Finite element solution of the unsteady Navier-Stokes equations by a fractional step method

A finite element method is presented for the numerical simulation of time-dependent incompressible viscous flows. The method is based on a fractional step approach to the time integration of the Navier-Stokes equations in which only the incompressibility condition is treated implicitly. This leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost-effective solutions of large-scale problems. Numerical results indicate the versatility and effectiveness of the proposed method.     

Application of a fractional step method for the numerical solution of the shallow water waves in a rotating rectangular basin

A numerical method is applied to the problem of an incompressible fluid in a slowly rotating rectangular basin for the simulation of wave propagation in shallow water. The present work is a complete study of the wave motion through evaluation of the wave height and the velocity components. The results are found by the application of a fractional step method and illustrated graphically. The technique is applied by splitting the shallow water equations and successive integration in every direction along the characteristics using the Riemann invariants associated with cubic spline interpolation. It has the advantage of reducing the multidimensional matrix inversion problem into an equivalent one-dimensional problem. Numerical results are represented in three dimensions for the velocity components at different times. The distribution of temperature and concentration are also calculated and plotted.     

Emulsion droplet deformation and breakup with Lattice Boltzmann model

In this paper we have performed an extensive study of the effects of various dimensionless numerical parameters used in the Lattice Boltzmann implementation of the diffuse interface model describing deformation and breakup of an emulsion droplet in 2D. Such an extensive study on these parameter is absent in scientific literature of diffuse interface models. We have found that parameters like the dimensionless interface thickness and the Peclet number have to be within certain ranges for correct physical behavior. Outside these ranges droplets either dissolve, show incorrect Laplace pressures, or do not deform to stable shapes at subcritical capillary numbers. Furthermore, we have found that droplet breakup is sensitive to these parameters.     

A variational approach to multi-phase motion of gas, liquid and solid based on the level set method

We propose a simple and robust numerical algorithm to deal with multi-phase motion of gas, liquid and solid based on the level set method [S. Osher, J.A. Sethian, Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulation, J. Comput. Phys. 79 (1988) 12; M. Sussman, P. Smereka, S. Osher, A level set approach for capturing solution to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146; J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999; S. Osher, R. Fedkiw, Level Set Methods and Dynamics Implicit Surface, Applied Mathematical Sciences, vol. 153, Springer, 2003]. In Eulerian framework, to simulate interaction between a moving solid object and an interfacial flow, we need to define at least two functions (level set functions) to distinguish three materials. In such simulations, in general two functions overlap and/or disagree due to numerical errors such as numerical diffusion. In this paper, we resolved the problem using the idea of the active contour model [M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision 1 (1988) 321; V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, International Journal of Computer Vision 22 (1997) 61; G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001; R. Kimmel, Numerical Geometry of Images: Theory, Algorithms, and Applications, Springer-Verlag, 2003] introduced in the field of image processing.     

Cluster integration method in Lagrangian particle dynamics

An efficient and robust approach is proposed in order to conduct numerical simulations of collisional particle dynamics in the Lagrangian framework. Clusters of particles are made of particles that interact or may interact during the next global time-step. Potential collision partners are found by performing a test move, that follows the patterns of a hard-sphere model. The clusters are integrated separately and the collisional forces between particles are given by a soft-sphere collision model. However, the present approach also allows longer range inter-particle forces. The integration of the clusters can be done by any one-step ordinary differential equation solver, but for dilute particle systems, the variable step-size Runge-Kutta solvers as the Dormand and Prince scheme [J. Comput. Appl. Math. 6 (1980) 19] are superior. The cluster integration method is applied on sedimentation of 5000 particles in a two-dimensional box. A significant speed-up is achieved. Compared to a traditional discrete element method with the forward Euler scheme, a speed-up factor of three orders of magnitude in the dilute regime and two orders of magnitude in the dense regime were observed. As long as the particles are dilute, the Dormand and Prince scheme is ten times faster than the classical fourth-order Runge-Kutta solver with fixed step size.     

Fast multipole methods on a cluster of GPUs for the meshless simulation of turbulence

Recent advances in the parallelizability of fast N-body algorithms, and the programmability of graphics processing units (GPUs) have opened a new path for particle based simulations. For the simulation of turbulence, vortex methods can now be considered as an interesting alternative to finite difference and spectral methods. The present study focuses on the efficient implementation of the fast multipole method and pseudo-particle method on a cluster of NVIDIA GeForce 8800 GT GPUs, and applies this to a vortex method calculation of homogeneous isotropic turbulence. The results of the present vortex method agree quantitatively with that of the reference calculation using a spectral method. We achieved a maximum speed of 7.48 TFlops using 64 GPUs, and the cost performance was near $9.4/GFlops. The calculation of the present vortex method on 64 GPUs took 4120 s, while the spectral method on 32 CPUs took 4910 s.     

