Element Method Time Research Articles

Fast and accurate methodologies for the eddy current losses analysis in axially laminated rotor of synchronous reluctance machines under different loads

AbstractThe authors present two fast and accurate methodologies for the computation of eddy current losses in the axially laminated rotor of a synchronous reluctance machine. The methodologies are based on different combinations of the finite element method in time and frequency domains with 2D and 3D formulations. First, a comparative study of the 2D and 3D formulations for loss calculation is presented, considering various load angles of the machine to illustrate the problem of eddy‐current losses in this type of machines and its dependence on the load angle. The influence of the iron saturation on the loss calculation is also evaluated in these computations. A novel correction factor based on the computations at two load angles is proposed to convert the losses computed from a 2D model to match those computed from a 3D model. For the sake of generality, investigations are also conducted for various thicknesses of the lamination layers and different machine lengths, and an analytical method to describe the dependency of eddy‐current losses on the load angle of the machine is introduced. Moreover, a simplified method is proposed for modelling eddy currents in the frequency domain and calculating losses in an axially laminated structure based solely on the results of a magnetostatic solution. The results obtained by the simplified model demonstrate excellent agreement with the full 3D magneto‐dynamic simulation. Overall, the findings contribute to understanding and accurately characterising the eddy current losses in axially laminated rotors, offering potential insights for designing and optimising axially laminated synchronous reluctance electric machines.

The Finite Element Method in Time for Multibody Dynamics

Abstract The generalized-α scheme has become the approach of choice for the time integration of the equations of motion of multibody systems. Despite its simplicity, this scheme presents drawbacks: the time-step size cannot be changed easily, making it difficult to implement time adaptivity, and the solution of periodic problems cannot be found easily. This paper explores an alternative approach based on the finite element method in time. The basic principles underpinning the approach are presented and both time-continuous and time-discontinuous approaches are investigated. Two types of Galerkin schemes will be presented here: the time-continuous and the time-discontinuous schemes. In the former, the displacement field is continuous across interelement boundaries, whereas discontinuities or “jumps” are allowed across interelement boundaries for the latter. Simple problems are treated to identify the best schemes. Families of schemes of various accuracies are presented. The first family, based on time-continuous elements, features schemes that do not present numerical dissipation. Asymptotic annihilation is achieved by the time-discontinuous elements that form the second family. The problem of kinematic constraints is treated within the framework of the finite element method in time. Special emphasis is devoted to the satisfaction of the kinematic constraints and their time derivative within a time element.

Numerical valuation of European and American options under Merton's model

In this paper, a new combination of time and spatial discretization is proposed for a partial integro-differential equation (PIDE) arising in the valuation of European options under Merton's model. We first present a high-order compact (HOC) difference scheme in space based on a uniform mesh to obtain a highly accurate result, and the discontinuous Galerkin (DG) finite element method in time is introduced that can deal with the loss of the time analyticity. A penalty method is proposed for a partial integro-differential complementarity problem arising in the valuation of the American put option. Numerical experiments are performed to verify the accuracy and efficiency of the proposed method.

A CG–DG method for Maxwell’s equations in Cole–Cole dispersive media

The numerical approximation of Maxwell’s equations in Cole–Cole dispersive media is studied for two and three-dimensional cases. We construct a combined scheme using the standard continuous Galerkin (CG) finite element method in time and discontinuous Galerkin (DG) method in space to discretize the Cole–Cole dispersive model. The L2-stability and error estimate of the proposed scheme are analyzed. Furthermore, we use sum-of-exponential approximation for the convolution kernel to speed up the evaluation of the Caputo derivative. Numerical examples are provided to demonstrate the performance of the proposed numerical methods.

On variational and symplectic time integrators for Hamiltonian systems

Various systems in nature have a Hamiltonian structure and therefore accurate time integrators for those systems are of great practical use. In this paper, a finite element method will be explored to derive symplectic time stepping schemes for (non-)autonomous systems in a systematic way. The technique used is a variational discontinuous Galerkin finite element method in time. This approach provides a unified framework to derive known and new symplectic time integrators. An extended analysis for the new time integrators will be provided. The analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties. These new time stepping schemes are necessary to get accurate and stable simulations of (forced) water waves and other non-autonomous variational systems, which we illustrate in our numerical results.

Numerical analysis solving the elastic dynamic problem

The purpose of this work is to present a numerical method for solving a problem of dynamic structures. For that, we use the Newmark algorithm and the finite element method for time and space discretisations, respectively. We prove the efficiency of the Newmark scheme in which the sum of elastic and kinetic energies is conserved. We conduct also numerical computations aiming to determine the influence of numerical parameters in terms of time step and mesh size on energy conservation.

Momentum–time flux conservation method for one-dimensional wave equations

We present a conservation element and solution element method in time and momentum space. Several paradigmatic wave problems including simple wave equation, convection–diffusion equation, driven harmonic oscillating charge and nonlinear Korteweg–de Vries (KdV) equation are solved with this method and calibrated with known solutions to demonstrate its use. With this method, time marching scheme is explicit, and the nonreflecting boundary condition is automatically fulfilled. Compared to other solution methods in coordinate space, this method preserves the complete information of the wave during time evolution which is an useful feature especially for scattering problems.

