Three dimensional magnetic field computation by a coupledvector-scalar potential method in brushless DC motors with skewedpermanent magnet mounts-the formulation and FE grids

A three dimensional finite element (3D-FE) method for the computation of global distributions of 3D magnetic fields in electric machines containing permanent magnets is presented. The formulation of this 3D-FE method including 3D permanent magnet modeling, which is based on a coupled magnetic vector potential-magnetic scalar potential (CMVP-MSP) approach, is given. The development of the necessary 3D-FE grids and algorithms for the application of the method to an example brushless DC motor, whose field is three dimensional due to the skewed permanent magnet mounts on its rotor, is also given here. A complete set of results of application of the method to the computation of the global 3D field distributions and associated motor parameters under no-load and load conditions are detailed in a companion paper     

Geometrically Non-Linear Bending Analysis of Piezoelectric Fiber-Reinforced Composite (MFC/AFC) Cross-Ply Plate Under Hygrothermal Environment

A finite element formulation for geometrically non-linear bending behavior of smart structures integrated with piezoelectric fiber-reinforced composite (AFC/MFC) layer acting as distributed actuator is presented in this article. The formulation of this hygro-thermo-electro-mechanical coupled problem is based on the first-order shear deformation theory (FSDT) and the von Kármán type geometric non-linearity. Resulting non-linear algebraic governing equations are linearized by Newton–Raphson iterative method. A developed finite element formulation is initially solved for validation and comparison with existing results. A wide variety of numerical examples considering cross-ply laminated substrates subjected to combined mechanical, hygrothermal and electrical loads are presented. The effect of piezoelectric fiber orientation in an actuator to counteract non-linear deflections is analyzed.     

Nonlinear Vector Potential Formulation and Experimental Verification of Newton-Raphson Solution of Three Dimensional Magnetostatic Fields in Electrical Devices

An iterative technique, based on magnetic vector potential formulation and the Newton-Raphson method, for the determination of the three dimensional magnetostatic field distributions in electrical devices is given. The proper degrees of magnetic saturation in the various materials within a given volume under consideration are obtained by repeated evaluation of the reluctivities in that volume, using a cubic spline representation of the B-H magnetization characteristics of composite materials (laminations). The formulation has been applied to a practical example of determining the field in and around a shell type 1.5 kva single phase transformer. The convergence and implementation characteristics of the developed method are given in this paper which show a saving of about 34% in CPU solution time in comparison with previously published methods. Experimental verification is given in terms of a comparison between computed and experimentally obtained values of flux densities surrounding the transformer core and winding, under heavily saturated conditions. Excellent agreement between test and calculated flux densities was achieved. This method is thus quite applicable to the solution of a wide class of three dimensional magnetostatic field problems associated with electrical apparatus.     

Linear Unsteady Aerodynamic Forces on Vibrating Annular Cascade Blades

The paper presents the formulation to compute numerically the unsteady aerodynamic forces on the vibrating annular cascade blades. The formulation is based on the finite volume method. By applying the TVD scheme to the linear unsteady calculations, the precise calculation of the peak of unsteady aerodynamic forces at the shock wave location like the delta function singularity becomes possible without empirical constants. As a further feature of the present paper, results of the present numerical calculation are compared with those of the double linearization theory (DLT), which assumes small unsteady and steady disturbances but the unsteady disturbances are much smaller than the steady disturbances. Since DLT requires far less computational resources than the present numerical calculation, the validation of DLT is quite important from the engineering point of view. Under the conditions of small steady disturbances, a good agreement between these two results is observed, so that the two codes are cross-v     

Shell analysis of thin-walled pipes. Part II – Finite element formulation

A finite element formulation is developed for the analysis of thin-walled pipes based on thin shell theory. The formulation starts with a Fourier series solution of the equilibrium equations developed in a companion paper and develops a family of exact shape functions for each mode. The shape functions developed are used in conjunction with the principle of stationary potential energy and yield a finite element that is exact within the assumptions of the underlying shell formulation. The stiffness matrix contribution for each mode n is observed to be fully uncoupled from those based on other modes m ≠ n. The resulting finite element is shown to be free from discretization errors normally occurring in conventional finite elements. The applicability of the solution is illustrated through examples with various loading cases and boundary conditions. A comparison with other finite element and closed form solutions demonstrates the validity and accuracy of the current finite element.     

