A study of the stability of subcycling algorithms in structural dynamics

Algorithms for explicit integration of structural dynamics problems with multiple time steps (subcycling) are investigated. Only one such algorithm, due to Smolinski and Sleith has proved to be stable in a classical sense. A simplified version of this algorithm that retains its stability is presented. However, as with the original version, it can be shown to sacrifice accuracy to achieve stability. Another algorithm in use is shown to be only statistically stable, in that a probability of stability can be assigned if appropriate time step limits are observed. This probability improves rapidly with the number of degrees of freedom in a finite element model. The stability problems are shown to be a property of the central difference method itself, which is modified to give the subcycling algorithm. A related problem is shown to arise when a constraint equation in time is introduced into a time-continuous space-time finite element model. (C) 1998 Elsevier Science S.A.

Finite element analysis of flow patterns near geological lenses in hydrodynamic and hydrothermal systems

We use theoretical and numerical methods to investigate the general pore-fluid flow patterns near geological lenses in hydrodynamic and hydrothermal systems respectively. Analytical solutions have been rigorously derived for the pore-fluid velocity, stream function and excess pore-fluid pressure near a circular lens in a hydrodynamic system. These analytical solutions provide not only a better understanding of the physics behind the problem, but also a valuable benchmark solution for validating any numerical method. Since a geological lens is surrounded by a medium of large extent in nature and the finite element method is efficient at modelling only media of finite size, the determination of the size of the computational domain of a finite element model, which is often overlooked by numerical analysts, is very important in order to ensure both the efficiency of the method and the accuracy of the numerical solution obtained. To highlight this issue, we use the derived analytical solutions to deduce a rigorous mathematical formula for designing the computational domain size of a finite element model. The proposed mathematical formula has indicated that, no matter how fine the mesh or how high the order of elements, the desired accuracy of a finite element solution for pore-fluid flow near a geological lens cannot be achieved unless the size of the finite element model is determined appropriately. Once the finite element computational model has been appropriately designed and validated in a hydrodynamic system, it is used to examine general pore-fluid flow patterns near geological lenses in hydrothermal systems. Some interesting conclusions on the behaviour of geological lenses in hydrodynamic and hydrothermal systems have been reached through the analytical and numerical analyses carried out in this paper.

Finite element analysis of heat transfer and mineralization in layered hydrothermal systems with upward throughflow

We use the finite element method to solve the coupled problem between convective pore-fluid flow, heat transfer and mineralization in layered hydrothermal systems with upward throughflow. In particular, we present the improved rock alteration index (IRAI) concept for predicting the most probable precipitation and dissolution regions of gold (Au) minerals in the systems. To validate the numerical method used in the computation, analytical solutions to a benchmark problem have been derived. After the numerical method is validated, it is used to investigate the pattern of pore-fluid Aom, the distribution of temperature and the mineralization pattern of gold minerals in a layered hydrothermal system with upward throughflow. The related numerical results have demonstrated that the present concept of IRAI is useful and applicable for predicting the most probable precipitation and dissolution regions of gold (Au) minerals in hydrothermal systems. (C) 2000 Elsevier Science S.A. All rights reserved.

Finite element modelling of rock alteration and metamorphic process in hydrothermal systems

We use the finite element method to simulate the rock alteration and metamorphic process in hydrothermal systems. In particular, we consider the fluid-rock interaction problems in pore-fluid saturated porous rocks. Since the fluid rock interaction takes place at the contact interface between the pore-fluid and solid minerals, it is governed by the chemical reaction which usually takes place very slowly at this contact interface, from the geochemical point of view. Due to the relative slowness of the rate of the chemical reaction to the velocity of the pore-fluid flow in the hydrothermal system to be considered, there exists a retardation zone, in which the conventional static theory in geochemistry does not hold true. Since this issue is often overlooked by some purely numerical modellers, it is emphasized in this paper. The related results from a typical rock alteration and metamorphic problem in a hydrothermal system have shown not only the detailed rock alteration and metamorphic process, but also the size of the retardation zone in the hydrothermal system. Copyright (C) 2001 John Wiley & Sons, Ltd.

Finite element modelling of heat transfer through permeable cracks in hydrothermal systems with upward throughflow

We use the finite element method to model the heat transfer phenomenon through permeable cracks in hydrothermal systems with upward throughflow. Since the finite element method is an approximate numerical method, the method must be validated before it is used to soh,e any new, kind of problem. However, the analytical solution, which can be used to validate the finite element method and other numerical methods, is rather limited in the literature, especially, for the problem considered here. Keeping this in mind, we have derived analytical solutions for the temperature distribution along the vertical axis of a crack in a fluid-saturated porous layer. After the finite element method is validated by comparing the numerical solution with the analytical solution for the same benchmark problem, it is used to investigate the pore-fluid flow and heat transfer in layered hydrothermal systems with vertical permeable cracks. The related analytical and numerical results have demonstrated that vertical cracks are effective and efficient members to transfer heat energy from the bottom section to the top section in hydrothermal systems with upward throughflow.

Universal simulation of Hamiltonian dynamics for quantum systems with finite-dimensional state spaces

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd [Phys. Rev. A 65, 040301(R) (2002)] provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems are qudits, that is, have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries.

Anisotropy-based robust performance analysis of finite horizon linear discrete time varying systems

We consider a problem of robust performance analysis of linear discrete time varying systems on a bounded time interval. The system is represented in the state-space form. It is driven by a random input disturbance with imprecisely known probability distribution; this distributional uncertainty is described in terms of entropy. The worst-case performance of the system is quantified by its a-anisotropic norm. Computing the anisotropic norm is reduced to solving a set of difference Riccati and Lyapunov equations and a special form equation.