Traditional Finite Element Method Research Articles

Overlapping finite elements for the Navier-Stokes equations

The purpose of this paper is to investigate the performance of a certain family of low-order quadrilateral finite elements in the solution of the stationary Navier-Stokes equations governing incompressible fluid flows. These finite elements are derived from the ‘overlapping finite elements’, first developed for the solution of problems in solid mechanics [5,9,48] and incorporate features of both meshfree and traditional finite element methods. One of their most remarkable properties is the insensitivity to mesh distortions. Also, since the Shepard functions are replaced by a suitable interpolation, the resulting basis functions are entirely polynomial, which allows the numerical integration of the weak forms to be performed with few integration points [6,49]. We also discuss the theoretical reasons for the stability of the solutions, that is, the absence of spurious pressure modes, and show that the proposed discretization scheme passes the relevant inf-sup test.

Study on the impact of ultrasonic vibration-assisted grinding of glass-ceramics on surface/subsurface damage mechanism

PurposeAn investigation was conducted into the impact of various process parameters on the surface and subsurface quality of glass-ceramic materials, as well as the mechanism of material removal and crack formation, through the use of ultrasonic-assisted grinding.Design/methodology/approachA mathematical model of crack propagation in ultrasonic-assisted grinding was established, and the mechanism of crack formation was described through the model. A series of simulations and experiments were conducted to investigate the impact of process parameters on crack depth, surface roughness, and surface topography during ultrasonic-assisted surface and axial grinding. Additionally, the mechanism of crack formation was explored.FindingsDuring ultrasonic-assisted grinding, the average grinding forces are between 0.4–1.0 N, which is much smaller than that of ordinary grinding (1.0–3.5 N). In surface grinding, the maximum surface stresses between the workpiece and the tool gradually decrease with the tool speed. The surface stresses of the workpiece increase with the grinding depth, and the depth of subsurface cracks increases with the grinding depth. With the increase of the axial grinding speed, the subsurface damage depth increases. The roughness increases from 0.780um/1.433um.Originality/valueA mathematical model of crack propagation in ultrasonic-assisted grinding was established, and the mechanism of crack formation was described through the model. The deformation involved in the grinding process is large, and the FEM-SPH modeling method is used to solve the problem that the results of the traditional finite element method are not convergent and the calculation efficiency is low.

The adaptive hygrothermo–magneto–electro–elastic coupling improved enriched finite element method for functionally graded magneto–electro–elastic structures

In this study, the hygrothermo–magneto–electro–elastic coupling improved enriched finite element method (HC-IEFEM) is proposed to analyze functionally graded magneto–electro–elastic (FG-MEE) structures. The static behavior of FG-MEE structures in the hygrothermal environment is studied. In the enriched finite element method (EFEM) enhanced by interpolation cover functions, the improved shape functions are added to solve the rank defect problem. Quadrilateral elements are used in the analysis of FG-MEE structures in this study. This improved method has proven to be relatively more efficient and resistant to mesh distortion than the traditional finite element method (FEM). Besides, the adaptive mesh refinement (AMR) scheme is used to refine the mesh in local areas to improve numerical calculation efficiency. Numerical examples show that the adaptive HC-IEFEM can achieve relatively high accuracy for analyzing FG-MEE structures. The proposed HC-IEFEM can obtain high accuracy in the hygrothermal environment using a relatively coarse mesh through the improved shape function and AMR scheme. Therefore, the significant potential is demonstrated in analyzing FG-MEE structures in the hygrothermal environment using the proposed adaptive HC-IEFEM.

