Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

We focus on a simple implicit time integration scheme for the transient response solution of structures when large deformations and long time durations are considered. Our aim is to have a practical method of implicit time integration for analyses in which the widely used Newmark time integration procedure is not conserving energy and momentum, and is unstable. The method of time integration discussed in this paper is performing well and is a good candidate for practical analyses.     

One- and two-equation turbulence models for the prediction of complex cascade flows using unstructured grids

One- and two-equation, low-Reynolds eddy-viscosity turbulence models are employed in the context of a primitive variable, finite volume, Navier-Stokes solver for unstructured grids. Through the study of the complex flow in a controlled-diffusion compressor cascade at off-design conditions, the ability of the models under consideration to predict the laminar separation bubble close to the leading edge and the boundary layer development is investigated. In order to control the unphysical growth of turbulent kinetic energy near the leading edge stagnation point, appropriate modifications to the conventional models are employed and tested. All of them improve the leading edge flow patterns and significantly affect the size of the predicted laminar separation bubble. The use of an adequately refined mesh around the airfoil, that is formed by triangles placed in a quasi-structured way, allows for the generation of grid elements of moderate aspect ratios. This helps to readily overcome any relevant problems of accuracy; a second-order upwind scheme without flux limiters or least squares approximations is successfully employed for the gradients. The test case includes quasi-3D effects by considering the streamtube thickness variation in the governing equations.     

Performance of implicit A-stable time integration methods for multibody system dynamics

Multibody System Dynamics - This paper illustrates the performance of several representative implicit A-stable time integration methods with algorithmic dissipation for multibody system dynamics,...     

A high-order time integration scheme for open channel flow

The time integration of the linearized, first-order hyperbolic equations for open channel flow usually follows some iterative techniques, the finite difference or the finite element method. Here the matrix exponential approach is used, which can be written in a form to allow for generating an arbitrary high-order approximation to the exact solution.     

A time semi-exponentially fitted scheme for chemotaxis-growth models

In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il’in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation.     

On a fourth-order unconditionally stable scheme for damped second-order systems

We consider the damped second-order system $\text{M}\text{u}\text{?}\text{+ C/.u + Ku = F(t)}$ (M, C, K symmetric, positive definite n × n matrices) by conversion to an equivalent first-order system

$\text{I}\text{O}\text{O}\text{M}\text{d}\text{d}\text{t}\text{O}\text{?K}\text{I}\text{?C}\text{u}\text{u}\text{?}\text{+}\text{o}\text{F(t)}$

We demonstrate that an algorithm proposed by Fairweather for the implementation of the (2, 2) Padé approximation of the exponential matrix for approximating the solution of homogeneous first-order systems extends advantageously to this case, yielding an unconditionally stable fourth-order scheme with the feature that the approximating equations decouple. As a result we are required only to solve one symmetric complex n × n system of linear algebraic equations at each time step, with a fixed matrix which may be decomposed into triangular factors at the outset. We also consider iterative schemes involving only real, positive definite, symmetric n × n matrices. Numerical results are included.     

Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs

The methods for the dynamical simulation of multi-body systems in real-time applications have to guarantee that the time integration of the equations of motion is always successfully completed within an a priori fixed sampling time interval, typically in the range of 1.0–10.0 ms. Model structure, model complexity and numerical solution methods have to be adapted to the needs of real-time simulation. Standard solvers for stiff and for constrained mechanical systems are implicit and cannot be used straightforwardly in real-time applications because of their iterative strategies to solve the nonlinear corrector equations and because of adaptive strategies for stepsize and order selection. As an alternative, we consider in the present paper noniterative fixed stepsize time integration methods for stiff ordinary differential equations (ODEs) resulting from tree-structured multi-body system models and for differential algebraic equations (DAEs) that result from multi-body system models with loop-closing constraints.     

VR-CAD integration: Multimodal immersive interaction and advanced haptic paradigms for implicit edition of CAD models

This paper presents an approach for the integration of Virtual Reality (VR) and Computer-Aided Design (CAD). Our general goal is to develop a VR-CAD framework making possible intuitive and direct 3D edition on CAD objects within Virtual Environments (VE). Such a framework can be applied to collaborative part design activities and to immersive project reviews. The cornerstone of our approach is a model that manages implicit editing of CAD objects. This model uses a naming technique of B-Rep components and a set of logical rules to provide straight access to the operators of Construction History Graphs (CHG). Another set of logical rules and the replay capacities of CHG make it possible to modify in real-time the parameters of these operators according to the user’s 3D interactions. A demonstrator of our model has been developed on the OpenCASCADE geometric kernel, but we explain how it can be applied to more standard CAD systems such as CATIA. We combined our VR-CAD framework with multimodal immersive interaction (using 6 DoF tracking, speech and gesture recognition systems) to gain direct and intuitive deformation of the objects’ shapes within a VE, thus avoiding explicit interactions with the CHG within a classical WIMP interface. In addition, we present several haptic paradigms specially conceptualized and evaluated to provide an accurate perception of B-Rep components and to help the user during his/her 3D interactions. Finally, we conclude on some issues for future researches in the field of VR-CAD integration.     

The extrapolated fully implicit scheme for the diffusion equation

The numerical solution of the diffusion equation with homogeneous boundary conditions is discussed with emphasis on the L-stable Lawson-Morris method (LMM) derived by the extrapolation of the fully implicit backward difference method. This type of scheme is particularly suitable when there is an initial/boundary discontinuity. In this paper, a scheme which improves the accuracy of LMM is proposed and discussed.     

