Showing papers on "Smoothed finite element method published in 2022"

Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element

Abstract: In this paper, the free vibration of the functionally graded porous (FGP) non-uniform annular-nanoplates lying on Winkler foundation (WF) is studied by using the smoothed finite element method based on the first-order shear deformation theory (FSDT). The combination of the mixed interpolation of the tensorial components for the three-node triangular element (MITC3 element) and the edge-based smoothed finite element method (ES-FEM) creates the ES-MITC3 element. This element is employed to avoid the shear locking problem as well as to improve the accuracy of the original MITC3 element. The small-scale effect is considered based on the nonlocal theory. Applying Hamilton's principle, the governing equation of the FGP non-uniform thickness annular-nanoplate is derived. Material properties of the nanoplate are characterized by two parameters: power-law index ( k ) and maximum porosity distributions ( Ω ) in the forms of cosine functions. The results of the present work are compared with other published work to verify accuracy and reliability. Moreover, the effects of geometry parameters and material properties on the free vibration of FGP non-uniform annular-nanoplates are comprehensively investigated.

Eighty Years of the Finite Element Method: Birth, Evolution, and Future

Abstract: Abstract This document presents comprehensive historical accounts on the developments of finite element methods (FEM) since 1941, with a specific emphasis on developments related to solid mechanics. We present a historical overview beginning with the theoretical formulations and origins of the FEM, while discussing important developments that have enabled the FEM to become the numerical method of choice for so many problems rooted in solid mechanics.

A selective smoothed finite element method for 3D explicit dynamic analysis of the human annulus fibrosus with modified composite-based constitutive model

Abstract: The human annulus fibrosus (HAF) is the major component in response to external forces for the intervertebral disk (IVD), which maintains the stability and flexibility of human spine. It can be assumed to be an anisotropic nearly incompressible hyperelastic composite consisting of collagen fibers and matrix in the numerical simulations of biomechanics. However, due to the geometric complexity and material nonlinearity of HAF, the conventional Finite Element Method (FEM) often gets into difficulties in mesh generation and uncertainty of accuracy control. In this paper, a modified composite-based constitutive model, which considers the slight compressibility of ground substance and the shear interaction between collagen fibers and matrix, is developed to describe the mechanical behavior of HAF. In addition, based on the gradient smoothing techniques, the selective 3D-edge-based and node-based smoothed finite element method (Selective 3D-ES/NS-FEM) is developed to alleviate volume locking and improve the accuracy of linear four-node tetrahedral (TET4) elements. Combined with the modified constitutive model, the Selective 3D-ES/NS-FEM is applied into the explicit dynamic analysis of HAF undergoing large deformation. By comparing with the experiment data in the literatures and the numerical results produced by conventional FEM, the presented approach is proved to possess excellent accuracy and efficiency in predicting the nonlinear mechanical behavior of HAF, as well as the orientation change of the collagen fibers. Moreover, the Selective 3D-ES/NS-FEM is demonstrated to have robust capability in handling element distortion, even with the simplest TET4 mesh. This study is significant to the biomechanical research of HAF, and has potential value for guiding the prevention and treatment of low back pain.

An efficient and accurate numerical method for the heat conduction problems of thermal metamaterials based on edge-based smoothed finite element method

Abstract: Thermal metamaterials are a class of artificial composite structures with special thermal conduction properties. Due to the existence of complex multi-phase materials arrangement and interfaces in the thermal metamaterials, an efficient and accurate numerical method is on-demand to predict their temperature distribution and performances. In this paper, we studied the edge-based smoothed finite element method (ES-FEM) for simulating the heat conduction of thermal metamaterials in both steady-state and transient circumstances. Benefit from the adopted smoothing technique, the numerical errors caused by the complex interface are further eliminated. The most significant advantage of the proposed method is that one can directly use the automatically generated coarse triangle mesh to reduce the computational costs while ensuring accuracy. Two numerical examples are presented, in which two typical thermal metamaterials, the thermal cloak and thermal concentration device, are considered respectively in both steady-state and transient cases. The results illustrate that the ES-FEM based approach has good efficiency and accuracy in the heat conduction simulation of the thermal metamaterials.

