Time-stepping Algorithms Research Articles

An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics

AbstractWe present in this paper a new finite element formulation of geometrically exact rod models in the three‐dimensional dynamic elastic range. The proposed formulation leads to an objective (or frame‐indifferent under superposed rigid body motions) approximation of the strain measures of the rod involving finite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod's cross‐section. Furthermore, the proposed framework allows the development of time‐stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties. Copyright © 2002 John Wiley & Sons, Ltd.

Cost Function for Dynamic Load Balancing of Explicit Parallel CFD Solvers with Variable Time-stepping Strategies

Parallel computing has been the main approach for solving large-size computational problems as in computational fluid dynamics, electromagnetics, structural mechanics, computational chemistry, etc. Due to the rapid advance of networking technology and wide availability of workstations in many institutions, networked heterogeneous workstations become attractive for parallel computations. One inherent problem for parallel computing on networked heterogeneous workstations is load balancing. Since parallel processes exchange information periodically, fast processes frequently wait for needed information from slow processes. Therefore, the slowest process among all parallel computation processes dictates the computation time of the whole program execution. The main task of load balancing is to reduce the processing time for the slowest process and to reduce the communication time among processes. It is not very difficult to find a reasonably good optimization algorithm for load balancing. However, the main difficulty for load balancing is how to find an accurate optimization cost function that can predict the program execution time for any given load distribution in a multi-user heterogeneous network environment. In this paper, we describe a practical approach for the derivation of a cost function for load balancing of an explicit parallel CFD solver using variable time-stepping algorithms. In this approach, the relative CPU speeds of computers and networks are measured since they are usually unknown to typical computer users. Then the elapsed program execution time on each processor is derived. Experimental results demonstrate that the proposed cost function is reasonably accurate. This cost function can be used for other parallel computation applications that are based on the domain decomposition approach.

On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods

We present in this paper the formulation of a new high-frequency dissipative time-stepping algorithm for nonlinear elastodynamics that is second-order accurate in time. The new scheme exhibits unconditional energy dissipation and momentum conservation (and thus the given name of energy-dissipative, momentum-conserving second-order scheme (EDMC-2)), leading also to the conservation of the relative equilibria of the underlying physical system. The unconditional character of these properties applies not only with respect to the time step size but, equally important, with respect to the considered elastic potential. Moreover, the dissipation properties are fully controlled through an algorithmic parameter, reducing to existing fully conserving schemes, if desired. The design of the new algorithm is described in detail, including a complete analysis of its dissipation/conservation properties in the fully nonlinear range of finite elasticity. To motivate the different constructions that lead to the dissipative properties of the final scheme, the same arguments are used first in the construction of new linear time-stepping algorithms for the system of linear elastodynamics, including first- and second-order schemes. The new schemes exhibit a rigorous decay of the physical energy, with the second-order schemes formulated in a general two-stage formula accommodating the aforementioned control parameter in the dissipation of the energy. A complete spectral analysis of the new schemes is presented in this linear range to evaluate their different numerical properties. In particular, the dissipative character of the proposed schemes in the high-frequency range is fully demonstrated. In fact, it is shown that the new second-order scheme is L-stable. Most remarkably, the extension of these ideas to the nonlinear range is accomplished avoiding the use of multi-stage formulae, given the freedom gained in using general nonlinear relations, while preserving the conservation laws of the momenta and the corresponding relative equilibria. Several representative numerical simulations are presented in the context of nonlinear elastodynamics to evaluate the performance of the newly proposed schemes.

Parametric identification of viscoelastic materials from time and frequency domain data

In this paper a parametric model of the elasticity modulus of a viscoelastic material (polypropylene) is determined by measured strains from wave propagation experiments. The identification of the model parameters is carried out both in the time and in the frequency domain. The advantages and limitations of either approach is discussed. Our main result is that a good representation of the elasticity modulus can be obtained by a general standard linear solid with three elements or seven free parameters. The model is valid over the same frequency range as that of the data, which in the present case extends from 100 Hz to 10 kHz. Such a model makes it possible to perform direct time simulations by time-stepping algorithms.

