General Numerical Scheme Research Articles

NUMERICAL CALCULATION OF MAGNETOBREMSSTRAHLUNG EMISSION AND ABSORPTION COEFFICIENTS

Magnetobremsstrahlung (MBS) emission and absorption play a role in many astronomical systems. We describe a general numerical scheme for evaluating MBS emission and absorption coefficients for both polarized and unpolarized light in a plasma with a general distribution function. Along the way we provide an accurate scheme for evaluating Bessel functions of high order. We use our scheme to evaluate the accuracy of earlier fitting formulae and approximations. We also provide an accurate fitting formula for mildly relativistic (kT /(mec 2 ) 0.5) thermal electron emission (and therefore absorption). Our scheme is too slow, at present, for direct use in radiative transfer calculations but will be useful for anyone seeking to fit emission or absorption coefficients in a particular regime.

Numerical Approximation of Black-Scholes Equation

This study deals with well-known Black-Scholes model in a complete financial market. We obtain numerical methods for european and exotic options, for one asset and for two assets models. Introduction. Options are used on markets and exchanges. The Black- Scholes model is a convenient way to calculate the price of an option. In this study numerical methods will be processed to solve that equation. The numerical methods are based on finite differences. We test fully implicit, semi-implicit and explicit methods. For the discretized problem, a linear algebraic system, we test direct and iterative methods. This way we intend to create a general numerical scheme for different types of options. Section 1 deals with the Black-Scholes model. In this part we show some options used, European call and put option, and the mathematical properties of these options. Section 2 shows our numerical setup for the equation presented in Sec- tion 1. This section deals with the space discretization of the parabolic differential equation with one underlying asset, using finite difference meth- ods. We obtain an explicit and two implicit methods (fully implicit and semi-implicit method) to solve the discretized system. Numerical experi- ments are related for each method.

Transient elastohydrodynamic lubrication in artificial knee joint with non-Newtonian fluids

This paper presents the transient analysis of a human artificial knee joint under elastohydrodynamic lubrication (EHL) for point contact with non-Newtonian lubricants. The artificial knee joints use ultra high molecular weight polyethylene (UHMWPE) against metal with time-varying speed and load during walking. This numerical simulation employed a perturbation method, Newton Raphson method and multigrid method with full approximation technique to solve simultaneously both the time-dependent Reynolds equation, with non-Newtonian fluid based on a Carreau model, and the elasticity equation. The general numerical schemes are implemented to investigate the characteristics of elastohydrodynamic lubrication in human artificial knee joints; profiles of pressure and film thickness are determined, with varying material and lubricant properties, applied loads and speeds. The results show that the elastohydrodynamic fluid film thickness between the metallic component of the artificial knee joint and the soft polyethylene bearing becomes larger as the contact area increases and the fluid film pressure decreases. At the beginning of the first walking cycle, the film thickness is lower than in subsequent cycles because of the time required to develop the fluid film; after the first cycle, the fluid film is similar for every cycle and is dependent on transient applied load and speed during human movement.

A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.

A new numerical approach to the design of foundation piles in laterally spreading liquefied soils

In this contribution, a general numerical scheme is proposed aimed at simulating the lateral spreading of a liquefied soil mass, modelled as a viscoplastic Bingham medium, along with evaluating its impact on neighbouring structures by calculating the lateral loading exerted by the liquefied ground. An illustrative application of this approach implemented in a finite element code using the decomposition-coordination method by augmented Lagrangian is also presented.

A Formulation and Numerical Scheme for Fractional Optimal Control Problems

This article presents a general formulation and general numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented, and the resulting equations, are very similar to those that appear in classical optimal control theory. Thus, the present formulation essentially extends classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme for finding the approximate numerical solution of the resulting equations is presented. For a linear system, this method results in a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach the classical solutions as the order of the fractional derivatives approaches 1. The formulation presented is simple, and can be extended to other FOCPs.

