Time Finite Element Method Research Articles

Enriched space–time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics

SUMMARYThe main objective of this paper is to present an enriched version of the space–time FEM method to incorporate multiple temporal scale features with a focus on dynamics problems. The method is established by integrating the basic framework of the space–time discontinuous Galerkin method with the extended finite element method. Two versions of the method have been developed: one at the continuum scale (elastodynamics) and the other focuses on the dynamics at the molecular level. After an initial outline of the formulation, we explore the incorporation of different types of enrichments based on the length scale of interest. The effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice. The robustness of the method is evaluated in terms of convergence and the ability to capture the fine scale features. It is shown that the enriched space–time FEM leads to an improvement in the convergence properties over the traditional space–time FEM for problems with multiple temporal features. It is also highly effective in integrating atomistic with continuum representations with a coupled framework. Copyright © 2012 John Wiley & Sons, Ltd.

Design and convergence analysis for an adaptive discretization of the heat equation

We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space–time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space–time finite element method).

Effective tunability and realistic estimates of group index in plasmonic metamaterials exhibiting electromagnetically induced transparency

We study the phenomenon of the electromagnetically induced transparency in planar and stacked plasmonic metamaterials (MMs) using the finite integration time domain and finite element methods. For such structures, the dependence of the tunability of the inherent structural resonances on geometry design is clarified. We also analyze the performance of recently demonstrated MM designs in terms of the achievable group refractive index and losses, which are of great interest for slowing light applications.

Verification of a one-dimensional finite element method for modeling blood flow in the cardiovascular system incorporating a viscoelastic wall model

In this study we present the implementation and verification of a space–time finite element method for solving the nonlinear one-dimensional (1-D) equations of blood flow incorporating a viscoelastic arterial wall model. The viscoelastic model used is based on the generalized Maxwell model and thin-walled tube theory assumptions. Verification of the implementation was conducted using two different methods: (1) the analytic solution to the linearized 1-D equations of blood flow and (2) the Method of Manufactured Solutions (MMS). MMS enables verification for conditions such as tapered vessel geometries with spatially varying wall properties which cannot be verified with the analytic solution. RMS error values for all variables of interest (blood flow rate, pressure and vessel cross-sectional area) were calculated for simulations that utilized different mesh and time step sizes. The error values decreased with each mesh and time step size refinement. A theoretical analysis of numerical dissipation was also carried out for the linearized 1-D equations of blood flow. Dissipation in simulations due to different time step sizes matched theoretical estimations for given frequencies. This dissipation analysis can provide one way to estimate mesh and time step sizes to control numerical dissipation in the nonlinear simulations. The presence of wall viscoelasticity in all simulations is clearly seen in the hysteresis (pressure–area) loops.

Nonlinear characteristics of vortex-induced vibration at low Reynolds number

Numerical simulations of vortex-induced vibration of a two-dimensional elastic circular cylinder under the uniform flow are calculated when Reynolds number is 200. In order to achieve the vortex-induced vibration, two-dimensional incompressible Navier–Stokes equations are solved with the space–time finite element method, the equations of the cylinder motion are solved with the new explicit integral method and the emeshing is achieved by the spring analogy technology. Considering vortex-induced vibration with the low reduced damping parameters, the variety trends of the lift coefficient, the drag coefficient, the displacement of cylinder are analyzed under different oscillating frequencies of cylinder. The nonlinear phenomena of locked-in, beat and phaseswith are captured successfully. The limit cycle and bifurcation of lift coefficient and displacement are analyzed. Besides, the Poincare sections of the lift coefficient are used for discussing the bifurcation of periodic solution. There are some differences in nonlinear characteristics between the results of the one degree of freedom cylinder model and those of the two degrees of freedom cylinder model. The streamwise vibration has a certain effect on the lateral vibration.

Hybridized enriched space–time finite element method for analysis of thin-walled structures immersed in generalized Newtonian fluids

The paper addresses the numerical treatment of a specific class of fluid-structure interaction problems: flow-immersed thin structures undergoing considerable motion and deformation. The simultaneous solution procedure uses a mixed-hybrid velocity-based formulation of both fluid and structure discretized by a stabilized time-discontinuous space-time finite element method. The continuity at the interface is ensured by a localized mixed-hybrid interface method avoiding Lagrange multipliers and penalty approaches. The XFEM is utilized for enrichment of the approximation space of the fluid variables in order to represent non-smooth (discontinuous) solution features resulting from immersing a thin structure in a fluid.

