Traditional Finite Element Method Research Articles

Bimaterial interface crack analysis using an extended consecutive-interpolation quadrilateral element

A very important problem in the research of layer structures is the modeling of cracks on the material interface. Due to the complex physical and mechanical properties of this structure, the simulation of discontinuities such as cracks and material interface by traditional finite element methods requires a very fine mesh density. Furthermore, mesh smoothing requires a really large amount of computational resources. Therefore, the extended algorithm which does not require the remeshing technique was born to solve the crack problems. In this paper, the extended consecutive-interpolation finite element method (XCFEM) is employed to modeling the mix-mode interface cracks between two dissimilar isotropic materials. The XCFEM using 4-node consecutive-interpolation quadrilateral element (XCQ4) provides continuity of nodal gradient due to the concept of “consecutive-interpolation” so that the stress and strain fields derived from XCQ4 is smoother than that obtained by the classical FEM element. The accuracy and effectiveness of the method are demonstrated via various numerical examples and compared with other researches.

A mixed selective edge-based smoothed PFEM with second-order cone programming for geotechnical large deformation analysis

Traditional particle finite element method (PFEM) with a six-node triangular (T6) element using an iteration algorithm often suffers from high computational costs and non-convergence problems. This paper proposes a novel dynamic implicit optimisation based smoothed PFEM for solving geotechnical large deformation problems. The governing equations are treated as an equivalent min–max optimisation that is recast as a second-order cone programming (SOCP) problem. Owing to the particularity of the min–max optimisation problem, which needs to simultaneously interpolate displacement and stress field, a novel mixed constant-stress selective edge-based smoothed strain element is proposed in which strain and stress are constant in the smoothing domain. The two-level mesh repartitioning scheme is implemented into the proposed method to overcome volumetric locking. Next, a mass lumping scheme using nodal integration is developed to improve computational efficiency. An efficient and simple variable mapping scheme is finally proposed and implemented into PFEM to tackle large deformation problems. Several classical statics and dynamics examples are used to validate the proposed method. All results demonstrate that the proposed method only using the lower-order element offers high accuracy, high efficiency, mesh insensitivity and is free of volumetric locking for geotechnical problems under both small and large deformation.

Fast electrothermal coupling calculation method for supporting digital twin construction of electrical equipment

AbstractThe failure of electrical equipment directly or indirectly caused by overheating has become one of the main reasons for equipment accidents. The real‐time condition monitoring method of electrical equipment based on digital twin (DT) has received extensive attention and is considered as a technology with great engineering values and excellent application prospects. However, the current calculation of DT mostly relies on the traditional finite element method (CFEM). This method becomes less computationally efficient as the size of the DT increases, especially for electrical equipment with high complexity. It is difficult to meet the requirements real‐time calculation in DT. Therefore, starting from the algorithm optimization and parallel architecture, based on the weighted residual method and stabilised conforming nodal integration, a novel discrete method of electrothermal coupling equation is proposed and the data storage structure of the algorithm from the bottom layer is redesigned, and the corresponding GPU and multi‐core CPU parallel framework are proposed. Finally, taking the high voltage bushing as an example, the correctness of the method in this paper is verified by the CFEM code and commercial software ABAQUS. Under the same grid and accuracy requirement, the calculation time is shortened by at least five times than ABAQUS. And the method is easy to extend to other types of multi‐physics calculations.

Approximate natural frequencies of AFGM beam

This article gives the approximate natural frequencies of axial functionally graded material (AFGM) beams. Based on Matlab software and the traditional finite element method, we can estimate the natural frequencies for this structure under four types of boundary conditions. Some examples are given which prove that this way is simple and applicable.

A two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential

AbstractIn this paper, we present a two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen‐Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two‐grid finite element method. The optimal convergence order of the two‐grid finite element method in H1 norm is achieved when the mesh sizes satisfy h = O(H2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two‐grid method can save the CPU time while keeping the same convergence rate.

Transfer Matrix Method for Calculating the Transverse Load Distribution of Articulated Slab Bridges

Articulated slab bridges have been widely used by transportation administration for short-to-medium span bridges because of their good economy, convenient construction, and environmental advantages, while the presence of shear keys increases the complexity of structural behavior. Developing more reasonable analysis approaches of quick assessment, pre-design, and hand calculations for the articulated slab bridges is a challenge because of the peculiar shear key mechanism. This paper is devoted to presenting a recursive algorithm, based on the force equilibrium conditions of each individual slab, thus resulting in simultaneous equations of the transfer matrix method (TMM). In this procedure, the state vector is an array composed of vertical displacement, shear force, unit constant; and the transfer matrix contains the bending and torsional stiffness parameters of simply supported slabs. Then, the influence line of transverse load distribution (TLD) is calculated for each slab by introducing boundary conditions. To validate and verify the efficiency of the TMM algorithm, a transversely prefabricated void slab bridge with a span of 20 m is considered as a case study. The traditional force (FM) and finite element (FEM) methods are used for comparison and validation. It is demonstrated that the TMM can provide good results with higher algorithm efficiency by exempting the modeling tasks in FM and FEM and capture variations in TLD along the bridge’s span. In addition, the influence of the span length and relative stiffness coefficient of slabs on the TLD of articulated slab bridges are analyzed from the parametric analysis.

