Finite Element Method Equations Research Articles

Screening effect of pile rows for ground vibration induced by moving loads

This study aims to predict the screening efficiency of pile rows in transversely isotropic (TI) saturated soils for ground vibrations subjected to moving loads. First, based on the fundamental solutions of TI saturated media, the boundary element method (BEM) was adopted to derive the force-deformation equation of the soil-pile boundary. The finite element method (FEM) was then applied to establish the stiffness matrix equation of the Euler-based pile rows. Subsequently, by coupling the BEM and FEM equations, the screening efficiency of the pile rows was developed according to the barrier isolation theory. The proposed approach was validated using published numerical results. Finally, the effect of soil stratification on the screening effect of pile rows in TI saturated soils was investigated.

Plane crack problems within strain gradient elasticity and mixed finite element implementation

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

On the modelling of ground-board mounted microphones for outdoor noise measurements

This paper presents a boundary element method for calculating the sound pressure level on and above a ground-board placed on an infinite ground plane with constant impedance. This method uses the Green's function for the Helmholtz equation in a half-space bounded by an infinite impedance plane and therefore avoids the complications which occur when modeling the ground as a finite impedance plane (which results in diffraction from the edges of the ground plane). Two formulations are presented: the first one models a finite thickness ground-board, whilst the second one models a zero-thickness ground-board with the upper surface coplanar with the ground plane. The method is validated against an equivalent finite element method simulation and example calculations are presented for typical ground-board designs used for outdoor noise measurements on different ground impedance planes and incidence angles. It is shown that the sound pressure level on the upper surface of a ground-board placed on a non-rigid ground can vary significantly and that this variation is dependant on the ground impedance, angle of incidence and frequency. These findings are consistent with previous investigations and highlight the fact that noise measurements made using ground-board mounted microphones on a non-rigid ground are dependant on the impedance of the surrounding ground surface. The method presented in this paper could be used to develop corrections to straightforwardly account for this effect.

Hybrid Layer Modeling Method With Dual-Leakage Flux Paths for Heavily Saturated Iron-Cored Linear Permanent Magnet Synchronous Motors

This paper presents the dual-leakage hybrid layer modeling (DL-HLM) method for the linear permanent-magnet synchronous motors (LPMSMs) in a heavily saturated state. The proposed hybrid model consists of the nonlinear dual-leakage flux-tube model for the iron-cored armature where the geometric saliency exists, the Maxwell model for the PM track with uniform layers, and the hybrid input-output conversion method for the iteration between two models. As the minimum number of leakage flux paths, the dual-leakage flux-tube network is selected, considering the opposite-direction leakage fluxes generated by the flux-steering effect in heavily saturated iron-core teeth. The novel reluctance adaptation methods are also proposed, which allow the DL-HLM to accurately capture significant nonlinearities even with the minimum number of leakage flux paths, thereby correctly estimating motor forces in high-performance LPMSMs. The proposed DL-HLM is validated comparing to the equivalent finite element method (FEM) in terms of the magnetic fields in the working air-gap and motor force performances. The proposed modeling method estimates the magnetic fields and forces with an accuracy of higher than 95% even at significantly high saturation levels while simultaneously achieving multiple-orders-of-magnitude faster computation time as compared to the FEM.

Dynamic response of the periodic pile-reinforced composite foundation to harmonic seismic waves

The dynamic analysis of the pile-reinforced composite foundation (PCF) under seismic waves is important for the design of the PCF and corresponding upper structures. Owing to huge number of degrees of freedom (DOFs) in the modeling, it is still difficult to simulate the dynamic response of the PCF to seismic waves directly by means of traditional numerical methods. To overcome this difficulty, the PCF is simplified as the periodic pile-reinforced composite foundation (PPCF) in this study, and the harmonic wave finite element (HWFE) model for the PPCF under seismic waves is proposed. To establish the HWFE model, the response of the PPCF is decomposed into a series of horizontally propagating harmonic waves with amplitudes varying with the depth. Based on the principle of virtual work (PVW), the finite element method (FEM) equations for each harmonic wave are established. To impose the boundary conditions and input seismic waves at the bottom of the PPCF, the traction-displacement relation at the surface of the underlying half-space bedrock is derived. Solving the FEM equations for the PPCF and synthesizing the resulting solutions for the harmonic waves via the DFT method, dynamic responses of the PPCF to plane harmonic seismic waves are obtained. To verify the developed HWFE model, results obtained by the proposed HWFE model are compared with those by the propagator matrix method (PMM). The numerical results show that the resonance frequency of the PPCF increases with decreasing pile spacing; also, the presence of the stiffer soil layers will enhance the resonance frequency of the PPCF.