On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics

We present in this paper the development of a class of time-stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high-frequency range required for the robust solution of the resulting numerically stiff systems. To motivate and illustrate better the developments in this general case, we present first the formulation and analysis of these methods for two simple model problems. Namely, we consider a nonlinear elastic spring/mass system and a simplified model of thin elastic beams. As it is discussed in detail in this paper, the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic Hamiltonian systems with symmetry is of the main importance. These conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place, even in the presence of the desired and controlled numerical dissipation of the energy. This situation is contrasted with traditional “dissipative” numerical schemes, which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range, but also the aforementioned conservation properties, thus leading to a qualitatively distorted approximation of the phase dynamics. The key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems. The algorithms presented in this work accomplish these goals. The focus in this first part is given to first-order methods. Representative numerical simulations, ranging from applications in nonlinear structural dynamics to nonlinear continuum three-dimensional elastodynamics, are presented in the context of the finite element method to illustrate these ideas and results.

Communication cost estimation for parallel CFD using variable time-stepping algorithms

Variable time-stepping algorithm can greatly reduce the execution time of parallel CFD by allowing each parallel CFD process to select its own computation and communication time-steps according to the computational stability conditions. However, this algorithm significantly complicates the task of deriving the cost function for computer load balancing. Computer load balancing for parallel CFD requires a cost function that generally uses the elapsed CFD execution time as the cost. The cost function consists of three components: the computation cost, the communication cost, and the idle cost. Since the execution of parallel CFD using variable time-stepping approach causes the change the load of of computer and network even in a single user environment, the cost function should be derived based on the measurement of the parallel CFD execution time. The derived cost function can be used for predicting the elapsed time of future program execution. In this paper, the derivation of communication cost function for a parallel CFD using variable time-stepping algorithm is described. The accuracy of the communication cost function is demonstrated by two test cases and has been successfully used in dynamic computer load balancing.

Contact et frottement en dynamique des systèmes de corps rigides

ABSTRACT A formulation of the dynamics of rigid body collections, taking into account the unilateral character of non-interpenetrability constraints, dry friction at possible contact points and certain restitution laws in the event of collision is presented. It results in time-stepping algorithms of the implicit type for calculating evolution, presently enabling one to treat systems with several ten thousands of contacts on a micro-computer.

Dynamical Systems and Adaptive Timestepping in ODE Solvers

Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.

Phase Space Error Control for Dynamical Systems

Variable time-stepping algorithms for initial value ordinary differential equations are traditionally designed to solve a problem for a fixed initial condition and over a finite time. It can be shown that these algorithms may perform poorly for long time computations with initial conditions that lie in a small neighborhood of a fixed point. In this regime there are orbits that are bounded in space but unbounded in time, and the classical error-per-step or error-per-unit-step philosophy may be improved upon. A new error criterion is introduced that essentially bounds the truncation error at each step by a fraction of the solution arc length over the corresponding time interval. This new control can be incorporated within a standard algorithm as an additional constraint at negligible additional computational cost. It is shown that this new criterion has a positive effect on the linear stability properties and hence improves behavior in the neighborhood of stable fixed points. Furthermore, spurious fixed points and period two solutions are prevented. The new criterion is shown to be admissible in the sense that it can always be satisfied with nonzero stepsizes. Implementation details and numerical results are given.

Elastodynamic direct boundary element methods with enhanced numerical stability properties

Evidence of numerical instabilities in two-dimensional time domain direct boundary element methods is presented. The e!ects of numerical versus analytical integration of spatial integrals on stability are shown, and two new time-stepping algorithms are introduced and compared to existing formulations. The so-called new &direct half-step' scheme and the &epsilon' scheme are shown to improve the numerical stability of direct boundary element methods. Copyright ( 1999 John Wiley & Sons, Ltd.

Biot consolidation analysis with automatic time stepping and error control Part 2: applications

The automatic time-stepping algorithms developed in a companion paper are used to study the behaviour of several problems involving the consolidation of porous media. The aim of these analyses is to demonstrate that the new procedures are robust, efficient, and can control the global temporal discretization (or time-stepping) error in the displacements to lie near a prescribed error tolerance. Copyright © 1999 John Wiley & Sons, Ltd.

New unconditionally stable staggered solution procedures for coupled soil-pore fluid dynamic problems

A new class of unconditionally stable staggered implicit–implicit time-stepping algorithms for coupled soil-pore fluid dynamic problems is presented. The proposed schemes are stabilized with a pressure correction method and the staggered procedure introduced earlier by the second author is a simplified case of the new algorithms. The stabilized procedure is efficient in numerical time domain analysis of soil-pore fluid interaction occurring in earthquake and consolidation problems. Several examples of linear and non-linear problems demonstrate the stability and accuracy of the proposed numerical solution algorithms. © 1998 John Wiley & Sons, Ltd.

Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems

This paper presents the formulation of conserving time-stepping algorithms for frictionless dynamic contact of solids. A new class of finite element methods is proposed for the solution of these problems that exhibit the same conservation laws as the underlying continuum dynamical system. The proposed methods are based on a penalty regularization of the constrained contact problem, and lead to full conservation of the total energy of the system (including the regularization penalty potential) during persistent contact, and restoration of the original energy upon release. Both linear and angular momenta are conserved by the scheme. Furthermore, the newly developed methods have the ability to enforce the associated constraints in the velocity besides the impenetrability constraint in the displacements, while preserving the conservation/restoration properties of the final numerical scheme. A modification of these schemes is described that assures positive energy dissipation if desired (even in the highly nonlinear setting of contact/impact problems), leading to contact schemes with high-frequency energy dissipation. Representative numerical simulations are presented illustrating the performance of the proposed numerical schemes.

Modelling subsurface coupled chemical reactions and fluid flow over long time periods

Simulation of coupled chemical reactions and fluid flow in porous sedimentary basins over long time periods is a numerical challenge. Most models representing such a physical problem are solved as PDEs where efficient timestepping with controlled error is difficult. We use the differential algebraic equation system approach where robust adaptive timestepping algorithms are available in the solvers, e.g., RADAU5 and DASSL. Mathematical and numerical models for coupled chemical reactions and fluid flow are derived. The models have several interesting properties, e.g., strong nonlinearities and stiffness, which are discussed. We test the performance of our code.

Implicit algorithms and their linearization for the transient incompressible Navier-Stokes equations

The issue of appropriate time discretization methods for the unsteady incompressible Navier-Stokes equations is considered from a practical perspective. Conventional implicit time-stepping algorithms are not feasible for long-time simulations since they inherit the quadratic nonlinearity of the steady-state equations. As a result, two new linearized versions of the 'pure' algorithms are analyzed herein. These have similar stability properties and comparable accuracy to the underlying nonlinear methods.

Mechanical integrators derived from a discrete variational principle

Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct time-stepping algorithms that approximate the flow of continuous ODEs for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We first present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing symplectic-momentum mechanical integrators for Lagrangian systems with holonomic constraints. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.

Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles

Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.

Sensitivity Methods for Time‐Continuous, Spatially Discrete Groundwater Contaminant Transport Models

Existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time‐stepping numerical algorithms and cannot be readily used with time‐continuous approaches to transport simulation, such as the Laplace transform Galerkin technique. We develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the Laplace domain and inverted numerically to the time domain. The methods are computationally efficient when used in conjunction with time‐continuous transport equations. The relative efficiency of the two methods depends on the number of model parameters, number of performance measures, and number of spatial discretization nodes. The adjoint method is favored when the number of performance measures is much smaller than the number of model parameters. The adjoint method is limited in that performance measures are restricted to being linear functions of state variables. A two‐dimensional transport example is developed in detail, and sensitivities with respect to nodal hydraulic conductivities are computed. In the problems analyzed, the direct and adjoint methods are 9 to 156 times faster than the perturbation method, with the computational savings increasing as the size of the problem is increased.

2D nonlinear finite element modelling of electromagnetic retarders using time-stepping algorithms, and the Petrov-Galerkin method with homogenization techniques

This paper describes a method for finite element modelling of transient 2D eddy currents in the nonlinear moving conductor of an electromagnetic retarder. The magnetic vector potential A formulation is used, with a general single-step time-marching algorithm, and the fixed and moving coordinate systems are compared. A homogenization technique, inspired by the lamination modelling, is used to make the geometry of a moving region invariant along the speed direction, in order to use a more direct method to compute the steady state eddy currents in the whole rotor. Results are compared to measurements.

EVALUATION OF SOME IMPLICIT TIME-STEPPING ALGORITHMS FOR PSEUDODYNAMIC TESTS

Two types of implicit time-stepping algorithms have been proposed recently for pseudodynamic tests. The first type consists of an algorithm which relies on Newton iterations to satisfy the equations of motion. The second type consists of an algorithm which is based on the Operator-Splitting technique and does not require any numerical iteration. While one or the other has been preferred by some researchers, these time-stepping algorithms have not been analysed and compared under a uniform setting. In this paper, a concise summary of these schemes is presented, and they are evaluated in a consistent manner in terms of numerical dissipation, frequency distortion and experimental errors. The analytical results are validated by numerical simulations as well as experimental results. It is shown that the algorithm based on Newton iterations can control experimental error effects effectively by means of an error-correction procedure. The algorithm based on the Operator-Splitting technique demonstrates similar performance provided the I-Modification is adopted.