Time Finite Element Research Articles

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

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.

Direct Transcription of Optimal Control Problems with Finite Elements on Bernstein Basis

The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with finite elements in time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with finite elements in time.

Time finite element based Moreau‐type integrators

SummaryWith the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite dimensional mechanical systems, which contain frictional contact interactions. Together with the contact and impact laws formulated as normal cone inclusions, the principle of virtual action directly leads to the measure differential inclusions commonly used in the dynamics of nonsmooth mechanical systems. The discretization of the principle of virtual action in its strong and weak variational form by local finite elements in time provides a structured way to derive various time‐stepping schemes. The constitutive laws for the impulsive and nonimpulsive contact forces, ie, the contact and impact laws, are treated on velocity‐level by using a discrete contact law for the percussion increments in the sense of Moreau. Using linear shape functions and different quadrature rules, we obtain three different stepping schemes. Besides the well‐established Moreau time‐stepping scheme, we can present two alternative integrators referred to as symmetric and variational Moreau‐type stepping schemes. A suitable benchmark example shows the superiority of the newly proposed integrators in terms of energy conservation properties, accuracy, and convergence.

Moreau‐type integrators based on the time finite element discretization of the virtual action

AbstractIn this paper, we derive and compare three integrators for nonsmooth mechanical systems by discretizing the principle of virtual action with finite elements in time. The weak as well as the strong variational form of the principle are discretized using a piecewise linear shape function and different quadrature rules. After introducing a suitable constitutive law for the contact forces arising in the discretized system, this approach leads to the well established time‐stepping scheme of Moreau [1], the variational Moreau‐type scheme derived in [3] and another related scheme, which we call the symmetric Moreau‐type scheme. It is shown using a benchmark system that the symmetric and the variatonal Moreau‐type schemes, in contrast to Moreau's scheme, show an excellent longterm energy behavior. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space

Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin–Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach–Nečas–Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

Piecewise constant control of a precise measurement platform

Control of a precise measurement platform is investigated in this paper. Piecewise constant control procedure on the basis of finite element in time and the method of least squares is proposed. It is demonstrated that the proposed technique can be used to follow not only a predetermined law of displacement variations, but also velocity variations. Computational experiments with a system comprising one degree of freedom are used to illustrate the efficiency of the proposed technique.

A Moreau‐type Variational Integrator

AbstractIn this paper we derive a variational integrator for nonsmooth mechanical systems by discretizing the principle of virtual action with finite elements in time. After the discretization with local finite elements, the constitutive laws for the contact forces are introduced as in Moreau's time stepping scheme. This derivation shows exemplary how variational integrators for systems with frictional unilateral constraints can be derived. The long‐time energy behavior of the presented scheme is compared with the behavior of Moreau's stepping scheme on an example system. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Improved Shaping Approach to the Preliminary Design of Low-Thrust Trajectories

This paper presents a general framework for the development of shape-based approaches to low-thrust trajectory design. A novel shaping method, based on a three-dimensional description of the trajectory in spherical coordinates, is developed within this general framework. Both the exponential sinusoid and the inverse polynomial shaping are demonstrated to be particular two-dimensional cases of the spherical one. The pseudoequinoctial shaping is revisited within the new framework, and the nonosculating nature of the pseudoequinoctial elements is analyzed. A two step approach is introduced to solve the time of flight constraint, related to the design of low-thrust arcs with boundary constraints for both spherical and pseudoequinoctial shaping. The solution derived from the shaping approach is improved with a feedback linear-quadratic controller and compared against a direct collocation method based on finite elements in time. The new shaping approach and the combination of shaping and linear-quadratic controller are tested on three case studies: a mission to Mars, a mission to asteroid 1989ML, a mission to comet Tempel-1, and a mission to Neptune.

A consistent time FE-method for large strain elasto-plasto-dynamics

In the present paper energy-consistent momentum-conserving time-stepping schemes for geometrically nonlinear multiplicative elasto-plasto-dynamics – referred to as ‘ECMC-cG methods’ – are developed. In this context, the entire discretisation procedure of the global equations of motion is based on Galerkin methods in space and time, whereby special emphasis is placed on the desired conservation properties related to the approximation of time-integrals. To guarantee energy-consistency also for higher-order Finite Elements in time, a thermodynamically motivated ‘enhanced stress tensor’, which defines a non-standard quadrature rule, is derived for elasto-plastic material behaviour. Concerning the integration of the local evolution equations in time, exemplarily, a well-established first-order accurate exponential update is applied and adapted according to the global time-stepping scheme based on linear Finite Elements in time. The performance of the resulting scheme is demonstrated by means of several representative numerical examples, whereby it is shown that global energy-consistency is guaranteed exactly within the calculation accuracy by applying the proposed concepts.

Discontinuous Galerkin finite element in time for solving periodic differential equations

The dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing. Discontinuous Galerkin (DG) methods which are known to be higher order accurate, A-stable and locally conservative are used to solve such equations in this work. DG methods are shown to find the steady state periodic solution of (a) constant coefficient structural dynamics problem with periodic forcing and (b) helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing. Convergence of the numerical solution obtained is studied, for both the above equations, while h and p are refined separately. It is found that the DG method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes. It is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p > 12. Two adaptive strategies based on the element level displacement jump norm, i.e., h and hp refinement, are proposed and demonstrated on problems with higher harmonic forcing, primary controls and parametric variations.

