Simple Explicit Method Research Articles

Comparison of Corrected and Uncorrected Enthalpy Methods for Solving Conduction-Driven Solid/Liquid Phase Change Problems

The numerical study of solid/liquid phase change problems represents a large and ongoing field of research with many applications. These simulations should run as fast and accurately as possible. Therefore, proceeding from previous work and findings from the literature, this study investigates enthalpy methods for solving solid/liquid phase change problems. The relationship between temperature and enthalpy is strongly non-linear and requires special treatment; iteratively corrected methods, as well as approaches that do not correct the temperature/enthalpy relationship at all or only once per time step, were considered for a one-dimensional test problem. Based on the results of this study, two solvers can be recommended, the so-called optimum approach and a simple explicit method; both provide accurate results. The explicit method is easy to program, but the optimum approach allows larger time steps and is, therefore, faster. The influence of several parameters was investigated. The mesh resolution strongly influenced the accuracy and the computational speed, and the time step size barely influenced the accuracy but did affect the computational speed. An artificial melting temperature range influenced the accuracy but had hardly any influence on the simulation speed. Higher-order time discretization schemes were not superior compared to the first-order implicit optimum approach.

Study on non-associated plasticity with various forward Euler stress integration algorithms and its prediction of earing in cylindrical cup drawing

Three anisotropic yield criteria, such as Hill's 48 [3], Yld2000-2d [6] as well as Drucker [32] under non-associated flow rule (simplified as non-AFR) are applied to predict earing profile after cylindrical cup drawing of aluminum alloy sheet AA2090-T3. Three forward Euler stress integration algorithms, i.e. tangent cutting plane explicit method (TCPE), semi explicit method (SEMIE) and simple explicit method (SIMPE) are implemented into user subroutines of ABAQUS/Explicit to simulation the deformation process. The predicted anisotropy coefficients including stress directionalities and r-values by single element tensile simulation are made comparisons with observed experimental results. The computation stability of different stress integration algorithms are also compared using specified time increments. The prediction of earing profile by anisotropic Hill's 48 and Yld2000-2d as well as Drucker yield functions with non-associated plasticity are compared with results of experiment to evaluate their accuracy. The computation times are also compared to evaluate the computing efficiency of these yield functions with different stress integration algorithms in numerical simulation of cup deep drawing. The comparison demonstrates that non-associated plastic constitutive model for Yld2000-2d using TCPE and SEMIE stress integration algorithms is the most accurate to simulate 6 earing profiles, while non-associated plastic constitutive model for Drucker using 3D solid elements is the most efficient. All the numerical simulation results indicate that no matter 2D shell or 3D solid elements are used in the simulation models, the TCPE stress integration algorithm is advised in consideration of computational efficiency and accuracy.

Explicit incipient motion of cohesive and non‐cohesive sediments using simple hydraulics

AbstractExisting dimensionless expressions that represent the incipient motion of sediments are based on studies of non‐cohesive sediments. Because of the complex behaviour of cohesive sediments, many simulators also assume non‐cohesiveness when simulating the erosion of cohesive sediments. However, studies show that the critical shear force needed for entrainment is much higher for consolidated cohesive sediments than for similarly sized non‐cohesive sediments. Treating cohesive sediments as non‐cohesive sediments thus will introduce a significant error with regard to quantifying the eroded sediment mass. On the other hand, the existing expressions of non‐cohesive sediments require relatively detailed hydraulic calculations to estimate the shear velocity or the bed shear stress and thus cannot be used with simplified simulators. Therefore, it is essential to have a versatile simple explicit method that estimates the incipient motion condition of both the consolidated cohesive and non‐cohesive sediments whenever needed. In this paper, explicit analytical expressions are proposed that simulate the incipient motion of consolidated cohesive and non‐cohesive sediments, based on the critical erosion curves of the Hjulström–Sundborg–Miedema diagram. The new method reproduces the latter diagram with high precision. It also mimics the critical incipient condition of non‐cohesive sediments determined by a well‐known analytical method for other experimental data sets and for the East Fork River without the need of an iterative solution. The new approach provides essential information for estimating the entrainment condition of pebbles or finer sediments. Besides, the use of the mean flow velocity and the flow depth as predictors of incipient condition allows for its easy and efficient implementation in conceptual simulators that do not perform detailed hydraulic calculations and for use by modelers that are not familiar with the hydrotechnical literature. It also reduces the computation time required for simulation.

Assessment of Basic Approaches to Numerical Modeling of Phase Change Problems—Accuracy, Efficiency, and Parallel Decomposition

The fast and accurate modeling of phase change is of a significant importance in many processes from steel casting to latent heat thermal energy storage. The paper presents a numerical case study on the transient 3D heat diffusion problem with phase change. Three different approaches to modeling of the solid–liquid phase change in combination with four commonly used numerical schemes are compared for their efficiency, accuracy, applicability, simplicity of implementation, and robustness. The possibility of parallel decomposition of the approaches is also discussed. The results indicate that the best accuracy was achieved with the second-order implicit methods, and the best efficiency was reached with the simple explicit methods.