Classical and Advanced Models for Laminated Plates with Piezoelectric Layers Actuated in Shear Mode

We discuss the two-dimensional modelling of laminated plates with embedded piezoelectric layers actuated in shear mode. Refined approaches are necessary to capture all relevant phenomena characterizing the electro-mechanical interactions. The framework of our approach is a previously established Unified Formulation, which permits the use of Equivalent Single Layer as well as Layer-Wise descriptions in conjunction with higher-order thickness approximations. Two distinct model types are proposed: classical models based on Hamilton's principle and advanced models based on a novel partially mixed four-field formulation. Advanced models are capable of a priori fulfilling the interlaminar continuity of all transverse fluxes. In this paper, we adopt an analytical Navier-type closed-form solution method to solve the resulting system of coupled partial differential equations governing the static and the free-vibration problems. Different interlaminar conditions and electrode configurations are addressed. The proposed 2D models recover accurately the global and local response of exact 3D solutions available in literature. An assessment of various 2D models confirms the superiority of the advanced models with respect to classical ones. By virtue of their accuracy, the analytical solutions obtained from our higher-order advanced models could be employed for providing reference solutions for more general numerical approximation schemes like the FEM which are more suitable for practically relevant applications.

Equation-free implementation of statistical moment closures

We present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex, large-scale, nonlinear systems. Building on recently developed equation-free methods, this approach numerically integrates the closure dynamics, the equations of which may not even be available in closed form. Although closure dynamics introduce statistical assumptions of unknown validity, they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model. The numerical closure approach can in principle be applied to a wide class of nonlinear problems, including strongly coupled systems (either deterministic or stochastic) for which there may be no scale separation. We demonstrate the equation-free approach for implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic partial differential equation.

Image analysis using p ‐Laplacian and geometrical PDEs

AbstractMinimizing the integral ∫Ω1/p |∇L |p d Ω for an image L under suitable boundary conditions gives PDEs that are well‐known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p ‐Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Contour dynamics in complex domains

This paper demonstrates that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected planar domain bounded by impenetrable walls. A general numerical scheme is presented based on this formulation. The algorithm makes use of conformal mappings and follows the evolution of a conformal pre-image of a given vortex patch in a canonical multiply connected circular pre-image region. The evolution of vortex patches can be computed given just the conformal map from this pre-image region to the physical fluid region. The efficacy of the scheme is demonstrated by illustrative examples.

WHAM: a WENO-based general relativistic numerical scheme - I. Hydrodynamics

Active galactic nuclei, X-ray binaries, pulsars and gamma-ray bursts are all believed to be powered by compact objects surrounded by relativistic plasma flows driving phenomena such as accretion, winds and jets. These flows are often accurately modelled by the relativistic magnetohydrodynamic (MHD) approximation. Time-dependent numerical MHD simulations have proven to be especially insightful, but one regime that remains difficult to simulate is when the energy scales (kinetic, thermal, magnetic) within the plasma become disparate. We develop a numerical scheme that significantly improves the accuracy and robustness of the solution in this regime. We use a modified form of the weighted essentially non-oscillatory (WENO) method to construct a finite-volume general relativistic hydrodynamics code called wham that converges at fifth order. We avoid (1) field-by-field decomposition by adaptively reducing down to two-point stencils near discontinuities for a more accurate treatment of shocks and (2) excessive reduction to low-order stencils, as in the standard WENO formalism, by maintaining high-order accuracy in smooth monotonic flows. Our scheme performs the proper surface integral of the fluxes, converts cell-averaged conserved quantities to point-conserved quantities before performing the reconstruction step, and correctly averages all source terms. We demonstrate that the scheme is robust in strong shocks, very accurate in smooth flows and maintains accuracy even when the energy scales in the flow are highly disparate.

Mesh-Free Simulation of Transport Phenomena in Continuous Castings of Aluminium Alloys

This paper introduces a general numerical scheme for solving convective-diffusive problems that appear in the solution of microscopic and macroscopic transport phenomena in continuous castings and the heat treatment of aluminium alloys. The numerical scheme is based on spatial discretisation that involves pointisation only. The solution is based on diffuse collocation with multi-quadric radial basis functions. The application of the method is demonstrated in a simplified model of a billet DC casting and verified by a comparison with the classical finite volume method.

A FORMULATION AND A NUMERICAL SCHEME FOR FRACTIONAL OPTIMAL CONTROL PROBLEMS

This paper presents a general formulation and a general numerical scheme for a class of Fractional Optimal Control Problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fractional Differential Equations (FDEs). The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An iterative numerical scheme is presented to find the approximate numerical solution of the resulting equations. For a linear system, this method results into a set of linear simultaneous equations, which can be solved directly. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

Assessing the Role of Curvature Effects in the Representation of Anisotropy Transport in Algebraic Reynolds-Stress Modelling Applied to Separated Flow