Compact Methodology for Computing Limit-Cycle Oscillations in Aeroelasticity

A technique for computing limit-cycle oscillations of aeroelastic systems is presented. This technique can be used also for constructing and identifying reduced-order models that characterize accurately limit-cycle oscillations. The models presented use a spatial model reduction combined with a nonlinear identification technique. The finite difference in time method, a methodology similar to a finite element in time, is discussed. The specific novel aspects of the proposed method are noted. Numerical examples analyzing the dynamics of a panel forced by a buffeting aerodynamic load demonstrate the performance of the proposed methods.

Stability of Interrupted Cutting by Temporal Finite Element Analysis

Chatter in milling and other interrupted cutting operations occurs at different combinations of speed and depth of cut from chatter in continuous cutting. Prediction of stability in interrupted cutting is complicated by two facts: (1) the equation of motion when cutting is not the same as the equation when the tool is free; (2) no exact analytical solution is known when the tool is in the cut. These problems are overcome by matching the free response with an approximate solution that is valid while the tool is cutting. An approximate solution, not restricted to small times in the cut, is obtained by the application of finite elements in time. The complete, combined solution is cast in the form of a discrete map that relates position and velocity at the beginning and end of each element to the corresponding values one period earlier. The eigenvalues of the linearized map are used to determine stability. This method can be used to predict stability for arbitrary times in the cut; the current method is applicable only to a single degree of freedom. Predictions of stability for a 1-degree of freedom case are confirmed by experiment.

Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories

In this paper a direct method based on a transcription by finite elements in time has been used to design optimal interplanetary trajectories, exploiting a combination of gravity assist maneuvers and low-thrust propulsion. A multiphase parametric approach has been used to introduce swing-bys, treated as coast phases between two thrusted or coasting trajectory arcs. Gravity maneuvers are at first modeled with a linked-conic approximation and then introduced through a full three-dimensional propagation including perturbations by the Sun. The method is successfully applied to the design of a mission to planet Mercury, for which different options corresponding to different sequences of gravity maneuvers or launch opportunities are presented.

Time finite elements and error computation for (visco)plasticity with hardening or softening

AbstractIn this paper we investigate the possibility for, and characteristics of, reliable and efficient a posteriori error computation at the integration of the constitutive relations pertinent to (visco)plasticity with softening/hardening. The variational structure admits the use of finite elements in time, which are based on the Discontinuous Galerkin method. An important feature is the possibility to select ‘goal‐oriented’ error measures with great freedom. A key task is to identify and solve an auxiliary problem, that is the dual of the actual (primal) problem. Different strategies for computing the dual solution accurately, yet avoiding excessive cost, are investigated in the paper. In particular, we consider the characteristics at the limit of rate‐independent plasticity. Copyright © 2003 John Wiley & Sons, Ltd.

Numerical instability of magnetic damping problem of elastic plate

Numerical instability occurs in an analysis of a vibration with magnetic damping, or an electromagnetic and structural coupled problem. In this paper, the numerical instability of the coupled analysis is examined by the finite element in time. It is confirmed that the simultaneous method is unconditionally stable even if the magnetic field and the time increment are large. For the staggered method, we obtain the conditions where the numerical instability occurs.

On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems

Energy preserving schemes achieve unconditional stability for nonlinear systems by establishing discrete energy preservation statements. Several schemes have been presented by various authors within drastically different frameworks: finite difference schemes based on a mid-point approximation, Galerkin and time discontinuous Galerkin approximations of the equations of motion written in the symmetric hyperbolic form, finite elements in time, and 2-stage FSAL Runge—Kutta methods. Furthermore, different types of parameterization of finite rotations were used in the various formulations. This paper presents a unified, finite difference framework which readily allows comparing the various schemes and their respective properties. Numerical examples are presented and show that the predictions of two of these schemes are in very close agreement with each other.

A new look at finite elements in time: a variational interpretation of Runge-Kutta methods

We show that integration schemes derived from some well known finite elements in time formulations are Runge-Kutta methods, and we discuss the implications of this finding.

A comprehensive unified set of single-step algorithms with controllable dissipation for dynamics Part I. Formulation

A new class of time integration algorithms for dynamics is presented. The algorithms are based on finite element in time. The equations of motion are expressed in first-order form and the variables are interpolated via time polynomials. The unknown nodal variables are solved by the generalized Galerkin method. Algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time. The salient feature of the algorithms is that it can be either asymptotic annihilating or non-dissipative. Besides, any desired algorithmic damping between these two extreme cases could be obtained. The algorithms are analysed in detail. The spectral characteristics equivalent to the upper diagonal, diagonal of the Padé approximations and other rational approximations are given.

Stick-Slip Motion of Frictionally Damped Turbine Airfoils: A Finite Element in Time (FET) Approach

A method is proposed for analyzing the periodic stick-slip motion of a single degree of freedom model of frictionally damped turbine blades. The method of finite elements in the time domain (FET) is based on a Hamilton’s weak principle, paralleling the variational methods in elastostatics. It permits a complete determination of the hysteretic friction force, and results in a set of highly structured system equations. The method has a number of unique features, which are utilized to provide a simple yet efficient approach for predicting the steady-state response. When applied to a number of example problems, including systems with static friction and the excitation of multiple discrete frequencies, the FET method is demonstrated to be an efficient and reliable alternative technique for nonlinear dynamic response analysis.