Time-stepping Finite Element Method Research Articles

An analysis of second-order sav-filtered time-stepping finite element method for unsteady natural convection problems

This paper presents an unconditionally stable time-filtering algorithm for natural convection equations. The algorithm is based on the scalar auxiliary variables in the exponential function and adopts a completely discrete Back-Euler combining time filter scheme. The proposed scheme requires minimal invasive modification of the existing program to improve the time accuracy from first-order to second-order without increasing the computational complexity, and we demonstrate the unconditional stability of the proposed algorithm and analyze its second-order convergence. In addition, due to the increasing demand for low-memory solvers, the application of a time-adaptive algorithm can improve the accuracy and efficiency of the proposed algorithm, so we extend the method to variable step sizes and construct an adaptive algorithm. Finally, the effectiveness of the proposed method and the accuracy of the theoretical results are verified by numerical experiments.

Analysis of DC bias characteristics of transformer by using fixed-point time-step FEM and dynamic J–A hysteresis model

The injection of the Direct Current (DC) bias component will lead to oversaturation of the transformer core and affect its safe operation. In this paper, the vector magnetic potential A and winding current I are used as the variables solved in the time-step finite element method to simulate the nonlinear transient magnetization characteristics of the transformer under DC bias, and the fixed-point reluctivity is introduced to deal with the nonlinear iterative problem. The field-circuit coupling relationship is established through the electromagnetic induction law, and the two-dimensional time-step finite element model of the transformer core is found. The dynamic J–A hysteresis model is used to deal with the hysteresis effect of iron cores. The fixed-point reluctivity of difference form is introduced to solve the problem of uneven transition and multi-value of magnetization curve with hysteresis in the iterative process. The determination scheme for the equivalent thickness of the core model and the initial iterative value are discussed. A laminated core transformer is made for experiments, and the accuracy and applicability of the model proposed in this paper are verified by comparing the calculation and experimental results. The influence of different voltages and different DC bias components on the electromagnetic characteristics of the transformer is analyzed.

Eddy Effect and Dynamic Response of High-Speed Solenoid Valve with Composite Iron Core.

To alleviate the Eddy effect of the high-speed solenoid valve (HSV) and improve its dynamic response speed, a novel HSV with a composite iron core is presented. The time-step finite element method is used to establish and verify the numerical simulation of HSV coupling multiple physical fields. Then, the Eddy effect and dynamic response characteristics of the conventional and composite HSVs are further compared and analyzed. The results showed that the Eddy current loss in the main pole was the largest for the conventional HSV, accounting for 72.5% and 64.4% in the actuation and release processes, respectively. It was found that the Eddy effect of the composite HSV was obviously weakened, and the total Eddy current losses in the actuation and release processes were reduced by 58.8% and 38.7%, respectively. Meanwhile, the actuation response time and release response time of the composite HSV were shortened by 15.6% and 18.5%, respectively. In addition, increasing the peak voltage further shortened the actuation response time of the composite HSV, but had no significant effect on the response time of the conventional HSV.

Analysis of Core Loss Characteristics of Linear Phase-Shifting Transformer

Linear phase-shifting transformer, as a new type of phase-shifting transformer, has the advantages of shifting phase at any angle, good expansibility, and easy modularization compared with the traditional core column phase-shifting transformer. To calculate the output performance such as the efficiency of linear phase-shifting transformer, the loss of linear phase-shifting transformer should be studied. The loss of linear phase-shifting transformer includes copper loss and core loss. Among them, the core loss has an important impact on linear phase-shifting transformer performance parameters such as efficiency and temperature rise, but there is a lack of relevant research at present. In this paper, based on the traditional separation model of ferromagnetic materials, a calculation method for linear phase-shifting transformer core loss is proposed. In this method, the loss coefficient of the linear phase-shifting transformer iron loss model is calculated by the least square method based on the calculation and analysis of the magnetic field law of the core by the time-step finite element method, and the loss coefficient is further modified by considering the influence of local hysteresis loop, non-sinusoidal magnetic field, and harmonic magnetic field. The linear phase-shifting transformer model based on finite element field-road coupling simulation is established, a low-power experimental prototype is designed, and a multi-stack inverter system platform is built. The theoretical calculation is verified by simulation and experiment. Simulation and experimental results demonstrate the accuracy of the core loss calculation method presented in this paper.

