Element Nodes Research Articles

3D Controlled‐source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

AbstractQuantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

Modeling of Ball Screw–Nut Interface Stiffness With Wear (Ball Screw Wear Dynamics)

Abstract The effects of wear, preload loss, and missing balls on the dynamics of ball screw drives in machine tools are modeled and incorporated into the finite element model of the drive assembly for condition monitoring. The contacts between the ball–nut and ball–screw are modeled using Hertzian springs, whose stiffnesses vary as a function of the worn contact area. These contact stiffnesses are then transformed to the finite element nodes on the nut and screw. The frequency response functions at the motor shaft and table, which can be measured by commercial computer numerical control (CNC), are predicted for various faults at different positions of the table. The experimentally validated model demonstrates that the faults primarily affect the first coupled torsional-axial mode of the ball screw drive and can be utilized for automated condition monitoring of ball screw drives.

Three-Dimensional MT Conductive Anisotropic and Magnetic Modeling Using A − ϕ Potentials Employing a Mixed Nodal and Edge-Based Element Method

Magnetotelluric (MT) sounding is a geophysical technique widely utilized in mineral resource surveys, where conductivity and magnetic permeability serve as essential physical parameters for forward modeling and inversion. However, the effects of conductive anisotropy and non-zero magnetic susceptibility are usually ignored. In this study, we present a three-dimensional (3D) MT modeling algorithm using Coulomb-gauged electromagnetic potentials, incorporating a mixed nodal and edge-based finite element method capable of simulating MT responses for conductive anisotropic and magnetic anomalies. Subsequently, the algorithm’s accuracy was validated in two steps: first, it was compared with analytical solutions for a 1D magnetic model; then, a comparison was made with previously published numerical results for a 3D generalized conductive anisotropic model. The results of two tests show that the maximum relative error is below 0.5% for both models. Furthermore, representative models were computed to comprehensively analyze the responses of MT. The findings illustrate the relationship between anisotropic parameters and electric fields and emphasize the significance of considering the impact of magnetic susceptibility in magnetite-rich regions.

Modelling of courtyard acoustics using the nodal discontinuous Galerkin finite element method to validate a newly developed PML

Evaluating the acoustics of courtyards has gained increasing attention from a sustainable urban design perspective, which requires numerical predictions tools for estimation of noise pollution and assessment of the influence of acoustic conditions. Accurate modeling of the acoustics of a courtyard is scarce because of the variety of factors to include: meteorological effects, building material properties, and geometry of the inner façades. We consider the nodal Discontinuous Galerkin finite element method framework in a 3D static and uniform medium and use it to model the sound propagation in a courtyard. A new and efficient Perfectly Matched Layer formulation is implemented at the open boundary of the courtyard model to absorb outgoing acoustic waves and this way limiting the size of the computational domain. The numerical results are validated against experimental measurements carried out in a scaled model of a courtyard with an elementary design to implement an accurate corresponding numerical model. The experiments were conducted in an anechoic chamber where the meteorological effects are controlled, and the courtyard facades are modelled as flat surfaces with known material properties.

Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems

Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed.

Using wave based SEA methods to model noise and vibration in underwater structures across a broad frequency range

Modelling noise and vibration transmission in underwater structures is often challenging due to the large size of the structures and the need for analysis across a broad frequency range. Finite Element and Boundary Element methods are useful at lower frequencies but are typically not computationally tractable for system level models at mid and high frequencies. Statistical Energy Analysis (SEA) methods are widely used for modeling structures in air at mid and high frequencies. However, traditional SEA methods often do not contain enough detail to describe wave propagation in heavy fluid loaded structures in underwater applications. This paper discusses the use of a more general 'wave based' SEA method that uses periodic structure theory to accurately calculate the wavetypes of typical underwater structures. The paper illustrates the method by initially considering simple structural sections and showing the effects of: curvature, heavy fluid-loading, panel stiffeners/ribs. The method is then applied to a simplified full scale submarine model and comparisons are made with a nodal finite element model. The domain of validity and the hypotheses of the method are then discussed.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

Damage of thin target plates impacted by conical projectiles with varying apex angles in ballistic application: experimental, FEA and ANN integrated approach

