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A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities.

A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation.

Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency.

Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

The paper aims to predict longitudinal deformation of a tunnel caused by grouting under the tunnel bottom in advance according to the grouting parameters, which can ensure the safety of the tunnel structure during the grouting process and also help to design the grouting parameters.

The paper adopted the analytical approach for calculating the longitudinal deformation of a shield tunnel caused by grouting under a tunnel, including usage of the Mindlin’s solution, the minimum potential energy principle and case validation.

The paper provides a variational method for calculating the longitudinal deformation of a shield tunnel in soft soil caused by grouting under the tunnel, which has high computational efficiency and accuracy.

This paper fulfils an identified need to study how the longitudinal deformation of a shield tunnel in soft soil caused by grouting under the tunnel can be calculated.

The purpose of this paper is to exploit the cyclic symmetry to reduce the computational expense of scaled boundary method (SBM) which may cumber its further application.

A partitioning EFG-SB (Element-free Galerkin-SB) algorithm is proposed for the two-dimensional elastic analysis of cyclically symmetric structures.

By utilizing the cyclic symmetry and partitioning algorithm, the whole computational cost can be significantly reduced. Three numerical examples are given to illustrate the advantages of the proposed algorithm.

It is proved that the matrices of eigenvalue and system equations of EFG-SBM for cyclically symmetric structures are block-circulant so long as a kind of symmetry-adapted reference coordinate system is adopted. No matter whether displacement constraints are cyclically symmetric or not, the partition is available for the eigenvalue equations. Therefore the major computational cost can be saved via the proposed partitioning algorithm. This paper provides an efficient algorithm for the two-dimensional elastic analysis of cyclically symmetric structures using EFG-SBM. A higher computing efficiency can be expected since the proposed partitioning algorithm facilitates parallel processing.

The meshfree node-based smoothed point interpolation method (NS-PIM) is extended to the forward and inversion analysis of a high gravelly soil core rock-fill dam during construction periods.

As one member of the meshfree methods, the NS-PIM has the advantages of “softer” stiffness and adaptability to large deformations which is quite indispensable for the stability analysis of rock-fill dams. In this work, the present method contains a reconstruction procedure to deal with the existence or nonexistence of the construction layers. After verifying the validity of the NS-PIM method for nonlinear elastic model during construction period, the convergence features of the NS-PIM and FEM methods are further investigated with different mesh schemes. Furthermore, the NS-PIM and FEM methods are applied for the forward analysis of a high gravelly soil core rock-fill dam and the convergence features under complex stress conditions are also studied using the rock-fill dam model. Finally, the NS-PIM method is used to calculate the Duncan–Chang parameters of the deep overburden under the high gravelly soil core rock-fill dam based on the back-propagation neural network method.

The results show that: the NS-PIM solution for construction analysis still possesses the property of upper bound solution even under complex stress conditions and can provide comparatively more conservative results for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution; the inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction period and the calculated results show good agreement with the measured displacement values and it is feasible to apply the NS-PIM to the forward and inversion analysis of high rock-fill dams on deep overburden during construction periods.

In further study, the feasibility of three-dimensional problems, elastic–plastic problems, contact problems and multipoint inversion can still be probed in the NS-PIM solution for the forward and inversion analysis of high rock-fill dams on deep overburden.

This paper introduced a method for the forward and inversion analysis of high rock-fill dams during construction period using the NS-PIM solution. The property of upper bound solution ensures that the NS-PIM can provide more conservative results for safety evaluation. The inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction periods.

First, the analysis from forward to inversion for high rock-fill dams during construction period using the NS-PIM solution is accomplished in this work. A procedure dealing with the existence or nonexistence of the construction layers is also developed for the construction analysis. Second, it is confirmed in this work that the NS-PIM still possesses the property of upper bound solution even under complex stress conditions (the forward analysis of high rock-fill dams during construction period). Thus, more conservative results can be provided for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution. Third, the calculated material parameters of the deep overburden in this work can be used for further studies of the high rock-fill dam.

The purpose of this paper is to propose a hybridization numerical method to solve the plastic deformation of metal working based on the flow function method and meshless method.