An Electromagnetic Design of Slotless Variable Reluctance PM-Resolver

This paper proposes a novel design of the variable reluctance permanent magnet (VRPM) resolvers. It utilizes a slotless stator to reduce the resolver volume. The proposed design overcomes the complex structure of windings by replacing them with magnetic flux measurement units (MFMU). The new design eliminates the computationally-expensive demodulation process of output signals by relying on none-modulated signals. An analytical model based on the nonlinear magnetic equivalent circuit (MEC) method is used to extract the output signals of the proposed resolver design. The accuracy of applied analytical model is verified against the time-stepping finite-element method (TSFEM). An optimization is implemented to best improve the resolver performance by saturating the stator ribs targeting a minimum possible PM volume. The design aims high flux density of iron to make the resolver robust against external distortions, but at the same time, the flux density of core is kept within the linear region of the B-H curve to avoid saturation. TSFEM is used to further analyze the performance of the optimal slotless VRPM-Resolver in details. The simulation results are verified by experimental tests on a prototyped resolver design.

Analysis of a Filtered Time-Stepping Finite Element Method for Natural Convection Problems

In this paper, we present, analyze, and test a novel low-complexity time-stepping finite element method for natural convection problems utilizing a time filter (TF). First, via a TF to postprocess the solutions of backward Euler (BE) schemes, we make a minimally intrusive modification to the existing codes to improve the time accuracy by one order. This also provides, at no extra complexity, an estimate of the temporal error,which is easy to construct a novel adaptive algorithm. Additionally, the TF can remove the overdamping of the BE scheme while remaining unconditionally energy stable. Hence, this paper addresses the question, how can one improve the time accuracy without increasing computational and cognitive complexity? Then long time stability and error estimates of BE plus time filter (BETF) with constant time stepsize are proved. Moreover, we construct adaptive algorithms by extending the approach to variable time stepsize, and we extend the methods to higher order algorithms. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

A Novel Structure of an Eddy Current Sensor for Speed Measurement of Rotating Shafts

This paper presents a novel structure of an eddy current sensor for the measurement of rotational speed. The sensor has a perpendicular configuration of the excitation coil and the pick-up coil. Its maximum nonlinearity error is 0.15%. The excitation coils and the pick-up coils are divided into two identical parts with a coil span of 180 degrees. Various solid iron cylinders with different electrical and magnetic characteristics, one solid aluminum cylinder and one solid copper cylinder, were used as the rotating shafts. A novel 2D analytical method in cylindrical coordinates has been developed for fast parametric analysis of this sensor. 2D and 3D time-stepping finite-element methods that take into consideration the shaft motion are used for a performance analysis of the eddy current sensor. The measurements of the speed sensor are performed at various frequencies and speeds and are compared with the calculations for detailed analysis. The effects of the rotating shaft materials are evaluated both in the measurements and in the calculations.

A Normalized Terzaghi Model and Time-Step FEA for Predicting the Adsorption of a Cylindrical Object in Subsea Salvage

This paper proposes a normalized Terzaghi model modified based on finite element analysis to predict the adsorption force of a cylindrical object for salvage from the seabed. The maximum relative error is less than 5% compared with finite element analysis. Furthermore, the time-step finite element method is adopted to analyze the effects of the lifting force and bury depth. With increased lifting force, the critical displacement is reduced slightly, soil separation on the bottom of the object occurs earlier, and the velocity increases more quickly at the same burial depth. In addition, the soil displacement on the bottom stops increasing earlier, and the off-mud process is completed earlier. With increased burial depth, soil separation takes considerably longer, velocity increases more slowly, the maximum soil displacement is increased, and the off-mud process takes longer to complete.

Speed Measurement of Rotating Hollow Shafts Using an Eddy Current Sensor

Rotating speed measurements of conductive hollow shafts using a new eddy current sensor are presented in this article. Two serially connected excitation coils are used, which are 180° geometrically shifted relative to each other. The pick-up coils have the same configuration as the excitation coils with a 90° geometrically shifted angle relative to the excitation coils. Parametric analyses of the speed sensor are performed using a novel developed analytical method. Detailed calculations of the sensor are presented using the 3-D time-stepping finite-element method. The performance of the eddy current speed sensor is measured up to 3000 r/min at different excitation frequencies. The rotating shafts are two hollow iron cylinders and three hollow aluminum cylinders with different shell thicknesses. The effects of the electrical conductivity and magnetic permeability of hollow shafts on the speed sensor performance are assessed. The measurement results are compared with the calculations for different shafts. The nonlinearity measured error is 0.15%.