In the current study, an experimental setup consisting of smoothbore 30-caliber powder gun was employed to launch spherical projectiles (3 g/ϕ9) in the ballistic range of velocities from 500 to 1700 m/s and obliquity (0, 33 and 65°), impacting 2 mm thin Steel 1006 target plates. Crater dimensions (major and minor dia.) obtained from series of FE simulations run for the above configuaration was validated by corresponding experimental data. Subsequently, a fullscale 3D FE model considering 8-noded hexahedral elements was used to discretize SS304 conical projectiles (replacing spherical projectile) with various apex angles viz. 40°, 60°, 80° and 100° impacting on single (3 mm) and double layer (1.5 mm each) steel 1006 targets (replacing 2 mm target) in LS dyna. The material behavior was characterized using J–C strength and damage models, along with Gruneisen Equation of State. An erosion algorithm was used in the explicit FE code LS-DYNA to remove undesirable elements. In the next stage, Adam and Nadam optimizers have been employed in ANN models developed using Python code within the Tensor-Flow framework. The Keras Tuner library in the Tensor-Flow framework was used for hyper parameter tuning. The ANN models trained using simulation data successfully predicted the residual KE of conical projectiles with intermediate apex angles of 90°, 70°, and 50° across twelve different impact scenarios. The models with Adam and Nadam optimizers achieved mean squared error (MSE)/coefficient of determination (R2)/mean absolute percentage error (MAPE) values of 109.44/0.998/6.72 and 91.19/0.998/7.76, respectively.

Fatigue life prediction method for bolted joints based on equivalent structural stress under tensile–compressive loading

In this study, load amplitude–life (Fa–N) curves were obtained through tensile–compressive fatigue tests of bolted joints. It was observed that the correlation coefficient squared (R2) value of the Fa–N curve with the same geometric and pre-tightening parameters was high, but the R2 value of the Fa–N curve with all the parameters was low, indicating poor correlation and inability to meet the engineering requirements. Therefore, an equivalent structural stress model for the bolted joint was first developed based on a mechanical model with a strict mathematical definition to normalize these Fa–N curves, which considered the bolted joint loads as the input conditions and integrated the geometric and pre-tightening parameters. Subsequently, a classical beam–shell equivalent finite element model of the bolted joint was constructed. The nodal loads in the bolted connection zone were coupled with the forces and moments of the beam element nodes through finite element simulation, and the equivalent structural stress (σs) of the bolted joint was then obtained based on the equivalent structural stress model. Consequently, the equivalent structural stress–life (Ss–N) curve and probabilistic stress–life (P–Ss–N) curve normalized for different Fa–N curves were obtained by fitting the data of σs and N. Lastly, the accuracy of the fatigue life prediction method based on equivalent structural stress was verified by conducting the vibration fatigue test on the bolted joint structure of the subway antenna bracket.

Multi‐parameter design optimization of crash box for crashworthiness analysis

AbstractThis study focuses on the optimization process of crash box design. The design optimization process is resource‐intensive and requires multiple dynamic simulations. Numerous design parameters can be varied to satisfy the crashworthiness objectives. Therefore an intelligent and robust machine learning (ML) framework has been developed. This framework can be employed to assist in the optimization of various crashworthiness components with different crashworthiness objectives.Here, a reinforcement learning‐based (RL) machine learning framework is developed. It consists of a finite element method (FEM) surrogate and RL environment. The FEM surrogate is trained using data from FEM simulations as well as synthetic data generated by a Generative Adversarial Network (GAN). An inverse problem is solved for optimizing the geometrical parameters of the boundary value problem while the RL is receiving input from the FEM surrogate. RL has proven to be accurate in many fields due to its ability to explore and exploit the learned dynamics. Conventional algorithms require numerous iterations and tuned functions to achieve appropriate results and need to be initialized after a slight change in the problem. Due to an optimization of input parameters in an inverse problem, RL is chosen for the present investigation.For the simulation data needed, the crash box is designed with linear first‐order accurate four nodal shell elements. Also, bilinear elastoplastic material along with geometrical nonlinearity is used to simulate the dynamic deformation process.The RL agents learn to vary the crash box design parameters and search for the optimal parameters. The optimal parameters are based on the user‐defined crashworthiness objectives.

