Finite Element Method Research Articles

A novel Fourier domain scheme for three‐dimensional magnetotelluric modelling in anisotropic media

SUMMARYThis study presents a novel algorithm that combines the Lorenz gauge equations with the Fourier domain technique to simulate magnetotelluric responses in three‐dimensional conductivity structures with general anisotropy. The method initially converts the Helmholtz equations governing vector potentials into one‐dimensional differential equations in the wave number domain via the horizontal two‐dimensional Fourier transform. Subsequently, a one‐dimensional finite element method employing quadratic interpolation is applied to obtain three five‐diagonal linear equation systems. Upon solving these equations, the spatial domain fields are obtained via the inverse Fourier transform. This process guarantees the computational efficiency, memory efficiency and high parallelization of the algorithm. Moreover, an anisotropic medium iteration operator guarantees stable convergence of the method. The correctness, competence and applicability of the algorithm are verified using some synthetic models. The results demonstrate that the new method is efficient and performs well in anisotropic undulating terrain and complex structures. Compared to other Fourier domain methods and the latest edge‐based finite element algorithm, the proposed method exhibits superior computing performance. Finally, the impact of the Euler angles on the magnetotelluric responses is analysed.

A Study on the Attenuation Patterns of Underground Blasting Vibration and Their Impact on Nearby Tunnels

The natural caving method, as a new technique in underground mining, has been promoted and applied in several countries worldwide. The destruction of the bottom rock mass structure directly impacts the structural stability of underground engineering, resulting in damage and collapse of underground tunnels. Therefore, based on the principles of explosion theory and field monitoring data, a scaled three-dimensional numerical simulation model of underground blasting was constructed using LS-DYNA19.0 software to investigate the attenuation patterns of underground blasting vibrations and their impact on nearby tunnels. The results show that the relative error range between the simulated blasting vibration velocities based on the FEM-SPH (Finite Element Method–Smoothed Particle Hydrodynamics) algorithm and the measured values is between 7.75% and 9.85%, validating the feasibility of this method. Significant fluctuations in blasting vibration velocities occur when the blast center increases to within a range of 10–20 m. As the blast center distance exceeds 25 m, the vibration velocities are increasingly influenced by the surrounding stress. Additionally, greater stress results in higher blasting vibration velocities and stress wave intensities. Fitting the blasting vibration velocities of various measurement points using the Sadovsky formula yields fitting correlation coefficients ranging between 0.92 and 0.97, enabling the prediction of on-site blasting vibration velocities based on research findings. Changes in propagation paths lead to localized fluctuations in the numerical values of stress waves. These research findings are crucial for a deeper understanding of underground blasting vibration patterns and for enhancing blasting safety.

A proof of convergence and equivalence for 1D finite element methods and ReLU neural networks

This paper investigates the convergence and equivalence properties of the Finite Element Method (FEM) and Rectified Linear Unit Neural Networks (ReLU NNs) in solving differential equations. We provide a detailed comparison of the two approaches, highlighting their mutual capabilities in function space approximation. Our analysis proves the subset and superset inclusions that establish the equivalence between FEM and ReLU NNs for approximating piecewise linear functions. Furthermore, a comprehensive numerical evaluation is presented, demonstrating the error convergence behavior of ReLU NNs as the number of neurons per basis function varies. Our results show that while increasing the number of neurons improves approximation accuracy, this benefit diminishes beyond a certain threshold. The maximum observed error between FEM and ReLU NNs is 10-4, reflecting excellent accuracy in solving partial differential equations (PDEs). These findings lay the groundwork for integrating FEM and ReLU NNs, with important implications for computational mathematics and engineering applications.

Large deformation simulation of uprooting of trees with complex root system architectures using material point method with embedded truss elements

Abstract Background and aims Urban trees in coastal cities like Hong Kong may suffer from an uprooting failure when subjected to extreme winds. A proper numerical model for tree uprooting simulation can help to select tree species or soil types that better resist uprooting failure. However, modeling tree uprooting is challenging as it is a cross-disciplinary problem involving complex root system architectures (RSAs) and large deformation of both roots and soils. This study aims to develop a hybrid numerical model that combines truss elements and material point method (MPM) to simulate the entire large-deformation uprooting process of trees with complex RSAs. Methods The tree uprooting model is developed by coupling truss elements in finite element method (FEM) with MPM. Laboratory pull-out tests using artificial roots and real root cuttings are adopted to validate the developed model. A comparative study is performed to investigate the difference between using complex and simplified RSAs in tree uprooting simulations. Results The developed model provides consistent predictions of peak load, critical displacement and failure mode when compared with results from laboratory tests. Moreover, the comparative study shows that the uprooting resistance obtained with a complex RSA is higher than that with a simplified RSA. The difference varies with the soil and root mechanical properties. Conclusion The developed hybrid model offers a novel way for simulating an entire tree uprooting process involving large deformations and complex RSAs. The study shows that using a simplified RSA to approximate the complex RSA might result in misleading failure modes.

