Iterative Finite Element Method Research Articles

Tunnel shape optimization method considering the interaction between lining and soil

Recent tunnel shape optimization considers only the stress distribution in the soil around the tunnel or the lining’s mechanical performance and frequently ignores the influence of the interaction between the lining and soil. Thus, this study presents a tunnel shape optimization method considering this interaction. First, the concept of a lining rational arch axis is employed to derive an analytical expression for the tunnel shape, which is subjected to earth pressure recommended by the International Tunneling Association (ITA). Further, the rationality of the obtained tunnel shape is verified using a finite element method (FEM) simulation, which considers the interaction between the lining and soil. Thus, an iterative FEM is further proposed to minimize the lining’s eccentricity. The tunnel shape is gradually modified via the iterative FEM by offsetting the lining axis along the normal vector using the corresponding eccentricity to obtain the optimal tunnel shape. The results reveal that, with the optimal tunnel shape based on the rational arch axis, the tunnel lining is mainly subjected to axial compressive force and exhibits good performance in terms of deformation and bending moment.

The iterative multiscale finite element method for sandwich beams and plates

AbstractWe present an adaptation of the multiscale finite element method to the analysis of sandwich beams and plates with complex lattice layers. The proposed modification significantly reduces the number of degrees of freedom (even by four orders) due to the anisotropic higher‐order coarse‐scale approximation and the novel shape functions that take into account the microscale boundary conditions. Moreover, the local iterative corrector scheme Nguyen and Schillinger (2019) adapted for the bending‐dominated responses of sandwich structures provides converges of the coarse mesh approximation to the best possible fine‐mesh solution. Several numerical examples are presented to demonstrate the capabilities of the method. We found that the proposed modifications of the shape functions and the higher‐order coarse mesh approximation increase the convergence rate. Finally, we validated the proposed model by comparison of the numerical results with experimental ones for a sandwich panel with a dual corrugated high‐density fiberboard core. Very good consistency of both results was observed.

Numerical Simulation of Some Steel Structural Elements with Uncertain Initial Porosity

The main research purpose of this work was to study the elasto-plastic responses of some fundamental steel structural elements exhibiting stochastic volumetric microvoids. The constitutive model of the steel material was consistent with the deterministic Gurson–Tvergaard–Needleman (GTN) porous plasticity model, where some of the microvoids parameters have additionally been defined as Gaussian random variables. The iterative stochastic finite element method implemented based on the-tenth order stochastic perturbation technique was utilized in numerical experiments. An interoperability of the computer algebra system MAPLE 2019 and the finite element method system ABAQUS was employed to study the influence of the initial microvoids f0 with uncertainty in the structural steel on the statistical scattering of the resulting stresses and deformations. The basic probabilistic characteristics of the structural response were computed and contrasted with statistical estimators inherent in the Monte–Carlo simulation and also with the results obtained from the semi-analytical probabilistic method. Reliability indices according to the first-order reliability method (FORM) were also calculated. Two numerical illustrations included the (i) tension test of the round cylindrical steel rebar and the (ii) bending test of the steel I-beam restrained at both its ends. Expectations and coefficients of variation of the structural responses confirmed here the importance of the microvoids for the stochastic elasto-plastic behavior of some basic engineering structures, where tensile stress plays an important role in designing procedures.

The Effect of Residual Stresses on Stress–Strain Curves Obtained via Profilometry‐Based Inverse Finite Element Method Indentation Plastometry

This paper concerns, the effect of (unknown) residual stresses in the near‐surface region of a sample on outcomes of an indentation plastometry technique for obtaining stress–strain relationships, using relatively large spherical indenters. The technique is based on the iterative finite element method (FEM) simulation of indentation, so as to infer the true stress–strain curve of the material (with the target outcome being the profile of the residual indent). It is expected that residual stress levels (if significant compared with the yield stress) are likely to influence the outcome, and indeed this forms the basis of many attempts to measure residual stresses in this way (using a known stress–strain curve). However, there are important issues of sensitivity here, which affect the reliability of such procedures and are also relevant to the accuracy of stress–strain curves inferred from measurements made on samples containing unknown levels of residual stress. Herein, both experimental work on samples with (equal‐biaxial) residual stresses created by the application of external loads and extensive FEM modeling to explore various scenarios are covered. The main conclusion is that the sensitivities involved are in general very low, particularly with relatively deep indentation. Inferred stress–strain curves are thus likely to be quite accurate, even in the presence of relatively high levels of residual stress. Conversely, the measurement of residual stress levels via this procedure is likely to be rather inaccurate, although the reliability is improved using shallow penetration (low ratio of depth‐to‐indenter radius). It is also noted that tensile residual stresses tend to influence outcomes more strongly than compressive ones.