Finite element method formulation in polar coordinates for transient heat conduction problems

The aim of this paper is the formulation of the finite element method in polar coordinates to solve transient heat conduction problems. It is hard to find in the literature a formulation of the finite element method (FEM) in polar or cylindrical coordinates for the solution of heat transfer problems. This document shows how to apply the most often used boundary conditions. The global equation system is solved by the Crank-Nicolson method. The proposed algorithm is verified in three numerical tests. In the first example, the obtained transient temperature distribution is compared with the temperature obtained from the presented analytical solution. In the second numerical example, the variable boundary condition is assumed. In the last numerical example the component with the shape different than cylindrical is used. All examples show that the introduction of the polar coordinate system gives better results than in the Cartesian coordinate system. The finite element method formulation in polar coordinates is valuable since it provides a higher accuracy of the calculations without compacting the mesh in cylindrical or similar to tubular components. The proposed method can be applied for circular elements such as boiler drums, outlet headers, flux tubes. This algorithm can be useful during the solution of inverse problems, which do not allow for high density grid. This method can calculate the temperature distribution in the bodies of different properties in the circumferential and the radial direction. The presented algorithm can be developed for other coordinate systems. The examples demonstrate a good accuracy and stability of the proposed method.     

Transient heat flow analysis using the fundamental collocation method

The fundamental collocation method for general two dimensional transient heat flow problems in solids is presented. The numerical procedure is applied directly to heat conduction in the time domain and is illustrated using several example problems. The method is capable of handling problems of arbitrary shapes subjected to arbitrary initial conditions and mixed time-dependent boundary conditions. Its simplicity and minimal computational effort combined with accuracy and stability of the results make the method an excellent alternate approach to domain methods such as the finite difference and finite element methods.     

Modal analysis of shell of revolution on elastic foundation

Static and dynamic analyses of the shell of revolution on elastic foundation have been analysed by a finite element method based on variational formulation. The general loading consists of laterally distributed forces and ring forces on the meridian and normal to the neutral plane directions and moments. The thermal load is also assumed to act in the meridian direction and the thermal gradient through the thickness. The elastic foundation is assumed to be either in distributed form extending over the entire surface of the element or in ring form applied at a nodal point. The dynamic solution of the resulting finite element equation is obtained using the theoretical modal analysis technique based on mode displacement and mode acceleration methods.     

Wave packet enriched finite element for generalized thermoelasticity theories for thermal shock wave problems

A unified enriched finite element (FE) formulation for two generalized thermoelsaticity theories is developed for the transient thermal shock problems in one and two dimensional domains. Both the displacement and temperature field interpolations are enriched with harmonic functions defined in the local element coordinates. The coupled field finite element equations are derived using the generalized Hamilton’s principle and solved directly in time domain using the standard Newmark-β time integration technique as opposed to the transform techniques usually adopted for thermal shock problems. The method is assessed in comparison with the Laplace transform based analytical solutions and FE solutions with dynamic meshing available in the literature. It is shown that the present solution with a static uniform mesh captures the sharp discontinuities in the temperature and displacement fields and the wave properties of heat conduction very accurately, practically eliminating the severe drawbacks of the conventional FE solutions such as the spurious undulations and flattening out, while maintaining better computational e?ciency.     