Quadtree-polygonal smoothed finite element method for adaptive brittle fracture problems

Abstract: In this work, a quadtree-polygonal smoothed finite element method is proposed for adaptive consistent framework of phase field model on brittle fracture problems. A Smoothed Galerkin Weak form aided with the gradient smoothing technique is formulated to construct the variational formulations for both displacement and phase field. Staggered scheme is employed to solve the coupled phase and displacement field, in which the displacement field is obtained by Newton iterating and central difference method for implicit and explicit dynamic, respectively, while the phase field is solved directly with a linear equation. The critical history energy in the phase field fracture model is obtained with spectral decomposition. In order to acquire high efficiency without accuracy loss, a novel quadtree-based adaptive algorithm is developed for phase field fracture model, and the nodal phase field value is adopted as the direct indicator for mesh refinement. In this way, mesh local refinement is implemented with quadtree subdivision when the nodal phase field value is achieved the given threshold. Meanwhile, arbitrary sided polygonal elements provide an effective way to connect different mesh regions with different sizes. In other words, there is no hanging node but the connecting node on polygonal elements. Several numerical examples are performed for validating the feasibility of the proposed approach, in which the adaptive quadtree-polygonal method can save much computational costs without accuracy loss.

A coupling scheme based on the partition of unity finite element method-finite element method for 3D vibro-acoustic problems

Abstract: • The coupling of non-matching meshes is realized by dividing virtual elements, implementing coordinate transformation. • A novel efficient 3D vibro-acoustic analysis method PUFEM-FEM is developed. • The PUFEM-FEM not only inherits the simplicity and good adaptability of the FEM-FEM, but also has better accuracy, efficiency and convergence. The accuracy and efficiency of the finite element method-finite element method (FEM-FEM) for 3D vibro-acoustic problems degrade gradually with the increase of the excitation frequency due to the “dispersion error”. In this study, a novel efficient 3D vibro-acoustic analysis method, the partition of unity finite element method-finite element method (PUFEM-FEM), is proposed based on the PUFEM applicable to acoustics and the FEM for thin plates, where the coupling of non-matching meshes is realized by dividing virtual elements, implementing coordinate transformation and introducing continuity conditions at the interface. The proposed method not only fully exploits the advantages of the PUFEM in suppressing “dispersion error” and reducing computational size, but also takes full use of its ability to analyze multiple frequencies without the need for repeated meshing, while retaining the FEM's ability to model the complex structures in great detail. Numerical examples demonstrate that the PUFEM-FEM has better accuracy, efficiency and convergence than the FEM-FEM for 3D vibro-acoustic problems.

An element‐based formulation for ES‐FEM and FS‐FEM models for implementation in standard solid mechanics finite element codes for 2D and 3D static analysis

Abstract: Edge‐based and face‐based smoothed finite element methods (ES‐FEM and FS‐FEM, respectively) are modified versions of the finite element method allowing to achieve more accurate results and to reduce sensitivity to mesh distortion, at least for linear elements. These properties make the two methods very attractive. However, their implementation in a standard finite element code is nontrivial because it requires heavy and extensive modifications to the code architecture. In this article, we present an element‐based formulation of ES‐FEM and FS‐FEM methods allowing to implement the two methods in a standard finite element code with no modifications to its architecture. Moreover, the element‐based formulation permits to easily manage any type of element, especially in 3D models where, to the best of the authors' knowledge, only tetrahedral elements are used in FS‐FEM applications found in the literature. Shape functions for non‐simplex 3D elements are proposed in order to apply FS‐FEM to any standard finite element.

