Rectangular Finite Element Research Articles

Thermo-elastic response of SUS316-Al2O3 functionally graded beams under various heat loads

Geometric nonlinearity and temperature dependent material properties are accounted for in the theoretical analysis of time dependent thermo-elastic response of thin functionally graded material (FGM) SUS316-Al2O3 beam subjected to various heat loads. A two dimensional Lagrangian rectangular finite element is used to obtain the temperature distribution on the transverse plane of the beam. Nonlinear thermo-elastic deflection and thermal stresses are evaluated for various structural and thermal boundary conditions. Thermo-elastic oscillations are observed in case of beams subjected to step, concentrated line and shock heat load whereas thermo-elastic deflection is observed for beams subjected to moving heat load. As the thermal load increases, the nonlinear thermal deflection of FGM beam are higher compared to linear analysis. In general, temperature dependency of material properties influence the amplitude of thermal oscillations. High thermal stresses are induced in beams with pin-pin and clamp-pin boundary condition as compared to hinge-hinge beam.

A generic type of frequency-domain spectral element model for the dynamics of a laminated composite plate

Various solution techniques have been developed in the last decades for accurate prediction of the dynamic responses of a laminated composite structure. The spectral element method (SEM) is well known as an exact solution method that provides extremely accurate dynamic responses even in the high-frequency region. In this study, we develop a spectral element model for a rectangular finite composite plate element. The present spectral element model is developed by modifying the boundary splitting method introduced in the previous studies of the authors. As a result, the four corner nodes of the rectangular finite composite plate element, which were inactive (fixed) in the previous studies, become active. Thus, the present spectral element model can be used as a generic type of finite element, which can be applied to any laminated composite plates with arbitrary boundary conditions. The accuracy and efficiency of the present spectral element model are evaluated by comparing its results with exact solutions and solutions using the commercial finite element analysis package, ANSYS. In addition, the vibration and wave characteristics are numerically investigated by varying the lay-ups of some examples of laminated composite plates with various geometries and boundary conditions.

A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

AbstractA nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

Local Multilevel Methods with Rectangular Finite Elements for the Biharmonic Problem

In this paper, some local multilevel methods are presented to solve the linear algebraic systems resulting from the application of adaptive conforming Bogner--Fox--Schmit rectangular element and nonconforming Adini rectangular element approximations to the biharmonic problem. The abstract Schwarz framework is applied to verify the uniform convergence of the local multilevel methods featuring Jacobi and Gauss--Seidel smoothing only on local nodes associated with the local refinements. By the abstract framework, a convergence estimate may also be derived from the stability of the space splitting and its strengthened Cauchy--Schwarz inequality. We demonstrate the optimality of the proposed algorithms by extensive numerical experiments.

Conforming shear-locking-free four-node rectangular finite element of moderately thick plate

AbstractAn outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

On Improved Thin Plate Bending Rectangular Finite Element Based on the Strain Approach

We propose in this paper the development of a new rectangular finite element for thin plate bending based on the strain approach with linear elastic behavior. An analytical integration is used to evaluate the element stiffness matrix. The present element possesses the three main degrees of freedom (d.o.f) per node, namely, one transverse displacement (w) and two normal rotations about x and y axis respectively (Ɵx, Ɵy). The proposed displacement field represents exactly the rigid body motion and satisfies the compatibility equations. The numerical results converges rapidly to the Kirchhoff solution for thin plates, this makes the present element robust, better suitable for computations, and particularly interesting in modeling this type of structures.

Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

In this paper, implicit–explicit multistep Galerkin methods are studied for two-dimensional nonlinear Schrödinger equations and coupled nonlinear Schrödinger equations. The spatial discretization is based on Galerkin method using linear and quadratic basis functions on triangular and rectangular finite elements. And the implicit–explicit multistep method is used for temporal discretization. Linear and nonlinear numerical tests are presented to verify the validity and efficiency of the numerical methods. The numerical results record that the optimal order of the error in L2 and L∞ norm can be reached.