An immersed finite element method for interfacial damage and debonding

An immersed finite element method for interfacial damage and debonding

Stress Field and Crack Pattern Interpretation by Deep Learning in a 2D Solid

ABSTRACTA nonlinear variational auto‐encoder (NLVAE) is developed to reconstruct the plane strain stress field in a solid with embedded cracks subjected to uniaxial tension, uniaxial compression, and shear loading paths. Latent features are sampled from a skew‐normal distribution, which allows encoding marked variations of the features of the stress field across the load steps. The NLVAE is trained and tested based upon stress maps generated with the finite element method (FEM) with cohesive zone elements (CZEs). The NLVAE successfully captures stress concentrations that develop across the loading steps as a result of crack propagation, especially when enhanced disentanglement is emphasized during training. Some latent variables consistently emerge as significant across various microstructure descriptors and loading paths. Correlations observed between the evolution of fabric descriptors and that of their significant stress latent features indicate that the NLVAE can capture important microstructure transitions during the loading process. Crack connectivity, crack eccentricity, and the distribution of zones of highly connected opened cracks versus zones with no cracks are the fabric descriptors that best explain the sequences of latent features that are the most important for the reconstruction of the stress field. Notably, the distributional shape, tail behavior, and symmetry of microstructure descriptor distributions have more influence on the stress field than basic measures of central tendency and spread.

Optimization of heat transfer in a double lid-driven cavity with isoperimetric heated blocks using GFEM.

This article is concerned with the examination of flow dynamics and heat transfer characteristics in a 1:4 double lid driven cavity in presence of isoperimetric heated blocks of various shapes. The focus is to identify the optimal shape that enhances the heat transfer in a tall cavity. The parametric settings are chosen in such a way that all the convection regimes including natural, forced and mixed convection could be generated. This cavity has lids positioned at the top and bottom, moving in opposite directions along the x-axis. The physical system is represented as a set of coupled partial differential equations incorporating the rheological properties of the power-law fluids (PL). The governing equations in conjunction with various non-dimensional physical parameters are simulated via Galerkin's Finite Element Method (GFEM) on a very fine hybrid grid. The study includes the computation of the Kinetic Energy and Average Nusselt number to determine the optimal shape. It is concluded that the circular block is superior to the other two in terms of heat transmission efficiency.

Research on Typical Decay-like Fracture Defects of Composite Insulators Based on Electro-Thermal Coupling

In response to the typical decay and fracture defects of composite insulators, a three-dimensional electrically and thermally coupled simulation physical model was constructed based on the finite element method, and the local electric field distortion and temperature rise were analyzed. The study confirms that the insulator interface’s axial electric and thermal fields show a U-shaped curve; the interface field strength is the largest. There is an electric field gradient difference between the mandrel and the sheath, and the thermal field is concentrated at the mandrel and the interface. The field strength at the edge of the defect is the largest, the aberrant electric field at the defect shows a sawtooth shape, and the temperature rise is concentrated in the defect area. The degradation is fast in the air gap, the etching hole diameter direction, and the carbonation channel axial direction. The larger the defect volume, the larger the aberration in the electric field and temperature rise. Water vapor air gaps, breakdown holes, and carbonized channels have the most pronounced electric field and temperature changes. The functional relationship between electric field aberration, temperature rise, and defect volume is established. The results provide a basis for the protection of insulator decay-like fracture.

Study on Bandgap of Two‐Dimensional Square Lattice Acoustic Metamaterial with Multiembedded Cores

Acoustic metamaterials have been widely studied for their excellent ability of vibration isolation. In this article, a 2D square lattice acoustic metamaterial structure with multiembedded cores is proposed. Based on Bloch's theorem and finite element method, the band structures and bandgap properties of the structure are calculated and analyzed, and the vibration isolation ability of the structure is investigated. The effects of structural parameters on the bandgap property are analyzed. Thus, the anisotropy of the structure is also analyzed by the contours of phased velocity and group velocity. The results show that the structure has significant bandgap properties and vibration isolation. The second‐order similarity ratio has significant effects on the bandgap property. Via increasing the second‐order similarity ratio, the total bandgap width and low‐frequency bandgap width are widened. In addition, the structure has significant anisotropy, and the energy of elastic wave propagates at the direction of 45° and 135°. The research in this article supplements the design and research of 2D lattice acoustic metamaterials and makes a beneficial exploration for future engineering applications.