Dynamic procedures of image reconstruction from projections (computer tomography)

A method of reconstruction of images from projections is suggested. In contrast to the static procedure conventionally used for the solution of problems of image reconstruction from projections, this method presupposes the use of a dynamic procedure. Such an approach, in combination with the use of special noise-immune algorithms of image reconstruction, permits obtaining images of the internal structure of the object under study of a high quality by irradiating it sequentially in time by the flows of photons of a small intensity. The effectiveness of the suggested method is illustrated by the example of results of a computer experiment.     

Theory of dynamic entropy-operator systems and its applications

The phenomenological and mathematical definitions of a class of dynamic systems with an entropy operator are formulated. Dynamic systems with an entropy operator are classified and the main theoretical results pertaining to the properties of entropy operators and these dynamic systems are studied within this classification. By way of examples, restoration of monochromatic images and modeling of labor market are examined.     

Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

This paper presents a computational technique based on the collocation method and Müntz polynomials for the solution of fractional differential equations. An appropriate representation of the solution via the Müntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations. The main advantage of the present method is its superior accuracy and exponential convergence. Consequently, one can obtain good results even by using a small number of collocation points. The accuracy and performance of the proposed method are examined by means of some numerical experiments.     

Numerical simulation of the flow field in a model of the nasal cavity

Results of a numerical simulation of the flow in a model of the human nasal cavity using an AUSM-based method of second-order accuracy on a multi-block structured grid are presented and compared with experimental data. Computations are performed for inspiration and expiration at rest with Reynolds numbers Re=1560 and Re=1230 at the nostril, respectively. The comparison shows good agreement with experimental findings.     

Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials

In practice, computer simulations cannot be perfectly controlled because of the inherent uncertainty caused by variability in the environment (e.g., demand rate in the inventory management). Ignoring this source of variability may result in sub-optimality or infeasibility of optimal solutions. This paper aims at proposing a new method for simulation–optimization when limited knowledge on the probability distribution of uncertain variables is available and also limited budget for computation is allowed. The proposed method uses the Taguchi robust terminology and the crossed array design when its statistical techniques are replaced by design and analysis of computer experiments and Kriging. This method offers a new approach for weighting uncertainty scenarios for such a case when probability distributions of uncertain variables are unknown without available historical data. We apply a particular bootstrapping technique when the number of simulation runs is much less compared to the common bootstrapping techniques. In this case, bootstrapping is undertaken by employing original (i.e., non-bootstrapped) data, and thus, it does not result in a computationally expensive task. The applicability of the proposed method is illustrated through the Economic Order Quantity (EOQ) inventory problem, according to uncertainty in the demand rate and holding cost.

     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.     

Numerical solution of transport equations for plasmas with transport barriers

An approach to solve numerically transport equations for plasmas with spontaneously arising and arbitrarily located transport barriers, regions with a strongly reduced transfer of energy, is proposed. The transport equations are written in a form conserving heat flux and solved numerically by using piecewisely exact analytical solutions of linear differential equations. Compared to standard methods, this approach allows to reduce significantly the number of operations required to obtain a converged solution with a heat conductivity changing abruptly at a critical temperature gradient and to use large time steps in modeling the formation and dynamics of transport barriers. Computations for the tokamak JET are done.     

Numerical solution of a shallow dam problem by a boundary element method

For a certain class of surface profiles in a shallow dam problem, the number and approximate positions of the points of separation can be determined. The application of a boundary element method can then produce more accurate results efficiently, as the problem of interest involves the boundary of the region under consideration.     

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the mth-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations

The main purpose of this paper is to utilize the collocation method based on fractional Genocchi functions to approximate the solution of variable-order fractional partial integro-differential equations. In the beginning, the pseudo-operational matrix of integration and derivative has been presented. Then, using these matrices, the proposed equation has been reduced to an algebraic system. Error estimate for the presented technique is discussed and has been implemented the error algorithm on an example. At last, several examples have been illustrated to justify the accuracy and efficiency of the method.

     

How would you integrate the equations of motion in dissipative particle dynamics simulations?

In this work we assess the quality and performance of several novel dissipative particle dynamics integration schemes that have not previously been tested independently. Based on a thorough comparison we identify the respective methods of Lowe and Shardlow as particularly promising candidates for future studies of large-scale properties of soft matter systems.     

Lattice Boltzmann method for variable density shallow water equations

A lattice Boltzmann method is developed for solution of a form of the shallow water equations that is suitable for flows which are fully mixed in the vertical direction but have variable density in the horizontal plane. In the present approach, double distribution functions are applied: one for the shallow water flows and the other for the mass transport. Direct coupling between the water flow and mass transport is achieved by updating the flow density from the concentration during simulation. Accuracy and applicability of the model are demonstrated by two numerical tests: the stationary hydrostatic equilibrium of liquid of variable density in a tank with non-uniform bed terrain, and the horizontal diffusion of species with an initial Gaussian distribution of concentration in a uniform flow field.