Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.

SELF-ADAPTIVE STRATEGY FOR ONE-DIMENSIONAL FINITE ELEMENT METHOD BASED ON ELEMENT ENERGY PROJECTION METHOD

Based on the newly-developed element energy projection （EEP） method for computation of super-convergent results in one-dimensional finite element method （FEM）, the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

Implementation of a particle-in-cell method for the energy solver in 3D spherical geodynamic modeling

The thermal evolution of the Earth’s interior and its dynamic effects are the focus of Earth sciences.However,the commonly adopted grid-based temperature solver is usually prone to numerical oscillations,especially in the presence of sharp thermal gradients,such as when modeling subducting slabs and rising plumes.This phenomenon prohibits the correct representation of thermal evolution and may cause incorrect implications of geodynamic processes.After examining several approaches for removing these numerical oscillations,we show that the Lagrangian method provides an ideal way to solve this problem.In this study,we propose a particle-in-cell method as a strategy for improving the solution to the energy equation and demonstrate its effectiveness in both one-dimensional and three-dimensional thermal problems,as well as in a global spherical simulation with data assimilation.We have implemented this method in the open-source finite-element code CitcomS,which features a spherical coordinate system,distributed memory parallel computing,and data assimilation algorithms.

COARSE-MESH-ACCURACY IMPROVEMENT OF BILINEAR Q_4-PLANE ELEMENT BY THE COMBINED HYBRID FINITE ELEMENT METHOD

The combined hybrid finite element method is of an intrinsic mechanism of enhancing coarse-mesh-accuracy of lower order displacement schemes. It was confirmed that the combined hybrid scheme without energy error leads to enhancement of accuracy at coarse meshes, and that the combination parameter plays an important role in the enhancement. As an improvement of conforming bilinear Q(4)-plane element, the combined hybrid method adopted the most convenient quadrilateral displacements-stress mode, i.e.,the mode of compatible isoparametric bilinear displacements and pure constant stresses. By adjusting the combined parameter, the optimized version of the combined hybrid element was obtained and numerical tests indicated that this parameter-adjusted version behaves much better than Q(4)-element and is of high accuracy at coarse meshes. Due to elimination of stress parameters at the elemental level, this combined hybrid version is of the same computational cost as that of Q(4)-element.

SPACE-TIME FINITE ELEMENT METHOD FOR SCHRDINGER EQUATION AND ITS CONSERVATION

Energy conservation of nonlinear Schrodinger ordinary differential equation was proved through using continuous finite element methods of ordinary differential equation; Energy integration conservation was proved through using space-time continuous fully discrete finite element methods and the electron nearly conservation with higher order error was obtained through using time discontinuous only space continuous finite element methods of nonlinear Schrodinger partial equation. The numerical results are in accordance with the theory.

Mathematical analysis of EEP method for one-dimensional finite element postprocessing

For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.

NONLINEAR QUASI-CONFORMING FINITE ELEMENT METHOD

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.

Continuous finite element methods for Hamiltonian systems

By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.

Finite Element Analysis on a Square Canister Piezoelectric Energy Harvester in Asphalt Pavement

A novel square canister piezoelectric energy harvester was proposed for harvesting energy from asphalt pavement. The square of the harvester was of great advantage to compose the harvester array for harvesting energy from the asphalt pavement in a large scale. The open circuit voltage of the harvester was obtained by the piezoelectric constant d33 of the piezoelectric ceramic. The harvester is different from the cymbal harvester which works by the piezoelectric constant d31. The finite element model of the single harvester was constructed. The open circuit voltage increased with increase of the outer load. The finite element model of the single harvester buried in the asphalt pavement was built. The open circuit voltage, the deformation difference percent and the stress of the ceramic of the harvester were obtained with different buried depth. The open circuit voltage decreased when the buried depth was increased. The proper buried depth of the harvester should be selected as 30 - 50 mm. The effects of structure parameters on the open circuit voltage were gotten. The output voltage about 64.442 V could be obtained from a single harvester buried under 40 mm pavement at the vehicle load of 0.7 MPa. 0.047 mJ electric energy could be gotten in the harvester. The output power was about 0.705 mW at 15 Hz vehicle load frequency.