激光激发Lamb波的有限元时域和频域数值分析

激光激发Lamb波的有限元时域和频域数值分析

Fuzzy stability analysis of regenerative chatter in milling

During machining, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece. Consequently it is desirable to predict, and hence avoid the onset of this instability. Regenerative chatter is a function of empirical cutting coefficients, and the structural dynamics of the machine-tool system. There can be significant uncertainties in the underlying parameters, so the predicted stability limits do not necessarily agree with those found in practice. In the present study, fuzzy arithmetic techniques are applied to the chatter stability problem. It is first shown that techniques based upon interval arithmetic are not suitable for this problem due to the issue of recursiveness. An implementation of fuzzy arithmetic is then developed based upon the work of Hanss and Klimke. The arithmetic is then applied to two techniques for predicting milling chatter stability: the classical approach of Altintas, and the time-finite element method of Mann. It is shown that for some cases careful programming can reduce the computational effort to acceptable levels. The problem of milling chatter uncertainty is then considered within the framework of Ben-Haim's information-gap theory. It is shown that the presented approach can be used to solve process design problems with robustness to the uncertain parameters. The fuzzy stability bounds are then compared to previously published data, to investigate how uncertainty propagation techniques can offer more insight into the accuracy of chatter predictions.

An integral equation based numerical solution for nanoparticles illuminated with collimated and focused light

To address the large number of parameters involved in nano-optical problems, a more efficient computational method is necessary. An integral equation based numerical solution is developed when the particles are illuminated with collimated and focused incident beams. The solution procedure uses the method of weighted residuals, in which the integral equation is reduced to a matrix equation and then solved for the unknown electric field distribution. In the solution procedure, the effects of the surrounding medium and boundaries are taken into account using a Green's function formulation. Therefore, there is no additional error due to artificial boundary conditions unlike differential equation based techniques, such as finite difference time domain and finite element method. In this formulation, only the scattering nano-particle is discretized. Such an approach results in a lesser number of unknowns in the resulting matrix equation. The results are compared to the analytical Mie series solution for spherical particles, as well as to the finite element method for rectangular metallic particles. The Richards-Wolf vector field equations are combined with the integral equation based formulation to model the interaction of nanoparticles with linearly and radially polarized incident focused beams.

Virtual functions of the space–time finite element method in moving mass problems

Classical time integration schemes fail in vibration analysis of complex problems with moving concentrated parameters. Moving mass problems and moving support problems belong to this group. Commercial systems of dynamic simulations do not support such an analysis. Moreover, the classical finite element method with the Newmark-type time integration method does not allow us to obtain convergent results at all. The reason lies in the impossibility of full mathematical consideration of the time integration stage and the analysis of inertial terms of a travelling mass. Both of them, unfortunately, are decoupled. In this paper we propose an efficient and exact numerical approach to the problem by using the space–time finite element method. We derive characteristic matrices of the discrete element of the string and the Bernoulli–Euler beam that carry the concentrated mass. We present four types of virtual functions in time and we apply two of them to the practical analysis. Displacements in time obtained numerically are compared with semi-analytical results. Almost perfect coincidence proves the efficiency of the approach.

An Integral Equation based Numerical Solution for Nanoparticles Illuminated with Collimated and Focused Light

AbstractAn integral equation based numerical solution is developed when the particles are illuminated with collimated and focused incident beams. The solution procedure uses the method of weighted residuals, in which the integral equation is reduced to a matrix equation and then solved for the unknown electric field distribution. In the solution procedure, the effects of the surrounding medium and boundaries are taken into account using a Green’s function formulation. Therefore, there is no additional error due to artificial boundary conditions unlike differential equation based techniques, such as finite difference time domain and finite element method. In this formulation, only the scattering nano-particle is discretized. The results are compared to the analytical Mie series solution for spherical particles, as well as to the finite element method for rectangular metallic particles. The Richards-Wolf vector field equations are combined with the integral equation based formulation to model the interaction of nanoparticles with linearly and radially polarized incident focused beams.