Numerical Investigations on Operation Modes and Transients of IPM Machines with Dual Windings

This paper investigates the flux-weakening characteristics and transient response of a dual winding interior permanent magnet (DWIPM) machine designed using the 2010 Toyota Prius IPM Machine specifications. In order to thoroughly adjust the main flux, dual windings are fed by separate identical inverters. Thus, significantly high-torque for high-speed requirements can be simply achieved by making one of the inverters open-circuited. Transient response of the DWIPM machine, including torque and current pulsations, occurring during normal drive or deactivation of one of the inverters has also been investigated for different operation modes. The flux-weakening performance characteristics of the proposed DWIPM machine have been compared with those of its single-fed counterpart in order to highlight the benefits and drawbacks of the dual-winding topology. The steady-state performance characteristics, torque/speed and power/speed curves, efficiency maps, machine transient response, different machine design options with various numbers of turns per phase and current amplitudes, and the importance of the electromagnetic coupling between dual windings have all been addressed. For transient and steady-state analyses, 2D, nonlinear, time-stepping finite element method (FEM) has been employed. It has been revealed that the proposed DWIPM machine exhibit significantly improved flux-weakening characteristics, particularly high-power at constant power region and high efficiency at constant torque region, and quite low current and torque pulsations occurring during operation mode change.

Experimental study on the influence of high frequency PWM harmonics on the losses of induction motor

For the purpose of studying the influence of high frequency PWM harmonics on the losses of induction motor, this paper first introduces the latest international standard IEC/TS 60034-2-3 for the separation of the losses and the measurement of the efficiency of induction motor under PWM power supply. Then, considering that the skin effect of windings in induction motors under the condition of PWM power supply more apparent, this paper uses the stator AC resistance to replace the stator DC resistance in the IEC/TS 60034-2-3 standard to obtain more accurate separation results. On this basis, an improved method of loss separation for converter-fed induction motor is proposed by using the accurate time-step finite element method to calculate the rotor copper loss under no-load condition which cannot be considered in IEC/TS 60034-2-3 standard. Finally, this method is used to measure and separate the losses of a 5.5 kW converter-fed motor under different operating conditions. The results show that the loss of induction motor which is driven by PWM converter is obviously greater than that of sinusoidal drive, especially under light load or no load condition. The whole no-load losses of the motor driven by PWM converter increases by more than 20.0% compared with that of the sinusoidal driving condition.

Effect of Slot Opening on Permanent Magnet Power Loss of a Permanent Magnet Synchronous Machine Driven by PWM

In this study, the influence of slot opening on the leakage reactance, power factors, winding current, and air gap flux density, as well as other parameters of the surface-mounted permanent magnet synchronous motor driven by an inverter is analyzed using the two-dimensional time-step finite element method. Thus, the law of permanent magnet power loss caused by slot opening under PWM qa obtained. The results show that the leakage reactance of the motor increased when the slot opening decreased from 7 mm to 2 mm. Then, the armature ripple current caused by the inverter was found to be suppressed well, and the high harmonic current was filtered well. This reduced the amplitude of the high-order harmonic of the air gap flux density, and the permanent magnet power loss was reduced by nearly 75%. Finally, two prototypes with different slot openings were tested for permanent magnet temperature rise, which verified the analysis method and calculation results proven in this paper.

On approximation classes for adaptive time-stepping finite element methods

Abstract We study approximation classes for adaptive time-stepping finite element methods for time-dependent partial differential equations. We measure the approximation error in $L_2([0,T)\times \varOmega )$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a by-product we define anisotropic Besov spaces for Banach-space-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.

Optimal design of q ‐axis field winding structure of dual‐excited synchronous condenser

The dual-excited synchronous condenser (DESC) has two field windings whose axes are located at direct and quadrature axes respectively. The DESC has more capacity of absorbing reactive power than that of the traditional synchronous condenser (TSC). However, the q-axis field winding distributed on the d-axis magnetic pole surface can decrease the magnetic circuit area of the rotor core of DESC, thus it may result in the increase of the total harmonic distortion (THD) of the magnetic field and the oversaturation of the rotor magnetic circuit. In order to obtain better electromagnetic characteristics of DESC, a q-axis field winding structure is proposed by optimising the number of slots, slot pitch and slot dimensions. The influences of the number and pitch of slot in q-axis on the air-gap radial flux density and THD of DESC are studied under the d-, q- and dual-axis excitation modes. The dimensions of the q-axis slots are optimised by combining the time-stepping finite element method and Taguchi method, and the optimal structure of q-axis field winding is obtained. A 10-kVar DESC is developed to verify the simulation results. This study provides theoretical basis for the design and manufacture of the DESC.

Optimal design of a short primary double‐sided linear induction motor based on derived Quasi‐3D equivalent circuit model using an improved differential evolution

Due to its simple construction, the linear induction motor (LIM) provides a linear driving force without any intermediate motion translation system. LIMs are widely used in various industrial applications, including maglev rail transit and the national defense industry. However, LIMs are affected by the end effect and suffer from problems such as low efficiencies and low power factors. To make improvements, in this paper, an ensemble multi-objective optimal design method for a short primary double-sided linear induction motor (SP-DLIM) is proposed. First, a simplified Quasi-3D equivalent circuit model (ECM) for an SP-DLIM applicable to the model in this paper is derived. The 3-D transient finite element method and an experimental prototype are utilised to prove that the derived ECM is accurate enough to solve the SP-DLIM optimisation problem. Second, an ensemble multi-objective optimal design method of SP-DLIM is presented, with proposed design constraints and four different optimisation problems. Then, an improved differential evolutionary (IDE) algorithm is proposed to optimise the efficiency, power factor, and tooth weight of the motor. The three-dimensional time-stepping finite element method is utilised to verify the validity of the optimisation method. Further, a comparison of the results suggests that the IDE yields the best performance to those of other advanced heuristic algorithms.