A general finite element method: Extension of variational analysis for nonlinear heat conduction with temperature-dependent properties and boundary conditions, and its implementation as local refinement

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence between the two analyses. A general FEM, extension of variational analysis, is derived as general form of conventional FEM modeling nonlinear heat conduction. Numerical examples demonstrate that the general FEM produces results with considerably higher accuracy and stability and also possesses higher performances on conforming with both two analyses. Since general FEM degenerates to conventional FEM if derivatives are of small-amplitude or zero and its direct implementation to the entire domain is costive, general FEM is alternatively utilized as local refinement of governing equation only to points with significant derivatives. The strategy of local refinement is optimized to enhance efficiency.

A posteriori error estimation for the spectral deferred correction method

The spectral deferred correction method is a variant of the deferred correction method for solving ordinary differential equations. A benefit of this method is that is uses low order schemes iteratively to produce a high order approximation. In this paper we consider adjoint-based a posteriori analysis to estimate the error in a quantity of interest of the solution. This error formula is derived by first developing a nodally equivalent finite element method to the spectral deferred correction method. The error formula is then split into various terms, each of which characterizes a different component of the error. These components may be used to determine the optimal strategy for changing the method parameters to best improve the error.

Influence of tip relief in spur gears dynamic using multibody models with movable teeth

Tip relief modifications are a common design feature used in industrial practice to reduce dynamic tooth loading, noise, vibrations and wear in high-performance gears. This paper presents a novel implementation of a specific multibody model to include the effect of tip relief micro-modification on spur gears. The model considers every tooth attached to the wheel hub with a revolute joint and an equivalent torsion spring. A penalty formulation is adopted to solve the teeth contact (2D). The proposed methodology offers a trade-off between the accuracy of an equivalent finite element method model and the advantage of reducing the degrees of freedom allowing for fast dynamic simulations. Moreover, there is no need to preliminary predict mesh stiffness or static transmission error as a source of excitation for dynamic simulation, because these are direct outputs of the model simulation given the modified tooth geometry. Model validation is done comparing numerical results with experimental data taken from literature. The effects of geometry modification are studied both in quasi-static and dynamic conditions. Finally, the model has adopted to assess the off-design performance of spur gears.

Stability and convergence of iterative finite element methods for the thermally coupled incompressible MHD flow

Purpose The purpose of this paper is to propose three iterative finite element methods for equations of thermally coupled incompressible magneto-hydrodynamics (MHD) on 2D/3D bounded domain. The detailed theoretical analysis and some numerical results are presented. The main results show that the Stokes iterative method has the strictest restrictions on the physical parameters, and the Newton’s iterative method has the higher accuracy and the Oseen iterative method is stable unconditionally. Design/methodology/approach Three iterative finite element methods have been designed for the thermally coupled incompressible MHD flow on 2D/3D bounded domain. The Oseen iterative scheme includes solving a linearized steady MHD and Oseen equations; unconditional stability and optimal error estimates of numerical approximations at each iterative step are established under the uniqueness condition. Stability and convergence of numerical solutions in Newton and Stokes’ iterative schemes are also analyzed under some strong uniqueness conditions. Findings This work was supported by the NSF of China (No. 11971152). Originality/value This paper presents the best choice for solving the steady thermally coupled MHD equations with different physical parameters.