Two dimensional computational model of calcium and IP3 interacting system dynamics in an obese hepatocyte cell

Abstract Calcium ion (Ca$^{2+}$) signaling is crucial in regulating numerous cellular processes vital for preserving structural integrity and functional equilibrium across diverse cell types. Both the calcium stores and mitochondria play significant roles in this signaling pathway. The calcium source may be in the form of a blip or a puff depending on the various conditions of the cellular systems. The one dimensional model of calcium dynamics with IP$_{3}$ gives crucial insight of feedback mechanisms influencing calcium homeostasis. In order to obtain deeper insights of local impacts of various mechanisms and feedbacks in hepatocyte cell, it is necessary to develop the models in higher dimensions. In order to get more deeper insights, two dimensional model is proposed assuming the phenomena to be uniform along z dimension. This research presents a two-dimensional computational model to analyse the interactive system dynamics of inositol 1,4,5-trisphosphate (IP$_{3}$) and Ca$^{2+}$, aiming to assess how these signaling patterns influence hepatocyte functionality which allows to incorporate puff type of calcium source under both obese and normal physiological states. It further examines the implications of calcium signaling on NADH synthesis, ATP production, and degradation rates. Numerical simulations are executed utilising the Crank-Nicolson method for temporal analysis and the Linear Finite Element Method for spatial analysis. Additionally, the study conducts a comparative analysis of calcium signaling between obese and normal hepatocyte. The findings offer enhanced insights into the interactive system dynamics of IP$_{3}$ and Ca$^{2+}$ in hepatocytes, elucidating the effects of various parameter alterations on cellular behaviour in both states.

A FEM-based model for failure initiation in the microstructure of gray cast iron under uniaxial compression

PurposeThe purpose of this study was to validate the feasibility of the proposed microstructure-based model by comparing the simulation results with experimental data. The study also aimed to investigate the relationship between the orientation of graphite flakes and the failure behavior of the material under compressive loads as well as the effect of image size on the accuracy of stress–strain behavior predictions.Design/methodology/approachThis paper presents a microstructure-based model that utilizes the finite element method (FEM) combined with representative volume elements (RVE) to simulate the hardening and failure behavior of ferrite-pearlite matrix gray cast iron under uniaxial loading conditions. The material was first analyzed using optical microscopy, scanning electron microscopy (SEM), energy-dispersive X-ray spectroscopy (EDS) and X-ray diffraction (XRD) to identify the different phases and their characteristics. High-resolution SEM images of the undeformed material microstructure were then converted into finite element meshes using OOF2 software. The Johnson–Cook (J–C) model, along with a damage model, was employed in Abaqus FEA software to estimate the elastic and elastoplastic behavior under assumed plane stress conditions.FindingsThe findings indicate that crack initiation and propagation in gray cast iron begin at the interface between graphite particles and the pearlitic matrix, with microcrack networks extending into the metal matrix, eventually coalescing to cause material failure. The ferritic phase within the material contributes some ductility, thereby delaying crack initiation.Originality/valueThis study introduces a novel approach by integrating microstructural analysis with FEM and RVE techniques to accurately model the hardening and failure behavior of gray cast iron under uniaxial loading. The incorporation of high-resolution SEM images into finite element meshes, combined with the J–C model and damage assessment in Abaqus, provides a comprehensive method for predicting material performance. This approach enhances the understanding of the microstructural influences on crack initiation and propagation, offering valuable insights for improving the design and durability of gray cast iron components.

AFM Nanoindentation of Stiff Inhomogeneous Layer onPolymeric Substrate.

Analysis of indentation data of heterogeneous material, in particular, layer on an elastic substrate requires information about the contact area that is essential for calculating mechanical properties. The actual shape of the AFM-tip is not described by simple body of revolution. In this work, the indentation of a stiff layer on a hyperelastic substrate by a truncated conical tip is studied using finite element methods. The size of the tip, elastic modulus, and thickness of the layer are varied. The obtained model dependences of loading and contact area versus the depth of indentation are used in analysis of the experimental data of AFM indentation of stiff inhomogeneous nanolayers of polyurethane surface. Thickness and elastic modulus of the layers, fracture properties of the surface are studied. The obtained results can find application in the study of mechanical properties and fracture toughness of thin flexible films on an elastic substrate.

