Finite Element Solution Research Articles

A deep learning method for solving thermoelastic coupling problem

Abstract The study of thermoelasticity problems holds significant importance in the field of engineering. When analyzing non-Fourier thermoelastic problems, it was found that as the thermal relaxation time increases, the finite element solution will face convergence difficulties. Therefore, it is necessary to use alternative methods to solve. This paper proposes a physics-informed neural network (PINN) based on the DeepXDE deep learning library to analyze thermoelastic problems, including classical thermoelastic problems, thermoelastic coupling problems, and generalized thermoelastic problems. The loss function is constructed based on equations, initial conditions, and boundary conditions. Unlike traditional data-driven methods, this approach does not rely on known solutions. By comparing with analytical and finite element solutions, the applicability and accuracy of the deep learning method have been validated, providing new insights for the study of thermoelastic problems.

A finite element method for stochastic diffusion equations using fluctuating hydrodynamics

We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the fluctuating forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free of the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.

New exact free vibration solution of two-dimensional decagonal quasicrystal cylindrical shells in the symplectic framework

An accurate free vibration analysis of two-dimensional decagonal quasicrystal (QC) cylindrical shells is performed under the framework of symplectic mechanics. Unlike the classical Lagrangain system, the governing equations are reduced into a set of lower order ordinary differential equations, and therefore exact solutions of displacements, rotation angles, internal forces and bending moments in the phonon and phason fields are simultaneously obtained and expressed in terms of symplectic vectors. The present results are compared with existing literature and finite element solutions, and excellent agreements are observed. Furthermore, the effects of key influencing factors on the free vibration characteristics are revealed also.

Dissipative Non-Slip MHD Nanofluid Flow with Variable Viscousity and Thermal Conductivity in the Presence of Arrhenius Chemical Reaction

This research investigates the intricate dynamics of dissipative non-slip magnetohydrodynamic (MHD) nanofluid flow, characterized by variable viscosity and thermal conductivity, under the influence of an Arrhenius chemical reaction. The inclusion of the Arrhenius chemical reaction adds complexity through heat generation or absorption, impacting temperature and concentration gradients. The study is motivated by the extensive applications of nanofluids in engineering and industrial processes, where precise control of heat and mass transfer is critical. We develop a comprehensive mathematical model that incorporates the variable properties of the nanofluid, the effects of the Lorentz force due to the applied magnetic field, and the temperature-dependent reaction rates dictated by the Arrhenius equation. The formulated governing equations were non-dimensionalised to identify the flow governing parameters. Finite Element Method (FEM), grid generation, solution algorithms, and post-processing to analyse velocity, temperature, and concentration distributions were used to obtain the numerical methods to solve fluid flow problems based on the Navier-Stokes equations, involving concepts of discretization. pdsolve subpackage in Maple 2023 was used to numerically solve PDEs with specific initial and boundary conditions, incorporating the plot and display commands for graphical analysis, and the results are presented and discussed. The findings reveal that the interplay between parameters like Hartmann number, Darcy parameter, and heat generation or absorption profoundly influences flow behaviour and thermal characteristics. The reactivity parameter is crucial, dictating the rate of chemical reactions and affecting system dynamics. This research enhances understanding of the interdependencies among fluid properties, chemical reactions, and external parameters in nanofluid flows.

On a Stokes hemivariational inequality for incompressible fluid flows with damping

In this paper, a Stokes hemivariational inequality is studied for incompressible fluid flows with the damping effect. The hemivariational inequality feature is caused by the presence of a nonsmooth slip boundary condition of friction type. Well-posedness of the Stokes hemivariational inequality is established through the consideration of a minimization problem. Mixed finite element methods are introduced to solve the Stokes hemivariational inequality and error estimates are derived for the mixed finite element solutions. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is introduced to solve the mixed finite element system. Numerical results are reported on the performance of the proposed algorithm and the numerical convergence orders of the finite element solutions.