High fidelity FEM based on deep learning for arbitrary composite material structure

Due to the outstanding performance, composite materials are widely used and analyzing their properties and designing them based on performance has become a crucial task in the field of many manufacturing industries. Composite materials possess complex multiscale structures, and traditional fine-scale finite element modeling and analysis may lead to severe computational resource challenges. To overcome this difficulty, breakthroughs in key technologies of multiscale accelerated analysis algorithms are required. In this study, an innovative approach based on artificial intelligence and multiscale finite element method is presented. This approach involves partitioning the entire composite material structure into coarse grids that resemble homogenous structures of similar size, providing results consistent to fine-grid finite element analysis. By utilizing CNN for image feature recognition and employing the CGAN adversarial method, coarse-grid equivalent stiffness matrices and multiscale shape functions from completely random microstructures of composite materials can be obtained. Consequently, this enables a rapid response process from microstructure to low-resolution grid to high-resolution physical field, with remarkably accurate physical field results. Moreover, compared to traditional fine-grid finite element methods, this approach significantly reduces memory usage and computation time. This method is applicable to composite materials with varying shaped inclusions, different component properties, and diverse geometric distributions, allowing these materials to perform high-fidelity finite element calculations on coarse grids and predict their mechanical behavior. Furthermore, this breakthrough opens avenues for accelerating the optimization design of composite materials with diverse mechanical functionalities, by employing a bottom-up approach.

An extended finite element method for the Nernst-Planck-Poisson equations

The Nernst-Planck-Poisson (NPP) system of equations is used to study the ion diffusion mechanism in materials, commonly used in storage batteries and solid oxide fuel cells. In particular, NPP equations are used to predict the concentration of charged defects and electric potential. Depending on the size of the computational domain, the distributions exhibit steep gradients near the boundaries. The traditional finite element method, when employed, requires extremely refined mesh to capture the steep gradient as they employ simple polynomials. To alleviate the mesh dependence, in this work, we propose to augment the traditional finite element approximation space with a suitable ansatz to capture the steep gradient within the framework of the extended finite element method. The robustness and the accuracy of the proposed framework is demonstrated by comparing it with an overkill finite element solution.

Seepage propagation simulation of a tunnel gasketed joint using the cohesive zone model

The gasketed joint plays a critical role in the waterproof or water-leakage resistance performance of tunnel segmental linings. This study presents a novel numerical simulation method based on the cohesive zone model (CZM) to accurately analyze the seepage phenomena along the interface of elastic sealing gaskets in shield tunnel joints. While traditional finite element methods are widely used, they cannot simulate interface opening, limiting their applicability. To overcome this challenge, the proposed method leverages the concept of hydraulic fracturing and redefines the seepage problem as a seepage problem of elastic impermeable media. By establishing a two-dimensional hydraulic fracturing model using the CZM, this study conducts a comprehensive sensitivity analysis of the cohesive element parameters and validates the accuracy and effectiveness of simulating seepage along the interface of elastic impermeable media. Based on the insights gained from the sensitivity analysis, a seepage calculation model is successfully developed, considering the unique characteristics of elastic sealing gaskets. The model is applied to three cross-sectional shapes of elastic sealing gaskets, and their seepage characteristics are analyzed based on the interface opening displacement and liquid pressure distributions during seepage. The results highlight the SC3 cross-sectional shape, which exhibits minimal opening displacement and the highest net water pressure at the moment of seepage occurrence, demonstrating superior waterproof performance compared to other forms. Through the introduction of the CZM, comprehensive parameter analysis, validation, and practical application, this study overcomes the limitations of traditional finite element methods in simulating interface opening. The proposed model, which more accurately captures seepage characteristics than other methods, provides an optimized simulation method for analyzing hydraulic coupling issues related to seepage at shield tunnel joints and is a significant contribution to the field, enhancing our understanding and enabling more accurate analyses of seepage phenomena in shield tunnel joints using elastic sealing gaskets.