The proposed method is named as flow function-element free Galerkin (F-EFG) method. It uses the flow function as the basic unknown quantity to get the basic control equation, the compactly supported approximate function to establish a local approximate flow function by means of moving least square approximation, and the element free Galerkin (EFG) method to solve variational equation. The F-EFG method takes the upper limit method essence of flow function method, and the convergence, stability, and error characteristics of EFG method.

The steady extrusion process of the axisymmetric extrusion problems as well as the extrusion deformation law and main field variables are subjects in the modeling and simulation analysis using F-EFG method. The results show that the F-EFG method has good computational efficiency and accuracy.

The F-EFG method proposed in this paper has the advantages of high-solution precision of flow function method and large deformation solution of element free method. It overcomes the difficulties in global flow function establishment in flow function method and low-solution efficiency in element free method. The method is beneficial to the development of flow function method and element free method.

This article aims to develop an accurate and efficient meshfree Galerkin method based on the strain smoothing technique for linear elastic continuous and fracture problems.

This paper proposed a generalized linear smoothed meshfree method (LSMM), in which the compatible strain is reconstructed by the linear smoothed strains. Based on the idea of the weighted residual method and employing three linearly independent weight functions, the linear smoothed strains can be created easily in a smoothing domain. Using various types of basic functions, LSMM can solve the linear elastic continuous and fracture problems in a unified way.

On the one hand, the LSMM inherits the properties of high efficiency and stability from the stabilized conforming nodal integration (SCNI). On the other hand, the LSMM is more accurate than the SCNI, because it can produce continuous strains instead of the piece-wise strains obtained by SCNI. Those excellent performances ensure that the LSMM has the capability to precisely track the crack propagation problems. Several numerical examples are investigated to verify the accurate, convergence rate and robustness of the present LSMM.

This study provides an accurate and efficient meshfree method for simulating crack growth.

This paper aims to introduce a method to couple truss finite elements to the material point method (MPM). It presents modeling reinforced material using MPM and describes how to consider the bond behavior between the reinforcement and the continuum.

The embedded approach is used for coupling reinforcement bars with continuum elements. This description is achieved by coupling continuum elements in the background mesh to the reinforcement bars, which are described using truss- finite elements. The coupling is implemented between the truss elements and the continuum elements in the background mesh through bond elements that allow for freely distributed truss elements independent of the continuum element discretization. The bond elements allow for modeling the bond behavior between the reinforcement and the continuum.

The paper introduces a novel method to include the reinforcement bars in the MPM applications. The reinforcement bars can be modeled without any constraints with a bond-slip constitutive model being considered.

As modeling of reinforced materials is required in a wide range of applications, a method to include the reinforcement into the MPM framework is required. The proposed approach allows for modeling reinforced material within MPM applications.

This study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C¹ basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.

The weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C⁰ non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C¹) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.

It is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.

Fourth-order PFM is implemented using C¹ non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996‐1999.

The state space method (SSM) is good at analyzing the interfacial physical quantities in laminated materials, while it has difficulty in calculating the mechanical quantities of interior points, which can be easily evaluated by the boundary element method (BEM). However, the material has to be divided into many subdomains when the traditional BEM is applied to analyze the functionally graded material (FGM), so that the computational amount will be increased enormously. This study aims to couple these two methods to strengthen their advantages and overcome their disadvantages.

Herein, a state space BEM in which the SSM is coupled by the BEM is proposed to analyze the elasticity of FGMs, where one BEM domain is set and the others belong to SSM domains. The discretized elements occur only on the boundary of the BEM domain and at the interfaces between different SSM domains. In SSM domains, the horizontal interfaces of FGMs are discretized by linear elements and the variables along the vertical direction are yielded by the precise integration method.

The accuracy of the proposed method is verified by comparing the present results with the ones from the finite element method (FEM). It is found that the present method can provide accurate displacements and stresses in the FGMs by fewer freedom degrees in comparison with the FEM. In addition, the present method can provide continuous interfacial stresses at the interfaces between different material domains, while the interfacial stresses by the FEM are discontinuous.

The system equations of the state space BEM are built by combining the boundary integral equation with the state equations according to the continuity conditions at the interfaces. The mechanical parameters of any inner point can be evaluated by the boundary integral equation after the unknowns on the boundaries and interfaces are determined by the system equation.