Version Of Finite Element Method Research Articles

Local meshless methods for second order elliptic interface problems with sharp corners

In the present paper, we develop a local meshless procedure for solving a steady state two-dimensional interface problem having discontinuous coefficients and curved interfaces with sharp corners. The proposed local meshless methods are based on three types of radial basis functions (RBFs): a local meshless method based on multiquadric RBF (LMM1P), a local meshless method based on integrated multiquadric RBF (LMM2P) and a local meshless method based on hybrid Gaussian-Cubic RBF (LMM3P). Stencils are designed at the interface and interior regions to cope with discontinuities and sharp corners. Due to the localized nature of the procedure and a sparse matrix representation, the local meshless methods become computationally less expensive than global meshless methods. The methods are augmented with linear polynomial to improve accuracy and ensure stable computation. Comparison with some existing versions of finite element methods is also performed to show better accuracy of the proposed meshless methods. Accuracies of the proposed local meshless methods are also compared among themselves. Flexibility of the meshless methods with respect to complex geometries, adapting to different shapes of the interfaces and selection of the shape parameter is also considered.

An adaptive p-FEM for three-dimensional concrete aggregate models

Numerical simulation for concrete aggregate models (CAMs) with different shape aggregates usually requires high accuracy and convergence near the material interfaces. But high memory usage will be needed for those traditional finite element methods such as the method by using mesh refinement throughout the domain. Thus, an adaptive [Formula: see text]-version finite element method ([Formula: see text]-FEM) is proposed in this paper for the solution of 3D CAM problems, and meanwhile the resulting adaptive computational algorithm and post-processing program are presented. We firstly focused two typical 3D weak discontinuity problems on the influence of different convergence criterions for the computational results of each point on the interface in order to verify the efficiency and convergence of the resulting [Formula: see text]-FEM, and then this method is successfully applied to the numerical simulation of CAMs with different shape aggregates. In addition, an efficient hybrid realization method which combines ANSYS and Hypermesh software is also presented in order to quickly establish the geometric models of 3D CAMs. The numerical results have been shown that the proposed [Formula: see text]-FEM can efficiently solve the concrete-like particle-reinforced composite problems and more accurate numerical results can be obtained under the case of fewer elements used in simulation of CAMs, even there being some elements with poor quality.

On Solutions of Emden-Fowler Equation

Finite Element Method (FEM), based on p and h versions approach, and the Adomians decomposition algorithm (ADM) are introduced for solving the Emden-Fowler Equation. A number of special cases of p and h versions of FEM are introduced. Several iterated forms of the ADM are considered also. To demonstrate the efficiency of both methods, the numerical solutions of different examples are compared for both methods with the analytical solutions. It is observed that the results obtained by FEM are quite satisfactory and more accurate than ADM. Moreover, the FEM method is applicable for a wide range of classes including the singularity cases with the given special treatments by the FEM. Comparing the results with the existing true solutions shows that the FEM approach is highly accurate and converges rapidly.

Finite element approximation of reaction–diffusion problems using an exponentially graded mesh

We present the analysis of an h version Finite Element Method for the approximation of the solution to singularly perturbed reaction–diffusion problems posed in smooth domains Ω⊂R2. The method uses piecewise polynomials of degree p in each variable, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We establish robust, optimal convergence rates in a variety of norms and illustrate our theoretical findings through numerical computations.

An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions

In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation $-_{0}D_{x}^{\alpha }u+Au=f(x)$ with Dirichlet boundary condition. First, we introduce the existence and uniqueness of the considered problem and show the wellposedness of the corresponding weak form. To solve the mentioned equation, the classical hierarchical polynomials are employed as the trial basis functions. Then, we develop a kind of novel test basis functions for the Petrov-Galerkin finite element method such that the stiffness matrix becomes an identity matrix and the coefficient matrix often has a small condition number. Moreover, we give some properties of the developed test basis functions, and discuss the implementation of the developed finite element method in detail. It is shown that the implementation of our method is easier than that of other finite (and spectral) element methods. Finally, we give a numerical example, and the numerical results show the effectiveness of the develped method.