A mixed time integration scheme for virtual fabrication of steel plate girders

In this paper the use of mixed time integration is proposed to increase the efficiency of welding simulation. A technique similar to the one commonly used in sheet metal forming is applied to the welding of steel plates. The welding procedure can be divided into two very distinguishable parts with significantly different characteristics. The first part, the phase of the actual welding, is a fast paced, rapidly changing process that involves highly nonlinear material behavior. The second part, the phase of cooling down, is a slow paced, slowly changing process that does not result in dramatic changes in the material behavior. In this project, explicit time integration was used for the thermo-mechanical analysis of welding and implicit time integration was proposed for the subsequent cooling phase. This paper presents examples based on the conventional fully implicit solution and the proposed explicit integration solution using the finite element codes ABAQUS/Standard and ABAQUS/Explicit, respectively.     

NICE h : a higher-order explicit numerical scheme for integration of constitutive models in plasticity

The article introduces, as a result of further development of the first-order scheme NICE, a simple and efficient higher-order explicit numerical scheme for the integration of a system of ordinary differential equations which is constrained by an algebraic condition (DAE). The scheme is based on the truncated Taylor expansion of the constraint equation with order h of the scheme being determined by the highest exponent in the truncated Taylor series. The integration scheme thus conceived will be named NICE ^h , considering both principal premises of its construction. In conjunction with a direct solution technique used to solve the boundary value problem, the NICE ^h scheme is very convenient for integrating constitutive models in plasticity. The plasticity models are defined mostly by a system of algebraic and differential equations in which the yield criterion represents the constraint condition. To study the properties of the new integration scheme, which, like the forward-Euler scheme, is characterised by its implementation simplicity due to the explicitness of its formulations, a damage constitutive model (Gurson–Tvergaard–Needleman model) is considered. The general opinion that the implicit backward-Euler scheme is much more accurate than the thus-far known explicit schemes is challenged by the introduction of the NICE ^h scheme. The accuracy of the higher-order explicit scheme in the studied cases is significantly higher than the accuracy of the classical backward-Euler scheme, if we compare them under the condition of a similar CPU time consumption.     

Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques

This paper deals with the implementation techniques of an implicit integrator to achieve fast and accurate analyses of spacecraft dynamics. For this purpose, the pseudospectral method is adopted to directly integrate the second-order system of equations for both the spacecraft dynamics and corresponding state transition matrix. Various implementation techniques are proposed to enhance the numerical efficiency and integration accuracy, which include a moving horizon approach, the decoupled integration of the second-order dynamics for the state transition matrix, and the grid adaptation method. The numerical features of the proposed techniques are investigated through their applications to a spacecraft’s motion around highly eccentric elliptic orbits, and the resultant numerical errors and computing times are compared with those from the Runge-Kutta method to show the relative efficiency and accuracy of the presented methods. In addition, an optimal two-impulse orbit transfer from the Earth to the Moon is analyzed by implementing the proposed methods using a multiple-shooting framework. The results show that the proposed techniques are extremely effective for dynamical problems requiring intensive and accurate time integrations, and can provide much better accuracy and efficiency than the explicit Runge-Kutta integrator.     

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.     

Applicability and evaluation of an implicit self-starting unconditionally stable methodology for the dynamics of structures

The applicability and evaluation of a new self-starting, unconditionally stable, implicit methodology of computation for the dynamics of structures is described. The methodology offers different perspectives and architecture for structural dynamics compared with the tranditional (widely advocated and commonly used) time integration methods. It is based on velocity representations and architecture and uses finite elements as the principal analysis tool for structural dynamic modeling/analysis. In particular, the dynamics of beam-type flexural models are considered, and comparative results validate and support the proposed use of the self-starting methodology of computation for the dynamics of linear/nonlinear structures. The overall effectiveness and elegance strongly support its use in most existing commercial codes.     

A new model-dependent time integration scheme with effective numerical damping for dynamic analysis

This paper presents a new semi-explicit dissipative model-dependent time integration algorithm for solving structural dynamics problems. Motivated by the superior properties of the composite time-stepping scheme, the proposed method is designed, so that it fully inherits the numerical characteristics of its parent algorithm, namely the Bathe method. The algorithm design procedure is carried out by assuming unknown integration parameters for the proposed method. Afterwards, by time discretization of an SDOF model equation, the unknown parameters can be obtained explicitly by solving nonlinear system of equations. Some numerical examples are analyzed by the presented technique and comparisons are also made with two other dissipative model-dependent time integration algorithms as well as the Bathe method. Results demonstrate that the suggested technique can effectively damp out the spurious oscillations of the high-frequency modes, while the other schemes exhibit significant overshoot in the calculated responses. Furthermore, it is also observed that numerical results of the presented method totally coincide with the parent algorithm. While the Bathe method subdivides each time increment into two sub-steps, the proposed algorithm is single-step, non-iterative and does not involve any time-step subdividing.

     

An implicit enthalpy scheme for one-phase Stefan problems

A compact finite-difference scheme to solve one-phase Stefan problems in one dimension is described. Numerical experiments indicate that the moving interface is obtained withO(t) accuracy when 3–4 iterations per time step are used to solve the nonlinear implicit scheme. The scheme can be adapted to ADI methods in higher dimensions.     

An implicit MacCormack scheme for unsteady flow calculations

This paper describes the implicit MacCormack scheme [1] in finite volume formulation. Unsteady flows with moving boundaries are considered using arbitrary Lagrangian–Eulerian approach.The scheme is unconditionally stable and does not require solution of large systems of linear equations. Moreover, the upgrade from explicit MacCormack scheme to implicit one is very simple and straightforward.Several computational results for 2D and 3D flows over profiles and wings are presented for the case of inviscid and viscous flows.