Several numerical examples are presented to demonstrate the accuracy of the present method.     

Floating stress‐point integration meshfree method for large deformation analysis of elastic and elastoplastic materials

A new meshfree formulation of stress‐point integration, called the floating stress‐point integration meshfree method, is proposed for the large deformation analysis of elastic and elastoplastic materials. This method is a Galerkin meshfree method with an updated Lagrangian procedure and a quasi‐implicit time‐advancing scheme without any background cell for domain integration. Its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation, which is not generally used in meshfree formulations. Hence, this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved. The details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method. Copyright © 2012 John Wiley & Sons, Ltd.     

Bending and vibration responses of laminated composite plates using an edge-based smoothing technique

In this paper, bending and vibration analysis of laminated composite plates is carried out using a novel triangular composite plate element based on an edge-based smoothing technique. The present formulation is based on the first-order shear deformation theory, and the discrete shear gap (DSG) method is employed to mitigate the shear locking. The smoothed Galerkin weak form is adopted to obtain the discretized system equations, and edge-based smoothing domains are used for the numerical integration to improve the accuracy and the convergence rate of the method. The present formulation is coded and used to solve various example problems of bending and free vibration of laminated composite plates. It is found that the present method can provide excellent results with a wide range of thickness and is free of shear locking.     

Limit analysis of plates using the EFG method and second‐order cone programming

The meshless element‐free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least‐squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non‐smooth minimization problem can be transformed into a form suitable for solution using second‐order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper‐bound solutions for benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Quasi-convex reproducing kernel meshfree method

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.     

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.     

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher‐order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum‐entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two‐dimensional and three‐dimensional elliptic (Poisson and linear elastostatic) boundary‐value problems that demonstrate the effectiveness of the proposed formulation are presented. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical integration of the Galerkin weak form in meshfree methods

The numerical integration of Galerkin weak forms for meshfree methods is investigated and some improvements are presented. The character of the shape functions in meshfree methods is reviewed and compared to those used in the Finite Element Method (FEM). Emphasis is placed on the relationship between the supports of the shape functions and the subdomains used to integrate the discrete equations. The construction of quadrature cells without regard to the local supports of the shape functions is shown to result in the possibility of considerable integration error. Numerical studies using the meshfree Element Free Galerkin (EFG) method illustrate the effect of these errors on solutions to elliptic problems. A construct for integration cells which reduces quadrature error is presented. The observations and conclusions apply to all Galerkin methods which use meshfree approximations.     

Reissner-Mindlin板壳无网格法的闭锁与灵敏度分析及优化的研究

该文利用罚函数法施加边界条件,建立了Reissner-Mindlin板壳无网格法的离散形式,通过数值锁死试验,探讨了EFG法、RPIM以及基于节点积分的无网格法在解决Reissner-Mindlin板壳闭锁问题中所存在的优缺点。所得结果表明,基于匹配近似场和节点积分方案的无网格法在处理剪切闭锁问题时具有优越性。然后以SCNI-MLS无网格法为基础,对Reissner-Mindlin板壳结构的尺寸、形状和轮廓设计进行了统一的设计灵敏度分析,结合约束变尺度序列二次规划法,完成了SCNI-MLS无网格法壳结构优化设计的算例,算例结果验证了所建立灵敏度分析的精度和优化方法的可行性。     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

Meshfree simulations of thermo-mechanical ductile fracture

In this work, a meshfree method is used to simulate thermo-mechanical ductile fracture under finite deformation. A Galerkin meshfree formulation incorporating the Johnson-Cook damage model is implemented in numerical computations. We are interested in the simulation of thermo-mechanical effects on ductile fracture under large scale yielding. A rate form adiabatic split is proposed in the constitutive update. Meshfree techniques, such as the visibility criterion, are used to modify the particle connectivity based on evolving crack surface morphology. The numerical results have shown that the proposed meshfree algorithm works well, the meshfree crack adaptivity and re-interpolation procedure is versatile in numerical simulations, and it enables us to predict thermo-mechanical effects on ductile fracture.     

An accelerated,convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics

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. Copyright © 2015 John Wiley & Sons, Ltd.     

Meshfree Galerkin method for a rotating Euler-Bernoulli beam

In this article, we solve the free vibration problem of a rotating Euler-Bernoulli beam using the meshfree Galerkin method. Radial basis functions are used for interpolation. An improved formulation of the rotating Euler-Bernoulli beam free vibration problem with the Galerkin method is explained for the first time, which results into a symmetric stiffness matrix and gives a significant computational advantage over the formulation given in the existing literature. A conventional hp-version of Galerkin finite element method is used for comparison. Results are obtained at different non-dimensional rotating speeds of a rotating beam. Results show excellent agreement with the existing literature.