High-Precision Calculation of Electro-Thermal Coupling Field Based on Smooth Finite Element Method

Abstract: Triangular (2D)/tetrahedral (3D) meshes have become the most common type of finite element method due to their fast mesh speed, strong geometric adaptability, and automatic mesh. However, in the calculation of electro-thermal coupling, the conventional finite element method uses the equal field strength/equal heat flow approximation method in the triangular/tetrahedral element, and the approximation accuracy is low. This paper proposes a high-precision smooth finite element for electrothermal coupling based on triangular/tetrahedral mesh. This method divides multiple smooth domains in the solution domain through the edges of the element. The calculation of the electric/heat conduction matrix is performed in the smooth domain instead of the grid element, which can avoid element mapping and complicated integration of surface (2D)/volume (3D). The discrete method based on the Joule thermoelectric thermal coupling equation based on the smooth domain is proposed, and the electrothermal coupling smooth finite element solver is developed. Finally, the solution accuracy of the method is verified and compared through the slice test and the numerical example.

Finite element methods

Abstract: Chapter 7 further examines finite element techniques as an alternative numerical approach for the solution of ordinary and partial differential equations. This technique is employed in various engineering disciplines. It is specifically used in the COMSOL commercial software package for multi-physics applications. The general theory of the finite element approach is further detailed.

Stability Analysis of Circular Tunnels in Cohesive-Frictional Soil Using the Node-Based Smoothed Finite Element Method (NS-FEM)

Abstract: In this chapter, the stability of a circular tunnel and dual circular tunnels in cohesive-frictional soils subjected to surcharge loading is investigated by using the node-based smoothed finite element method (NS-FEM). In the NS-FEM, the smoothing strain is calculated over smoothing domains associated with the elements’ nodes. The soil is assumed as a uniform Mohr-Coulomb material, and it obeys an associated flow rule. By using the second-order cone programming (SOCP) for solving the optimization problems, the ultimate load and failure mechanisms of the circular tunnel are considered. This chapter discusses the influence of the soil weight γD/c, the tunnel diameter ratio to its depth H/D, the vertical and horizontal spacing ratio (L/D, S/D) of two tunnels and soil internal friction angle ϕ on the stability numbers σs/c are calculated. The stability numbers obtained from the present approach are compared with the available literature for tunnels.

Comparison of Finite Difference, Finite Element and Boundary Element Methods for electrostatic charged particle optics

Abstract: The speed and accuracy of three computational methods used for solving problems in electrostatic charged particle optics have been compared. The methods considered are the Finite Difference, Finite Element and Boundary Element Methods, as applied to two dimensional systems. ‘Benchmark’ problems, all of which have accurately known solutions, are used for these comparisons

A topology optimization method based on the edge-based smoothed finite element method

Abstract: In this paper, we combined the edge-based smoothed finite element method (ES-FEM) with topology optimization. The edge-based gradient smoothing operation was introduced to overcome the accuracy-loss of the classical finite element method raised by the coarse mesh and “overly stiff” phenomenon. By employing the ES-FEM, design variables can be related to the smoothed edge, thus more design variables can be adaptively obtained without additional remeshing. Two classical topology optimization problems were considered, namely compliance minimization and stress-constrained topology optimization. We presented several numerical examples, among which the compliance minimization examples illustrated the potential of the proposed method, and the advantages of applying such a numerical method in topology optimization were demonstrated through the stress-constrained topology optimization.

FEM‐DEM bridging coupling for the modeling of gouge

Abstract: We discuss the multiscale modeling of a granular material trapped between continuum elastic domains. The amorphous granular region, usually termed “gouge”, is under high confinement pressure, to represent the loading of faults at depth. We model the granularity of gouge using the Discrete Element Method (DEM), while the elastic regions surrounding it are represented with two continuum domains modeled with the Finite Element Method (FEM). We resort to a concurrent coupling of the discrete and continuum domains for a proper transmission of waves between the discrete and continuum domains. The confinement pressure results in the appearance of a new kind of ghost forces, which we address via two different overlapping coupling strategies. The first one is a generalization to granular materials of the Bridging Method , which was originally introduced to couple continuum domains to regular atomic lattices. This method imposes a strong formulation for the Lagrange constraints at the coupling interface. The second strategy considers a weak formulation. Different DEM samples sizes are tested in order to determine at which scale a convergence of the elastic properties is reached. This scale sets the minimal mesh element size in the DEM/FEM interface necessary to avoid undesirable effects due to an elastic properties mismatch. Then, the two DEM/FEM strategies are compared for a system initially at equilibrium. While the performance of both strategies is adequate, we show that the strong coupling is the most stable one as it generates the least spurious numerical noise. Finally, as a practical example for the strong coupling approach, we analyze the propagation of pressure and shear waves through the FEM/DEM interface and discuss dispersion as function of the incoming wave frequency. presence of to the decreases as the the propagation of waves confirms that a mesh size of least is to avoid a mismatch of material properties between FEM and DEM. the results confirm the aptitude of our numerical framework in the LibMultiScale software to satisfactorily different between the two