Nonlinear Numerical Simulation of Reinforced Concrete Columns Under Cyclic Biaxial Bending Moment and Axial Loading

A nonlinear finite element algorithm is proposed to analyze the reinforced concrete (RC) columns subjected to cyclic biaxial bending moment and axial loading. In the proposed algorithm, the following parameters are considered: uniaxial behavior of concrete and steel elements, the pseudo-plastic hinge produced in the critical sections, and global behavior of the columns. In the proposed numerical simulation, the column is discretized into two macro-elements located between the pseudo-plastic hinges at critical sections and the inflection point. The critical sections are discretized into fixed rectangular finite elements. The basic equilibrium is justified over a critical hypothetical cross section assuming the kinematics Navier’s hypothesis with an average curvature. The method used qualifies as a “strain plane control process” that requires the resolution of a quasi-static simultaneous equation system using a triple iteration process over the strains in each section. To reach equilibrium, three main strain parameters (the strains in the extreme compressive point, the strains in the extreme tensile point and the strains in another corner of the section) are used as the three main variables. The proposed algorithm has been validated by the results of tests carried out on full-scale RC columns. The application of the components effects combination method is also compared with the proposed simultaneous direct method. The results obtained show the necessity of applying SDM for the post-elastic phase, which occurs frequently during earthquake loading.

Nonlinear Static-Oriented Pushover Analysis of Reinforced Concrete Columns Using Variable Oblique Finite-Element Discretization

A fast converging and fairly accurate nonlinear simulation method to assess the behavior of reinforced concrete columns subjected to static-oriented pushover force and axial loading (sections under biaxial-bending moment and axial loading) is proposed. In the proposed method, the sections of column are discretized into “Variable Oblique Finite Elements” (VOFE). By applying the proposed oblique discretization method, the time of calculation is significantly decreased, and since VOFE are always parallel to neutral axis, a uniform stress distribution along each oblique element is established. Consequently, the variations of stress distribution across an element are quite small which increases the accuracy of the calculations. In the discretization of section, the number of VOFE is significantly smaller than the number of “Fixed Rectangular Finite Elements” (FRFE). The advantages of using VOFE compared to FRFE are faster convergence and more accurate results. The nonlinear local degradation of materials and the pseudo-plastic hinge produced in the critical sections of the column are also considered in the proposed simulation method. A computer program is developed to calculate the local and global behavior of reinforced concrete columns under static-oriented pushover and cyclic loading. The proposed simulation method is validated by the results of tests carried out on the full-scale reinforced concrete columns. The application of the “Components Effects Combination Method” is compared with the proposed “Simultaneous Direct Method” (SDM). The obtained results show the necessity of applying SDM for nonlinear calculations. Especially, during the post-elastic phase, which occurs frequently during earthquake loading.

Design and Analysis of MEMS Linear Phased Array.

A structure of micro-electro-mechanical system (MEMS) linear phased array based on “multi-cell” element is designed to increase radiation sound pressure of transducer working in bending vibration mode at high frequency. In order to more accurately predict the resonant frequency of an element, the theoretical analysis of the dynamic equation of a fixed rectangular composite plate and finite element method simulation are adopted. The effects of the parameters both in the lateral and elevation direction on the three-dimensional beam directivity characteristics are comprehensively analyzed. The key parameters in the analysis include the “cell” number of element, “cell” size, “inter-cell” spacing and the number of elements, element width. The simulation results show that optimizing the linear array parameters both in the lateral and elevation direction can greatly improve the three-dimensional beam focusing for MEMS linear phased array, which is obviously different from the traditional linear array.

Streamline-diffusion method of a lowest order nonconforming rectangular finite element for convection-diffusion problem

The streamline-diffusion method of the lowest order nonconforming rectangular finite element is proposed for convection-diffusion problem. By making full use of the element’s special property, the same convergence order as the previous literature is obtained. In which, the jump terms on the boundary are added to bilinear form with simple user-chosen parameter δK which has nothing to do with perturbation parameter e appeared in the problem under considered, the subdivision mesh size hK and the inverse estimate coefficient μ in finite element space.

The Highest Superconvergence of the Tri-linear Element for Schr $$\ddot{\text {o}}$$ o ¨ dinger Operator with Singularity

In this paper, the eigenvalues for Schr$$\ddot{\text {o}}$$o¨dinger operator with singularity are analyzed. A special piecewise uniform rectangular partition is constructed and it has been proven that, under this partition, the tri-linear rectangular finite element method has the highest possible superconvergence rate for eigenvalue.

Modal Analysis of a Plate Compliant in Transverse Shear

A finite-element solution of the problem on natural vibrations of a plate compliant in transverse shear is considered. A four-node rectangular finite element whose basic nodal kinematic parameters include the angles of transverse shear deformations is used. A comparative analysis of frequencies and modes of natural vibrations of composite and sandwich plates is performed for two variants of boundary conditions on their contour: the classical clamping and clamping with free transverse shear deformations.

