Time Step Iteration Research Articles

An improved moving-boundary heat exchanger model using the variable time step method

Moving-boundary heat exchanger model is a fast low order model. In this paper, an improved moving-boundary heat exchanger model is developed to provide a stable fast model for the research on control strategy of refrigeration system. In this model, the treatment of each iterative time step is added. Before the calculation of the state variables of the next step, the maximum time step of the next iterative is determined by the change state variable values in per unit time of the current iterate. Then, the time step of the next iterative is determined by the maximum time step and the setting time step. Meanwhile, in order to meet the need of the research on control strategy, the independent variable is employed with temperature variable instead of pressure and enthalpy variable in the calculation of refrigerant state parameters. The simulation results shows that the present model can well predict the transient behaviours of heat exchangers using any setting value of time step, and the computing speed of the present model is faster than the speed of original model.

Simulation of ultrasonic inspection of composite using bulk waves: Application to curved components

The increasing use of composite materials in aircraft, particularly carbon-fiber reinforced samples, presents new challenges for Non-Destructive Testing (NDT) operations to ensure reliable inspections. The aircraft industry is showing a growing interest in the simulation of NDT techniques applied to such materials, which exhibit strong anisotropy. Simulation tools are needed to predict beam deviations that may arise for curved or tilted parts in complex specimen. A main issue is the modelling of the wave propagation and flaw scattering inside a homogeneous medium considering a continuously variable anisotropic orientation. Previous works have presented a way to model these media. This method consists in decomposing them into a set of homogeneous sub-sections with constant orientations. This approximation allows us to use straight paths over homogeneous sections but the main drawback is that it introduces artificial boundaries leading to numerical instabilities. The new model deals with a continuously variable anisotropy. It is based on a ray calculation for an inhomogeneous medium using an iterative time-step method. This method consists in solving for each time step a system of differential equations taking into account the continuous anisotropic orientation variation along the trajectory.

Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions

This paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems. It is shown that the Newton–Raphson (NR) method converges for any time step size, if the flux function is convex, concave, or linear, which is, in general, the case for CFD problems. In some problems, e.g., multiphase flow in porous media, the nonlinear flux function is S-shaped (not uniformly convex or concave); as a result, a standard NR iteration can diverge for large time steps, even if an implicit discretization scheme is used to solve the nonlinear system of equations. In practice, when such convergence difficulties are encountered, the current time step is cut, previous iterations are discarded, a smaller time step size is tried, and the NR process is repeated. The criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and/or NR iteration. Here, we propose a simple modification to the NR iteration scheme for conservation laws with S-shaped flux functions that converges for any time step size. The new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver. The proposed method can be implemented in an existing simulator, e.g., for CO 2 sequestration or reservoir flow modeling, quite easily. The numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media. The analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems.

Numerical simulation of the magnetization of high-temperature superconductors: a 3Dfinite element method using a single time-step iteration

In this paper, we report progress towards a 3D finite element model for the magnetizationof a high-temperature superconductor (HTS): we suggest a method that takes intoaccount a power law conductivity and demagnetization effects, while neglecting theeffects associated with currents that are not perpendicular to the local magneticinduction. We consider samples that are subjected to a uniform magnetic fieldvarying linearly with time. Their magnetization is calculated by means of a weakformulation in the magnetostatic approximation of the Maxwell equations (A–ϕ formulation). An implicit method is used for the temporal resolution (backwardEuler scheme) and is solved with the open source solver GetDP. Fixedpoint iterations are used to deal with the power law conductivity of HTS.The finite element formulation is validated for an HTS tube with largen value by comparing with results obtained with other well-established methods. We showthat carrying out the calculations with a single time-step (as opposed to many smalltime-steps) produces results with excellent accuracy in a drastically reduced simulationtime. The numerical method is extended to the study of the trapped magnetization ofcylinders that are drilled with different arrays of columnar holes arranged parallel to thecylinder axis.

Combining a polarizable force‐field and a coarse‐grained polarizable solvent model: Application to long dynamics simulations of bovine pancreatic trypsin inhibitor