On the Efficient and Accurate Non-linear Computational Modeling of Multilayered Bending Plates. State of the Art and a Novel Proposal: The 2D+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ ext {D}+$$\\end{document} Multiscale Approach

After conducting a comprehensive historical review of presently established methods for computational modeling of multilayered bending plates, the present work introduces a novel 2D multiscale strategy, termed the 2D+ approach. The proposed approach is based on the computational homogenization formalism and is envisaged to serve as an appealing alternative to current methodologies for modeling multilayered plates in bending-dominated situations. Such structural elements involve modern and relevant materials, such as laminated composites characterized by the heterogeneous distribution of low-aspect-ratio layers showing substantial non-linear mechanical behavior across their thickness.Within this proposed approach, the 2D plate mid-plane constitutes the macroscopic scale, while a 1D filament-like Representative Volume Element (RVE), orthogonal to the plate mid-plane and spanning the plate thickness, represents the mesoscopic scale. Such RVE, in turn, is capturing the non-linear mechanical behavior throughout the plate thickness at each integration point of the 2D plate-midplane finite element mesh. The chosen kinematics and discretization at the considered scales are particularly selected to (1) effectively capture relevant aspects of non-linear mechanical behavior in multilayered plates under bending-dominated scenarios, (2) achieve affordable computational times (computational efficiency), and (3) provide accurate stress distributions compared to the corresponding high-fidelity 3D simulations (computational accuracy).The proposed strategy aligns with the standard, first-order, hierarchical multiscale setting, involving the linearization of the macro-scale displacement field along the thickness. It employs an additional fluctuating displacement field in the RVE to capture higher-order behavior, which is computed through a local 1D finite element solution of a Boundary Value Problem (BVP) at the RVE. A notable feature of the presented 2D+ approach is the application of the Hill–Mandel principle, grounded in the well-established physical assumption imposing mechanical energy equivalence in the macro and meso scales. This links the 2D macroscopic plate and the set of 1D mesoscopic filaments, in a weakly-coupled manner, and yields remarkable computational savings in comparison with standard 3D modeling. Additionally, solving the resulting RVE problem in terms of the fluctuating displacement field allows the enforcement of an additional condition: fulfillment of linear momentum balance (equilibrium equations). This results in a physically meaningful 2D-like computational setting, in the considered structural object (multilayered plates in bending-dominated situations), which provides accurate stress distributions, typical of full 3D models, at the computational cost of 2D models.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

Shape optimization of embedded solids using implicit Vertex-Morphing

One of the biggest challenges in optimizing the shape of complex solids is the requirement to maintain a reasonable mesh quality not only at the boundary but also for the bulk discretization of the interior. Thus, additional regularization and, in many cases, re-meshing of the structure during the iterative process is unavoidable with a Lagrangian description. By tracking the shape update using an Eulerian representation, embedded boundary methods are a promising technique for eliminating mesh distortion problems.This work consistently combines the unique features of implicit Vertex-Morphing and embedded boundary methods, facilitating the node-based shape optimization of solids with industrial complexity. One of the crucial elements for solving the primal problem on a fixed background grid is an efficient and robust quadrature scheme. To this end, we incorporate the open-source C++ framework QuESo (https://github.com/manuelmessmer/QuESo) developed for the numerical integration of arbitrarily complex embedded solids defined by oriented boundary meshes, e.g., in STereoLithography (STL) format. Meanwhile, applying the Helmholtz/Sobolev-based (implicit) filter to the vertices of the embedded boundary mesh not only exploits the extensive design space of node-based optimization but also ensures robust control over the feature size. To realize the above methodology, this work introduces a novel sensitivity analysis that yields mesh-independent shape gradients with respect to the immersed boundary. Through a specific sensitivity weighting, we recover a continuous gradient field from discrete values calculated at the nodes of the background grid. In addition, the presented workflow ensures robust enforcement of challenging geometrical constraints, including minimum wall thicknesses and design space limitations.We critically assess our approach with benchmarks and structures of industrial relevance. In all examples, trivariate B-Spline bases span the background grid, providing highly accurate finite element solutions and shape sensitivities at every iteration. Moreover, the elimination of mesh distortion problems enables a successful termination at the local optimum, even for large shape modifications.