Application of the H-P version Finite Element Method in Analysis of Hydraulic Structures

In this paper, we consider the application of the h-p version finite element method in analysis of hydraulic structures by using the FEM computational software StressCheck of American ESRD (Engineering Software Research and Development) company as the computational tool. We choose a three dimensional concrete gravity dam as the computational model and compute the maximum displacement and the first principal stress of the concrete gravity dam. The results show that the h-p version finite element method is effective and reliable in analysis of hydraulic structures for the model chosen, and we obtained faster convergence rate, higher precision and smaller error by using the h-p version finite element method than by using p-version and h-version finite element method.

Spectral element simulations of three dimensional convective heat transfer

This paper presents new formulation for simulating three-dimensional convective heat transfer by incorporating the complete viscous dissipation function in the energy equation to account for viscous heating and adopting the Boussinesq approximation for the thermal buoyancy term in the Navier-Stokes equations. In addition, new implementation of fourth order Stiffly Stable Schemes was achieved and tested for time integration. In order to provide dual-level mesh refinement, the hp-refinement, for flexible spatial resolutions, a modal spectral element method was used to solve these equations in three dimensions. Simulation results were compared with exact solutions or higher order solutions and good agreement were accomplished. This demonstrates that the new formulation and implementation are accurate enough for investigating convective heat transfer with viscous heating subject to complex thermal and flow boundary conditions in three dimensional irregular domains using the high order version of finite element method.

Isogeometric analysis of free vibration of framed structures: comparative problems

PurposeThe purpose of this paper is to check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element method (GFEM) and other enriched versions of FEM through numerical examples of free vibration problems.Design/methodology/approachSince its conception, IGA was widely applied in several problems. In this paper, IGA is applied for free vibration of elastic rods, beams and trusses. The results are compared with FEM, GFEM and the enriched methods, concerning frequency spectra and convergence rates.FindingsThe results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies.Originality/valueIsogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.

Finite element approximation of convection–diffusion problems using an exponentially graded mesh

We present the analysis of an h version Finite Element Method for the approximation of the solution to convection–diffusion problems. The method uses piece-wise polynomials of degree p≥1, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We consider a model convection–diffusion problem, posed on the unit square and establish robust, optimal convergence rates in the energy and in the maximum norm. We also present the results of some numerical computations that illustrate our theoretical findings and compare the proposed method with others found in the literature.

Forced vibration analysis of composite cylindrical shells using spline finite strip method

In this study, forced vibration behavior of thin-walled composite circular cylindrical shells is investigated using the spline finite strip method (spline FSM). Spline FSM is one of the versions of finite element method (FEM) employing a special element called finite strip. The shells considered in this study are assumed to be thin; therefore, the classical bending theory of shells and Sanders–Koiter's strain–displacement relation are used in the theoretical formulation of developed method. Time-history response of shells concerning arbitrary type of forces, boundary conditions and damping effects are obtained using developed spline FSM. To check the validity of results a comparison has been performed with the results obtained by conventional finite element method using ANSYS software. As far as the time history response of cylindrical shells is available the natural frequencies are extracted using Fast Fourier Transform (FFT) approach. The evaluated natural frequencies of composite shells are then compared with those obtained from an eigenvalue analysis. The comparison of results revealed the fact that developed spline FSM is an accurate tool that can be used in transient and harmonic analyses of composite laminated cylindrical shells.

Free geometrically nonlinear oscillations of perfect and imperfect laminates with curved fibres by the shooting method