The dynamic coupling between a polarizable protein force field and a particle-based implicit solvent model is described. The polarizable force field, TCPEp, developed recently to simulate protein systems, is characterized by a reduced number of polarizable sites, with a substantial gain in efficiency for an equal chemical accuracy. The Polarizable Pseudo-Particle (PPP) solvent model represents the macroscopic solvent polarization by induced dipoles placed on mobile Lennard-Jones pseudo-particles. The solvent-induced dipoles are sensitive to the solute electric field, but not to each other, so that the computational cost of solvent-solvent interactions is basically negligible. The solute and solvent induced dipoles are determined self-consistently and the equations of motion are solved using an efficient iterative multiple time step procedure. The solvation cost with respect to vacuum simulations is shown to decrease with solute size: the estimated multiplicative factor is 2.5 for a protein containing about 1000 atoms, and as low as 1.15 for 8000 atoms. The model is tested for six 20 ns molecular dynamics trajectories of a traditional benchmark system: the hydrated Bovine Pancreatic Trypsin Inhibitor (BPTI). Even though the TCPEp parameters have not been refined to be used with the solvent PPP model, we observe a good conservation of the BPTI structure along the trajectories. Moreover, our approach is able to provide a description of the protein solvation thermodynamic at the same accuracy as the standard Poisson-Boltzman continuum methods. It provides in addition a good description of the microscopic structural aspects concerning the solute/solvent interaction.

Approaches for modeling longitudinal dispersion in pore-networks

The purpose of this study is to quantify the dispersivity in the longitudinal direction by upscaling pore scale mixing over a network domain and to verify the dispersivity with that obtained through the more rigorous upscaling technique, the Brownian particle tracking model (BPTM). We model a porous medium with a network of pore-units that are comprised of pore bodies and bonds of finite volume. Such a pore-unit is assumed to be a mixing cell with the steady state flow condition for a single fluid. Dispersivity can be obtained by solving the mixing cell model (MCM) for the concentration in each pore-unit and by averaging the concentrations for a large number of pore units (as a function of time and space). A minimal size of network that ascertains an asymptotic value of dispersivity was determined and verified with large size pore networks. This numerically computed dispersivity is compared with the results from the BPTM for the same porous medium and flow conditions. We show that the dispersivity obtained from the MCM is equally reliable for the heterogeneous pore-networks and can be estimated as a function of pore size heterogeneity. For homogeneous networks with the MCM, the iteration time step plays an important role. On the other hand, for networks with the BPTM, the assumption of intra-bond velocity profile affects the results.

Preconditioning for incompressible flows with free-surfaces and two-fluid interfaces

We investigate preconditioners for solving steady or implicit-unsteady arbitrary-Lagrangian–Eulerian moving-mesh formulations of incompressible free-surface and interfacial flow problems. The solution for the flow is obtained using the artificial compressibility method combined with a multigrid cycle. To find the surface positions in a free-surface problem, we use an iteration that is quasi-physical with the position of the free-surface driven by the kinematic condition. By modifying the fluxes through the free-surface and also using an iterative free-surface time step that is different from the flow time-step, we are able to obtain convergence rates that are almost indistinguishable from those of a flow without a free-surface. This is true in the limits of both large and small surface stresses such as gravity or surface tension and also for all flow conditions (high or low Reynolds number or small implicit Courant–Friedrichs–Lewy number). For interfacial flows, the free-surface preconditioner is extended by using a mass averaging technique. The iterative flow time-step at the interface is a mass average of the time step for the flow conditions on either side of the interface and the iterative time step for the kinematic condition is also based on mass-average quantities. We verify that this technique produces the correct results in the limits of identical fluids or a free-surface flow. We also verify that it produces well-conditioned results when the characteristic time scales of the two fluids are very different.

Distributed parameter thermal controllability: a numerical method for solving the inverse heat conduction problem

AbstractThis paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes. A numerical control algorithm is developed for distributed parameter conduction systems, based on Galerkin optimization of an energy index employing Green's functions. Various temperature profiles of variable complexity are studied, using the proposed technique, in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body. Furthermore, the effect of altering the iterative time step and duration of processing time, on the convergence of the solution generated by the aforementioned method is investigated. It is proved that despite the variations in numerical processing, the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials. Copyright © 2004 John Wiley & Sons, Ltd.

Transmission‐line matrix models for solving transient problems of diffusion with recombination

AbstractStarting from the general telegrapher's equation, we investigate two nodal network constructions for modelling diffusion with recombination by means of the transmission‐line matrix (TLM) method. The diffusion effect is modelled by the series and shunt capacitance in one approach, and by the series inductance and shunt resistance in the other. Both approaches use the series and shunt resistances to model the recombination effect. The constraint of using both TLM networks for solving transient problems of diffusion with recombination is found to be identical in terms of the physics behind the numerical routines. A practical way of determining the spatial resolution and iteration time step for accurate TLM numerical computations is suggested based on a simple frequency analysis. Copyright © 2003 John Wiley & Sons, Ltd.