Stiffness Evaluation of the Welded Connection between Guide Thimbles and the Spacer Grids for 16 × 16 Fuel Assemblies Types, Using the Finite Element Method

The present work aims to evaluate the increase in the number of spot welds in the 16 × 16 type fuel assembly structure that connects guide thimbles and spacer grids, in order to provide a proper joint for this connection. This new and improved process can provide more stiffness to the whole structure, since the number of spots raised from four to eight. A 3-D geometric model of a guide thimble section was generated in a CAD （computer aided design） program （SolidWorks）. After that, the geometric model was imported to a CAE （computer aided engineering） program （ANSYS Mechanical APDL, Release 14.0）, where the finite element model was built, considering the guide thimble geometry assembled with the spacer grid through the welded connections. Boundaries conditions were implemented in the model in order to simulate the correct physical behavior due to the operation of the fuel assembly inside the reactor. The analysis covered specific loads and displacements acting on the entire structure. The method used to solve this finite element analysis was a linear static simulation in order to perform the connection between a spacer grid cell and a guide thimble section. Hence, four models was evaluated, differing on the spot weld number in the spacer grid and guide thimble connection. The rotational stiffness results of each model were compared. The results acquired from four and eight spot weld were validated with physical test results. The behavior of the structure under the acting force/displacement and the related results of the analysis, mainly the stiffness, were satisfied. The results of this analysis were used to prove that the increasing spot welds number is an improvement in the dimensional stability when submitted to loads and displacements required on the fuel assembly design. This analysis aid to get more information of extreme importance such as, the pursuance to develop better manufacturing process and to improve the fuel assembly performance due to the increasing of the bum-up.

Finite element analysis of stress and strain distributions in InAs/GaAs quantum dots

In this paper, we perform systematic calculations of the stress and strain distributions in InAs/GaAs truncated pyramidal quantum dots （QDs） with different wetting layer （WL） thickness, using the finite element method （FEM）. The stresses and strains are concentrated at the boundaries of the WL and QDs, are reduced gradually from the boundaries to the interior, and tend to a uniform state for the positions away from the boundaries. The maximal strain energy density occurs at the vicinity of the interface between the WL and the substrate. The stresses, strains and released strain energy are reduced gradually with increasing WL thickness. The above results show that a critical WL thickness may exist, and the stress and strain distributions can make the growth of QDs a growth of strained three-dimensional island when the WL thickness is above the critical value, and FEM can be applied to investigate such nanosystems, QDs, and the relevant results are supported by the experiments.

Simulated Experiments for Teaching CAD Techniques Using Analytic and Finite Element Solutions of Electromagnetic Two-Dimensional Problems with Longitudinal Symmetry

The paper describes an approach to teaching low-frequency electromagnetic CAD techniques to undergraduate students pursuing a degree course in electrical engineering. The simulated experiments make use of a two-dimensional open-access software based on the finite-element method. At the laboratory meetings, the problems are initially solved analytically. Upon this, students learn how to create the numeric model and how to define the sequence of field problems that lead to the required solution. Simulation tasks based on a force-producing electromagnet are used to introduce numeric techniques to determine magnetic field distribution, evaluation of energy storage and generation of magnetic forces. The nature of the magnetic force generated in the air gaps of the C-core electromagnet is explained in detail. Magnetic forces are calculated by the classical and weighted versions of the method of Maxwell stress tensor. The paper provides all the basic elements required for further exploration of devices with longitudinal symmetry.

Energy density response prediction of structures with uncertainty using interval analysis method

Prediction of vibration energy responses of structures with uncertainties is of interest in many fields. The energy density control equation for one-dimensional structure is provided firstly. Interval analysis method is applied to the control equation to obtain the range of energy density responses of structures with interval parameters. A cantilever beam with interval-valued damping coefficient is exemplified to carry out a simulation. The result shows that the mean value of energy density from the interval analysis method is the same as that from a probabilistic method which validates the interval analysis method. Besides, the response range from the interval analysis method is wider and includes that from the probabilistic method which indicates the interval analysis method is a more conservative method and is safer in realistic engineering structures.

Force-Based Quadrilateral Plate Bending Element for Plate Using Large Increment Method

A force-based quadrilateral plate element( 4NQP13) for the analysis of the plate bending problems using large increment method( LIM) was proposed. The LIM, a force-based finite element method( FEM),has been successfully developed for the analysis of truss,beam,frame,and 2D continua problems. In these analyses,LIMcan provide more precise stress results and less computational time consumption compared with displacement-based FEM. The plate element was based on the Mindlin-Reissner plate theory which took into account the transverse shear effects.Numerical examples were presented to study its performance including accuracy and convergence behavior,and the results were compared with the results have been obtained from the displacementbased quadrilateral plate elements and the analytical solutions. The4NQP13 element can analyze the moderately thick plates and the thin plates using LIMand is free from spurious zero energy modes and free from shear locking for thin plate analysis.