Helical Motion Wound-Rotor Resolver

In this paper an integrated core, high reliability, linear-rotational resolver with a simple slotless configuration is proposed. The moving part of the proposed resolver has only one helical winding. While, its stator needs at least two individual windings to be able to determine the linear and angular position in simultaneous linear and rotational, helical, motions. Three different configurations are proposed for the stator’s windings: (1) helical and horizontally skewed winding (2) helical and vertically skewed winding (3) horizontally and vertically skewed windings. The influence of the mentioned configurations is discussed in the terms of sensor’s accuracy in independent and simultaneous linear and rotational motions. Furthermore, a decoupling technique is proposed for the helical motion. All the studies are done using three-dimensional (3-D) time stepping finite element method (TSFEM).

Slotless Disk Type Resolver: A Solution to Improve the Accuracy of Multi-Speed Wound Rotor Resolvers

A common solution for increasing the accuracy of wound-rotor (WR) resolvers is to increase the number of poles. To achieve a compact design, fractional slot concentrated winding (FSCW) is used for multi-pole WR resolvers. However, employing such windings leads to rich sub-harmonics in the induced voltages. To suppress these undesirable sub-harmonics, multi-layer windings with an appropriate shifting angle between the layers are suggested. However, the said shifting angle is limited by the slot pitch so that the capability of the multi-layer winding is not utilized fully. To overcome the mentioned deficiency, a slotless disk type configuration is proposed in this article. Then, an optimization based on the winding function (WF) model of the sensor is carried out to achieve an optimal winding configuration. 3-D time-stepping finite element method (FEM) is used to verify the success of the proposed analytical model and the performed optimization. Finally, a prototype of the optimal configuration is built for experimental measurements.

Use of Resolvers in Limited Rotations

The electromagnetic structure of the resolvers makes them excellent for difficult applications. Among them, axial flux resolvers show better performance against mechanical faults. However, a complicated resolver-to-digital converter (RDC) overshadows the whole system's reliability. Both of the conventional software and hardware decoding methods suffer from deficiencies, such as difficulty to implement, poor performance against noise, expensive equipment, etc. To overcome these challenges any easy-to-implement software-based RDC is suggested in this paper that is used for rotational forth-and-back motions. In fact, instead of the conventional arc-tangent method to extract the position information from SIN and COS signals, three transfer functions are proposed to estimate the position. Tylors series are used to linearize the transfer functions over the zero position and a transfer function with minimum linearization error over the whole region is selected as opt one. The design process of axial flux resolvers is presented as well. 3-D time-stepping finite element method (FEM) is used to verify the success of the proposed analytical calculations. Finally, prototypes of the different configurations are tested for experimental measurements.

A time-stepping finite element method for a time-fractional partial differential equation of hidden-memory space-time variable order

We analyze a time-stepping finite element method for a time-fractional partial differential equation with hidden-memory space-time variable order. Due to the coupling of the space-dependent variable order with the finite element formulation and the hidden memory, the variable fractional order cannot be split from the space and destroys the monotonicity in the time-stepping discretization. Because of these difficulties, the numerical analysis of a fully-discrete finite element method of the proposed model remained untreated in the literature. We develop an alternative analysis to address these issues and to prove an optimal-order error estimate of the fully-discrete finite element scheme without any assumption on the regularity of the true solution and perform numerical experiments to substantiate the theoretical findings.

Dynamic Phasor Finite Element Modeling of Grid-Connected DFIG Considering Winding Space Harmonics

This paper presents a new technique for quantifying the winding MMF space harmonics of wind-driven Doubly-Fed Induction Generator (DFIG) for power quality assessments. Instead of following the commercial electromagnetic transient programs in modeling electric machines using lumped-parameters models, the paper uses a combined Dynamic Phase modeling and Finite Element Method (FEM) to model the generator accurately while reducing the computational burden and simulation time associated with the conventional time-stepped finite element analysis. The technique has the ability to efficiently model core nonlinearity and stator/rotor MMF space harmonics. The results of a C++ code written by the authors of the new dynamic phasors FEM technique are compared to the conventional time-stepping FEM to show the capability of the method to produce comparable results in shorter simulation time.