Proposal of analysis method of global-local finite element method for finite strain elastic-plastic fracture mechanics analysis

Abstract: The purpose of this study is to make it possible to perform a large number of paratric analyzes on the position and shape of cracks under finite elasto-plastic deformation. In this report, we propose a one-way coupled analysis method using displacement interpolation between a global finite element method analysis model created without assuming cracks and a local finite element method analysis model assuming cracks. The proposed method is based on the superconvergent patch recovery, and uses the least-squares method to apply the nodal displacements of the global finite element model to the nodes of the local finite element model as displacement boundary conditions. In addition, we will create an merged finite element method analysis model assuming a crack directly in the global finite element method analysis model and identify the application conditions that can be applied to the problem without loss of accuracy through relative comparison by J-integral calculation at the crack tip which is nonlinear fracture mechanics parameter.

Application of Finite Element Modeling and Simulation Aimed at Solving Students’ Learning Problems

Abstract: Aiming at solving a difficult problem raised by the students, this study explores the teaching method of Engineering Mechanics. The combination of finite element simulation and integral derivation is attempted to be applied. The relevant finite element model is established and simulated on a computer. The simulated U3 nephogram shows the magnitude of strain with different colors (the redder the color, the greater the strain). These visual materials provide support for the students to understand the abstract concepts. Subsequently, the detailed integral calculation process is sorted out by the teachers. The result of integral calculation is 0.06385 mm, which is very close to the result of the finite element model (0.06271 mm). Finite element analysis and mathematical integral calculation support each other, which can help the students understand relevant knowledge points of Engineering Mechanics more deeply. The application of finite element modeling and simulation is an important innovation in vocational education teaching methods, which should be concerned by more scholars.

Magnetic Flux Density Calculation Accuracy Improvement Of Edge Finite Element Based On Superconvergent Patch Recovery Method

Abstract: In this paper, the Superconvergent Patch Recovery method (SPR) is applied to the electromagnetic field calculation problem. On the basis of the double curl equation of vector magnetic potential, the magnetic flux density is solved by the edge finite element method using the tetrahedral mesh. According to the Superconvergent Patch Recovery technology of finite element method, all elements around the certain element are formed into element group. By the interpolation of the superconvergence point field intensity in the element, the calculation accuracy of magnetic flux density is improved.

Nonstandard Finite Element Methods

Abstract: Finite element methodologies dominate the computational approaches for the solution to partial differential equations and nonstandard finite element schemes most urgently require mathematical insight in their design. The hybrid workshop vividly enlightened and discussed innovative nonconforming and polyhedral methods, discrete complex-based finite element methods for tensor-problems, fast solvers and adaptivity, as well as applications to challenging ill-posed and nonlinear problems.

Nonlinear Stochastic Finite Element Method

Abstract: Considering the influence of random factors on the structure, three stochastic finite element methods for general nonlinear problems are proposed. They are Taylor expansion method, perturbation method and Neumann expansion method. The mean value of displacement is obtained by the tangent stiffness method or the initial stress method of nonlinear finite elements. Nonlinear stochastic finite element is transformed into linear stochastic finite element. The mean values of displacement and stress are obtained by the incremental tangent stiffness method and the initial stress method of the finite element of elastic-plastic problems. The stochastic finite element of elastic- plastic problems can be calculated by the linear stochastic finite element method.