Harmonic viscoelastic response of 3D histology-informed white matter model

White matter (WM) consists of bundles of long axons embedded in a glial matrix, which lead to anisotropic mechanical properties of brain tissue, and this complicates direct numerical simulations of WM viscoelastic response. The detailed axonal geometry contains scales that range from axonal diameter (microscale) to many diameters (mesoscale) imposing an additional challenge to numerical simulations. Here we describe the development of a 3D homogenization model for the central nervous system (CNS) that accounts for the anisotropy introduced by the axon/neuroglia composite, the axonal trace curvature, and the tissue dynamic response in the frequency domain. Homogenized models that allow the incorporation of all the above factors are important for accurately simulating the tissue's mechanical behavior, and this in turn is essential in interpreting non-invasive elastography measurements.Geometric and material parameters affect the material properties and thus the response of the brain tissue. More complex, orthotropic, or anisotropic material properties are to be considered as necessitated by the 3D tissue structure. An assembly of micro-scale 3D representative elemental volumes (REVs) is constructed, leading to an integrated mesoscale WM finite element model. Assemblies of microscopic REVs, with orientations based on geometrical reconstructions driven by confocal microscopy data are employed to form the elements of the WM model. Each REV carries local material properties based on a finite element model of biphasic (axon-glial matrix) unidirectional composite. The viscoelastic response of the microscopic REVs is extracted based on geometric information and fiber volume fractions calculated from the relative distance between the local elements and global axonal trace. The response of the WM tissue model is homogenized by averaging the shear moduli over the total volume (thus deriving effective properties) under realistic external loading conditions. Under harmonic shear loading, it is proven that that the effective transverse shear moduli are higher than the axial moduli when the axon moduli are higher than the glial. Methodologically, the process of using micro-scale 3D REVs to describe more complex axon geometries avoids the partition process in traditional composite finite element methods (based on partition of finite element grids) and constitutes a robust algorithm to automatically build a WM model based on available axonal trace information. Analytically, the model provides unmatched simulation flexibility and computational power as the position, orientation, and the magnitude of each tissue building block is calculated using real tissue data, as are the training and testing processes at each level of the multiscale WM tissue.

Efficient EDM-PO Method for the Scattering From Electrically Large Objects With the High-Order Impedance Boundary Condition

This work proposes an efficient hybrid method consisting of the efficient iterative method of moments–physical optics (EI-MoM-PO) and the equivalent dipole moment (EDM) with the high-order impedance boundary condition (HOIBC) to explore the electromagnetic scattering from the object with an isotropic or anisotropic surface coating. By introducing the HOIBC into the EDM method, the relationship between the equivalent electric and magnetic dipole moments is established. Therefore, the EDM method is extended to general cases. For the composite scattering from electrically large objects including MoM and PO regions, the EDM method is implemented to fill the MoM impedance matrix and to accelerate the evaluation of the coupling between the MoM and PO regions, which considerably improves the computing efficiency. The numerical results indicate that the proposed hybrid method allows for a substantial decrease in computation time. Also, they also agree well with the traditional finite element method, MoM, or MoM-PO method, which proves the validity of the proposed method.

A combined multiscale finite element method based on the LOD technique for the multiscale elliptic problems with singularities

In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the LOD-based MsFEM on a coarse mesh of the other part. The key point is how to define local correctors for the basis functions of the elements near the coarse and fine mesh interface, which require meticulous treatment. The proposed method takes advantages of the traditional FEM and the LOD-based MsFEM, which uses much less DOFs than the standard FEM and may be more accurate than the LOD-based MsFEM for problems with singularities. The error analysis is carried out for highly varying coefficients, without any assumptions on scale separation or periodicity. Numerical examples with periodic and random highly oscillating coefficients, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high-contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method.

