FEM Solutions Research Articles

Quantitative Analysis of Magnetic Force of Axial Symmetry Permanent Magnet Structure Using Hybrid Boundary Element Method

This paper investigates the forces generated by axially magnetized ring permanent magnets with trapezoidal cross-sections when placed near a soft magnetic cylinder. Utilizing the Hybrid Boundary Element Method (HBEM), this study models interactions in magnetic configurations, aiming to improve force calculation efficiency and accuracy compared to traditional finite element methods (FEMM 4.2 software program). The influence of the permanent magnet and the soft magnetic cylinder is approximated with a system of thin toroidal sources on the surfaces of the magnet and the cylinder, which significantly reduces the computation time for the force calculation. The approach is validated by comparing results with FEM solutions, revealing high precision with a much faster computation. Additionally, this study explores the influence of various parameters, including magnet size, separation distance, and magnetic permeability of the cylinder, on the magnetic force. The results demonstrate that the HBEM approach is effective for analyzing complex magnetic configurations, particularly in applications requiring efficient parametric studies. This approach can be adapted for other geometries, such as truncated cones or rectangular cross-section ring magnets. The findings contribute valuable insights into designing efficient magnetic systems and optimizing force calculations for varied magnet geometries and configurations, including the atypical ones.

Extending the Natural Neighbour Radial Point Interpolation Meshless Method to the Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core

This work investigates the mechanical behaviour of sandwich beams with cellular cores using a multiscale approach combined with a meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). The analysis is divided into two steps, aiming to analyse the efficiency of NNRPIM formulation when combined with homogenisation techniques for a multiscale computational framework of large-scale sandwich beam problems. In the first step, the cellular core material undergoes a controlled modification process in which circular holes are introduced into bulk polyurethane foam (PUF) to create materials with varying volume fractions. Subsequently, a homogenisation technique is combined with NNRPIM to determine the homogenised mechanical properties of these PUF materials with different porosities. In this step, NNRPIM solutions are compared with high-order FEM simulations. While the results demonstrate that RPIM can approximate high-order FEM solutions, it is observed that the computational cost increases significantly when aiming for comparable smoothness in the approximations. The second step applies the homogenised mechanical properties obtained in the first step to analyse large-scale sandwich beam problems with both homogeneous and functionally graded cores. The results reveal the capability of NNRPIM to closely replicate the solutions obtained from FEM analyses. Furthermore, an analysis of stress distributions along the beam thickness highlights a tendency for some NNRPIM formulations to yield slightly lower stress values near the domain boundaries. However, convergence towards agreement among different formulations is observed with mesh refinement. The findings of this study show that NNRPIM can be used as an alternative numerical method to FEM for analysing sandwich structures.

A hybrid radiative energy transfer and image source method for high-frequency vibrational coupled plates

A hybrid approach composed of the radiative energy transfer method (RETM) and image source method (ISM) is proposed in this study for estimating the vibrational energy of coupled plates loaded by high-frequency point excitation. The vibrational energy at a receiver is represented by energy density and intensity, which are superposed by incoherent rays emitted by the load point in the analyzed domain and reflected/transmitted by the domain boundaries. The energy rays reflected/transmitted at the boundaries are described by two modes: diffuse reflection/transmission, which is a basic assumption in RETM, and the other obeys Snell’s law of reflection/refraction, which is deduced by Fermat’s principle and often applied in ISM. The local energy response distribution in the loading subsystem is described by ISM. The energies in other subsystems are described by RETM. The energy transfer relationship between subsystems is expressed by the energy transfer coefficient. At the boundary of coupled plates, a Fredholm equation of the second type is established through the balance among the outgoing energy of the diffuse reflection fictitious sources and the incident energy of actual sources, other diffuse reflection sources, and specular reflection image sources, which can be used to determine the intensity of the diffuse reflection fictitious sources. The average energy transfer coefficient in the transmission direction half-space is used to express the energy transfer relationship associated with diffuse reflection sources, while the energy transfer relationship associated with specular reflection sources is expressed directly by the energy transfer coefficient. Numerical tests show that the energy distributions and energy flow fields of typical coupled plates are well predicted by the proposed hybrid RETM-ISM approach. Compared with RETM, hybrid RETM-ISM is more consistent with the FEM solution.

First numerical evidence of the two-close frequency heating effect on electron cyclotron resonance ion sources

The two-close frequency heating (TCFH) is a new implementation of the well-known two frequency heating. In TCFH, the two frequencies differ around 200-300 MHz each other in order to establish two contiguous ECR resonance zones. TCFH has been proved to be a powerful technique to suppress plasma instabilities in Electron Cyclotron Resonance Ion Sources (ECRIS), as well as to improve their performances. Its beneficial effect, compared to the application of a single frequency, is always deduced from the extracted charge states distributions and from the detection of the plasma self-emission in the X-ray and microwave ranges. This paper presents the first approach to a numerical description of the two-close frequency effect, based on the relevant plasma parameters of the ECRIS setup operating at ATOMKI-Debrecen. Simulations have been performed by our PIC-Full Wave code, joining electron kinetics and FEM solution of Maxwell equations in a cold plasma model. Results on plasma electron density and energy distribution will be shown, together with a direct comparison with the already published data on X ray emission.