A Linearly Convergent Dual-Based Gradient Projection Algorithm for Quadratically Constrained Convex Minimization

This paper presents a new dual formulation for quadratically constrained convex programs. The special structure of the derived dual problem allows us to apply the gradient projection algorithm to produce a simple explicit method involving only elementary vector-matrix operations, proven to converge at a linear rate.

Improving simple explicit methods for unsteady open channel and river flow

AbstractA rigorous study of the explicit Lax–Friedrichs scheme for its application to one‐dimensional shallow water flows is presented. The deficiencies of this method are identified and the way to overcome them are presented. It is compared to the explicit first order upwind scheme and to the explicit second order Lax–Wendroff scheme by means of the simulation of several test cases with exact solution. All three schemes in their best balanced version are applied to the simulation of a real river flood wave leading to very satisfactory results. Copyright © 2004 John Wiley & Sons, Ltd.

Analytic Prediction of Surface Pressures over a Hemisphere-Cylinder at Incidence

The utility of the hemisphere cylinder cone guration lends itself to widespread application, ranging from measurement probes to underwater vehicles. A simple explicit method to estimate the surface pressures over this cone guration would be of use for both conceptual design and for understanding the relation of the design variables. Consequently, an explicit equation is developed that requires only the hemisphere cylinder incidence and the location of the point of interest to yield the e nal surface e ow. Surface pressures are calculated using perturbation potentials calculated from least-square curve e ts to numerically estimated potentials. Comparisons of the expression with experimental data show good agreement.

Fast approximate maximum likelihood algorithm for single source localisation

The authors present an approximation for the deterministic maximum likelihood (ML) method for estimating the direction of arrival of a signal from a single source. To apply the proposed approximate ML (AML) method one has to compute the principal eigenvector of the sample covariance matrix, i.e. the unit eigenvector corresponding to the largest eigenvalue. Surprisingly, the AML method coincides with the principal eigenvector method of Evans, Johnson and Sun (1982) that was derived based on the unnecessary assumption of high signal-to-noise ratio. Next, the authors present for the AML method a fast and simple explicit method for computing the principal eigenvector of the sample covariance matrix as well as an explicit estimator for the signal-to-noise ratio. The proposed explicit AML (EAML) method is faster than the ML method in carrying out the maximisation step. Simulations reveal that AML and ML methods have similar performance, while EAML performance is only slightly inferior to the ML method.

Numerical Methods for Partial Differential Equations.

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

Numerical and laboratory modeling of propagation over troughs and ridges

Numerical predictions and laboratory measurements of the underwater sound pressure field are presented for four different bathymetry profiles: a range-independent control case, two trough shapes, and a ridge. Differences between numerical and experimental transmission loss are shown to be consistent with small differences in assumed water depth and these differences are used to invert the measured data to extract the actual bathymetry, using a simple explicit method applicable to slowly varying but otherwise arbitrary bottom profiles. In retrospect, measurement of the position of interference fringes provides a means of inferring the water depth profile.

A Simple Explicit form of Finite Differences to Compute the Effective Gain in Degenerate Two-wave Mixing; Comparison with the Shooting Method

Abstract A numerical study is presented of the effective gain in degenerate two-wave mixing, for a reflection geometry in a photorefractive crystal. Apart from the so-called ‘shooting method’ we employ a simple explicit method of ‘finite differences’ to obtain numerical solutions of the nonlinear coupled-wave equations. A comparison between the two indicates that the computed results from the finite differences method are still in reasonable agreement with those obtained from the shooting method. At the same time the finite differences method is more straightforward and usually found to be faster than the shooting method for a small number of discrete points. The dependence of the effective gain on various parameters is shown at the saturation as well as non-saturation (i.e. below saturation) stage of the refractive-index modulation. A significant difference in gain is found to exist between these two regimes of modulation. Numerical results are compared and presented as graphs.

A new very simple explicit method for the integration of mildly stiff ordinary differential equations

A new explicit, first-order method with improved stability properties for the integration of stiff ODE systems is developed. The new method is very simple and requires only two derivative evaluations per step (like the second-order explicit Runge—Kutta method) but has a stability region that is almost four times larger than that of RK2. Furthermore, error control/variable-step integration for the new method is easy. The new method is more efficient than RK2 for typical stiff problems (including method-of-lines problems). It is most efficient on mildly and moderately stiff problems when low or intermediate accuracy is desired. Furthermore, when limited computer memory is available, the new method offers a valuable alternative to the implicit methods (especially for very large stiff systems). In addition, the use of global extrapolation is shown to improve the accuracy of the new method and to offer global error estimates.