The numerical simulation of a typical fluid-structure interaction problem

To determine the dynamic response of a structure under the influence of the fluid flow one must solve a coupled computational fluid dynamics (CFD) and computational structural dynamics (CSD) mathematical problem. This paper presents the comparison of two methods for the calculation of the fluid-structure interaction. The first one is of explicit-implicit type and uses a staggered time advancement of the fluid and structure problems. The second uses a fully implicit discretization in the physical time of the fluid-structure equations and an explicit advancement in the dual-time. The physical fluid-structure problem is accompanied by the equations of the mesh motion, which are written as for a pseudo-structural system with its own dynamics. Representative numerical results are presented for the two degrees of freedom tipical section in unsteady transonic flow. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparative analysis of numerical methods for MHD simulation on parallel computer systems

Translated from Vychislitel'nye Sistemy i Voprosy Prinyatiya Reshenii, pp. 28–33, 1991.     

Efficient decoupling technique applied to the numerical time integration of advanced interaction models for railway dynamics

Railway interaction is characterised by the coupling between the train and the track introduced through the wheel/rail contact. The introduction of the flexibility in the wheelset and the track through the finite element (FE) method in the last four decades has permitted to study high-frequency phenomena such as rolling noise and squeal, whose origin lies in the strongly non-steady state and non-linear behaviour of the contact forces that arise from the small contact area. In order to address models with a large number of degrees of freedom, innovative Eulerian-modal models for wheelsets with rotation and cyclic tracks have been developed in recent years. The aim of this paper is to extend the resulting formulation to an uncoupled linear matrix equation of motion that allows solving each equation independently for each time step, considerably reducing the associated computational cost. The decoupling integration method proposed is compared in terms of computational performance with Newmark and Runge-Kutta schemes, commonly used in vehicle dynamics, for simulations with the leading wheelset negotiating a tangent track and accounting the rail roughness.     

On large Eddy simulation and variational multiscale methods in the numerical simulation of turbulent incompressible flows

Numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory), and the use of a turbulence model becomes necessary. The paper will discuss several aspects of two approaches of turbulent modeling—Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the detailed derivation of these models, the analysis of commutation errors in LES models as well as other results from mathematical analysis.     

Longitudinal track-structure interaction on railway bridges

In this paper, a computational model for the correct simulation of an arbitrary sequence of longitudinal loading cases on coupled track-bridge systems is proposed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The application of time-domain numerical simulation methods to the microwave heating of foods

Simulation of the microwave processing of food and other materialsin general involves two distinct but interdependent processes.These are the determination of the electromagnetic field profileand the resulting temperature profile as a function of timeinside the load. Both these processes are described in thispaper from the point of view of transmission-line modelling.The techniques described are related to other methods such asthe finite-difference time-domain method. Typical results ofsimulations are presented and outstanding problems are identified.     

On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems

In this paper, an extremely accurate numerical algorithm, namely the “clean numerical simulation” (CNS), is proposed to accurately simulate the propagation of micro-level inherent physical uncertainty of chaotic dynamic systems. The chaotic Hamiltonian Hénon–Heiles system for motion of stars orbiting in a plane about the galactic center is used as an example to show its basic ideas and validity. Based on Taylor expansion at rather high-order and MP (multiple precision) data in very high accuracy, the CNS approach can provide reliable trajectories of the chaotic system in a finite interval t ∈ [0, T_c], together with an explicit estimation of the critical time T_c. Besides, the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion. In this way, the numerical noises of the CNS can be reduced to a required level, i.e. the CNS is a rigorous algorithm. It is illustrated that, for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10⁻¹²⁴⁴ and 10⁻¹⁰⁰⁰, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10⁻⁶⁰) of the Hénon–Heiles system can be investigated accurately. It is found that, due to the sensitive dependence on initial condition (SDIC) of chaos, the micro-level inherent physical uncertainty of the position and velocity of a star transfers into the macroscopic randomness of motion. Thus, chaos might be a bridge from the micro-level inherent physical uncertainty to the macroscopic randomness in nature. This might provide us a new explanation to the SDIC of chaos from the physical viewpoint.     

Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations

Summary. This paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the Hartree-Fock equations. Upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues.     