Finite deformation micropolar peridynamic theory: Variational consistency of wryness measure

This paper concerns developments in both classical and peridynamic (PD) finite deformation micropolar elasticity theory. The first part presents an alternative proof of a theorem that demonstrates variational consistency of the wryness measure in classical finite deformation micropolar elasticity which is one of the novel contributions of this work. The linearized version of the proposed wryness measure is compared with the wryness measure of small deformation micropolar elasticity. In the second part of this work, a finite deformation micropolar PD theory is presented. After proposing an additional integro-differential equation along with the standard PD equation of motion, the global balance of angular momentum is proved to be satisfied. The balance of virtual work is derived for the micropolar PD theory. Next, constitutive correspondence approach is used to relate PD force and moment states with their classical counterparts. The nonlocal versions of the strain and the proposed wryness measure are used in the constitutive correspondence. The constitutive equations of the classical micropolar theory are presented and how they can be incorporated in PD framework is discussed. We introduce a new bond breaking criterion for PD micropolar materials based on critical stretch and critical relative rotation. Quasi-static numerical simulations on deformations of plate with a hole and fracture of a double notched sheet are presented. Dynamic simulations concern wave propagation through a solid specimen and plate with a hole. Validation of the simulation results with the finite element solutions, boundary element solutions, and experimental observations demonstrate the potential of our approach.

Cross-sectional analysis of initially curved composite beams with rectangular and circular sections using an asymptotic meshless method

An analytical meshless cross-sectional dimensional reduction method is developed to perform structural analysis of initially curved composite beams. By introducing the Pascal polynomials in the Cartesian coordinates system for beams with rectangular cross-sections and in the polar coordinates system for the beams with circular cross-sections to the warping field, a three-dimensional beam problem can be split into a two-dimensional cross-sectional analysis and a one-dimensional along the reference line of the beam. The obtained cross-sectional stiffness matrix based on higher-order of the strain energy can be incorporated into a one-dimensional finite element solution for initially curved composite beams, for the purpose of strain and stress analysis. Finally, the three-dimensional strain through the thickness of the beam can be achieved by the recovery analysis. The present method takes advantage of Pascal polynomials so that the solution procedure will be more simplified and computationally more efficient compared to three-dimensional finite element (3D FE) method. The proposed method for different isotropic and anisotropic beams is examined and compared with the literature, 3D FE, and Variational Asymptotic Beam Sectional (VABS) package which is a finite element based cross-sectional analysis tool for composite beams.

Dynamic characteristics of density-graded cellular materials for impact mitigation

Due to their exceptional impact resistance capabilities, density-graded cellular materials have immense potential in applications where crashworthiness requirements are of prime importance. Under impact loading, the deformation in these materials is characterized by compaction front propagation. Previous studies have utilized numerical techniques to solve the equations governing compaction shock propagation for cellular materials with density gradation. In this study, analytical solutions are formulated using compaction front position as the independent variable. The expressions for the velocity of the impinging rigid mass, energy absorption, incident stress, and transmitted stress are determined. The analytical solutions are shown to be in excellent agreement with the cell-based finite element solutions. The effect of density gradient on energy absorption and stresses is studied. The impact resistance factor is employed to assess different density-graded cellular materials for their effectiveness in impact mitigation. This study demonstrates that density-graded cellular materials can offer superior impact protection to objects at both the incident and transmitted ends. Lower density toward the incident end enhances impact resistance to objects located there, and likewise, lower density at the transmitted end offers more impact resistance to objects at that end.