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

AbstractIn recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space–time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave‐like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space–time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.

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.

Simulation on the thermal cycle of a welding process by space–time convection–diffusion finite element analysis

Welding plays an important role in manufacturing. But difficulties still exist for simulation of the welding process of large welded structures, due to the limitation of computer capacity, mathematical models and software. This paper is devoted to developing an algorithm that tries to simulate the thermal cycle during welding efficiently and accurately. A space–time finite element method (FEM) is proposed to solve the transient convection–diffusion thermal equation. The method has been applied to the steady-state thermal analysis of welds. A moving coordinate frame (Eulerian frame), in which the heat source is stationary, is used to improve the spatial resolution of a numerical analysis for the thermal cycle of welds effectively, as well as to incorporate the addition of the filler metal naturally. This method is suitable for the thermal analysis in the weld pool or/and weld joint region including starting and stopping transients.

Vibration analysis of plane elasticity problems by the [formula omitted]-continuous time stepping finite element method

This paper proposes a C 0 -continuous time stepping finite element method to solve vibration problems of plane elasticity. In the time direction, unlike the existing methods [F. Costanzo, H. Huang, Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics, Comput. Methods Appl. Mech. Engrg. 194 (2005) 2059–2076; D.A. French, A space–time finite element method for the wave equation, Comput. Methods Appl. Mech. Engrg. 107 (1993) 145–157; H. Huang, F. Costanzo, On the use of space–time finite elements in the solution of elasto-dynamic problems with strain discontinuities, Comput. Methods Appl. Mech. Engrg. 191 (2002) 5315–5343; T.J.R. Hughes, G. Hulbert, Space–time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988) 339–363; G. Hulbert, T.J.R. Hughes, Space–time finite element methods for second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg. 84 (1990) 327–348; C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129; X.D. Li, N.E. Wiberg, Structural dynamic analysis by a time-discontinous Galerkin finite element method, Int. J. Numer. Methods Engrg. 39 (1996) 2131–2152; X.D. Li, N.E. Wiberg, Implementation and adaptivity of a space–time finite element method for structural dynamics, Comput. Methods Appl. Mech. Engrg. 156 (1998) 211–229], this method does not use the discontinuous Galerkin (DG) method to simultaneously discretize the displacement and velocity fields, but only use the C 0 -continuous Galerkin method to discretize the displacement field instead. This greatly reduces the size of the linear system to be solved at each time step. The finite element in the space directions is taken as the usual P r − 1 -conforming element with r ⩾ 2 . It is proved that the error of the method in the energy norm is O ( h r − 1 + k 3 ) , where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. Some numerical tests are included to show the computational performance of the method.

Analysis of temperature and deformation fields in an optical coating sample

By solving thermal conduction and thermo-elastic equations with an integral-transform method, a theory to describe the three-dimensional temperature and deformation distributions in an optical coating sample irradiated by a pulsed or square-wave modulated laser is developed for both transient and steady state cases. The theoretical simulations are compared with corresponding results of the finite-element method in time and space domains and good agreements are obtained. In addition, the influences of heating-beam radius and modulation frequency on the temperature and deformation distributions are discussed. Applications of the theoretical model to absorptance and deformation measurements of optical coatings are discussed.

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

AbstractThe paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects.The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.

A space–time formulation and improved spatial reconstruction for the “divide-and-conquer” multiscale method

A time-discontinuous space–time formulation for the “divide-and-conquer” multiscale method is presented. This multiscale method aims at stable and accurate solutions of transient convection–diffusion-reaction problems. The method is constituted by a time-discontinuous space–time finite element method on the fine-scale level, an explicit procedure for advancing the coarse-scale solution on the respective level, and inter-scale operators providing the data transfer between the coarse- and fine-scale level in both directions. The results from three numerical test cases show that for both problematic flow regimes, that is, the regime of dominant convection and the regime of dominant convection and absorption, the proposed method provides completely stable solutions, which are not achieved by standard stabilized methods, particularly for the later regime. For remedying a still to be noted shortcoming of the proposed method, that is, a too smooth resolution of sharp-gradient regions in the solution field, considerable improvement can be provided by the use of cubic spline interpolation as the spatial reconstruction (i.e., the spatial coarse-to-fine inter-scale) operator.

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.