3D numerical simulation on fluid-structure interaction of structure subjected to underwater explosion with cavitation

In the underwater-shock environment,cavitation occurs near the structural surface.The dynamic response of fluid-structure interactions is influenced seriously by the cavitation effects.It is also the d...     

CHARACTERISTICS-FINITE ELEMENT METHODS FOR SEAWATER INTRUSION NUMERICAL SIMULATION AND THEORETICAL ANALYSIS

0.IntroductionSeawaterintrusionisaproblemwhichhasmuchtodowithnaturalresourcesandenvironmentinmodernsociety,andisgettingmoreandmoreseriousinmanyajreasofourcountryandmailyothercountriesintheworldsuchasUSA,NetherlandsandIsrael.Theharmbroughtaboutbyseawaterintrusionisimmeasurable.Theareasintrudedbyseawaterareusuallycoastalalluvialplainswherethesoilusedtobefertile,groundwaterreservesarerichandagricultureiswell-developed.Becauseofseawaterintrusion,groundwaterbecomessaltedandunfitforpeopleandanimal…     

Numerical simulation of fluid-structure interaction problems on hybrid meshes with algebraic multigrid methods

Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods.     

Feedback methods for inverse simulation of dynamic models for engineering systems applications

Inverse simulation is a form of inverse modelling in which computer simulation methods are used to find the time histories of input variables that, for a given model, match a set of required output responses. Conventional inverse simulation methods for dynamic models are computationally intensive and can present difficulties for high-speed applications. This article includes a review of established methods of inverse simulation, giving some emphasis on iterative techniques that were first developed for aeronautical applications. It goes on to discuss the application of a different approach that is based on feedback principles. This feedback method is suitable for a wide range of linear and non-linear dynamic models and involves two distinct stages. The first stage involves design of a feedback loop around the given simulation model, and in the second stage, that closed-loop system is used for inversion of the model. Issues of robustness within closed-loop systems used in inverse simulation are not significant as there are no plant uncertainties or external disturbances. Thus, the process is simpler than that required for the development of a control system of equivalent complexity. Engineering applications of this feedback approach to inverse simulation are described through case studies that put particular emphasis on non-linear and multi-input multi-output models.     

Some remarks on the efficient algorithms of numerical statistical simulation

Some numerical algorithms for the simulation of random variables and functions, as well as for parametrical numerical statistical analysis are considered. Specifications and explanations of the algorithms important for practical purposes are given.     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

Structural stability on basins for numerical methods

In this paper, we show that a flow with a hyperbolic compact attracting set is structurally stable on the basin of attraction with respect to numerical methods. The result is a generalized version of earlier results by Garay, Li, Pugh, and Shub. The proof relies heavily on the usual invariant manifold theory elaborated by Hirsch, Pugh, and Shub (1977), and by Robinson (1976).

     

Unified formulation of numerical methods

A summary of the unified theory of numerical methods recently published by the author is presented [5–8]. In the theory, a direct method of analysis is developed applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions. The scheme includes finite differences, finite elements and boundary methods. A fundamental ingredient of the procedure are general Green's formulas for operators defined in discontinuous fields.     

On numerical simulation of tsunami run-up

The improved adaptive grid method is presented for the numerical modelling of propagation of long surface waves in large water areas and their interaction with coasts. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The numerical methods in the field of operators

The approximate solution of a class of nonlinear differential equations, in the field of Mikusiński operators is constructed by using the Euler method. The obtained results are applied to a class of partial integro–differential equations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the numerical analysis of analytical nodal methods

This article presents the basic numerical analysis of the analytical nodal methods, which were originally developed in the late 1970s in relation with static and dynamic nuclear reactor calculations but are actually applicable to the numerical solution of partial differential equations (PDEs) in general, over fairly regular domains. The basic idea consists in “transverse integrating” the original PDE over all the variables minus one, leading to sets of 1D equations which are then solved in an “analytical” way, using fundamentals as well as particular solutions of the corresponding 1D operators. After examining the existing analytical methods in a critical way, we propose a more satisfactory extended analytical formalism. Superconvergence results finally lead us to useful conclusions with respect to the choice of a particular scheme.