Evaluation of hydrogen-induced cracking in high-strength steel welded joints by acoustic emission technique

Hydrogen-induced cracking behavior in high-strength steel mainly composed of martensite was analyzed by acoustic emission (AE) technique and finite element method (FEM) in slow strain-rate tensile (SSRT) tests and welding tests. The crack initiation was detected by the AE signals, and the time evolution of stress concentration and hydrogen diffusion were calculated by FEM. The effect of hardness and plastic strain on the hydrogen diffusion coefficient was explicitly introduced into the governing equation in FEM. The criterion and indicator parameter for the crack initiation were derived as a function of maximum principal stress and locally accumulated hydrogen concentration. The results showed that the cracking criterion derived by AE and FEM is useful for predicting the cold cracking behavior and determining the critical preheat temperature to prevent hydrogen-induced cracking.

Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.

Size Dependent Thermo-Piezoelectricity for In-Plane Cracks

The finite element method (FEM) is developed to analyse the size effect (flexoeletricity) for 2-D crack problems in thermo-piezoelectricity. Flexoelectricity is observed in micro/nanoelectronic structures, where large strain gradients destroy the symmetric structure of atoms in crystals and thereby causing polarization, even in dielectric materials. In contrast to using classical Fourier heat conduction theory, a finite speed of the thermal wave is considered in the higher order transport equation. The variational principle is applied to derive the FEM equations and C1-continuous elements are employed in the implementation of the FEM. An example is presented to demonstrate the effect of the characteristic time parameter on the crack opening displacement and temperature distribution.

A coupling FEM/BEM method with linear continuous elements for acoustic-structural interaction problems

The effects of dynamical load from acoustic waves to structural responses cannot be neglected for some cases such as a thin structure submerged in a heavy fluid. To deal with this kind of problem, a hybrid numerical method consisting of finite element method (FEM) and boundary element method (BEM) for the acoustic-structural coupling problems is proposed. The FEM is built on the first-order Reissner-Mindlin plate theory and the discrete shear gap (DSG) method is applied to construct a locking-free FEM. A collocation BEM with linear continuous element based on Burton-Miller formulation is adopted with an aim to produce a meshing conforming coupling method with the FEM not only on geometry but also on physical nodal quantities at the interface. In addition, analytical expressions of the singular integrals appeared in the Burton-Miller formulations are available to eliminate the numerical difficulties in the implementation of the BEM. Furthermore, the coupling matrix at the interface based on the conforming linear triangular mesh is obtained analytically to avoid the numerical quadrature. A solution scheme by absorbing the FEM equations to the BEM’s is proposed to take the advantage of less memory cost of the linear BEM. Optimized algorithms are designed to reduce the memory cost while keep the accuracy of the coupling method during the absorption of the structure effects to the acoustic domain. Numerical examples are setup to validate the accuracy and demonstrate the potential capability of the proposed method.

Comparative assessment of electromagnetic simulation tools for use in microstrip antenna design: Experimental demonstrations

AbstractThis paper presents a better understanding of the use of finite integration techniques (FIT) and finite element method (FEM) in different types of microstrip antennas in order to determine which numerical method gives relatively more accurate results. Although the theoretical formulation based on Maxwell's equations of both FEM and FIT are approached from different aspects in the literature, there is still a lack of comparison of the same antenna type using different numerical methods employing FEM and FIT. Therefore, in this study, FEM and FIT were applied to two different types of microstrip antennas, and their simulation and experimental results was compared. For the first antenna demonstration, a multilayer structure was chosen to achieve one of the significant parameters. Then, a microstrip antenna with a compact structure was used in the second demonstration. Using these two antennas, the accuracy of FEM and FIT in different structures were compared and all simulated return loss and gain results were verified by the measured results. The experimental demonstrations show that FEM performs better for both types of microstrip antennas while FIT provides an adequate result for two‐layer microstrip antennas.

The discrete element method with deformable particles

SummaryThis work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform‐sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro‐macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.