Nonlinear bending characteristics of multidirectional functionally graded porous panels using higher-order finite element solution with parametric optimization

The current study aims to investigate the nonlinear bending behavior of unidirectional (1D) and multidirectional (2D/3D) functionally graded porous shell panels. Here, the material properties of the multidirectional functionally graded (FG)-panels are estimated by employing the representative volume element method and Voigt’s micromechanical homogenization scheme. The nonlinear mathematical model of the FG-panel is developed by incorporating Green-Lagrange type nonlinear strains in higher-order-midplane kinematics. The minimum potential energy principle is used to obtain the final form of the equilibrium equation for the multidirectional FG shell panel and further solved by using Picard’s iterative-based 2D-isoparametric finite element method via quadrilateral Lagrangian elements. The efficacy and robustness of the present model are checked by comparing its results with previously reported literature. Numerous examples are solved to analyze the impact of material parameters, such as volume fraction index and porosity index, and geometrical parameters, such as aspect ratio and support conditions, on the nonlinear deformation behavior of multidirectional FG shell panels. In addition, parametric optimization is carried out to minimize the flexural response by using the response surface methodology and, hence, determining the optimal values of multidirectional porous FG-panel. The results demonstrate that, within the defined ranges of FG-panel, lower values of porosity index and volume fraction index yield lesser deflection.

NeuFENet: neural finite element solutions with theoretical bounds for parametric PDEs

NeuFENet: neural finite element solutions with theoretical bounds for parametric PDEs

A novel approach for generalized Green-Naghdi-type electro-magneto-thermo-hyperelasticity wave propagation and reflection investigations in near-incompressible layers under shock loads

A novel nonlinear coupled finite-strain electro-magneto-thermo-hyperelasticity (EMTHE) model is developed for the first- and second-sound wave propagation and reflection investigations in media that are exposed to electromagnetic fields and mechanical shocks. In contrast to the previous studies which studied behaviors of the finite-strain elastic or incompressible materials with small deformations, the current research proposes a much more complicated but much more accurate novel practical model for wave propagation and reflection analyses in near-incompressible finite-strain materials. Furthermore, to evaluate the effects of the electro-magneto-thermomechanical coupling, the strain energy density function of the hyperelastic material is expanded in a new way. The governing equations are obtained according to a nonlinear version of the Helmholtz free energy. The energy equations comprise the first- and second-order time rates of the temperature to enable the modeling of the finite-speed heat transfer; i.e., the establishment of a second-sound model. A nonlinear iterative finite element solution algorithm is proposed and implemented for the resulting coupled time-dependent generalized electro-magneto-thermo-hyperelasticity equations. The results show significant differences between the predicted wave propagation and reflection characteristics and behaviors of the near-incompressible and incompressible finite-strain models.

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Detecting Shoreline Changes on the Beaches of Hainan Island (China) for the Period 2013–2023 Using Multi-Source Data

This study presents an in-depth analysis of the dynamic beach landscapes of Hainan Island, which is located at the southernmost tip of China. Home to over a hundred natural and predominantly sandy beaches, Hainan Island confronts significant challenges posed by frequent marine natural disasters and human activities. Addressing the urgent need for long-term studies of beach dynamics, this research involved the use of CoastSat to extract and analyze shoreline data from 20 representative beaches and calculate the slopes of 119 sandy beaches around the island for the period from 2013 to 2023. The objective was to delineate the patterns of beach evolution that contribute to the prevention of sediment loss, the mitigation of coastal hazards, and the promotion of sustainable coastal zone management. By employing multi-source remote sensing imagery and the CoastSat tool, this investigation validated slope measurements across selected beaches, demonstrating consistency between the calculated and actual distances despite minor anomalies. The effective use of the finite element solution (FES) in the 2014 global tidal model for tidal corrections further aligned the coastlines with the mean shoreline, underscoring CoastSat’s utility in enabling precise coastal studies. The analysis revealed significant seasonal variations in shoreline positions, with approximately half of the monitored sites showing a seaward progression in summer and a retreat in winter, which were linked to variations in wave height. The southern beaches exhibited distinct seasonal variations, which contrasted with the general trend due to differing wave impacts. The western and southern shores showed erosion, while the northern and eastern shores displayed accretion. The calculated slopes across the island indicated that the southern beaches had steeper slopes, while the northern areas exhibited more pronounced slope variations due to wave and tidal impacts. These findings highlight the critical role of integrated coastal management and erosion control strategies in safeguarding Hainan Island’s beaches. By understanding the mechanisms driving seasonal and regional shoreline changes, effective measures can be developed to mitigate the impacts of erosion and enhance the resilience of coastal ecosystems amidst changing environmental conditions. This research provides a foundational basis for future efforts aimed at the sustainable development and utilization of coastal resources on Hainan Island.