A “curvature correction”, in the sense used herein, is model fragment that is intended to account for some proportion of the error that is introduced into explicit algebraic Reynolds-stress models as a consequence of ignoring the convective transport of stress anisotropy within a framework in which the velocity vector and stress tensor are represented in terms of Cartesian components in highly-curved flow conditions. The present paper examines the ability of such corrections to represent this error, based on a-priori investigations of representative 2-D and 3-D massively separated flows for which the ‘exact’ level of the stress transport is known. In essence, the corrections reflect the assumption that the convection of the Reynolds-stress components, or the associated strain-tensor components, expressed in terms of curvature-oriented coordinates, is negligible. The analysis shows, first, that in general recirculating flows, the contribution of anisotropy transport to the stress balance is generally small, so that any form of related correction is of little consequence. Second, the variants of curvature correction examined correlate poorly with the real anisotropy convection. Thus, while these curvature corrections are useful in very particular conditions, such as flow in highly-curved ducts, they are not generality effective – indeed, possibly counterproductive – and cannot be recommended for inclusion in general numerical schemes.

Parallel 3-d simulations for porous media models in soil mechanics

Numerical simulations in 3-d for porous media models in soil mechanics are a difficult task for the engineering modelling as well as for the numerical realization. Here, we present a general numerical scheme for the simulation of two-phase models in combination with an abstract material model via the stress response with a specialized parallel saddle point solver. Therefore, we give a brief introduction into the theoretical background of the Theory of Porous Media and constitute a two-phase model consisting of a porous solid skeleton saturated by a viscous pore-fluid. The material behaviour of the skeleton is assumed to be elasto-viscoplastic. The governing equations are transfered to a weak formulation suitable for the application of the finite element method. Introducing an abstract formulation in terms of the stress response, we define a clear interface between the assembling process and the parallel solver modules. We demonstrate the efficiency of this approach by challenging numerical experiments realized on the Linux Cluster in Chemnitz.

Breaking and merging of liquid sheets and filaments

The surface of a sheet of liquid which contracts due to surface tension, breaks, and then pulls apart into two pieces, is calculated. Before breaking, the flow is the self-similar one found by Keller, Milewski and Vanden-Broeck. After breaking, it is the self-similar flow found by Keller and Miksis. A general numerical scheme, which includes the previous ones, is presented and new numerical results are discussed. There is an analogous flow of an axially symmetric liquid filament, but it is not calculated.

Calculation of intrinsic hysteresis loops of ferroelectric thin films

A general numerical scheme is applied to calculate the dielectric equation of state (mean polarisation versus applied field) and resulting hysteresis loop for ferroelectric films. The scheme is sufficiently general to apply to films in which the phase transition from the paraelectric state is either first or second order, and possible variation of the polarisation P near the surfaces is included. It is shown that by design of the film thickness and of the variation of P the hysteresis loop in a film of first-order material can be varied between forms which resemble those of the bulk material in various temperature intervals.

First-order phase transitions in ferroelectric films

We discuss a version of the Landau–Devonshire model for films undergoing a first-order phase transition. The spatial variation P( z) of the order parameter is described by an Euler–Lagrange equation with associated boundary condition. We define a general numerical scheme for finding P( z) and evaluating the resulting thermodynamic functions. Results are presented for the thickness, temperature and boundary-condition dependence of P( z), the free energy and entropy and the superheating, supercooling and thermodynamic critical temperatures.

Implicit Bidiagonal Scheme for Depth-Averaged Free-Surface Flow Equations

A general fast implicit bidiagonal numerical scheme, based on the MacCormack's predictor-corrector technique requiring the inversion of only block bidiagonal matrices, has been developed and subsequently applied for subcritical and supercritical free-surface flow calculations. The model has been applied to depth-averaged steady flows. There are two main advantages of the proposed method: the technique has fast convergence and utilizes a body fitted nonorthogonal local coordinate system to simulate irregular geometry flows. The model is used to analyze a wide variety of hydraulic engineering problems including flows in a converging-diverging subcritical flume, supercritical expansions at various Froude numbers, and supercritical converging chutes. For each of these test cases, the calculated results are compared with experimental data. The comparisons with measurements as well as with other numerical solutions show that the proposed method is comparatively accurate, fast, and reliable.

On the importance of cross-sectional warping in solid composite beams

The relative importance of the cross-sectional warping components in composite beams is studied and demonstrated using an exact solution for solid orthotropic beam of arbitrary cross-sectional geometry that undergoes a bending moment. In light of the effort required for warping modeling in general numerical schemes of composite beams, the present study contributes to the understanding of the importance of modeling the in-plane and the out-of-plane warping components.