Explicit finite element analysis of slope stability by strength reduction

The construction of stable slopes and vertical cuts is an important step in many geotechnical projects. Limit equilibrium methods (LEMs) are well-accepted procedures to compute factors of safety (FoS); however, they fail to provide any information about the distribution of the field variables within the soil mass because they do not include any stress-strain relationship in their formulation. On the other hand, the iterative finite element method (FEM/I) can estimate the field variables, but in the current study, we show that, for steep slopes and vertical cuts, it underestimates the FoS compared to the LEM. To overcome the obstacles that exist in this method, this study proposes a new approach to define the initiation of instability based on an abrupt change in the kinetic energy of the system. We also suggest a procedure to calculate the minimum FoS based on the explicit finite element method (FEM/E). Comparison of the results obtained from the proposed method, LEM, and FEM/I revealed that the FoS computed by the proposed method is in good agreement with the results of the LEM for a wide range of material parameters, geometries and external loading conditions, while no assumption regarding the critical slip surface needs to be made.

Steady-state heat transfer analysis in a spherical domain revisited

The paper discusses the numerical solution of the one-dimensional radially axi-symmetric non-linear second-order differential equation to model the conduction and radiation transfer through a spherical domain as a result of an exothermic heat source. The equation is transformed to a non-dimensional form. The dimensionless numbers emanating from the transformation represent the effect of the reaction rate, reaction type, activation energy, radiation and the convection on the temperature. The non-dimensional differential equation for the temperature distribution was previously solved using the Runge-Kutta-Fehlberg method coupled with a Shooting technique. In this paper the solution of the non-dimensional differential equation using an iterative Galerkin finite element method approach employing the Picard method is described. The commercial finite element code Comsol is also employed to solve the non-dimensional differential equation. The current study was motivated by inconsistencies that were observed in the previous results that were presented. Although the assumed underlying physics is used to evaluate the results, the study focuses purely on the numerical solution of the non-dimensional differential equation. The results obtained by the Galerkin finite element code and Comsol were found to be in exact agreement and also exhibit no inconsistencies.

A Simple Elasto-Plastic Iterative Method by Fem for the Analysis of Plane Articulated Truss: Case of a 3-Bar Truss

This paper presents a simplified numerical simulation tool for the elasto-plastic calculation of plane articulated truss by the finite element method (FEM) in MATLAB. The simplified approach consists of linearizing isotropic strain-hardening (to obtain a bilinear material law). The numerical implementation is built on the basis of the incremental and iterative FEM algorithms. The numerical resolution technique used is based on the projection methods of the modified Newton-Raphson solution. The MATLAB program is developed for the application of a 3-bar truss under monotonous quasi-static loading. Different values of the approximation error of the convergence criterion are used to study its impact on the quality of the algorithm. Numerical simulations have shown the reliability and quality of our simplified approach regardless of the approximation error.

A Parallel Iterative Finite Element Method for the Linear Elliptic Equations

By combining the two-grid discretization with the partition of unity method, a parallel iterative finite element method for the linear elliptic equations is proposed and investigated. Since the construction of the partition of unity is based on the coarse mesh triangulation, the computational domain of each subproblem can be divided automatically and the number of subproblems can be arbitrarily huge as the coarse mesh parameter H tends to zero. That means our method can be easily implemented in high performance supercomputers or cluster of workstations. Theoretical results based on a priori error estimation of the scheme are obtained, which indicate that our method can reach the optimal convergence orders within a few two-grid iterations. Numerical results are reported to assess the theoretical results.

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.

