First-order Finite Element Research Articles

Convergent finite element methods for antiferromagnetic and ferrimagnetic materials

We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations averaging the spins of two sublattices. For the static setting, which requires the solution of a constrained energy minimization problem, we introduce a discretization based on first-order finite elements and prove its Γ-convergence. Then, we propose and analyze two iterative algorithms for the computation of low-energy stationary points. The algorithms are obtained from (semi-)implicit time discretizations of gradient flows of the energy. Finally, we extend the algorithms to the dynamic setting, which consists of a nonlinear system of two Landau–Lifshitz–Gilbert equations solved by the two fields, and we prove unconditional stability and convergence of the finite element approximations toward a weak solution of the problem. Numerical experiments assess the performance of the algorithms and demonstrate their applicability for the simulation of physical processes involving antiferromagnetic and ferrimagnetic materials.

Modal analysis of plates resting on elastic foundation based on the first-order shear deformation theory and finite element method

This paper investigates the free vibration characteristics of plate structures supported by a Pasternak elastic foundation, utilizing the first-order shear deformation theory (FSDT). FSDT simplifies the plate theory by considering only first-order shear deformation, enhancing formulation simplicity. Additionally, employing plate theory reduces computational complexity, as 2D models entail fewer degrees of freedom compared to their 3D counterparts. The finite element method (FEM) with 8-node quadrilateral element is employed for computational analysis, implemented using MATLAB. First, a comparison is made with some existing data to show the accuracy and reliability of the research. Numerical examples are then presented of the influence of the effects of thickness variation, foundation parameters and boundary conditions on frequency are investigated. The results show that the method converges very fast and reliability when compared to other research findings. The results of the research can be applied to many different engineering applications related to plates resting on elastic foundation.

RFOR-DQHFEM: Hybrid relaxed first-Order reliability and differential quadrature hierarchical finite element method for multi-physics reliability analysis of conical shells

In this current work, a hybrid reliability analysis and theoretical frequency technique are suggested for the reliability response of conical shells. Two levels of analyses are proposed as the main loop of the reliability method for finding the failure probability and the second level applied in the main loop for giving the performance function of frequency applied in conical shell structures with multi-physics vibration analysis. A dynamical adjusting procedure is proposed for computing the relaxed factor using the enough descent condition inside the reliability method. The superior convergence rate is considered for selecting the relaxed factor of the proposed first-order reliability method named RFORM. An elastic-electro-mechanical model based on the Higher-Order Shear Deformation Theory (HSDT) is extended for frequency analysis of conical shells. The innovative numerical procedure named Differential Quadrature Hierarchical Finite Element Method (DQHFEM) as a robust framework for giving the vibration behavior of studied mechanical structures is applied for solving motion equations. The developing DQHFEM and RFORM are applied for the laminated, nanocomposite, and piezoelectric conical shell structures with multi-source uncertainties. Increasing the volume percentage of nanoparticles from 0% to 10% significantly enhances the reliability index, with carbon nanoparticles showing a 132% increase, silica nanoparticles showing a 97% increase, and other nanoparticles showing an approximate 40% increase. Also, as moisture content increases from 0% to 30%, the reliability index for a thickness-to-large-radius ratio of 0.2 drops by about five times. Excessive moisture levels (above 20%) result in a negative reliability index, indicating a hazardous condition.

A Novel Fully Decoupled Scheme for the MHD System with Variable Density

Abstract In this paper, we first establish a novel first-order, fully decoupled, unconditionally stable time discretization scheme for the MHD system with variable density. This scheme successfully decouples all the coupling terms by combining the gauge-Uzawa method and the scalar auxiliary variable (SAV) method. And we prove its unconditional energy stability. Then we give the first-order finite element scheme and its implementation. Furthermore, we perform a rigorous error analysis of the proposed numerical scheme. Finally, we perform some numerical experiments to demonstrate the effectiveness of the decoupling scheme.

Damped Vibration and Optimization of the Geometrical Parameters of FG Stiffened Cylindrical Shells Resting on Elastic Foundation Using GA

Functionally graded materials with their structural characteristics, such as the type of distribution, the size of phases, gradually change from one surface to another and provide a wide range of applications. In this research, the optimization of geometric parameters of FG structure rested on the elastic foundation, including damping effects is of interest. The cylindrical shell’s material properties are assumed to vary smoothly and continuously across the thickness according to the power law distribution of the volume fraction of constituents. The governing equations of stiffened functionally graded cylindrical shell resting on an elastic foundation are obtained using Hamilton’s principle, the first-order shear deformation theory and finite element method. The numerical results are obtained for studying the effect of various factors, such as distribution of volume fraction, length and thickness, different boundary conditions, stiffness and damping coefficients, and also the natural frequency of the dynamic responses of the cylindrical shell. The damping effects are considered which were not presented in previous studies. The frequency responses of the cylindrical FGM shell resting on elastic foundation were obtained by adding the damping term into the governing equations. Besides, analytical formulation results were considered as an objective function (input) for optimization in Genetic Algorithms. Finally, the compatibility of the analytical modeling results with the commercial finite element software was checked which showed good agreement.