An Analysis of Dynamic Recrystallization During the Reduction Pretreatment Process Using a Multiscale Model

In this study, a multiscale model is developed through secondary development (UMAT and UEXTERNALDB) in Abaqus with the objective of simulating the thermal deformation process with dynamic recrystallization behavior. The model couples the finite element method (FEM) with the multiphase field model (MPFM), thereby establishing bidirectional coupling between macroscopic mechanical behavior and microstructural evolution. A comparison between the single-element hot compression simulation and experimental results demonstrates that the model accurately simulates both the macroscopic mechanical behavior and microstructural evolution during the thermal deformation process, thereby exhibiting high precision. Simulations of the reduction pretreatment (RP) process under different reduction amounts and billet surface temperatures demonstrate that increasing the reduction amount and billet surface temperature significantly enhances both plastic deformation and the volume fraction of dynamic recrystallization in the billet core. This results in the closure of core voids and the refinement of the core microstructure, thereby providing valuable guidance for the development of optimal reduction pretreatment (RP) processes.

Mixed FEM implementation of three-point bending of the beam with an edge crack within strain gradient elasticity theory

AbstractThis paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory.

A novel gyroscope based on the slow surface acoustic wave in a phononic metamaterial.

Limited to the direct modulation on the surface acoustic wave (SAW) by the rotation, the conventional SAW gyroscopes incur weak Coriolis effects and gyroscopic effects. In this paper, we innovatively utilize a phononic metamaterial (PM) operated at whispering-gallery modes (WGMs) as the vehicle for the Coriolis effect rather than SAW itself. The gyroscopic effects of this PM are investigated, and a new SAW gyroscope is subsequently proposed based on the slow SAW in PM. We show, combining theoretical modeling and finite element method simulation, that the rate of rotation can linearly induce the splitting of WGMs and modulate the phase velocity of SAW down to 4600 m/s (initial phase velocity of 5355 m/s); the direction of rotation results in the chiral symmetry of the PM vibration and the asymmetric distribution of the transmissive SAW. Besides, the proposed SAW gyroscope measures the angular velocity by detecting the phase shift resulting from rotation-dependent slow SAW in PM, obtaining a sensitivity of 0.016 deg/Hz when 50-cell PM. Compared with the existing SAW gyroscopes based on phase velocity modulation, the gyroscopic gain factor in this paper is enhanced by 430-1600 times. This work jumps out of the framework of directly modulating SAW in gyroscopes and provides an innovative scheme of the indirect modulations from the rotation-dependent PM on SAW, showing excellent performance and potential for angular velocity measurement in extreme environments.

Mixed Finite Element Formulation in Nonlinear Geometrical Analysis of Space Trusses and Application to Trusses With Member Length Imperfection

Abstract Space trusses usually have a significant number of elements. As a result, it is inevitable that some elements have imperfections. In this study, the authors proposed a mixed finite element method for nonlinear geometrical analysis of space trusses. The post-buckling behavior of space trusses with length imperfections was predicted using a code list written in the programming software. The numerical results indicate that when the length imperfection approaches zero, the results converge to those obtained in studies on standard trusses. A second approach based on the displacement model is also proposed, in which the Lagrange multiplier method is used to deal with the member length imperfection. The results show that the two approaches proposed by the authors are effective, in which the mix formulation based on the finite element model is superior to that based on the displacement model. These results verify the accuracy of the method. Based on the proposed method, the effects of imperfections in the element layers on the truss were studied. The results show that the imperfections in the top two element layers have the maximum influence on the limit load value, whereas the remaining layers have a negligible influence. The combination of the imperfect lengths of these two layers of elements can produce results in which the limit load value is much larger than in the case of a perfect system. Based on the obtained results, it is possible to provide a solution for improving the overall stability of truss structures by creating reasonable imperfections.