Time-Frequency Domain Conversion of UHVDC Transmission Line Corona Current Based on Fourier Transform

In the power system, Ultra-High-Voltage (UHV) transmission is an effective means to improve and solve the imbalance between power supply and demand. However, with the continuous improvement of the voltage level, the adverse effects caused by the electromagnetic environment of the transmission lines are also growing. Audible noise produced by corona discharge of transmission lines has seriously interfered with the daily life of residents near the transmission lines, which can not meet the requirements of protecting residents and environmental friendliness. In this article, the background noise in the time-domain waveform of corona current is removed by wavelet analysis. The time domain waveform of corona current is transformed into the frequency domain by a fast Fourier transform. Through the analysis of the spectrum of corona current, it is found that the corona current attenuates with the increase of frequency, and most of the energy is concentrated in the first 1.5 MHz frequency band. The relationship between each frequency band of corona current and audible noise is obtained by the multiple linear regression, and the regression model is proved to be well fitted by testing. The experimental analysis shows that the algorithm in this article is 8% better than the traditional finite element method, and is closest to the actual measurement value, which proves that the actual effect of the research contained in this article is more effective. These achievements are helpful for the further study of the electromagnetic environment of UHV transmission lines, which is conducive to promoting the vigorous development of the UHV DC transmission technology in China, allocating resources more reasonably and reducing environmental pollution, and meeting the strategic requirements of energy and environment in China.

A Hybrid Finite Element Method–Analytical Model for Classifying the Effects of Cracks on Gear Train Systems Using Artificial Neural Networks

Rotating machinery is fundamental in industry, gearboxes especially. However, failures may occur in their transmission components due to regular usage over long periods of time, even when operations are not intense. To avoid such failures, Structural Health Monitoring (SHM) techniques for damage prediction and in-advance detection can be applied. In this regard, correlations between measured signal variations and damage can be inspected using Artificial Intelligence (AI), which demands large numbers of data for training. Since obtaining signal samples of damaged components experimentally is currently unviable for complex systems due to destructive test costs, model-based numerical approaches are to be explored to solve this problem. To address this issue, this work applied an innovative hybrid Finite Element Method (FEM)–analytical approach, reducing computational effort and increasing performance with respect to traditional FEM. With this methodology, a system can be simulated with accuracy and without geometrical simplifications for healthy and damaged cases. Indeed, considering different positions and dimensions of damages (e.g., cracks) on the tooth roots of gears can offer new ways of damage investigation. As a reference to validate healthy systems and damage cases in terms of eigenfrequencies, a back-to-back test rig was used. Numerical simulations were performed for different cases, resulting in vibrational spectra for systems with no damage, with damage, and with damage of different intensities. The vibration spectra were used as data to train an Artificial Neural Network (ANN) to predict the machine state by Condition Monitoring (CM) and Fault Diagnosis (FD). For predicting the health and the intensity of damage to a system, classification and multi-class classification methods were implemented, respectively. Both sets of classification results presented good prediction agreement.

Finite Element Implicit 3D Subsurface Structural Modeling

We introduce a method for 3D implicit geological structural modeling from sparse sample points, where several conformable geological surfaces are represented by one single scalar field. Laplacian and Hessian regularization energies are discretized on a tetrahedral mesh using finite elements. This scheme is believed to offer some geometrical flexibility as it is readily implemented on both structured and unstructured grids. While implicit modeling on unstructured grids is not new, methods based on finite elements have received little attention. The finite element method is routinely used to solve boundary value problems. However, because boundary conditions are typically unknown in implicit subsurface structural modeling, the traditional finite element method requires some adjustments. To this end, we present boundary free discretizations of the Laplacian and Hessian energies that do not assume vanishing Neumann boundary conditions, thereby eliminating the boundary artifacts usually associated with that assumption. Furthermore, we argue that while an appropriate discretization of the Laplacian can be used to minimize the curvature of a function on triangulated meshes, it may fail to do so on tetrahedral meshes. • A finite element discretization of the Laplacian without imposing any boundary condition. • A finite element discretization of the Hessian without imposing any boundary condition. • Building 3D subsurface numerical models using finite elements. • The Laplacian may not be as adequate for interpolation on tetrahedra as it is on triangles.