A Coupled Finite-Boundary Element Method for Efficient Dynamic Structure-Soil-Structure Interaction Modeling

This paper introduces an innovative approach to numerically model Structure–Soil-Structure Interaction (SSSI) by integrating the Boundary Element Method (BEM) and the Finite Element Method (FEM) in a coupled manner. To assess the accuracy of the proposed method, a comparative study is undertaken, comparing its outcomes with those generated by the conventional FEM technique. Alongside accuracy, the computational efficiency aspect is crucial for the analysis of large-scale SSSI problems. Hence, the computational performance of the coupled BEM–FEM method undergoes a thorough examination and is compared with that of the standalone FEM method. The results from these comparisons illustrate the superior capabilities of the proposed method in comparison to the FEM method. The novel approach provides more reliable results compared to traditional FEM methods, serving as a valuable tool for engineers and researchers involved in structural analysis and design.

Mechanical Analysis of Semi-Rigid Base Asphalt Pavement under the Influence of Groundwater with the Spectral Element Method

Over prolonged exposure to groundwater conditions, semi-rigid base asphalt pavements can undergo significant changes in their internal moisture field, resulting in substantial variations in the pavement’s stiffness and, consequently, affecting the overall load-bearing capacity and stability of the road structure. This paper employs FWD non-destructive testing equipment to assess its mechanical performance and conduct data analysis and conducts a mechanical response study of asphalt road surfaces, considering the influence of roadbed moisture levels. Using the dynamic analytical theory, the fundamental equations and stiffness matrices for a linear elastic half-space model were established, leading to the development of a computational model for the mechanical response of semi-rigid base asphalt pavements under FWD dynamic loading, with an examination of the surface deflection in relation to changes in groundwater levels. Numerical examples and engineering applications were employed to validate the proposed model. The research findings indicate: With the passage of time, surface deflection values initially increase and then decrease, exhibiting a sinusoidal variation pattern similar to that of the applied load. As the distance from the loading center increases, the moment of peak deflection continually lags behind. The average absolute relative error between the results obtained using the method proposed in this study and the traditional ABAQUS finite element method was only 0.70%. The correlation coefficient between the theoretically computed deflection curve and the measured deflection curve using the spectral element method was greater than 0.9, with an average absolute relative error of 4.92% between the theoretical peak deflection and the measured peak deflection. As the groundwater level rises, surface deflection noticeably increases, with an approximately 40% increase in deflection values at the loading center. These research findings can be utilized to analyze the dynamic deflection of semi-rigid base asphalt pavements under various groundwater conditions, providing significant practical value for assessing road structural performance and serviceability.

Deep-time temperature field simulation of hot dry rock: A deep learning method in both time and space dimensions

Hot dry rock has emerged as a crucial renewable resource with a potential to provide significant energy to China. Accurately simulating the temperature field of hot dry rocks in space is essential in estimating their reserves. Moreover, predicting the temperature change curve of hot dry rocks over time is crucial for research in the field. Although these tasks are interrelated and share a common ground of simulating the temperature of hot dry rocks, previous approaches have dealt with them separately, each being challenging. In this study, we propose a 3D U-shaped deep-time neural network that can extract the nonlinear relationship between the temperature field and rock strata parameters, thereby enabling the simulation of the temperature field of hot dry rocks in both space and time dimensions. The proposed neural network, which incorporates an attention mechanism and residual structure, can incorporate multiple rock parameters and geological structures, leading to more accurate simulation of the temperature field of hot dry rocks. To train the network, we constructed extensive datasets based on granite rock parameters and actual geological structures, with the temperature fields simulated by the finite element method serving as labels. The resulting trained network model can successfully emulate the complex hot dry rock model, reducing the workload by combining the two networks into one. Furthermore, the simulated temperature field allows the observation of the effect of different geological activities on hot dry rocks. Network simulation has shown notable advancements in efficiency compared to traditional finite element methods. Moreover, it offers the advantage of exploring the impact of distinct geological structures on geothermal reservoirs in a spatial context. Additionally, it allows for the observation of temporal variations in the temperature distribution across different geological strata during different time periods. Thereby laying the groundwork for further prediction and exploitation of hot dry rock geothermal reserves.

