Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps

General Relativistic Cosmological N-body Simulations I: time integration

This is the first in a series of papers devoted to fully general-relativistic $N$-body simulations applied to late-time cosmology. The purpose of this paper is to present the combination of a numerical relativity scheme, discretization method and time-integration algorithm that provides satisfyingly stable evolution. More precisely, we show that it is able to pass a robustness test and to follow scalar linear modes around an expanding homogeneous and isotropic space-time. Most importantly, it is able to evolve typical cosmological initial conditions on comoving scales down to tenths of megaparsecs with controlled constraint and energy-momentum conservation violations all the way down to the regime of strong inhomogeneity.Comment: 28 pages, 16 figure

TIME INTEGRATION: AGRIBUSINESS STRUCTURE FOR COMPETITIVE ADVANTAGE

Agribusiness,

Real time integration of user preferences into virtual prototypes

Within new product development (NPD), both virtual prototypes and physical prototypes play important roles in creating, testing and modifying designs. However, in the current design process, these two forms of prototyping methods are normally used independently and converted from one to the other during different design phases. This conversion process is time consuming and expensive and also introduces potential information loss/corruption problems. If the design process requires many iterations, it may simply be impractical to generate all the conversions that are theoretically required. Therefore, the integration of virtual and physical prototyping may offer a possible solution where the design definition is maintained simultaneously in both the virtual and physical environment. The overall aim of this research was to develop an interface or a tool that achieves real time integration of physical and virtual prototyping. “Real time integration” here means changes to the virtual prototypes will reflect any changes that have been made contemporaneously to the physical prototypes, and vice versa. Thus, conversion of the prototype from physical to virtual (or vice versa) will be achieved immediately, hence saving time and cost. A review of the literature was undertaken to determine what previous research has been conducted in this area. The result of the review shows the research in this area is still in its infancy. The research hypothesis was developed through the use of a questionnaire survey. Totally 102 questionnaires were sent to designers, design directors or design managers to address the issue: will industrial designers want to make use of real time integration and if so, how? The outcome from the literature review drove further development of the research hypothesis and an initial pilot experiment to test this. The pilot trial was designed to address the research questions: • Can real time physical and virtual prototyping integration be conveniently demonstrated? • Will designers and users be comfortable using the integration method? • Will users recognise the benefits of the integration? The results showed that real time integration between physical and virtual prototyping is necessary in helping designers develop new products and for getting users more closely involved. The future research suggested is that more investigations and experiments are needed to explore a proper method that simultaneously employing these two types of prototyping in product development process. Keywords: Physical Prototyping; Virtual Prototyping; Integration; Real Time.</p

Energy conserving time integration scheme for geometrically exact beam

An energy conserving finite-element formulation for the dynamic analysis of geometrically non-linear beam-like structures undergoing large overall motions has been developed. The formulation uses classical displacement-based planar beam finite elements described in an inertial frame. It takes into account finite axial, bending and shear strains. A theoretically consistent approach is used to derive a novel and simple energy conserving scheme, using the unconventional incremental strain update rather than the standard strong form. Numerical examples demonstrate perfect energy and momenta conservation, stability and robustness of the scheme, and good convergence properties in terms of both the Newton-Raphson method and time step size. (c) 2006 Elsevier B.V. All rights reserved

Time integration for diffuse interface models for two-phase flow

We propose a variant of the $\theta$-scheme for diffuse interface models for two-phase flow, together with three new linearization techniques for the surface tension. These involve either additional stabilizing force terms, or a fully implicit coupling of the Navier-Stokes and Cahn-Hilliard equation. In the common case that the equations for interface and flow are coupled explicitly, we find a time step restriction which is very different to other two-phase flow models and in particular is independent of the grid size. We also show that the proposed stabilization techniques can lift this time step restriction. Even more pronounced is the performance of the proposed fully implicit scheme which is stable for arbitrarily large time steps. We demonstrate in a Taylor flow application that this superior coupling between flow and interface equation can render diffuse interface models even computationally cheaper and faster than sharp interface models