Large-amplitude free vibrations of composite laminated plates with curvilinear fibres are studied. The fibre angle in a ply changes linearly in relation to one Cartesian coordinate. The plates are rectangular with geometric imperfections (out-of-planarity). The edges of the laminates are clamped, except in comparison studies, where simply supported conditions are applied. The displacement field is modelled by a third-order shear deformation theory, and the equations of motion (full model), in the time-domain, are obtained using a $$p$$ -version finite element method. When possible, the model is statically condensed neglecting in-plane inertias but still taking into account the in-plane displacements. The condensed model is transformed to modal coordinates in order to have a model with fewer degrees of freedom (reduced model). Backbone curves are found by the shooting method, using Runge–Kutta–Fehlberg method modified with Cash–Karp method to control the error with adaptive stepsize. Backbone curves of composite laminates with different curvilinear fibre angles are plotted and compared. In addition to the fundamental backbone curve, the method is able to find bifurcations leading to other branches, as shown in some examples. Oscillations of some points are studied in detail using phase-plane plots and Fourier spectra of deflection. Finally, the effects of geometrical imperfection on the backbone curves are analysed.

Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems

Abstract We present the mathematical analysis for the convergence of an h version Finite Element Method (FEM) with piecewise polynomials of degree p ≥ 1, defined on an exponentially graded mesh. The analysis is presented for a singularly perturbed reaction-diffusion and a convection-diffusion equation in one dimension. We prove convergence of optimal order and independent of the singular perturbation parameter, when the error is measured in the natural energy norm associated with each problem. Numerical results comparing the exponential mesh with the Bakhvalov–Shishkin mesh from the literature are also presented.

Improved Finite Element Approach for Modeling Three-Dimensional Linear-Elastic Bodies

Objectives: This paper proposes a general formula for computation of the stiffness matrix of three-dimensional linearelastic bodies by the method of Moment Schema of Finite Elements (MSFE). Methods: Numerical methods based on Lagrange variational principle, such as Finite Element Method (FEM) and the method of Moment Schema of Finite Elements (MSFE). The paper focuses on the problem of obtaining the stiffness matrices of Finite Elements (FE), which ensure effectiveness of the developed version of FEM. Variational principle for calculation of energy functional of rectangular three-dimensional finite element is used. Results: Analyses of various problems of mechanics of deformable solids shows slow convergence of traditional FEM, especially for rigid bodies and surfaces of complex curved shapes. This paper proposes a scheme for inference for FEM ratios, which takes into account the basic properties of rigid displacements for isoparametric and curvilinear FE. The presented version of FEM allows us to obtain the stiffness matrix of FE, taking into account the effect of a false shift and low compressibility of elastomers. Conclusion/Application: The general method to find the potential energy of three-dimensional linear-elastic bodies is proposed, which allows us to solve the problems of mechanics of complex structures with higher accuracy.

New Development of the P and H-P Version Finite Element Method with Quasi-Uniform Meshes for Elliptic Problems

In recent three decades, the finite element method (FEM) has rapidly developed as an important numerical method and used widely to solve large-scale scientific and engineering problems. In the fields of structural mechanics such as civil engineering , automobile industry and aerospace industry, the finite element method has successfully solved many engineering practical problems, and it has penetrated almost every field of today's sciences and engineering, such as material science, electricmagnetic fields, fluid dynamics, biology, etc. In this paper, we will overview and summarize the development of the p and h-p version finite element method, and introduce some recent new development and our newest research results of the p and h-p version finite element method with quasi-uniform meshes in three dimensions for elliptic problems.

New Development of the P and H-P Version Finite Element Method with Quasi-Uniform Meshes for Elliptic Problems

New Development of the P and H-P Version Finite Element Method with Quasi-Uniform Meshes for Elliptic Problems

Quasi-optimal degree distribution for a quadratic programming problem arising from the [formula omitted]-version finite element method for a one-dimensional obstacle problem

We present a quadratic programming problem arising from the p-version for a finite element method with an obstacle condition prescribed in Gauss–Lobatto points. We show convergence of the approximate solution to the exact solution in the energy norm. We show an a-priori error estimate and derive an a-posteriori error estimate based on bubble functions which is used in an adaptive p-version. Numerical examples show the superiority of the p-version compared with the h-version.