3D meshless modeling of piezoelectric structure based on the radial point interpolation method

Piezoelectric materials find widespread applications in high-precision actuators and sensors. However, the traditional finite element method falls short in meeting the simulation needs of piezoelectric structures due to complexities in mesh generation and precision requirements for accurate simulations. This paper focuses on adapting and generalizing the meshless modeling technique based on the radial point interpolation method to address three-dimensional simulations of piezoelectric structures and accurately model positive and inverse piezoelectric behaviors. The piezoelectric body is discretized into point cloud data, and displacement and potential are approximated by interpolating values through the radial point interpolation method. Subsequently, a 3D meshless model for general piezoelectric structures is formulated based on the principle of minimum potential energy. The influences of key factors on interpolation accuracy and computational cost were discussed. Numerical examples, featuring a 3D piezoelectric bimorph beam as actuators and sensors, validate the efficacy of this extended approach. Results affirm that the three-dimensional meshless method, rooted in the radial point interpolation technique, achieves superior accuracy in predicting piezoelectric behavior compared to the traditional finite element method. This adaptation not only enhances the applicability of meshless modeling but also represents a significant advancement in accurately simulating complex three-dimensional piezoelectric structures.

Convergence analysis of virtual element method for the electric interface model on polygonal meshes with small edges

A conforming virtual element method is studied for approximating the solution of a pulsed electric interface model on polygonal meshes with small edges or faces, which traditional virtual and finite element methods cannot easily handle. One of the significant advantages of this virtual element method is to generate interface-conforming meshes efficiently exploiting polygonal meshes and hanging nodes. The virtual element method employs specific projection operators to establish optimal error estimates with reasonable assumptions regarding the solution's regularity. The voltage potential of the pulsed electric model through the physical media is approximated by using a fully discrete virtual element approach based on Crank-Nicolson temporal discretization. Numerical examples are used to illustrate the expected order of convergence. The efficiency and robustness of the proposed method are displayed on interface-independent/dependent background-fitted meshes with small edges and large magnitudes of discontinuities across the interface by these experiments.

Low order nonconforming finite element methods for nuclear reactor model

A uniform framework of the linearized fully decoupled fully discrete schemes is developed and investigated with low order nonconforming finite element methods (FEMs) for nuclear reactor model. On the one hand, a general scheme called Scheme I is constructed. Then, the modified Ritz projection and mathematical induction are used to obtain its unconditional optimal error estimate in the L2-norm and superclose estimates in the broken H1-norm, thereby correcting the mistake in previous literature and eliminating the need for the so-called popular time-space splitting method. On the other hand, a new positivity-preserving scheme named Scheme II, which combines the traditional FEMs with cut-off post-processing method, is designed to avoid the non-physical oscillation in numerical calculations. Based on the results of Scheme I, an optimal error estimate in the L2-norm for Scheme II is also derived. Finally, taking the nonconforming EQ1rot element as an example, some numerical experiments are provided to demonstrate the theoretical results. It is worth noting that the analysis presented herein is also suitable for simulating nuclear reactor model in the narrow channel with anisotropic meshes.

An Improved Decoupled Finite Element Analysis Method of Ultra-Large Offshore Wind Turbine Subjected to Combined Wind–Wave Actions

Abstract Owing to the simplification of the wind turbine, it is difficult to accurately simulate the interaction between the rotor system and the supporting structure using the decoupling finite element method. Therefore, when using this method for safety assessment and dynamic response research of the offshore wind turbine (OWT), there is a deviation between the simulation result and the real response of the OWT. In this study, an improved decoupled finite element analysis method is proposed based on a theoretical derivation. A simplified finite element model of a 10 MW jacket OWT with an equivalent substructure was established, and the dynamic response of the OWT under wind and waves was studied. By comparing the results of the fully coupled analysis method and the traditional finite element method, the applicability of the traditional finite element analysis method to the dynamic analysis of an OWT under typical winds and waves is discussed. The limitations of using the traditional finite element method to study or evaluate an OWT complex dynamic system were revealed, and the effectiveness and applicability of the improved method proposed in this study were qualitatively and quantitatively verified. Subsequently, based on the proposed improved decoupled finite element analysis method, a numerical calculation corresponding to a fully coupled test was performed. Compared with the numerical results obtained by the traditional finite element method, the improved decoupled finite element method proposed in this study obtained more consistent results with the fully coupled test.