Application of the soil parameter random field in the 3D random finite element analysis of Guanyinyan composite dam

Due to the inhomogeneous of dam materials, effect of the compaction environment and experimental data errors, the soil properties of the Guanyinyan composite dam present obvious spatial variability, resulting in the occurrence of obvious discordant settlements and cracks during the impounding period. The traditional finite element method (FEM) using inversion parameters that are modified by monitoring displacement back-analysis can neither simulate spatial variability in soil parameters nor predict dam safety before dam displacements occur. In this research, soil parameter random fields are developed considering spatial variability, and a three-dimensional random FEM (RFEM) analysis of the Guanyinyan composite dam is applied. According to soil dry density samples and experimental results, the autocorrelation of the same soil parameter in different positions in dam is determined. The cross-autocorrelation of different soil parameters is conducted using the Nataf transformation. The RFEM analysis shows that the mean values of maximum settlement, the mean values of maximum deformation gradient and area ratio with high hydraulic fracture probability exceeding 0.3 are close to those calculated by inversion parameters. The cumulative frequencies of maximum settlement and maximum deformation gradient show 51% and 33% probabilities of exceeding the inversion calculation results. The area ratio and distribution range of elements with hydraulic failure probabilities greater than 0.2 are significantly larger than those calculated by the inversion parameters. The traditional certain security indexes, which ignore spatial variability of soil parameters, have a certain probability to overestimate the security of the dam. The uncertain security index considering spatial variability of soil parameters can provide a more reasonable and accurate reference for dam safety evaluation before water impounding.

Analytical Determination of Static Deflection Shape of an Asymmetric Extradosed Cable-Stayed Bridge Using Ritz Method.

A practical method to analyze the mechanical behavior of the asymmetric extradosed cable-stayed (AECS) bridge is provided in this paper. The work includes the analysis of the equivalent membrane tension of the cables, the ratio of side-span cable force to middle-span cable force, and the deflection of the main girder subject to uniformly distributed load. The Ritz method is a simple and efficient way to solve composite structures, such as the AECS bridge, compared with the traditional force method, displacement method, or finite element method. The theoretical results obtained from the Ritz method are in good agreement with that from the finite element analysis, which shows the accuracy of this approach. Then, a parametric study of AECS bridges is carried out by using the proposed equations directly, instead of using the traditional finite element modeling process, which requires a lot of modeling work. As a result, reasonable values of very important parameters are suggested, which helps the readers reach a better understanding of the mechanical behavior of AECS bridges. More importantly, it helps the designers to enhance the efficiency in the stage of conceptual design.

CARBON NANOFIBER - REINFORCED MULTIFUNCTIONAL SHEET INTEGRATED WITH PIEZOELECTRIC CRYSTAL: AN OVERVIEW OF STRUCTURAL BEHAVIOR ANALYSIS METHODS

Most of the phenomena in life, such as heat transfer, wave propagation, thermoelectricity, sheet materials, multifunctional material properties, etc., can be expressed through mathematical equations or partial differential equations (PDEs). However, with complex differential equations, it is difficult or impossible to find solutions, especially differential equations for complex problems such as solid-liquid interactions, mechanical-thermal-electrical environments, and functional material plate environments in the multiphysics environment, etc. Since then, many numerical methods have been researched and developed to find approximate solutions to partial differential equations. Today, the finite element method (FEM) is the most widely used and effective among the commonly used numerical methods. However, with the continuous emergence of new complex problems in science and technology, especially problems in the multiphysics environment in sheet-material structures, the finite element method has revealed new challenges. There are certain limitations related to the element technique and the weak form discretization of the problem of sheet structure with many different degrees of freedom, significantly affecting the accuracy and efficiency of calculations. Therefore, it is always important to propose improvements to the traditional finite element method and the optimal method algorithm to meet the increasing requirements in the behavior analysis of sheet materials. This research direction is always topical and has received the attention of scientists around the world for many decades.

CARBON NANOFIBER - REINFORCED MULTIFUNCTIONAL SHEET INTEGRATED WITH PIEZOELECTRIC CRYSTAL: AN OVERVIEW OF STRUCTURAL BEHAVIOR ANALYSIS METHODS

Most of the phenomena in life, such as heat transfer, wave propagation, thermoelectricity, sheet materials, multifunctional material properties, etc., can be expressed through mathematical equations or partial differential equations (PDEs). However, with complex differential equations, it is difficult or impossible to find solutions, especially differential equations for complex problems such as solid-liquid interactions, mechanical-thermal-electrical environments, and functional material plate environments in the multiphysics environment, etc. Since then, many numerical methods have been researched and developed to find approximate solutions to partial differential equations. Today, the finite element method (FEM) is the most widely used and effective among the commonly used numerical methods. However, with the continuous emergence of new complex problems in science and technology, especially problems in the multiphysics environment in sheet-material structures, the finite element method has revealed new challenges. There are certain limitations related to the element technique and the weak form discretization of the problem of sheet structure with many different degrees of freedom, significantly affecting the accuracy and efficiency of calculations. Therefore, it is always important to propose improvements to the traditional finite element method and the optimal method algorithm to meet the increasing requirements in the behavior analysis of sheet materials. This research direction is always topical and has received the attention of scientists around the world for many decades.

