International Journal of Computational Methods

Within the framework of edge-based smoothed point interpolation method (ES-PIM), a fully automatic adaptive procedure has been proposed by introducing a novel local critical value definition. Owing to the softened stiffness, ES-PIM can generally provide much better results than the traditional finite element method using the simplest linear triangle elements. By comparing with most adaptive models which need to set a refinement rate manually, a novel local critical value considering the area-dependent average magnitude of error indicator for all the cells has been introduced in this work, which leads to an adaptive ES-PIM model without manual operation during the adaptive process. Numerical results have shown that the present adaptive model conducts the adaptive process dynamically, adds the new nodes properly and provides more accurate results with less field nodes than both the model with uniform refinement and the adaptive model using predefined refinement rate. Also considering linear triangular background elements used, the present method is supposed to have much potential for solving complicated engineering problems.

Smoothed finite element method (S-FEM) is a new general numerical method which has been applied to solve various practical engineering problems. It combines standard finite element method (FEM) and meshfree techniques based on the weaken-weak (W2) formulation. This project, for the first time, develops a preprocessor software package SFEM-Pre for creating types of two-dimensional (2D) and three-dimensional (3D) S-FEM models following strictly the S-FEM theory. Because the software architecture of our 3D processor is the same as our 2D preprocessor, we will mainly introduce the 2D preprocessor in terms of software design for easier description, but the examples will include both 2D and 3D cases to fully demonstrate and validate the whole preprocessor of S-FEM. Our 2D preprocessor package is equipped with a graphical user interface (GUI) for easy use, and with a connectivity database for efficient computation. Schemes are developed for not only automatically meshes the problem domains using our GUI, but also accepts various geometry files made available from some existing commercial software packages, such as ABAQUS®and HyperMesh®. In order to improve the efficiency of our preprocessor, a parallel triangulation mesh generator has also been developed based on the advancing front technique (AFT) to create triangular meshes for complex geometry, and at the same time to create six types of connectivity needed for various S-FEM models. In addition, a database is implemented in our code to record all these connectivity to avoid duplicated calculation. Finally, intensive numerical experiments are conducted to validate the efficiency, accuracy and stability of our preprocessor codes. It is shown that with our preprocessor, an S-FEM can be created automatically without much human intervention for geometry of arbitrary complexity.

A state-of-the-art tetrahedral smoothed finite element method, F-barES-FEM-T4, is demonstrated on viscoelastic large deformation problems. The stress relaxation of viscoelastic materials brings near incompressibility when the long-term Poisson’s ratio is close to 0.5. The conventional hybrid 4-node tetrahedral (T4) elements cannot avoid the shear locking and pressure checkerboarding issues, meanwhile F-barES-FEM-T4 can suppress these issues successfully by adopting the edge-based smoothed finite element method (ES-FEM) with the aid of the F-bar method and the cyclic smoothing procedure. A few examples of analyses verify that F-barES-FEM-T4 is locking-free and pressure oscillation-free in viscoelastic analyses as well as in nearly incompressible hyperelastic or elastoplastic analyses.

This paper reports a work to develop a general solver of smoothed finite element methods (S-FEMs) for stress analysis of 2D and 3D solid mechanics problems. In the solver, several efficient algorithms are proposed to construct the real smoothing domains and calculate all the connectivity for later computation. The present implementation of S-FEM distinguishes from the existing published implementation of S-FEM in terms of computing the smoothed strains. The existing implementation uses the volume/area-weighted average method to compute the smoothed strains for the smoothing domain. The present algorithm uses surface/line integrals to compute the smoothed strains strictly following the general ${\mathrm{\text{W}}}^2$ formulation of S-FEM. Therefore, the present solver is the most general, applicable to any polygon elements, and even to higher order interpolation including using radial basis functions. Numerical experiments for a number of 2D and 3D solid mechanics examples are carried out to demonstrate the effectiveness and reliability of our solver.