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.

Fuzzy Finite Elements Solution Describing Recession Flow in Unconfined Aquifers

In this work, a novel fuzzy FEM (Finite Elements Method) numerical solution describing the recession flow in unconfined aquifers is proposed. In general, recession flow and drainage problems can be described by the nonlinear Boussinesq equation, while the introduced hydraulic parameters (Conductivity K and Porosity S) present significant uncertainties for various reasons (e.g., spatial distribution, human errors, etc.). Considering the general lack of in situ measurements for these parameters as well as the certain spatial variability that they present in field scales, a fuzzy approach was adopted to include the problem uncertainties and cover the disadvantage of ground truth missing data. The overall problem is encountered with a new approximate fuzzy FEM numerical solution, leading to a system of crisp boundary value problems. To prove the validity and efficiency of the new fuzzy FEM, a comparative analysis between the proposed approach and other well-known and tested approximations was carried out. According to the results, the proposed FEM numerical solution agrees with Karadinumerical method for the crisp case and is in close agreement with the original analytical solution proposed by Boussinesq in 1904 with the absolute reduced error to be 4.6‰. Additionally, the possibility theory is applied, enabling the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of hydraulic properties (e.g., water level, outflow volume) and make the right decisions for rational and productive engineering studies.

Artificial neural networks-based J-integral prediction for cracked bodies under elasto-plastic deformation state –cyclic loading

The study builds upon a prior modeling framework designed for monotonic loading to compute the cyclic J-integral encompassing a crack tip experiencing elastic–plastic deformation due to cyclic loads. The proposed modeling approach extends the authors' earlier time-efficient method for computing the J-integral by combining artificial neural networks (ANNs) with finite element (FE) analyses. The ANN-FE integration allows efficient and accurate prediction of elastic–plastic stresses, strains and displacement fields for a crack body under cyclic loadings from the linear elastic FE solution. In the context of cyclic loading, stress, strain, and displacement fields near the crack tip of stainless steel (SS304) are determined by FE analyses under both elastic and elasto-plastic states for various crack sizes and R-ratios. ANNs models are developed to establish the nonlinear relationship between these two states. Consequently, the proposed approach enables the prediction of the elasto-plastic cyclic J-integral through the equivalent domain integral method, based on the elastic FE solution bypassing the need of a complicated elasto-plastic FE solution under cyclic loadings. The results show that this approach can effectively predict elasto-plastic cyclic J-integral while avoiding the complex computation of elastic–plastic deformation fields around the crack tip during cyclic loadings.

Prediction of laser-welded deformation using artificial neural networks

When predicting welding deformation of the laser-manufactured vehicles and aerospaces, analytical solutions or empirical formulas are not usually accessible in complex problems. Based on the inherent deformation method, a machine learning (ML) approach for predicting welding deformation of welded structures is proposed based on an artificial neural network (ANN). This method is a promising substitute for analytical, empirical, and finite element (FE) solutions due to its accuracy, easy-to-use, efficiency, and universality. First, the outputs of the ANN are determined via dimensionless analysis and comparison of numerical results, which are dimensionally independent. Then, based on the inherent deformation method, the training and validation sets of the ANN are generated through an elastic finite element analysis. At last, the structure of the ANN is determined by analyzing the ANN prediction accuracy with different hidden layers, numbers of neurons, and activation functions. The results show that the ML solutions are in good agreement with the FE results, verifying the effectiveness and generalization ability of the proposed method.