Nonlinear numerical simulation of RC columns subjected to cyclic oriented lateral force and axial loading

A nonlinear Finite Element (FE) algorithm is proposed to analyze the Reinforced Concrete (RC) columns subjected to Cyclic Loading (CL), Cyclic Oriented Lateral Force and Axial Loading (COLFAL), Monotonic Loading (ML) or Oriented Pushover Force and Axial Loading (OPFAL) in any direction. In the proposed algorithm, the following parameters are considered: uniaxial behavior of concrete and steel elements, the pseudo-plastic hinge produced in the critical sections, and global behavior of RC columns. In the proposed numerical simulation, the column is discretized into two Macro-Elements (ME) located between the pseudo-plastic hinges at critical sections and the inflection point. The critical sections are discretized into Fixed Rectangular Finite Elements (FRFE) in general cases of CL, COLFAL or ML and are discretized into Variable Oblique Finite Elements (VOFE) in the particular cases of ML or OPFAL. For pushover particular case, a fairly fast converging and properly accurate nonlinear simulation method is proposed to assess the behavior of RC columns. The proposed algorithm has been validated by the results of tests carried out on full-scale RC columns.

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.

Telescopic Hybrid Fast Solver for 3D Elliptic Problems with Point Singularities

Abstract This paper describes a telescopic solver for two dimensional h adaptive grids with point singularities. The input for the telescopic solver is an h refined two dimensional computational mesh with rectangular finite elements. The candidates for point singularities are first localized over the mesh by using a greedy algorithm. Having the candidates for point singularities, we execute either a direct solver, that performs multiple refinements towards selected point singularities and executes a parallel direct solver algorithm which has logarithmic cost with respect to refinement level. The direct solvers executed over each candidate for point singularity return local Schur complement matrices that can be merged together and submitted to iterative solver. In this paper we utilize a parallel multi-thread GALOIS solver as a direct solver. We use Incomplete LU Preconditioned Conjugated Gradients (ILUPCG) as an iterative solver. We also show that elimination of point singularities from the refined mesh reduces significantly the number of iterations to be performed by the ILUPCG iterative solver.

Transverse Vibration of the Thin Plates: Frequency-Domain Spectral Element Modeling and Analysis

It has been well known that exact closed-form solutions are not available for non-Levy-type plates. Thus, more accurate and efficient computational methods have been required for the plates subjected to arbitrary boundary conditions. This paper presents a frequency-domain spectral element model for the rectangular finite plate element. The spectral element model is developed by using two methods in combination: (1) the boundary splitting and (2) the super spectral element method in which the Kantorovich method-based finite strip element method and the frequency-domain waveguide method are utilized. The present spectral element model has nodes on four edges of the finite plate element, but no nodes inside. This can reduce the total number of degrees of freedom a lot to improve the computational efficiency significantly, when compared with the standard finite element method (FEM). The high solution accuracy and computational efficiency of the present spectral element model are evaluated by the comparison with exact solutions and the solutions by the standard FEM.

농촌지역 저층 조적조 건축물의 벽체 및 옹벽의 진동 안정 해석 - 전북 정읍시 ◯◯면 농촌지역 사례를 중심으로 -

This paper deals with vibration of plates with concentrated mass on elastic foundation. The object of investigating natural frequencies of tapered thick plate on pasternak foundation by means of finite element method and providing kinetic design data for mat of building structures. Free vibration analysis that tapered thick plate in this paper. Finite element analysis of rectangular plate is done by use of rectangular finite element with 8-nodes. In order to analysis plate which is supported on pasternak foundation. The Winkler parameter is varied with 10, , and the shear foundation parameter is 5, 10. This paper is analyzed varying thickness by taper ratio. The taper ratio is applied as 0.0, 0.25, 0.5, 0.75, 1.0. And the Concentrated Mass is applied as P1, Pc, P2 respectively.

Buckling Analysis of a Plate Compliant in Transverse Shear

A finite-element solution to the buckling of a plate compliant in transverse shear is considered. A rectangular finite element with four nodes is used. The basic kinematic variables include the angles of transverse shear strains. A comparative analysis of the load-carrying ability of composite and sandwich plates compressed along one of plate sides is performed. Two variants of boundary conditions on the plate contour are compared, namely the classical fixed contour and a fixed contour with free transverse shear deformations.

Study on Factors of Axial Crushing of Thin-Walled High-Strength Steel Sections Used in Passenger Car

This paper firstly studied the structure of front longitudinal, then reviewing comprehensive literature draw the similarity of collision of the thin-walled tube and front longitudinal,finally ensure the study of collision of the thin-walled tube. And establishing three kinds of thin-walled welded rectangular beam finite element model of spot welding, seam welding and laser welding, arrive a more superior technology through researching and analyzing the model.