In this paper, transient heat transfer problems are analyzed using the smoothed finite element methods (S-FEMs) with explicit time integration. For a numerical method with spatial discretization, the computational cost per time step in the explicit method is less than that in the implicit method, but the time step is much smaller in the explicit analysis than that in the implicit analysis when the same mesh is used. This is because the stability is of essential importance. This work thus studies the stability of S-FEMs, when applied to transient heat transfer problems. Relationships are established between the critical time steps used in S-FEMs with the maximum eigenvalues of the thermal stiffness (conduction) matrix and mass matrix. It is found that the critical time step relates to the “softness” of the model. For example, node-based smoothed finite element method (NS-FEM) is softer than edge-based smoothed finite element method (ES-FEM), which leads to that the critical time step of NS-FEM is larger than that of ES-FEM. Because computing the eigenvalues and condition numbers of the stiffness matrices is very expensive but valuable for stability analysis, we proposed a concise and effective algorithm to estimate the maximum eigenvalue and condition number. Intensive numerical examples show that our scheme for computing the critical time step can work accurately and stably for the explicit method in FEM and S-FEMs.

Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ($\beta$FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter $\beta$ is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of $\beta$FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the $\beta$FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

In this work, a three-dimensional (3D) nonlinear smoothed finite element method (S-FEM) solver is developed for large deformation problems. Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements (NS-FEM-Te4 and FS-FEM-Te4) are adopted to find the solution bounds in strain energy. The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4, while the upper bound solutions are obtained using NS-FEM-Te4. A combined $\alpha$S-FEM-Te4 with a scaling factor $\alpha$ that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting $\alpha$. This is achieved using the property that a successive change of scaling factor $\alpha$ can make the model transform from “overly-stiff” to “overly-soft”. Considering the properties of FS-FEM and NS-FEM, a selective FS/NS-FEM-TE4 is also used to solve 3D nonlinear large deformation problems, which produces a lower bound in strain energy. Hyperelastic Mooney–Rivlin and Ogden materials are both used in this study. Numerical examples reveal that S-FEM-Te4 is an effective method for obtaining solution bounds together with the standard FEM, and the FS-FEM-Te4, NS-FEM-Te4 and selective FS/NS-FEM-TE4 are robust with the high accuracy and computational efficiency for large deformation nonlinear problems.

Smoothed finite element method (S-FEM) based on triangular elements has recently been widely used for solving solid mechanics problems. In this paper, a novel $\alpha$S-FEM using quadrilateral elements ($\alpha$SFEM-Q4) is proposed to obtain ultra-accurate solutions in the displacement and strain energy for solid mechanics problems. This method combines node-based S-FEM (NS-FEM), edge-based S-FEM (ES-FEM) and cell-based S-FEM (CS-FEM) equipped with a scale factor $\alpha$ that controls contribution from each of these three different S-FEM models. This novel combination makes the best use of the upper bound property of the NS-FEM and the lower bound property of the CS-FEM (with 4 or more sub-smoothing domains for each element), and establishes a continuous strain-energy function of a scale factor $\alpha$ for obtaining close-to-exact solutions. Our $\alpha$SFEM-Q4 also ensures the variational consistence and the compatibility of the displacement field, and hence guarantees to reproduce linear field exactly. Various solid mechanics problems are presented to validate the stability, effectiveness and ultra-accuracy of the proposed method.

A new concept of smoothed finite element method (S-FEM) using 10-node tetrahedral (T10) elements, CS-FEM-T10, is proposed. CS-FEM-T10 is a kind of cell-based S-FEM (CS-FEM) and thus it smooths the strain only within each T10 element. Unlike the other types of S-FEMs [node-based S-FEM (NS-FEM), edge-based S-FEM (ES-FEM), and face-based S-FEM (FS-FEM)], CS-FEM can be implemented in general finite element (FE) codes as a piece of the element library. Therefore, CS-FEM-T10 is also compatible with general FE codes as a T10 element. A concrete example of CS-FEM-T10 named SelectiveCS-FEM-T10 is introduced for large deformation problems of nearly incompressible solids. SelectiveCS-FEM-T10 subdivides each T10 element into 12 four-node tetrahedral (T4) subelements with an additional dummy node at the element center. Two types of strain smoothing are conducted for the deviatoric and hydrostatic stress evaluations and the selective reduced integration (SRI) technique is utilized for the stress integration. As a result, SelectiveCS-FEM-T10 avoids the shear/volumetric locking, pressure checkerboarding, and reaction force oscillation in nearly incompressible solids. In addition, SelectiveCS-FEM-T10 has a relatively long-lasting property in large deformation problems. A few examples of large deformation analyses of a hyperelastic material confirm the good accuracy and robustness of SelectiveCS-FEM-T10. Moreover, an implementation of SelectiveCS-FEM-T10 in the FE code ABAQUS as a user-defined element (UEL) is conducted and its capability is discussed.