Development of improved finite element formulations for pile group behavior analysis under cyclic loading

AbstractThe effect of cyclic loading is an essential factor leading to progressive soil strength degradation. Therefore, a comprehensive analysis of the pile‐soil system behavior under cyclic loading is required to ensure the stability of pile group. There is room for improvement in the inherent constraint of the conventional numerical model in terms of approximating the soil resistance distribution along the pile by point loads at element nodes, necessitating a specific element that integrates considerations of pile group effect and cyclic loading within a unified framework. This study aims to develop a newly specific type of element for efficiently predicting nonlinear behavior within the pile‐soil system, addressing simulations involving nonlinear pile‐soil interaction, pile group effect, and cyclic loading. Modified element formulations based on soil stiffness matrices and soil resistance vectors specifically address pile group effect and consider parameters that influence pile behavior under cyclic lateral loading. The numerical solution procedure with Newton‐Raphson iteration allows the calculation of pile responses in geometric and material nonlinear analyses. The validation of the proposed method includes several examples, comparing it with existing numerical solutions and experimental tests of single piles and pile groups under cyclic loading. These comparisons further support the consistency of the proposed method with measured data and validate its accuracy in considering group effect and cyclic loading. The parametric study illustrates the ability of the proposed method to capture cyclic loading parameters while considering the influence of the number and magnitude of load cycles, the cyclic load direction, and the installation methods.

Size-Dependent Finite Element Analysis of Functionally Graded Flexoelectric Shell Structures Based on Consistent Couple Stress Theory

The functionally graded (FG) flexoelectric material is a potential material to determine the structural morphing of aircrafts. This work proposes the penalty 20-node element based on the consistent couple stress theory for analyzing the FG flexoelectric plate and shell structures with complex geometric shapes and loading conditions. Several numerical examples are examined and prove that the new element can predict the size-dependent behaviors of FG flexoelectric plate and shell structures effectively, showing good convergence and robustness. Moreover, the numerical results reveal that FG flexoelectric material exhibits better bending performance and higher flexoelectric effect compared to homogeneous materials. Moreover, the increase in the material length scale parameter leads to a gradual increase in the natural frequencies of the out-of-plane modes of FG flexoelectric plate/shell, while the natural frequencies of the in-plane modes change minimally, resulting in the occurrence of mode-switching phenomena.

Computational modeling and uncertainty prediction of hyperelastic constitutive responses of damaged brain tissue under different temperature and strain rates.

The effect of strain rate and temperature on the hyperelastic material stress-strain characteristics of the damaged porcine brain tissue is evaluated in this present work. The desired constitutive responses are obtained using the commercially available finite element (FE) tool ABAQUS, utilizing 8-noded brick elements. The model's accuracy has been verified by comparing the results from the previously published literature. Further, the stress-strain behavior of the brain tissue is evaluated by varying the damages at various strain rates and temperatures (13, 20, 27, and 37°C) under compression test. Additionally, the sensitivity analysis of the model is computed to check the effect of input parameters, that is, the temperature, strain rate, and damages on the material properties (shear modulus). The modeling and discussion sections enumerate the inclusive features and model capabilities.

Buckling load optimization of laminated composite plates with elliptical hole under different non-uniform edge loads using bonobo optimizer algorithm

This present work investigates the buckling load optimization of the laminated composite plates with the elliptical hole under different non-uniform edge loads. The design objective is the maximization of the critical buckling load by determining the optimum fiber orientations in the layers. The stability equations are derived based on the first-order shear deformation theory (FSDT) of the laminated plates. The critical buckling loads are calculated using the finite element method with the nine noded rectangular element having five degrees of freedom per node. The bonobo optimizer (BO) algorithm is employed to optimize the laminated composite plates with the elliptical hole. The computer programming is developed in the MATLAB environment. Besides, the parametric studies are conducted for different types of the non-uniform edge loads, boundary conditions, cutout radius ratio and aspect ratios and the obtained numerical results are compared. Finally, the optimum results show that these factors play an imperative role in the optimum pattern and stacking sequence of the laminated composite plates with hole. The new significant findings can aid the designers in the structural design, and other industrial applications.

Finite element analysis of characterization parameters the double cracks in linear elastic DCFEM

In fracture mechanics, the lifespan of structures plays an important and effective role in terms of rigidity and resistance to secondary effects linked to direct friction and the effects of the atmosphere, in order to avoid any danger that could end to the life of the work. structure, and among these effects are those linked to internal and external cracks. Sometimes the structure may be exposed to double cracks resulting from external loads and influences, such as our study that we present in this paper. We must also not forget the structures which contain holes, which play an important role in tension with other elements and have an essential function in operation and resistance. We don't forget the basic dimensions between the crack and the other. These days, numerical modeling techniques, such the finite element method which is the most popular approach for determining characterisation parameters at the bottom of a crack are the foundation for fatigue problem analysis. This work aims to investigate the impact of two-dimensional parameters (b, c) and the crack length (a) on crack parameters, including the J-integral and stress intensity factor (SIF), of an aluminum alloy structure with a double starting fracture. The finite element method (DCFEM) has been used to study these characteristics under uniform tensile loading. After that, rupture happens in plane stress under mode I loading, and the modeling is done using 4-node CPS4R quadratic elements. Numerous instances have been provided. The optimal rupture resistance solution for plates with two cracks on each sides will be developed using the results obtained.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Layerwise third-order shear deformation theory with transverse shear stress continuity for piezolaminated plates