Error correction iterative method for the stationary incompressible MHD flow

A new iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations is derived in this paper. The method consists of two steps at each iteration step, we need first to solve the MHD equations by the Oseen‐type iterative scheme, and then an error correction strategy is applied to control the error arising from the linearization of the nonlinear MHD equations. The new method not only maintains the advantage of the standard Oseen‐type scheme but also possesses a rapid rate of convergence. It is proved that the convergence rate of the proposed method is increased greatly under the uniqueness condition. The uniform stability and convergence of the new scheme are analyzed. Ample numerical experiments are performed to validate the accuracy and the efficiency of the new numerical scheme.

An Extended Iterative Identification Method for the GISSMO Model

This study examines an extended method to obtain the parameters in the Generalized Incremental Stress State Dependent Damage (GISSMO) model. This method is based on an iterative Finite Element Method (FEM) method aiming at predicting the fracture behavior considering softening and failure. A large number of experimental tests have been conducted on four different alloys (7003 aluminum alloy, ADC12 aluminum alloy, ZK60 magnesium alloy and 20CrMnTiH Steel), here considering tests that span a wide range of stress triaxiality. The proposed method is compared with the two existing methods. Results show that the new extended Iterative FEM method gives the good estimate of the fracture behaviors for all four alloys considered.

The Arrow–Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow

In this paper, we propose and analyze an Arrow–Hurwicz iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations discretized by mixed element methods. Comparing with the other iterative methods, the significant feature of this iterative method is that it does not need to solve any saddle-point system at each iteration step except for determination of the initial functions. Under several reasonable conditions, it is proved that the solution solved by the proposed iterative method is uniformly bounded with respect to the mesh width h and the iteration number n and the method converges geometrically with a contraction number independent of the finite element mesh width h. Ample numerical experiments are performed to validate the accuracy and the efficiency of the numerical scheme.

An iterative finite element method for piezoelectric energy harvesting composite with implementation to lifting structures under Gust Load Conditions

In this paper, a novel iterative finite element method (FEM) for energy harvesting purpose is presented. In this iterative FEM, the electromechanical coupling in the governing piezoelectric energy harvesting equation is separated into two parts. One part is an actuator-like problem and can be evaluated with ease via commercial software. The solution of the other part, for voltage generation, is obtained straightforward utilising the structural response of the earlier part. These two parts are solved iteratively until the convergence in voltage and structural responses is reached. The new iterative FEM is implemented to solve the dynamic problems of piezoelectric-based energy harvester in the frequency and time domains. The application of the present iterative FEM to evaluate the piezoelectric energy harvesting of lifting structures under an aeroelastic condition, i.e., gust load, is shown in some details. The validation against other methods from the literature shows the robustness and capabilities of the iterative FEM.

A new iterative least square Chebyshev wavelet Galerkin FEM applied to dual phase lag model on microwave drying of foods

In this paper, we developed a dual-phase-lag (DPL) model of heat and mass transfer in presence of a source term for microwave drying foods of different geometrical configuration like slab, cylinder or sphere under the most generalized boundary conditions. The particular cases of present model are the diffusion model, Luikov model, Weng-Chang model and Andarwa-Tabrizi model. The proposed model is the most generalized boundary value problem of a coupled system of two hyperbolic second order partial differential equations. Stability analysis of present model is provided. A new iterative least square Chebyshev wavelet Galerkin finite element method provided for solution. The discretization in space and then application of Chebyshev wavelet Galerkin method converts our problem into a coupled system of two most generalized Sylvester equations. The iterative least square method provide solution of coupled system of Sylvester equations. Convergence and stability analysis of present method is discussed in detail. In a particular case of DPL model, the solution obtained by present method is compared with exact solution (Laplace transform technique) and are approximately same. Effect of relaxation time, Fourier number, phase change coefficient, thermo-gradient coefficient, heat generation, radius of the food and diffusion coefficient on heat and mass transfer are discussed in detail. This generalized model and its solution play important role in the study of microwave food drying.

Analysis of Local and Parallel Algorithm for Incompressible Magnetohydrodynamics Flows by Finite Element Iterative Method

Analysis of Local and Parallel Algorithm for Incompressible Magnetohydrodynamics Flows by Finite Element Iterative Method

Solution of the logarithmic Schrödinger equation with a Coulomb potential

The nonlinear logarithmic Schrödinger equation (log SE) appears in many branches of fundamental physics, ranging from macroscopic superfluids to quantum gravity. We consider here a model problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical solution for the ground state energy and wave function as a function of the strength of the logarithmic interaction. We develop an iterative finite element method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 1010. The excited states (n>1) are converged to better than one part in 108. We also construct a remarkably simple variational wave function, consisting of a sum of Gaussons with n free parameters. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. Although the Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force Hamiltonian.