Multilayered Equivalent Finite Element Method for Embedded Honeycomb Plates

To investigate the mechanical properties of embedded honeycomb plates with high efficiency and accuracy, a new multilayered equivalent finite element method (FEM) model is proposed. A series of FEM numerical studies (modal analysis, static analysis, and shock spectrum analysis) are performed. The goal is to compare the errors produced by the multilayered equivalent method and by existing equivalent approaches. The obtained results indicate that the proposed model shows good agreement with the original plate. Moreover, based on the new model, a parametric study correlating the microstructure parameters (embedded depth/cell size) to modal frequency is proposed, and a multiparameter equation for frequency and embedded depth/cell size is established to serve as a basis for structural optimization design.

Software complexes and new approaches to non-linear analysis of framed structures

There is a great number of non-linear tasks which can be solved with the use of computer-based simulation and finite elements method (FEM). But in order to ensure high speed and accuracy of calculations it is necessary to use new mathematical models and algorithms. The analysis of the known FEM forms and approaches to numerical solution of tasks based on force method is performed in this work. Represented in the document is formulation of the finite-element stress method and algorithmization of the classical force method for calculation of framed structures based on the equation of strain compatibility – contour forces method. Statement of problems which can be more effective from the point of view of elastic-plastic deformations at numerical solution is considered. In particular, the hybrid finite-element method based on the generalized Mohr formula for determination of long bar stiffness coefficient with account of non-linear behavior. The main relationships for creation of finite-element method equation system in the form of force method are represented.

Numerical study of smoothed particle hydrodynamics method in orthogonal cutting simulations – Effects of damage criteria and particle density

This paper conducts a numerical study of smoothed particle hydrodynamics (SPH), a mesh-free method, for orthogonal cutting simulations of both ductile and brittle materials. Finite element method (FEM) is commonly used for cutting simulations, but issues with excessive element deformation hinder its applications. SPH can be an alternative option because of the particle-based algorithm eliminating the use of volumetric elements. However, studies have reported inconsistent ways to set up damage definition and particle density. This paper instructs a method to build an SPH model and compares the results with an equivalent FEM, in terms of the effects of damage definition and particle density on chip morphology and cutting forces. The materials used for this study are aluminum, which represents a common ductile engineering material, and the cortical bone representing a brittle counterpart. The results reveal that in spite of the natural separation of SPH, proper damage criteria must be defined for the model accuracy. SPH tends to produce a lower cutting force than that of FEM. SPH and FEM have an opposite convergence trend as the particle density and mesh size reduce. SPH better simulates fragmented debris in brittle cuttings but fails to produce curled chips in ductile cuttings.

A hybrid finite volume – finite element method for bulk–surface coupled problems

The paper develops a hybrid method for solving a system of advection–diffusion equations in a bulk domain coupled to advection–diffusion equations on an embedded surface. A monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface. In our approach, the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way. Moreover, a triangulation of the surface into regular shaped elements is not required. The background mesh is an octree grid with cubic cells. As an example of an application, we consider the modeling of contaminant transport in fractured porous media. One standard model leads to a coupled system of advection–diffusion equations in a bulk (matrix) and along a surface (fracture). A series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach. The method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures.

Prototyping of Nonlinear Time-Stepped Finite Element Simulation for Linear Induction Machines on Parallel Reconfigurable Hardware

The finite element method (FEM) is widely used for accurate design and analysis of electric machines; however, it suffers from long execution time. In this paper, for the first time hardware acceleration of two-dimensional FEM for a single-sided linear induction motor on the field programmable gate array (FPGA) is proposed. The nonlinearity of the iron core as well as the movement are taken into consideration. A new sparse solver is proposed based on left-looking Gilbert–Peierls algorithm for the system of linear equations of FEM that need to be solved in different iterations and time steps. Implementation of the model is performed in a massively paralleled and deeply pipelined hardware architecture using VHDL coding with single precision floating-point number representation. The proposed emulation was performed at various time steps resulting in significant average speedup of 9.73 times in comparison with JMAG-Designer as a commercial finite element software, and the overall hardware latency of each time step for the emulation was 49.2 ms in average with minimum achievable FPGA clock of 5.59 ns.