Multi-Objective Optimization of Three-Stage Turbomachine Rotor Based on Complex Transfer Matrix Method

This study presents the complex transfer matrix method (CTMM) as an advanced mathematical model, providing significant advantages over the finite element method (FEM) by yielding rapid solutions for complex optimization problems. In order to design a more efficient structure of a three-stage turbomachine rotor, we integrated this method with various optimization algorithms, including genetic algorithm (GA), differential evolution (DE), simulated annealing (SA), gravitational search algorithm (GSA), black hole (BH), particle swarm optimization (PSO), Harris hawk optimization (HHO), artificial bee colony (ABC), and non-metaheuristic pattern search (PS). Thus, the best rotor geometry can be obtained fast with minimum bearing forces and disk deflections within design limits. In the results, the efficiency of the CTMM for achieving optimized designs is demonstrated. The CTMM outperformed the FEM in both speed and applicability for complex rotordynamic problems. The CTMM was found to deliver results of comparable quality much faster than the FEM, especially with higher element quality. The use of the CTMM in the iterative optimization process is shown to be highly advantageous. Furthermore, it is noted that among the different optimization algorithms, ABC provided the best results for this multi-objective optimization problem.

Modeling Electrochemical Impedance Spectroscopy Using Time-Dependent Finite Element Method

A time-dependent electrochemical impedance spectroscopy (EIS) model is presented using the finite element method (FEM) to simulate a 2D interdigitated electrode in an aqueous NaCl electrolyte. Developed in COMSOL Multiphysics, the model incorporates ion transport, electric field distribution, Stern layer effects, and electrode sheet resistance, governed by the Poisson and Nernst–Planck equations. This model can predict the transient current response to an applied excitation voltage, which gives information about the dynamics of the electrochemical system. The simulation results are compared with the experimental data, reproducing key features of the measurements. The transient current response indicates the need for multiple excitation cycles to stabilize the impedance measurement. At low frequencies (<1 kHz), the voltage drop at the Stern layer is significant, while at higher frequencies (>100 kHz), the voltage drop due to sheet resistance dominates. Moreover, the amplitude of the excitation voltage influences the EIS measurement, higher amplitudes (above 0.1 V) lead to non-linear impedance behavior, particularly at low ion concentrations. Discrepancies at low frequencies suggest that Faradaic processes may need to be incorporated for improved accuracy. Overall, this model provides quantitative insights for optimizing EIS sensor design and highlights critical factors for high-frequency and low-concentration conditions, laying the foundation for future biosensing applications with functionalized electrodes.

Research on the Geosynthetic-Encased Gravel Pile Composite Highway Foundation in Low-Temperature Stable Permafrost Regions

In low-temperature stable permafrost regions, both active and passive cooling measures are commonly employed to ensure the long-term stability of highway structures. However, despite adopting these measures, various types of structural issues caused by permafrost degradation remain prevalent in high-grade highways. This indicates that in addition to preventing permafrost melting, structural reinforcement of the foundation is still necessary. Based on the analysis of the long-term foundation temperature field and settlement using the finite element method, which was validated through an indoor top-down freeze–thaw cycle test, this paper explores, for the first time, the feasibility of applying geosynthetic-encased gravel pile composite highway foundations—previously commonly used for permafrost destruction—in low-temperature stable permafrost areas where permafrost protection is the primary principle. By analyzing the long-term temperature field, settlement behavior, and pile–soil stress ratios of permafrost foundations influenced by both the highway structure and composite foundation, it was found that when the pile diameter is 0.5 m, pile spacing is 2 m, and pile length is 11 m, the mean monthly ground temperature of the permafrost foundation will not be significantly affected. Therefore, the properly designed geosynthetic-encased gravel pile composite highway foundation can be adopted in low-temperature stable permafrost regions where permafrost protection, rather than destruction, is required.

Design and application of inclined tensioned steel cantilevered scaffolding.

In order to reduce the cost of scaffolding usage in engineering projects as much as possible under the premise of safety and rationality, while avoiding the risk of leakage caused by holes left in the building structure by the scaffolding, a type of inclined tensioned steel cantilevered scaffolding is proposed based on the stress characteristics of traditional cantilevered scaffolding. Considering the actual construction characteristics, a comprehensive approach involving theoretical analysis, on-site field tests, and finite element methods is adopted to clarify the full-process design method of the inclined tensioned steel cantilevered scaffolding. Additionally, recommendations are provided for the practical engineering design and application of the inclined tensioned steel cantilevered scaffolding. Finally, the inclined tensioned steel cantilevered scaffolding is applied in actual engineering. The research results indicate that the proposed full-process design method is feasible, and the inclined tensioned steel cantilevered scaffolding is more cost-effective than traditional cantilevered scaffolding.