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

Energy absorption and dissipation characteristic of FG honeycomb reinforced laminate embedded with viscoelastic material in hygrothermal conditions under low-velocity impact

A novelty FG honeycomb-reinforced laminate (FGHRL) with viscoelastic material is constructed for an industrial exoskeleton. The study examines how FGHRL dissipates energy under hygrothermal conditions. Low-velocity impacts on the exoskeleton’s load-bearing structure are simulated using a spherical object. Parameters for the FGHRL, made of a two-phase material, are estimated using the Halpin-Tsai model and macroscopically uniform theory. In-plane deformations are determined using Reddy’s HSDT, while the impact response is evaluated with Newmark-β and Wilson-θ methods. Energy dissipation properties are calculated using the energy balance model, validated through CUF-based FEM and LS-DYNA explicit solution, and factors influencing energy dissipation are investigated.

Fluid Focusing Contributes to the BM Vibration Amplification by Boosting the Pressure.

Two hydrodynamic effects are introduced in the standard transmission-line formalism, the focusing of the pressure and fluid velocity fields near the basilar membrane and the viscous damping at the fluid-basilar membrane interface, which significantly affect the cochlear response in the short-wave region. In this region, in which the wavelength is shorter than the cochlear duct height, only a layer of fluid of order of the wavelength is effectively involved in the traveling wave. This has been interpreted [8] as a reduced fluid contribution to the system inertia in the peak region, which is a viewpoint common to the 3-D FEM solutions. In this paper we propose an alternative approach, from a slightly different physical viewpoint. Invoking the fluid flux conservation along the traveling wave propagation direction, we can derive a rigorous propagation equation for the pressure integrated along the vertical axis. Consequently, the relation between the average pressure and the local pressure [4] at the fluid-BM interface can be written. The local pressure is amplified by a factor dependent on the local wavenumber with respect to the average pressure, a phenomenon we refer to as "fluid focusing", which plays a relevant role in the BM total amplification gain. This interpretation of the hydrodynamic boost to the pressure provides a physical justification to the strategy [10] of fitting the BM admittance with a polynomial containing both a conjugated pole and a zero. In the short-wave region, the sharp gradients of the velocity field yield a second important effect, a damping force on the BM motion, proportional to the local wavenumber, which stabilizes active models and shifts the peak of the response towards the base, with respect to the resonant place. This way, the peaked BM response is not that of a proper resonance, corresponding to a sharp maximum of the admittance, but rather a focusing-driven growth toward the resonant place, which is "aborted" before reaching it by the sharply increasing viscous losses. The large values of the wavenumber that ensure strong focusing are ultimately fueled, against viscosity, by the nonlinear OHC mechanism, hence the otherwise puzzling observation of a wide nonlinear gain dynamics with almost level-independent admittance.

A 3D distributed plasticity beam–column element for metal structures considering tangential stresses and warping with minimal DOFs

The fiber model evaluates the normal stress at a number of points over the section for a given strain increment following the plane section assumption and, by integration, axial force and bending moments. The interaction with tangential stresses is usually neglected at the point level due to the inaccurate tangential strains of the kinematics. This work proposes a generalization of the fiber model able to capture automatically the interaction among all stress components. A preliminary cross-section analysis based on the Saint Venant problem provides an accurate 3D strain as a function of the section generalized strains. This field, accurate also in the inelastic case, is exploited to impose at each section point a 3D von Mises elasto-plastic law, obtaining by integration all the resultants and moments with a full interaction. Non-uniform warping is also easily included. The section model is implemented in a mixed 3D beam–column finite element with equilibrated stress field, accurate with a minimal mesh. Numerical tests show the excellent prediction of the proposal compared to analytical and solid FEM solutions also for structures not flexure-dominated. Its efficiency, on the same order as a standard fiber model, makes the approach suitable also for large buildings.