Regarding laminated structures, an electromechanically coupled Finite Element (FE) model based on Layerwise Third-Order Shear Deformation (LW-TOSD) theory is proposed for static and dynamic analysis. LW-TOSD ensures the continuity of in-plane displacements and transverse shear stresses. The current LW-TOSD can be applied to arbitrary multi-layer laminated structures with only seven Degrees of Freedom (DOFs) for each element node and eliminates the use of the shear correction factors. Moreover, a shear penalty stiffness matrix is constructed to satisfy artificial constraints to optimize the structural shear strain. A dynamic finite element model is obtained based on LW-TOSD using the Hamilton’s principle. First, the accuracy of the current model is validated by comparing with literature and ABAQUS results. Then, this study carries out numerical investigations of piezolaminated structures for different width-to-thickness ratios, length-to-width ratios, penalty stiffness matrix, boundary conditions, electric fields and dynamics.

Exact conservation in transient electromagnetics using edge elements

In time domain to solve electromagnetic radiation and scattering problems, several numerical methods are available in literature. Among these numerical methods Finite Difference Time Domain (FDTD) method and Time Domain FEM (TDFEM) method are more popular. In the present work, we have proposed a quantity in continuum, which is conserved under certain boundary conditions. Additionally, we have also proposed a time marching scheme within edge element framework, which satisfy this conservation characteristic in algorithmic sense irrespective of the choice of time step. To facilitate this, we have used some conversion algorithm that transform nodal finite element data to edge finite element data.

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

AbstractIn this paper, the three‐dimensional stress field in nonlinear laminated composite shells with piezoelectric patches subjected to displacement‐dependent loads is analyzed. To solve the problem, the geometrically exact (GeX) hybrid‐mixed four‐node solid‐shell element has been developed. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms are taken exactly at the element nodes. The laminated solid‐shell element formulation is based on the choice of an arbitrary number of sampling surfaces (SaS) parallel to the middle surface and located at the Chebyshev polynomial nodes within the layers, in order to introduce the displacements and electric potentials of these surfaces as unknown functions. The outer surfaces and interfaces are also included into a set of SaS. Due to analytical integration utilized to evaluate the tangent stiffness matrix, the proposed GeX piezoelectric solid‐shell element under nonconservative loading exhibits superior performance in the case of coarse meshes and allows the use of only one load step in all considered benchmark problems. Therefore, it can be recommended for the large deformation analysis of doubly curved laminated composite shells with piezoelectric patches, since the SaS formulation allows one to find three‐dimensional solutions of electroelasticity with a prescribed accuracy.

Numerical analysis of downward progressive landslides in long natural slopes with sensitive clay

Landslides occurring in sensitive clay often result in widespread destruction, posing a significant risk to human lives and property due to the substantial decrease in undrained shear strength during deformation. Assessing the consequences of these landslides is challenging and necessitates robust numerical methods to comprehensively investigate their failure mechanisms. While studies have extensively explored upward progressive landslides in sensitive clays, understanding downward progressive cases remains limited. In this study, we utilised the nodal integration-based particle finite element method (N-PFEM) with a nonlinear strain-softening model to analyse downward progressive landslides in sensitive clay on elongated slopes, induced by surcharge loads near the crest. We focused on elucidating the underlying failure mechanisms and evaluating the effects of different soil parameters and strain-softening characteristics. The simulation results revealed the typical pattern for downward landslides, which typically start with a localised failure in proximity to the surcharge loads, followed by a combination of different types of failure mechanisms, including single flow slides, translational progressive landslides, progressive flow slides, and spread failures. Additionally, inclined shear bands occur within spread failures, often adopting distinctive ploughing patterns characterised by triangular shapes. The sensitive clay thickness at the base, the clay strength gradient, the sensitivity, and the softening rate significantly influence the failure mechanisms and the extent of diffused displacement. Remarkably, some of these effects mirror those observed in upward progressive landslides, underscoring the interconnectedness of these phenomena. This study contributes valuable insights into the complex dynamics of sensitive clay landslides, shedding light on the intricate interplay of factors governing their behaviour and progression.