Vectorial finite element method for neutron transport solving with preconditioning GMRES acceleration

This work proposed a vectorial finite element method (VFEM) for solving neutron transport equation with high accuracy and convergence. The Streamline Upwind Petrov–Galerkin method is used to establish the VFEM for accuracy, and the traditional discrete-ordinate neutron transport equation is organized in vectorial form for high stability and convergence. To further improve the convergence efficiency of VFEM, the preconditioning GMRES is introduced for solving the VFEM linear system, and different preconditioners are imposed, including the Jacobian and Block-Jacobian. Numerical results showed that the proposed VFEM has high accuracy and adaptability for multi-group and multi-dimensional neutron transport simulation. Further performance analyses showed that the proposed VFEM has higher efficiency than the traditional finite element method, while the application of preconditioning GMRESs can further improve the convergence efficiency. This work can provide an innovative and effective unstructured-mesh neutron transport solution format, and provide a valuable reference for further improvement of other numerical methods.

An Efficient Goal-Oriented Adaptive Finite Element Method for Accurate Simulation of Complex Electromagnetic Radiation Problems

A goal-oriented adaptive frequency domain finite element method for solving electromagnetic radiation problems including complex structures is presented in this paper. Compared with the traditional adaptive finite element method, the goal-oriented method can flexibly control the refined regions according to the parameters of interest; therefore it has better convergence and has made significant progress in scattering problems and eigenvalue problems. To simulate complex antennas, this paper proposes an error indicator with high accuracy and low computational cost, and it uses the adjoint problem to weight element residuals without additional degrees of freedom. Moreover, high-quality mesh refinement algorithms adapted to this indicator are developed using a suitable point insertion strategy for multiscale structures. By simulating two practical antennas, comparisons with the traditional goal-oriented FEM and the well-developed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -adaptive finite element method in commercial software demonstrate the accuracy and efficiency of the proposed method.

Research on rapid calculation method of wind turbine blade strain for digital twin

Digital twin (DT) technology provides unlimited possibilities for remote intelligent operation and maintenance of wind turbine blade (WTB) in operation. However, the urgent demand for high accuracy and real-time performance limit the engineering applications of this technology. This article presents a novel rapid calculation method of WTB strain, which aims to break the technical difficulties of real-time calculation in digital twin and realize synchronous mapping of WTB health status. Firstly, the load acting on the WTB is simplified according to the equivalent conversion theory. Then, a simplified finite element model, namely mechanism calculation model, of WTB is built based on mechanics. The multi-fidelity surrogate model is innovatively adopted to calibrate the results of the mechanism calculation via a small amount of data measured via sensors on the WTB. Based on the calibrated results, a unique strain field estimation model is built to provide more comprehensive information about WTB. Finally, the proposed method is verified via an experiment about WTB strain measurement and numerical simulation. The results demonstrate a considerable advancement in real-time performance with approximately 1444× compared to the traditional finite element method. The shortest elapsed time is 0.91s. Moreover, the minimum relative error is 0.11 % and the average relative error is less than 0.76 %.