A NON-SPURIOUS VECTOR SPECTRAL ELEMENT METHOD FOR MAXWELL'S EQUATIONS

In this paper, a non-spurious vector spectral element method is proposed to solve Maxwell's equations using E and H as variables. The mixed-order curl-conforming basis functions are used for both variables to facilitate applying boundary and interface conditions; and the interpolation degree of basis functions for E is set difierent from that for H to suppress the spurious modes. The proposed method can be utilized in both time domain and frequency domain, and it is very suitable for the future implementation of discontinuous Galerkin spectral element method. Numerical results demonstrate the property of spurious-free and the spectral accuracy of this method. The method has also been implemented for the more general flnite element method in time and frequency domains.

Space-time Discretization of an Integro-differential Equation Modeling Quasi-static Fractional-order Viscoelasticity

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind, with a weakly singular kernel in the time variable, and which also involves partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used at each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, which can be used as the basis for an adaptive strategy.

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is com...

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is computed by enforcing periodicity of the element set. Of particular interest are periodic forcing functions possessing high frequency content. To maintain the spectral accuracy for such forcing functions, an h-refinement scheme is employed near the semi-discontinuity without increasing the number of degrees of freedom.Time discretization by spectral elements is applied initially to a standard form of a set of linear, first-order differential equations subject to harmonic excitation and an excitation admitting rapid variation. Other case studies include the application of the SE approach to parabolic and hyperbolic partial differential equations. The first-order form of these equations is obtained through semi-discretization using conventional finite-element, spectral element and finite-difference schemes. Element clustering (h-refinement) is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation. The convergence in time of the method is demonstrated. In some cases, machine precision is obtained with 25 degrees of freedom per cycle. Finally the method is applied to a weakly nonlinear problem with time-periodic solution to demonstrate its future applicability to the analysis of limit-cycle oscillations in aeroelastic systems.

Modeling of non‐classical thermoelasticity

AbstractThe present contribution deals with computational modeling of non‐classical theories according to the approach of Green and Naghdi [1]. Their theory is subdivided into three types of heat conduction, labeled type I, II and III. Each of the three heat equations is coupled with the balance of momentum. The arising partial di.erential equations are discretized with a Galerkin finite element method in time as well as in space. All three types are applied to an isotropic and homogeneous elastic‐deformable continuum and studied by a numerical example. The reader is also referred to [2, 3] for a more detailed description of the introduced results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

ADAPTIVE DISCRETIZATION OF AN INTEGRO-DIFFERENTIAL EQUATION MODELING QUASI-STATIC FRACTIONAL ORDER VISCOELASTICITY

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind with a weakly singular kernel in the time variable involving also partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used in each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, and develop an adaptive strategy based on the a posteriori error estimate.

Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes

In the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite-dimensional Hamiltonian systems with symmetry is shown. The result of this development is the enhanced Galerkin (eG) finite element method in time. The conservation of the eG method is generally related to its collocation property. Total energy conservation, in particular, is obtained by a new projection technique. The eG method is, moreover, based on objective time discretization of the used strain measure. This paper is concerned with particle dynamics and semi-discrete non-linear elastodynamics. The related numerical examples show good performance in presence of stiffness as well as for calculating large-strain motions. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.

Adaptive discretization of fractional order viscoelasticity using sparse time history

An efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed. The method can handle variable time steps. To overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature. We use an internal variable formulation of the viscoelastic equations where the internal variable is of stress type. The rate equation that governs the evolution of the internal variable involves a fractional integral and can be identified as a Volterra integral equation of the second kind with a weakly singular kernel. For the numerical integration of the rate equation we adopt the finite element method in time, in particular the discontinuous Galerkin method with piecewise constant basis functions is used. A priori and a posteriori error estimates are proved. An adaptive strategy based on the a posteriori error estimate is developed. Finally, the precision and effectiveness of the method are demonstrated by comparing the numerical solutions with analytical solutions.

Extended GDQ and related discrete element analysis methods for transient analyses of continuum mechanics problems

The extended GDQ proposed by the author is used to develop solution algorithms for solving the discrete transient equation system of a continuum mechanics problem. It is a direct integration approach. Two integration methods are developed. They are time-element by time-element method and stages by stages method. These two time integration algorithms can be used to solve a generic discrete transient equation system of an originally discrete system or a discrete system resulting from the discretization of a transient system of continuum mechanics problems by using a certain discretization technique such as the DQEM, FEM, FDM, etc.

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.

Dynamic Response of Timoshenko Beam Structures Solved by DQEM Using EDQ

The dynamic response of Timoshenko beam structures is solved by using the DQEM to the space discretization and EDQ to the time discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.Copyright © 2002 by ASME

Dynamic Analysis of Flexible Mechanisms With Clearances

A finite element method in time is presented for the periodic solution of vibrating elastic mechanisms with clearances. The solution of motion is made possible by utilizing time finite elements which discretize the solution period into a number of time intervals. The periodic response is described in terms of a set of temporal nodes of all spatial degrees of freedom of the system, yielding a block-diagonal nonlinear algebraic system to be solved iteratively. The suggested method is applied to an example problem of cam-driven valve train, demonstrating the effectiveness of the method in dealing with multiple clearance nonlinearities.