Development of DKT Finite Element Formulation for Thermal Bending of Thin Plate

The finite element formulation for thermal bending analysis of a thin plate due to the temperature gradient through its thickness is presented. The formulation is developed for the Discrete Kirchhoff Triangle (DKT) element to analyze the plate bending behavior under the thermal loading. The finite element load vector is derived in closed-form expressions so that the numerical integration is not required. Solution accuracy of the developed DKT finite element is evaluated by several examples. Performance of the developed DKT finite element is also compared with the effective nonconforming triangular element (BCIZ element). Results show that the developed DKT finite element formulation provides high solution accuracy for analysis of plate bending problem under thermal loading.     

Finite element solution of radiative transfer across a slab with variable spatial refractive index

To avoid the complicated computation of ray trajectories, a finite element formulation is developed to solve the radiative transfer problem in a one-dimensional absorbing-emitting-scattering semitransparent slab with variable spatial refractive index. A problem of radiative equilibrium is taken as an example to verify this finite element formulation. The predicted temperature distributions are determined by the proposed method and compared with the data in references. The results show that the finite element formulation presented in this paper has good accuracy in solving the radiative transfer in one-dimensional absorbing-emitting-scattering semitransparent medium with variable spatial refractive index.     

An efficient algorithm for phase change problem in tumor treatment using αFEM

Cryosurgery is an effective treatment for killing tumor tissues. During the cryosurgery process, a phase transformation occurs in the undesired tissue. The most popular numerical method for simulation of phase change problem is fixed grid methods where the latent heat is function of temperature. In this paper, a fixed grid method using the alpha finite element (αFEM) formulation is presented to simulate the phase transformation and temperature field during the cryosurgery process for liver tumor treatment. The αFEM model is first established and tuned to have a close-to-exact stiffness compared with the standard finite element method (FEM). Three examples of liver tumor treatment including the single probe for regular shape of tumor, and multiple probes for regular and irregular shape of tumor are presented. The numerical results using alpha finite element method have demonstrated the effectiveness of the procedure.     

基于参数化建模的转子有限元剖分

分析汽轮机转子的结构特点,将汽轮机转子结构参数化,提出了汽轮机转子参数化建模的方法,提高了转子原始几何模型的输入效率和准确性。利用有限元网格生成的Delaunay三角化方法,得到了汽轮机转了二维有限元计算模型,生成的有限元网格大小均匀,疏密过渡平滑,没有奇异单元,从而保证了转子温度场和热应力有限元分析计算的精度。     

A FULL 3-DIMENSIONAL ADAPTIVE FINITE VOLUME SCHEME FOR TRANSPORT AND PHASE-CHANGE PROCESSES,PART II: APPLICATION TO CRYSTAL GROWTH

Application of the full three-dimensional (3-D) adaptive finite volume scheme as presented in a companion paper (Part I) to hydrothermal and two variants of Czochralski (Cz) crystal growth processes is presented. A composite fluid-superposed porous layer formulation with non-Darcian flow model is employed to model the hydrothermal system to demonstrate the versatility of this formulation. The unified formulation is then employed to model the Cz process, which is the main focus here. The model takes into account the irregular geometries, nonplanar and moving interfaces, and multiple driving forces that typically characterize a generic class of crystal growth processes. Simulation of the low and high pressure Cz growth of silicon and indium phosphide, respectively, reveal complex three dimensionality in flow, heat transfer, and interface dynamics, which have a profound effect on other quality affecting phenomena such as defect generation, segregation, and so on.     

BeamDyn: a high‐fidelity wind turbine blade solver in the FAST modular framework

This paper presents a numerical implementation of the geometrically exact beam theory based on the Legendre‐spectral‐finite‐element (LSFE) method. The displacement‐based geometrically exact beam theory is presented, and the special treatment of three‐dimensional rotation parameters is reviewed. An LSFE is a high‐order finite element with nodes located at the Gauss–Legendre–Lobatto points. These elements can be an order of magnitude more computationally efficient than low‐order finite elements for a given accuracy level. The new module, BeamDyn, is implemented in the FAST modularization framework for dynamic simulation of highly flexible composite‐material wind turbine blades within the FAST aeroelastic engineering model. The framework allows for fully interactive simulations of turbine blades in operating conditions. Numerical examples are provided to validate BeamDyn and examine the LSFE performance as well as the coupling algorithm in the FAST modularization framework. BeamDyn can also be used as a stand‐alone high‐fidelity beam tool. Copyright © 2017 John Wiley & Sons, Ltd.     