Nonlinear bifurcation theory is very powerful and efficient to deal with multifarious nonlinear phenomena emerging from physics, biology, engineering, and economics. However, the execution of augmented system for bifurcation points will be very complicated for a system with too many variables. The purpose of this paper is to give a new optimization based method for computation of fold and Hopf bifurcation points and discuss their applications in stability analysis of power system. The validity of the proposed method has been testified by a WSCC9-bus power system in which the load increment at any bus was taken as a parameter.

The smoothed finite element method (S-FEM) has been developed recent years and is increasingly used for stress analysis for engineering design of structures, due to its high computational accuracy and outstanding robustness in against mesh distortion. However, there is currently no commercial S-FEM software package available for convenient engineering applications. This paper aims to integrate S-FEM into the ${\text{ABAQUS}}^{\circledR }$ software, because it is most widely used in engineering analyses and well integrated in computer aided engineering (CAE). From a family of S-FEM models, the cell-based finite element method (CS-FEM) is chosen to be implemented in ABAQUS, because a smoothing cell in the CS-FEM involves only one element, and hence the implementation can be achieved via the use of the user-defined element library (UEL). Since only nodal displacement results can be extracted when UEL subroutine is used in ABAQUS, a post-processing program is also developed to compute nodal strains/stresses and strain energy results that are useful in structure analysis and CAE. Our CS-FEM UEL is validated using four numerical examples under plane stress conditions. Compared with standard ABAQUS, the CS-FEM in ABAQUS improves the solution accuracy remarkably, and we have also confirmed the robustness of CS-FEM against heavily distorted meshes.

In this paper, we discuss the implementation of a cell-based smoothed finite element method (CSFEM) within the commercial finite element software Abaqus. The salient feature of the CSFEM is that it does not require an explicit form of the derivative of the shape functions and there is no need for isoparametric mapping. This implementation is accomplished by employing the user element subroutine (UEL) feature in Abaqus. The details on the input data format together with the proposed user element subroutine, which forms the core of the finite element analysis are given. A few benchmark problems from linear elastostatics in both two and three dimensions are solved to validate the proposed implementation. The developed UELs and the associated input files can be downloaded from https://github.com/nsundar/SFEM_in_Abaqus.

The complex variable boundary element-free method (CVBEFM) is a meshless method that takes the advantages of both boundary integral equations (BIEs) in dimension reduction and the complex variable moving least squares (CVMLS) approximation in element elimination. The CVBEFM is developed in this paper for solving 3D problems. This paper is an attempt in applying complex variable meshless methods to 3D problems. Formulations of the CVMLS approximation on 3D surfaces and the CVBEFM for 3D potential and Helmholtz problems are given. In the current implementation, the CVMLS shape function of 3D problems is formed with 1D basis functions, and the boundary conditions in the CVBEFM can be applied directly and easily. Some numerical examples are presented to demonstrate the method.

In computational fluid dynamics there have been many attempts to combine the advantages of having a fixed mesh, on which to carry out spatial calculations, with using particles moving according to the velocity field. These ideas in fact go back to particle-in-cell methods, proposed about 60 years ago. Of course, some procedure is needed to transfer field information between particles and mesh. There are many possible choices for this “assignment”, or “projection”. Several requirements may guide this choice. Two well-known ones are conservativity and stability, which apply to volume integrals of the fields. An additional one is here considered: preservation of information. This means that assignment from the particles onto the mesh and back should yield the same field values when the particles and the mesh coincide in position. The resulting method is termed “mass” assignment, due to its strong similarities with the finite element method. Several procedures are tested, including the well-known FLIP, on three scenarios: simple 1D convection, 2D convection of Zalesak’s disk, and a CFD simulation of the Taylor–Green periodic vortex sheet. Mass assignment is seen to be clearly superior to other methods.