A neural network finite element method for contact mechanics

Computational biomechanical models of organ systems utilizing the finite element (FE) method have been extensively applied to the study of normal and pathophysiological function. While a robust method and providing for many unique pathophysiological insights, traditional FE methods remain prohibitively slow for real-time many-query clinical applications. To meet these demanding computational requirements, we have developed a general neural network finite element (NNFE) approach for hyperelastic soft tissue organ simulations that can produce equivalent accuracy within clinically relevant time frames. However, many organ systems involve contact between structural elements (e.g. heart valve leaflets), which has not been previously addressed in the NNFE and related approaches. In the present work, we developed a NNFE-based method for contact between elastic deformable bodies. We exploited the fact that stable equilibrium solutions in nonlinear hyperelasticity minimize the potential energy to train the neural network, with contact incorporated through an potential energy penalty. This penalty was discretized and computationally implemented in JAX (Google, Inc.) in a way that could be statically allocated and jit compiled for rapid computation on modern GPU hardware. Following our previous NNFE formulation, we represented the problem domain geometry and enforced the necessary boundary conditions using a Non-Uniform Rational B-Splines (NURBS) based geometry scheme. NURBS were chosen for their ability to smoothly represent body and surface geometries. To demonstrate the method, we developed and verified two basic contact problems using a hyperelastic material model: indentation and flap. Both problems were simulated with a fully connected neural network with a parallel linear layer. Training continued until the gradient of the energy with respect to network parameters dropped below 10−2, taking less than six hours in each case studied. Once trained, the NNFE approach was capable of accurately predicting the equilibrium configurations of representative contact problems. Moreover, the NNFE contact approach required nearly six order of magnitude less time to evaluate than an equivalent FEniCSx implementation that utilized a similar mesh of cubic order Lagrange elements (0.01 s compared to 1715 seconds, respectively). Moreover, the solutions were trained over full ranges of loading and were thus able to faithfully represent families of solutions without requiring retraining. While addressing basic problems in contact, this study serves as crucial stepping stone to generate NNFE organ simulations for rapid prediction of patient specific solutions that involve contact.

Contact Analysis for Cycloid Pinwheel Mechanism by Isogeometric Finite Element

Cycloid drives are generally used in precision machinery requiring high-reduction ratios, such as robot joint (RV) reducers. The contact stress of cycloidal gears greatly affects lifetime and transmission performance. Traditional finite element method (FEM) has less computational efficiency for contact analysis of complex surface. Therefore, in this paper, isogeometric analysis (IGA) was employed to explore the multi-tooth contact problem of the cycloid pinwheel drive. Based on the nonuniform rational B spline (NURBS) curved surface generation method, the NURBS tooth profile of the cycloid gear was reconstructed. In addition, the NURBS surface of the cycloid gear–pin tooth–output pin was generated via the element splicing method. A geometrical analysis model of cycloid pinwheel drive was established to solve the contact force of the meshing pair under different input angles and compared with the finite element method in terms of convergence, resultant accuracy, and solving timeliness. The results show that isogeometric analysis has higher accuracy and efficiency than the finite element method in calculating the contact stress and contact force. The error of the IGA is only 8.8% for 10 × 10 elements in contact, while the error of the finite element method reaches about 40%. The method can improve the contact simulation accuracy of the cycloid drive and provides a reference for the design evaluation of RV reducer.

A coupling FEM/BEM method for acoustic-structural interaction problems in uniform relative motion between structure source and fluid

When a vibrating thin structure moves in a heavy flow field, the effect of sound waves on the structure cannot be ignored. This effect is especially pronounced when the relative velocity of the structure and the flow field increases. The hybrid numerical method of finite element method (FEM) and convective boundary element method (BEM) is proposed to solve acoustic-structural interaction problems in uniform relative motion between structure source and fluid. Besides, an acoustic-analogy Lorentz transform is proposed to handle the direct convective boundary integral equation (BIE), which makes the convective BIE has the same form as traditional BIE so that there are no more convective terms and extra singular integrals. The continuity condition at the fluid-structure interface is also proposed to generate the coupling equation. Compared with the traditional FEM, this hybrid method has the advantages of greatly reducing the number of finite elements outside the structure as well as automatically satisfying the radiation condition at far field. And compared with the direct convective BIE, this method reduces the computational complexity and time. It is potential to deliver a solver with higher accuracy and less memory consumption. Some numerical examples are designed to study the influence of the convective effect. Keywords: Acoustic-structural interaction, Finite element method, Boundary element method, FEM-BEM coupling

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.