A one dimensional discrete approach for the determination of the cross sectional properties of composite rotor blades

This work presents the theoretical aspects of a one dimensional computational approach for the determination of the cross sectional stiffness of composite rotor blades. The method is based on a vector variant of the classical lamination theory embedded into a geometrically exact large deformation-small strain thin-walled beam formulation, which is naturally oriented to multibody problems. The procedures rely on a one-dimensional discretization of the aerodynamic profile of the blade; this generates groups of finite segments of composite laminates which are assembled to find the stiffness properties of the blade cross section. The formulation accounts for warping and transverse shear; the warping problem is solved numerically by means of a one dimensional finite element formulation. The numerical tests show that the formulation gives very accurate results.     

Design analysis of permanent magnet DC motors with differingarmature, magnet and yoke lengths

A modified magnetic circuit method and a 2-D finite element procedure are presented here for the analysis of DC permanent magnet motors with differing armature, magnet and yoke lengths including nonlinear material behavior of steel and permanent magnet. A 12 V, 120 W motor is analyzed through these methods and compared with experimental results of a prototype sample. In case of finite element analysis, commutation effects and short chording of the armature winding are treated appropriately, and cogging torque and voltage ripple are predicted     

基于解析位移形函数的改进风力机叶片梁单元

为提高空间梁单元在风力机叶片结构计算方面的精度,提出一种基于解析位移形函数的改进空间梁单元.首先考虑叶片截面偏心的影响,从空间Timoshenko梁的平衡方程出发,构建弯曲位移的解析解插值函数.根据弯曲位移与横向位移和剪切变形的关系,得到横向位移和剪切角的形函数.采用有限元理论,得到改进风力机叶片梁单元的刚度矩阵和质量...     

Control of macrosegregation during the solidification of alloys using magnetic fields

Numerical modeling of convection damping and macrosegregation suppression during solidification of alloys with prominent mushy zones through the use of tailored magnetic fields is demonstrated here. Macrosegregation leads to commonly observed defects such as freckles, channels and segregates in cast alloys that severely affect the performance and suitability of the alloy for further applications. The current work demonstrates the successful use of magnetic fields in suppressing thermosolutal convection and eliminating some of these defects in solidifying metallic alloys. The computational model presented utilizes volume-averaged governing transport equations and stabilized finite element techniques to discretize these equations. A finite-dimensional optimization problem, based on the continuum sensitivity method is considered to design the time history of the imposed magnetic field required to effectively damp convection. The coefficients that determine this time variation are the main design parameters of this optimization problem. Continuum sensitivity equations are derived by design-differentiating the governing equations of the direct problem. The cost functional here is given by the square of the L₂ norm of an expression representing the deviation of the volume-averaged velocity corresponding to conditions of convection less growth. The cost functional minimization process is realized through a non-linear conjugate gradient algorithm that utilizes finite element solutions of the continuum direct and sensitivity problems. Design of the time history of the imposed magnetic field is highlighted through different examples with the main objective being the suppression of convection and macrosegregation during alloy solidification.     

Tubular linear induction motor for hydraulic capsule pipeline. I.Finite element analysis

An axisymmetric model of the magnetic field of the tubular linear induction motor (TLIM) for application to hydraulic capsule pipelines is developed using finite element method (FEM) analysis. The FEM model is used to analyze a specific TLIM design at standstill for a given supply current. The finite element formulation of the field equations is discussed and the magnetic field contours at different instants of time are presented. FEM computation of the thrust compares very well with experimental data