Boundary Discontinuous Fourier analysis of clamped isotropic and cross-ply laminated plates via Unified Formulation

This paper presents an analytical solution for the static analysis of plates with clamped boundary conditions prescribed at the edges. The displacement field is expressed via the Carrera Unified Formulation (CUF) where an Equivalent-Single-Layer (ESL) approach is adopted. The governing equations are obtained by employing the principle of virtual displacements (PVD) statement. The main novelty is the use of the boundary-discontinuous Fourier-based approach to provide accurate numerical solutions. From thick to thin isotropic, cross-ply laminated and sandwich plates with different side-to-thickness ratios and stacking sequences are studied. Furthermore, the out-of-plane stresses are calculated via both the constitutive relation and the stress recovery technique. The accuracy of the proposed solution is verified by comparing the numerical results obtained with those from the literature and 3D FEM solutions. The present approach seems capable of handling not just fully clamped conditions but also mixed external conditions, which may include clamped and simply-supported edges. The solution approach provided in this article is unique, hence the proposed results might be useful as a benchmark for validating new plate theories and finite elements.

Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods

The need to design structures and structural elements that are more efficient in terms of performance is a key aspect of engineering. For a given material to be used at its maximum capacity, considering non-linear characteristics is mandatory. The non-linear regime is a subject of extreme interest for this reason and is an area with intense research activity. In this work, advanced discretization techniques (i.e., meshless methods) are applied in the elastoplastic analysis of 2D and 3D structural elements. The literature shows that meshless methods are capable of producing more accurate and smoother strain and stress fields, which are the variable fields required in the non-linear models describing elastoplasticity. Thus, in this study, the Radial Point Interpolation Method (RPIM) and the Natural Neighbor Radial Point Interpolation Method (NNRPIM) are combined with a non-linear iterative algorithm, fully developed by the authors, with the objective of analyzing for the first time the elastoplastic behavior of a two-bay asymmetric frame and bowstring bridge considering 2D and 3D analysis. The accuracy and robustness of the RPIM and the NNRPIM are shown in the end, comparing the obtained results with FEM solutions and the available literature.

Peridynamic modeling of elastic-plastic ductile fracture

A peridynamics-based framework is established for the elastic-plastic ductile fracture analysis. In this frame, a new state-based peridynamic elastic-plastic model is presented using the novel bond extension state forms of yield function and plastic flow rule. The nonlinear energy release rate is computed by the extended peridynamic finite crack extension (PFCE) method, and an energy dissipation rate-based bond failure criterion is proposed for ductile crack growth modeling. Numerical methods for plastic states update and yield function solution are also given. Then, examples of plates with a center hole or a center crack, and compact tension (CT) tests are analyzed by the proposed models, and compared to those from theoretical and FEM solutions. The results demonstrate that the proposed peridynamic models can well capture the characteristics of elastic-plastic deformation and ductile fracture, including the plastic shape and size, energy distributions, nonlinear energy release rate, load-displacement and resistance curves, etc.

An adaptive global–local generalized FEM for multiscale advection–diffusion problems

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEMgl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEMgl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEMgl closely matches reference fine-scale FEM solutions in the L∞ norm.

Sensitivity analysis of a layered piezoelectric system using ZFEM

AbstractThe complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two‐dimensional linear first‐order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady‐state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT‐4 and PZT‐5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well‐known benchmark problem. Comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT‐5 specifically in , , and .

Prediction of Temperature of Liquid Steel in Ladle Using Machine Learning Techniques

Maintaining the temperature of liquid steel in the ladle in the required range affects the quality of casted billets, reduces energy consumption, and guarantees smooth control of the melting sequence. Measuring its temperature is a challenging task in industrial settings, often hindered by safety concerns and the expensive nature of equipment. This paper presents models which enable the prediction of the cooling rate of liquid steel for variable production parameters, i.e., steel grade and weight of melt. The models were based on the FEM solution of the Fourier equation, and machine learning approaches such as decision trees, linear regression, and artificial neural networks are utilized. The parameters of the model were identified using data from the monitoring system and inverse analysis. The results of simulations were verified with measurements performed in the production line.

Equilibrium finite element and error estimation for frictional contact problem based on linear complementarity problem formulation

AbstractThis article proposes a novel equilibrium finite element method (EFEM) for equilibrated solution of frictional contact problem. The development of such EFEM is mainly two‐fold. On the one hand, the traction‐based equilibrium element is employed to construct the discrete equilibrated stress field. This equilibrium element, using edge tractions as basic variables, is found straightforward to deal with the friction and contact constraints on the contact surface. On the other hand, the Coulomb frictional contact problem is recast into a linear complementarity problem (LCP), with which the algebraic complementarity formulation can be obtained with the help of the constructed equilibrated stress field. A remarkable property of the proposed EFEM is that the equilibrated solution, in conjunction with the compatible solution of the traditional FEM, can provide reliable error estimation for frictional contact within the framework of dual analysis. Numerical examples are finally conducted to see the performance and the ability to reliable error estimation of the proposed EFEM.

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.