Numerical approach for the sensitivity of a high-frequency magnetic induction tomography system based on boundary elements and perturbation method

Magnetic induction tomography (MIT) is an imaging technique based on the measurement of the magnetic field perturbation due to eddy currents induced in conducting objects exposed to an external magnetic excitation field. In MIT, current-carrying coils are used to induce eddy currents in the object and the induced voltages are sensed with the receiving coils. When the driving frequency is significantly high relative to the frequency range in which MIT normally operates, metallic targets with high conductivity between the coils can be treated as perfect electric conductors (PEC) with negligible errors. In this scenario, the penetration depth of the magnetic field into the target is extremely small and the traditional versions of the finite element method (FEM) are not efficient for the calculation of the sensitivity and the forward problem due to the requirement for large number of elements to reach an acceptable computational precision. Other versions of FEMs (such as Hp-FEM), which have higher discretization efficiency and more advanced elements to satisfy the requirement, are exceptions. Nevertheless, the discretization regions for all FEMs have to extend beyond the region that contains the conducting object and volumetric elements are generally required for 3D problems. In contrast, the boundary element method (BEM) based on integral formulations becomes an effective way to analyze this kind of scattering problem since meshes are only required on the surface of the object. By point collocation, the boundary integral equations can be transformed into linear equations. Numerical methods are used to solve the linear equations and the solution of the original integral equations can be obtained. In this paper, we compute four typical sensitivity maps between the coil pairs in high-frequency MIT system due to a PEC perturbation. The magnetic scalar potential is used to improve the efficiency. Five PEC objects of different shapes are used in the simulation. The results have been compared with the experimental results and that obtained from the H · H formulations. We can know that the sensitivity maps derived by BEM are in good agreement with that from the experiment and theoretical solution. Overall, BEM is an effective way to calculate the sensitivity distributions of a high-frequency MIT system.

Numerical Simulation of Stress-Strain State for Nonhomogeneous Shell-Type Structures Based on the Finite Element Method

Projective-iterative version of finite element method has developed for numerical simulation of the stress-strain state of nonhomogeneous shell-type structures (shells with openings). Plastic deformation of the material is taken into account when using the method of elastic solutions that reduce the solution of elastoplastic problems to solution of elastic problems. Developed PIV’s significant savings of computer calculation has been compared with the calculation on a fine mesh of traditional FEM. Designed scheme allows analysis of the mutual influence of openings. Analysis of the transformation zone of plastic deformation is developed. For definiteness, the cylindrical shell structures with several rectangular openings are considered.

Local Jacobi Operators and Applications to thep-Version of Finite Element Method in Two Dimensions

Based on the Chebyshev projection-interpolation on each edge of elements and the Chebyshev projection on each element, we have designed the local Jacobi operators $\Pi_{\Omega_j}$ on the triangular or quadrilateral element $\Omega_j$, $1\leq j\leq J$ such that $\Pi_{\Omega_j}u$ is a polynomial of degree p on $\Omega_j$ which interpolates u at the vertices of $\Omega_j$, coincides with the Chebyshev projection-interpolation of u on the edges of $\Omega_j$, and possesses the best approximation to the smooth and singular functions u. By a simple assembly of $\Pi_{\Omega_j}u$, $1\leq j\leq J$ we construct a piecewise and globally continuous polynomial of degree p which has the best approximation error bound locally and globally for singular as well as smooth solutions on general quasi-uniform meshes and satisfies the homogeneous Dirichlet boundary conditions. An application of the local Jacobi operators to the p-version of the finite element method associated with general meshes composed of (curvilinear) triangular and quadrilateral elements for problems on polygonal domains leads to the optimal convergence, which has been an open problem for more than three decades.

Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

Ap-version shell finite element based on the so-called shallow shell theory is for the first time employed to study vibrations of deep cylindrical shells. The finite element formulation for deep shells is presented and the linear natural frequencies of different shells, with various boundary conditions, are computed. These linear natural frequencies are compared with published results and with results obtained using a commercial software finite element package; good agreement is found. External forces are applied and the displacements in the geometrically non-linear regime computed with thep-model are found to be close to the ones computed using a commercial FE package. In all numerical tests thep-FE model requires far fewer degrees of freedom than the regular FE models. A numerical study on the dynamic behaviour of deep shells is finally carried out.