Calculation of the loading capacity of high voltage cables laid in close proximity to heat pipelines using iterative finite-element method

The paper presents a new iterative finite-element model for the analysis of the thermal interference between a three-phase system of high voltage cables and heat pipelines buried in close proximity. A triangular formation of 110 kV cables and a pair of polyurethane insulated steel heat pipes are considered as a characteristic real-life example in authors’ country. The maximum allowable load magnitudes for cables in winter weekdays have been determined for various distances between cables and pipes. The proposed mathematical model takes into account the effects of conduction, and the heat exchange by convection and radiation on the pavement surface. In order to adequately model the convective heat transfer from the water in motion an iterative approach is elaborated based upon the properties of the Nusselt number. It was shown that the maximum allowable magnitudes of load current of cables should be decreased for app. 6% if the distance between the cables and the adjacent pipe is 3 m, which is the minimum distance usually tolerated in practice in authors’ country.

Three-Dimensional Elastic Analysis of a Structure with Holes Using Accelerated Coupling-Matrix-Free Iterative s-Version FEM

Some improvements of the coupling-matrix-free iterative s-version finite element method (FEM) to shorten its computational time are proposed. Then, the proposed method is applied to three-dimensional stress concentration problems. For sufficiently small computational time for practical use, two key techniques are introduced. First, the iteration is accelerated drastically by using the proposed convergence acceleration techniques. Secondly, stress transfers between global and local meshes are accelerated considerably by a bucket search algorithm. The proposed method was more than one hundred times faster than the straightforward algorithm of the coupling-matrix-free iterative s-version FEM.

Robust Reconstruction of Elasticity Using Ultrasound Imaging and Multi-Frequency Excitations.

Biomedical parameters of tissue can be important indicators for clinical diagnosis. One such parameter that reflects tissue stiffness is elasticity, the imaging of which is called elastography. In this paper, we use displacements from harmonic excitations to solve the inverse problem of elasticity based on a finite-element method (FEM) formulation. This leads to iterative solution of nonlinear and nonconvex problems. In this paper, we show the importance and selection of viable initializations in numerical simulation studies and propose techniques for the fusion of multiple initializations for ideal reconstructions of unknown tissue as well as combining information from excitations at multiple frequencies. Results show that our method leads up to 76% decrease in root-mean-squared error (RMSE) and 9.9 dB increase in contrast-to-noise ratio (CNR) in simulations with noise, when compared to conventional iterative FEM without multiple initializations and frequencies. As the wave patterns in individually selected frequencies may introduce artifacts, a joint inverse-problem solution of multi-frequency excitations is introduced as a robust solution, where CNR improvements of up to 11.9 dB are observed. We also present the methods on a tissue-mimicking gelatin phantom study using mechanical excitation and ultrafast plane-wave ultrasound imaging, where the RMSE was improved by up to 51%. An experiment of ablation via heating an ex-vivo bovine liver shows that reconstruction artifacts are reduced with our proposed method.

Calculation of Space Charge Density in Negative Corona Based on Finite-Element Iteration and Sound Pulse Method

Space charge distribution is a valuable indicator of ion flow field and is widely utilized to assess electromagnetic environment for transmission line. The finite-element method (FEM) is widely used to solve this problem. However, Kaptzov assumption in FEM assumes that onset voltage is unchangeable after corona occurs, which does not tally with the actual situation. This paper adopted the finite-element iteration method (FEIM), which uses empirical formula instead of assumptions to calculate the charge density. The surface field intensity is not constant in the empirical formula because the empirical formula takes the effects of space charge on surface field into consideration. So this is more consistent with actual situation. In order to verify the accuracy of the algorithm, sound pulse method is applied to measure the charge density around the conductor in the laboratory. Simulation results show that the charge density obtained in this paper agrees with the data in the experiment, which verifies the validity of the empirical formula for calculation of space charge density. Compared with a method considering Kaptzov assumption, FEIM is closer to the experiment in area around conductor line and near the ground. This indicates that empirical formula in FEIM is more effective than Kaptzov assumption in these two domains.