Semi-Analytical Solution for Modelling Moving Heat Sources in a Semi-Infinite Medium with Radiative and Convective Boundary Conditions

The weld quality is highly related to the thermal history of the weld and there have been many trials to monitor the quality using an infrared (IR) sensor. To obtain the real temperature of a surface based on the brightness temperature values measured by an IR camera, the emissivity value must be derived. For an accurate assessment of the emissivity, one must be aware of the melting point isotherm. The temperature profiles only depend on three factors during laser processing, specified as constants the characteristics of the material: laser beam speed (v), laser beam diameter (d), and power (P). Predicting the width of the melted zone reached during the welding process as the parameters vary is a tool for helping a quality laser processing and for determination of true temperature in laser welding using IR camera. This study describes the semi-analytical (SA) solution of the heat conduction equation for a localized moving Gaussian heat source with constant parameters on a semi-infinite medium. The solution, simple and quick to obtain, provides information on the width of the melted zone with an average error < 5 %. The outcome is assessed numerically and contrasted with FEM solutions for a Gaussian source, the latter having undergone experimental validation. With two distinct defocus values, def0 and def-6, and by varying the speed and power settings, two separate types of experiments were run. Thus, the SA solution was obtained and compared after the FEM solution had been obtained with a good approximation (max err 4.3 %, average err 2.7 %). Only in regard to the 1AL test is an error more than 5 % detected; in the other case, the average error is 3.75 %. Two more tests at the defocus values of def-4 and def-8 were conducted to confirm the model's validity as the parameters varied. Overall, the average error between the semi-analytical and the FEM solution is 4.1%. The SA solution may be used to effectively estimate the isotherms related to the melting point of aluminum (770 K). This allows to obtain a tool which helps restoring the real temperature based on the brightness values measured by the IR camera during laser welding. At the same time, this effective tool allows to investigate the importance of different processing parameters in laser manufacturing.

Modeling density-driven flow in porous media by physics-informed neural networks for CO2 sequestration

Accurate prediction of density-driven convection of CO2 overlaying brine in porous media is crucial to the applications of geological carbon sequestration. In this paper, we introduce the physics-informed neural network (PINN) method to investigate the mass transfer of CO2 injected into a homogeneous subsurface porous formation, where the concentration and the streamfunction are approximated by neural network representations. To mitigate the computational burden and training difficulties arising from learning the long-term diffusion and convection processes, we implement a time domain decomposition-based scheme so that the PINN solution can represent short-interval dynamics in a sequential manner. Furthermore, Fourier-type basis functions are encoded in the PINN architecture to map the spatial coordinates onto an embedding feature space characterized by various frequencies. The numerical results show that the use of Fourier feature embedding improves the training performance by handling the notorious spectral bias in deep neural networks and enhances the stability for long-term prediction of flow and transport. The proposed PINN method is applied to a benchmark problem to understand the effect of natural convection on the simulation performance. Compared to the high-fidelity stabilized FEM solution, the PINN approach presents agreeable statistical characteristics of CO2-rich brine fingers, demonstrating the effectiveness of PINNs in modeling density-driven flow.

Bio-nanoconvective Micropolar Fluid Flow in a Darcy Porous Medium Past a Cone with Second-Order Slips and Stefan Blowing: FEM Solution

Bio-nanoconvective Micropolar Fluid Flow in a Darcy Porous Medium Past a Cone with Second-Order Slips and Stefan Blowing: FEM Solution

Coupled Chemo-Mechanical Swelling Behavior of PH-Sensitive Hollow Cylinder Hydrogels under Extension–Torsion and Internal Pressure: Analytical and 3D FEM Solutions

The coupled transient chemo-mechanical behavior as well as the large deformation behavior under various complex load conditions must be taken into account when designing a functional responsive polymer actuator or sensor. One sort of deformation that can be used to characterize the properties of materials with complicated behavior, like soft hydrogels, is coupled extension and torsion with internal pressure. It is important to thoroughly research the complex kinetics of pH-hydrogels with coupled diffusion and massive deformation behavior. The transient behavior of cylindrical hydrogels under coupled extension–torsion with internal pressure under indifferent conditions is proposed in this work using a reliable semi-analytical method. In this regard, an analytical solution is offered to inspect this problem, which is used as a common experimental methodology for the characterization and modeling of polymeric materials. The results show that the rate of deformation and the physical characteristics of the material have a substantial impact on the cylindrical hydrogel’s transient behavior under coupled extension–torsion and internal pressure. For the same problem, a 3D finite element study was done to assess the analytical solution. The accuracy of our method is supported by the results’ agreement in both the FE analysis and the proposed approach. However, offering such a solution for this complex problem is of tremendous relevance given the significantly cheaper computational cost of analytical methods when compared to FEM. Additionally, the calculations indicate a complex reaction force and moment because the hydrogel experiences nonlinear Poynting-type effects in this deformation domain. The suggested semi-analytical procedure’s resilience behavior is demonstrated by the visualization of the effects of various material properties. This method can be used to calibrate constitutive models and to develop and improve hydrogel structures.