Traditional Finite Element Method Research Articles

Comparison of Continuous and Discrete Modeling Strategies for the Structural Assessment of a Masonry Vault Under Dynamic Seismic Loading

ABSTRACT To begin, the paper introduces methodologies employing both homogeneous continuous Finite Element Method (FEM) and block-by-block Discrete Element Method (DEM) to model masonry behaviors. In the continuous approach, the masonry is modeled as an anisotropic damageable material (homogeneous or not). In contrast, the block-based approach represents rigid blocks interacting through contact joints governed by frictional cohesive behaviors. The DEM framework allows for significant displacements, rotations, and complete detachments of blocks, aspects often overlooked in traditional FEM models. The primary application of this research involves the seismic assessment of a masonry cross vault. Notably, the numerical results exhibit the approach’s capability to provide realistic predictions of failure mechanisms, crucial for retrofitting efforts. This includes an accurate representation of the actual cracking pattern, accounting for significant displacements. The numerical implementation is accessible through the open-source LMGC90 software. Comparative analyses are then carried out, focusing on the strengths and weaknesses of FEM macro-modeling (continuous homogeneous description) and DEM utilizing rigid blocks.

Variable horizon ordinary state-based peridynamic analysis in ANSYS framework

This study presents the implementation of ordinary state-based peridynamic (OSB PD) analysis in a finite element framework using ANSYS, a commercial software. The novelty of the present PD formulation is its ability to accommodate nonuniform discretization of the PD solution domain, which may be crucial for the realistic modeling of complex structures. It enables the use of variable horizon while eliminating the need for surface and volumetric correction procedures. The PD governing equations are constructed by using MATRIX27, an element native to ANSYS. The PD bond breakage during the implicit solution of these equations is achieved through the EKILL option in ANSYS. The efficacy and robustness of the present PD approach are demonstrated by considering four distinct geometric configurations, each subjected to specific boundary conditions. The results from the present PD approach exhibit a good agreement with those obtained from traditional finite element method (FEM) and experimental data.

Scalar Field Prediction on Meshes Using Interpolated Multi-Resolution Convolutional Neural Networks

Abstract Scalar fields, such as stress or temperature fields, are often calculated in shape optimization and design problems in engineering. For complex problems where shapes have varying topology and cannot be parametrized, data-driven scalar field prediction can be faster than traditional finite element methods. However, current data-driven techniques to predict scalar fields are limited to a fixed grid domain, instead of arbitrary mesh structures. In this work, we propose a method to predict scalar fields on arbitrary meshes. It uses a convolutional neural network whose feature maps at multiple resolutions are interpolated to node positions before being fed into a multilayer perceptron to predict solutions to partial differential equations at mesh nodes. The model is trained on finite element von Mises stress fields, and once trained it can estimate stress values at each node on any input mesh. Two shape datasets are investigated, and the model has strong performance on both, with a median R-squared value of 0.91. We also demonstrate the model on a temperature field in a heat conduction problem, where its predictions have a median R-squared value of 0.99. Our method provides a potential flexible alternative to finite element analysis in engineering design contexts. Code and datasets are available at: https://github.com/kevinferg/sfp-cnn.

Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

AbstractImmersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

Investigation of Crack Propagation and Failure of Liquid-Filled Cylindrical Shells Damaged in High-Pressure Environments

Cylindrical shell structures have excellent structural properties and load-bearing capacities in fields such as aerospace, marine engineering, and nuclear power. However, under high-pressure conditions, cylindrical shells are prone to cracking due to impact, corrosion, and fatigue, leading to a reduction in structural strength or failure. This paper proposes a static modeling method for damaged liquid-filled cylindrical shells based on the extended finite element method (XFEM). It investigated the impact of different initial crack angles on the crack propagation path and failure process of liquid-filled cylindrical shells, overcoming the difficulties of accurately simulating stress concentration at crack tips and discontinuities in the propagation path encountered in traditional finite element methods. Additionally, based on fluid-structure interaction theory, a dynamic model for damaged liquid-filled cylindrical shells was established, analyzing the changes in pressure and flow state of the fluid during crack propagation. Experimental results showed that although the initial crack angle had a slight effect on the crack propagation path, the crack ultimately extended along both sides of the main axis of the cylindrical shell. When the initial crack angle was 0°, the crack propagation path was more likely to form a through-crack, with the highest penetration rate, whereas when the initial crack angle was 75°, the crack propagation speed was slower. After fluid entered the cylindrical shell, it spurted along the crack propagation path, forming a wave crest at the initial ejection position.

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.

Finite-element assembly approach of optical quantum walk networks

We present a finite-element approach for computing the aggregate scattering matrix of a network of linear coherent scatterers. These might be optical scatterers or more general scattering coins studied in quantum walk theory. While techniques exist for two-dimensional lattices of feed-forward scatterers, the present approach is applicable to any network configuration of any collection of scatterers. Unlike traditional finite-element methods in optics, this method does not directly solve Maxwell’s equations; instead it is used to assemble and solve a linear, coupled scattering problem that emerges after Maxwell’s equations are abstracted within the scattering matrix method. With this approach, a global unitary is assembled corresponding to one time step of the quantum walk on the network. After applying the relevant boundary conditions to this global matrix, the problem becomes non-unitary and possesses a steady-state solution that is the output scattering state. We provide an algorithm to obtain this steady-state solution exactly using a matrix inversion, yielding the scattering state without requiring a direct calculation of the eigenspectrum. The approach is then numerically validated on a coupled-cavity interferometer example that possesses a known, closed-form solution. Finally, the method is shown to be a generalization of the Redheffer star product, which describes scatterers on one-dimensional lattices (2-regular graphs) and is often applied to the design of thin-film optics, making the current approach an invaluable tool for the design and validation of high-dimensional phase-reprogrammable optical devices and study of quantum walks on arbitrary graphs.

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.