FINITE ELEMENT ANALYSIS OF TRASINENT TWO DIMENSIONAL CHANNEL FLOW WITH A SUDDEN EXPANSION

The finite element method is employed to simulate incompressible viscous unsteady flow in two dimensional channel with a sudden expansion. The streamline patterns and the velocity vector plots are presented at different expansion ratios and at different moment. The results obtained have certain significance to analyze the formation of eddies and energy loss in a sudden expansion channel flow and other complex channel flow.

Finite Element Modeling of Solar Powered Injera Baking Oven for Indoor Cooking

In this paper, a 2D transient finite element analysis was carried out for a new type of solar powered injera baking system. In the proposed system （currently under development）, heat transfer oil is heated using solar energy by parabolic trough and the oil circulates through the space below the baking pan in the kitchen. Based on previous finite element study on existing electric injera baking pans, a new type of baking pan made from ceramic with 8 mm thickness was manufactured and used for the proposed system. The model was further extended to study the heat up time and temperature distributions during initial heat up and cyclic baking of the new model. The proposed baking pan that uses solar energy gives acceptable heat up and baking time compared to existing conventional baking methods. Generally, the finite element model predicts well the temperature distributions during initial heat up and cyclic baking.

基于EFEM的5R机械臂刚柔耦合动力学分析

为分析复杂机械系统动力学特性,将系统划分为含有限刚性构件的刚柔耦合系统,基于等效元素集成法,对构件自身的动力学特性进行等效,组建成等效系统。将建模的关键归结为系统中二阶转换张量和相应的三阶转换张量的求解以及系统质量阵的组装,然后,按多柔体动力学的方式建立方程。建立了港口起重机5R机械臂刚柔耦合动力学模型,给出了动力响应曲线,并与刚性解进行了比较。结果表明,采用EFEM方法,动力学方程维数减少,系统质量阵阶数降低,模型建立简单快捷,规范化、程式化,解题效率和精度较高。

Modeling and power performance improvement of a piezoelectric energy harvester for low-frequency vibration environment

A novel oscillator structure, bimorph piezoelectric cantilever beam with two-stepped variable thicknesses,is proposed to improve the energy harvestingperformance of the vibration energy harvester (VEH) under low-frequency vibration environment. Firstly, the piezoelectric cantilever is segmented to obtain the energy functions based on the Euler-Bernoulli beam assumptions, and the Galerkin approach is utilized to discretize the energy functions. Applying boundary conditions and continuity conditions enforced at separation locations, the electromechanical coupled governing equations for the piezoelectric energy harvesterareintroduced by means of the Lagrange equations. Furthermore, the steady state response expressions are obtained for harmonic base excitations at arbitrary frequencies. Numerical results are computed and the effects ofthe lengths-ratio, thicknesses-ratio,end thicknessand load resistance on the output voltage, harvested power and power density are discussed. Moreover, to verify thecorrectness ofanalytical results, the finite element method (FEM)simulationis also conducted to analyze performance of the proposed VEH, where a good agreement is presented. All the results show thatthe present oscillator structureis moreefficient than the conventional uniform beam structure, specifically, for vibration energy harvesting in low-frequency environment.

Optimal Design of Multi-channel Water Cooled Radiator for Motor Controller of New Energy Vehicle

In order to improve the heat dissipation capability of motor controller for new energy vehicles,the water cooled radiator with multiple channels is optimized in this paper.The heat conduction between the heat source IGBT and the radiator,the convective heat transfer between the radiator and the coolant,the mechanical strength and the manufacturing cost are comprehensively considered during the optimization process.The power loss and thermal resistance of the IGBT unit are calculated at first,and finite element model of the radiator is established.On this basis,multi-physics coupling analysis of the water cooled radiator is carried out.Secondly,the sensitivity analysis is applied to verify the influence of structural parameters on the heat dissipation performance of the radiator system.The influence of coolant inlet velocity v,number of cooling ribs n,height of radiator ribs H on the maximum temperature rise T,the temperature difference ΔT between phase U and W,and the coolant pressure lossΔP are analyzed in depth,and the optimal range of the structural parameters for heat dissipation is obtained.Finally,an experimental platform was set up to verify the performance of the proposed structure of water cooled radiator for motor controller of new energy vehicle.The results show that the heat dissipation capability of the proposed radiator is improved compared with the initial design.