A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation

When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~ 200,000 or more arbitrarily selected observation points within approximately 5 s, with the solution time itself of 3 s. The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods—an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution. Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task. The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.

A comprehensive analysis of in-plane functionally graded plates using improved first-order mixed finite element model

In this study, the mechanical behaviors of the in-plane functionally graded plate (IFG) are comprehensively explored via a finite element (FE) model. The current FE model is developed base on improved first-order shear deformation theory (FSDT) in conjunction with the mixed FE formulation. The volume fraction as well as the material properties change smoothly along the width and length of the plates for six different patterns. A simple improved transverse shear stress is applied to modify its distribution across the thickness of the IFG plates. Hence, the transverse shear stress equals zeros on the lower and upper surfaces of the IFG plates. Besides, no shear correction factor is needed for calculation of shear strain energy. A four-node mixed plate element, IMQ4, is established via mixed FE formulation for mechanical analysis of the IFG plates. The accuracy and fast convergence rate are validated via some examples. A comprehensive parametric analysis is provided to demonstrate the roles of several parameters including the material gradient index, the side-to-thickness ratio, aspect ratio, and boundary conditions on the mechanical behaviors of the IFG plates.

Using a one-dimensional finite-element approximation of Webster's horn equation to estimate individual ear canal acoustic transfer from input impedances

Knowledge of the sound pressure transfer to the eardrum is important. The transfer is highly influenced by the shape of the ear canal and its acoustic properties, such as the acoustic impedance at the eardrum. Invasive procedures to measure the sound pressure at the eardrum are usually elaborate or costly. We propose a numerical method to estimate the transfer impedance of the ear canal given only input impedance measurements at the ear canal entrance, by using one-dimensional first-order finite elements and Nelder-Mead optimization algorithm. Estimations on the area function of the ear canal and the acoustic impedance at the eardrum are achieved. Results are validated through numerical simulations on ten different ear canal geometries and three different acoustic impedances at the eardrum, using synthetically generated data from three-dimensional finite element simulations.

Finite Element Model Updating of Steel Arch Bridge Based on First-Order Mode Test Data

In order to obtain an accurate finite element model of a steel arch bridge, a first-order modal finite element model updating method is proposed by using the measured first-order modal data of the bridge. Using the measured acceleration time history data under random excitation and the first-order mode updating method, the stiffness matrix of the finite element model is updated, and the first-order frequencies and first-order mode shapes before and after updating of the model are compared and analyzed. The state space method is used to compare and analyze the dynamic response and the reliability of the structure before and after the updating of the model. The results show that the difference of the first-order frequencies between the updated finite element model and the measured result is about 0.001, and the difference of the first-order mode shapes is less than 0.197, which meets the needs of engineering. Dynamic response values of the updated structural model are much larger than those of the structural model before updating. The theoretical model is different from the dynamic response of actual structure, so it is necessary to update the theoretical model. The finite element model updating method can provide a reliable analytical way for bridge structural health monitoring, state evaluation, and damage identification.

Strictly Uniform Exponential Decay of the Mixed-FEM Discretization for the Wave Equation With Boundary Dissipation

Uniform preservation of stability in approximations of wave equations is a long-standing issue. In this paper, a one-dimensional wave equation with a partially reflective boundary is approximated using a first-order mixed finite element method. The multiplier method is used to prove that the approximated systems are exponentially stable with a decay rate independent of the mesh size. Upper bounds on the exponential decay are obtained in terms of the physical parameters.

Two discretisations of the time-dependent Bingham problem

This paper introduces two methods for the fully discrete time-dependent Bingham problem in a three-dimensional domain and for the flow in a pipe also named after Mosolov. The first time discretisation is a generalised midpoint rule and the second time discretisation is a discontinuous Galerkin scheme. The space discretisation in both cases employs the non-conforming first-order finite elements of Crouzeix and Raviart. The a priori error analyses for both schemes yield certain convergence rates in time and optimal convergence rates in space. It guarantees convergence of the fully-discrete scheme with a discontinuous Galerkin time-discretisation for consistent initial conditions and a source term fin H^1(0,T;L^2(Omega )).

Optimal L2 error analysis of first-order Euler linearized finite element scheme for the 2D magnetohydrodynamics system with variable density

In this paper, a first-order Euler semi-implicit finite element scheme is proposed for solving the two-dimensional incompressible magnetohydrodynamics flows with variable density numerically. The proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing efficient algorithms for the multi-physical field problems. When the time step size τ and the mesh size h both are sufficiently small, optimal spatial error estimates in L 2 -norm are presented for the density, velocity and magnetic field by using energy techniques based on uniform regularities of solutions to the time-discrete scheme and the method of mathematical induction. Finally, numerical results are given to confirm the theoretical analysis.

Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects

We consider the numerical approximation of the inertial Landau–Lifshitz–Gilbert equation (iLLG), which describes the dynamics of the magnetisation in ferromagnetic materials at subpicosecond time scales. We propose and analyse two fully discrete numerical schemes: The first method is based on a reformulation of the problem as a linear constrained variational formulation for the linear velocity. The second method exploits a reformulation of the problem as a first order system in time for the magnetisation and the angular momentum. Both schemes are implicit, based on first-order finite elements, and generate approximations satisfying the unit-length constraint of iLLG at the vertices of the underlying mesh. For both methods, we prove convergence of the approximations towards a weak solution of the problem. Numerical experiments validate the theoretical results and show the applicability of the methods for the simulation of ultrafast magnetic processes.

Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau–Lifshitz equation, Quart. Appl. Math., 76, 383–405, 2018) proposed two novel predictor-corrector methods for the Landau–Lifshitz–Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau–Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

This paper emphasized the size-dependent thermo-elastic vibration characteristics of uncertain functionally graded (FG) porous nano-beams by employing the first-order perturbation theory (FOPT) with the finite element method. An isoparametric quadratic beam element having three nodes is used for the finite element analysis of the nano-beams with variation in degree of uncertainty in material properties, geometric configuration, and thermal environment. The power-law model is employed to obtain the effective material properties of the considered beams. The governing equations are presented according to Eringen’s elasticity theory and Reddy’s beam theory using the minimum potential energy. The computed FOPT results for the vibration statistics are assessed with the help of Monte Carlo simulation (MCS) and a well agreement is found. Further, results are presented by analyzing the effects of degree of source uncertainty, size-dependent parameter, volume fraction indices, porosity distribution, porosity index, thermal environment, and aspect ratios on the fundamental frequency of the FGM nano-beams. The numerical results revealed the significant influence of thermal environment and porosity on the vibration statistics of the nano-beams and provided guidance for the precise and reliable design of nano-devices for the application in thermal environment.

Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials.

One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylindrical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies.

A space-time discretization for an electromagnetic problem with moving non-magnetic conductor

This paper deals with a space-time discretization scheme for an eddy current problem in a multi-component domain with a moving non-magnetic conductor. We incorporate the Coulomb gauge to the formulation, then propose a fully-discrete finite element scheme combined with backward Euler's method to find an approximation of the solution to the variational system. The convergence of the scheme is proved, and the error estimates for the first-order Lagrangian finite elements are established. Under appropriate assumptions on the weak solution and the given initial data, we show the optimal convergence rate for the space discretization and the suboptimal rate for time discretization. Some numerical experiments are performed to support the obtained theoretical results.

A vertex scheme for two-phase flow in heterogeneous media

This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux upwinding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions. For both structured and unstructured meshes, we observe the high-accuracy wetting saturation profile that ensures minimal numerical diffusion at the front. Performing several examples of quarter-five spot problems in two and three dimensions, we show that the method can easily handle heterogeneities in the permeability field. Two distinct features that make the method appealing to reservoir simulators are: (i) maximum principle is satisfied, and (ii) local mass error is small and remains below nonlinear solver tolerance.

A comparative study on the metaheuristic-based optimization of skew composite laminates

Plate structures are the integral parts of any maritime engineering platform. With the recent focus on composite structures, the need for optimizing their design and functionality has now been tremendously realized. In this paper, a comprehensive study is carried out on the effectiveness and optimization performance of three metaheuristic algorithms in designing skew composite laminates under dynamic operational environments. The natural frequencies of the composite panels are measured using a first-order shear deformation theory-based finite element (FE) approach. The stacking sequence of the composite panels is optimized so that the natural frequency separation between the first two modes is maximized. The three metaheuristics considered here are genetic algorithm (GA), repulsive particle swarm optimization with local search and chaotic perturbation (RPSOLC), and co-evolutionary host-parasite (CHP) algorithm. It is observed that in general, the FE-coupled metaheuristic algorithms are quite capable to significantly improve the baseline designs. In particular, FE-CHP algorithm outperforms both the FE-GA and FE-RPSOLC algorithms with respect to accuracy, computational speed and solution reliability.

Linear Transient Dynamic Analysis of Plates With and Without Cutout

The present work exhibits the analysis of a plate with and without cutout considering the dynamic analysis in a transient zone with its application in the civil, structural, and aerospace industry. For this, first-order shear deformable C0 continuous Finite Element (FE) is implemented to examine transient response of plate. The governing ODE of motion’s is integrated using central difference explicit time integration scheme. A lumped mass matrix is obtained by a special diagonlization scheme and the total mass of the element is conserved by inducing effective rotary inertia terms. A FE program is developed based on derived FE formulation to investigate the effect of dynamic loading on such plates. The analysis exhibits the results in terms of displacement, forces, and bending moment of the plate for various aspect ratios and different parameters. The results are shown in terms of tables and graphs for smaller to larger cutout in plates with its different aspect ratio considering transient analysis. The results reveal that transient dynamics analysis shows a considerable difference in the behavior of plates with and without cutout in terms of time period, amplitude, and bending moment than that obtained from static analysis.