Displacements produced by linearly varying eigenstrains with application to isoparametric triangular inclusion

The planar Eshelby's inclusion model has been widely applied in exploring the micromechanical behavior of fiber-reinforced composites. Based on contour integral formulation, we first present a closed-form solution to the displacement of a polygonal inclusion subjected to linear eigenstrains. In view of isoparametric representation, the resulting contributions from a triangular inclusion are conceived as the elementary solution, which may be regarded as building blocks for analyzing an arbitrarily shaped inclusion with any distributed eigenstrains. The present numerical discretization is merely performed on the inclusion domain, as opposed to the traditional Finite Element Method where the infinitely extended matrix also requires tremendous meshing efforts.

Multibody system dynamic analysis and payload swing control of tower crane

A new payload swing control method considering the vibration of tower crane is proposed in this study. The coupling between the structural vibration and the payload swing is neglected in existing studies on tower crane swing suppression. Changes of sling length are considered as disturbances. They cause the existing methods of swing suppression method ineffectual in the actual working situation. In contrast, the structural vibration of the tower crane is taken into account in this study. At the same time, the sling length is regarded as a control variable to reduce the payload swing. A new swing suppression algorithm is proposed in conjunction with phase plane analysis method. In addition, a systematical tower crane multibody system dynamic analysis platform with changing of sling length is established to verify the effectiveness of the algorithm. The traditional finite element method is used to model the tower body and boom of tower crane while the Arbitrary Lagrange Euler Absolute Nodal Coordinate Formulation (ALE-ANCF) cable element is applied for modeling the sling. This two parts are connected with sliding joint obtaining the tower crane multibody system model. Numerical examples show that the proposed method can reduce the swing amplitude the payload effectively when considering the coupling between the structural vibration and the payload swing.

Unconditional optimal error estimates of a modified finite element fully discrete scheme for the complex Ginzburg-Landau equation

In this paper, based on the 2-step backward differentiation formula (BDF2 for short) in time and the nonconforming Wilson element in space, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE for short). On one hand, a modified Ritz projection operator Rh is introduced and analyzed, which plays an important role in getting the unconditional optimal error estimates. On the other hand, a time-discrete system is constructed with the linearized BDF2 and the regularity is derived with the temporal error results. Combining these two aspects, the errors between RhUn and Uhn with order O(h3+h2△t) in L2-norm and O(h2+h2△t) in the modified energy norm are deduced, where h is the subdivision parameter, △t is the time step, Un and Uhn denote the solutions of the time-discrete system and the BDF2-MG FEM respectively. Therefore the boundedness of ‖Uhn‖0,∞ is proven without any restriction on the time-space grid ratio. Furthermore, by using the properties of Rh, unconditional optimal error estimates of order O(h3+(△t)2) in L2-norm and O(h2+(△t)2) in the modified energy norm are obtained directly. It should point out the spatial discrete errors of the BDF2-MG FEM are all one order higher than that of the BDF2 traditional Galerkin finite element method with Wilson element for the GLE. At last, a numerical experiment is presented to verify the validity of the theoretical analysis.

Voronoi-Based Discrete Element Analyses to Assess the Influence of the Grain Size and Its Uniformity on the Apparent Fracture Toughness of Notched Rock Specimens

The fracture toughness reflects the rock resistance to crack propagation, and therefore represents an important parameter for rock fracture assessments. From a strict point of view, the real fracture toughness (it {K}_{mathrm{it IC}}) corresponds to a cracked situation in which the notch radius is theoretically equal to zero. However, most of the defects in rocks have a finite radius and, therefore, should be studied as notch-type defects. Here, the notch effect is numerically studied together with the influence of the grain size and the sorting coefficient (grain size uniformity) on the apparent fracture toughness (it {K}_{mathrm{it IN}}). To this end, several four-point bending tests with different U-shaped notch radii, mean grain sizes and degrees of uniformity in grain size and shape have been simulated using the Discrete Element Method. In order to represent the grains of the rocks, the Voronoi tessellation is used to create randomly sized and distributed polygonal blocks. These Voronoi polygons have been defined, on the one hand, by an average edge length of 1, 2 and 3 mm, and, on the other hand, by a different number of iterations (n) in the relaxation process during the generation of the polygons, which defines the grain size uniformity. The numerical analyses performed and the interpretation of the results show a clear notch effect in all the studied cases, as the apparent fracture toughness (it {K}_{mathrm{it IN}}) increases with notch radius. Finally, the obtained stress fields at the notch tip have been compared to those obtained from the traditional finite element method.