Elastic molecular dynamics with self-consistent flexible constraints

A new algorithm for constrained molecular dynamics is proposed. In contrast to the standard approach, the constrained bond-length/bond-angle value is adjusted at each time step so that total energy is minimized with respect to the constrained distances. This can be viewed as modifying the equilibrium bond-length/bond-angle according to external and centripetal forces. Two approaches are constructed to implement the algorithm. Method I includes all energy terms, but it is neither holonomic nor symplectic. Method II neglects a rotational kinetic energy term, resulting in a more expensive symplectic integrator. Both integrators are reversible and well conserve total energy. Due to the iterative nature of the optimization, which requires an energy evaluation at each iteration, these methods are intended for use in conjunction with other integration techniques that require an optimization at each time step iteration, such as the self-consistent electronic polarization methods or iterative combined quantum mechanics/molecular mechanics (QM/MM) hybrid simulations. Simulation results for collisions between two diatomic molecules, two water molecules, and for a periodic water box are compared and contrasted with fully constrained and free dynamics. These results demonstrate the utility and efficacy of these new methods.

Three-dimensional modelling of dissipative quantum transport in quantum dots and atomistic scale devices using nonHermitian generalized potentials

In this paper we introduce a phenomenology for inserting dissipation into the single-particle Schrödinger equation for carrier transport by utilizing appropriate nonHermitian additions to the Hamiltonian. The nonHermitian terms are determined by incorporating model particle trapping/de-trapping, momentum gain/loss, energy gain/loss into the quantum continuity equations derived within the Bohm picture and then reconstructing the full Hamiltonian by reversing the Bohm projection. The new phenomenology is designed to obtain quantum velocity flows using the Bohm projection of solutions to the nonHermitian Schrödinger equation for applications in 2D and 3D quantum dots and mesoscopic MOSFETs. For this purpose we introduce a novel fast algorithm to compute the wave function in 2D and 3D based on a two time step iteration and direct integration.

Discretization effects in TLM diffusion models

The standard methods for bringing a TLM diffusion model to convergence are necessary but tedious. However, much time and effort can be saved if the initial choices of elemental size and iteration timestep are appropriate. Here we consider the general form of the solution to the diffusion equation and sample the component dynamics to determine the impulse invariant response. We then derive a convenient method for charting the discrepancy between the coefficients of the continuous description and the impulse invariant transformation, as this discrepancy varies with discretization of space and time. The use of the curves in estimating appropriate elemental size and iteration timestep is described. Copyright © 2000 John Wiley & Sons, Ltd.

Particle-in-cell and finite difference approaches for the study of marginal ice zone problems

Eulerian Finite Difference (FD) and quasi-Lagrangian Particle-In-Cell (PIC) approaches are applied in the present paper to study Marginal Ice Zone (MIZ) problems. The elliptical, viscous-plastic, rheological model of Hibler (Hibler, W.D. III, 1979. A dynamic thermodynamic sea ice model. J. Phys. Oceanogr., Vol. 9, pp. 815–846) was used. The numerical implementation of the equations governing the ice motion is described in detail. A Successive Over-Relaxation (SOR) scheme is employed for the FD and PIC methods in solving the momentum equation. Following Zhang and Hibler (Zhang, J., Hibler, W.D. III, 1997. On an efficient numerical method for modeling sea ice dynamics. J. Geophys. Res., Vol. 102, pp. 8691–8702), a pseudo time stepping procedure was used to update the bulk and shear viscosities, and the other velocity dependent terms, while maintaining the ice strength and the old time step velocity appearing in the inertia term ∂ v /∂ t unchanged. By performing these pseudo time step iterations, it was possible to (1) ensure that the stresses corresponded to the elliptical yield criterion during plastic deformation, and (2) improve the accuracy of the velocity field and hence, the nonlinear terms involving velocities. Two geometrical configurations were examined: a straight MIZ in which the coast was modeled by a straight rigid boundary; and a MIZ in which the coast was considered to be a straight corner-shaped solid boundary. Special attention was given to the implementation of the boundary conditions for ice–water interfaces. This is an important element in MIZ problems where one can find complex moving ice–water interfaces, which are not usually observed in the Arctic basin. A simple way to calculate the ice compactness, which is particularly useful for the PIC method, is described. The present numerical results obtained from both the FD and the PIC methods are in general agreement. In particular, results obtained from these two approaches for the straight MIZ are almost identical. However, the PIC method has the advantage of providing an exact location of the free edge of the ice. It also appears to provide a more accurate and trouble-free means to predict deformations within the ice fields.

A three-level parallelization of spatial direct numerical simulation

The parallelization of a NASA Navier–Stokes simulation code is carefully and systematically investigated. Based on numerical, dependence and partition analysis, three different levels of parallelization have been proposed and implemented. The low-level parallelization is at the kernel level. A previously developed parallel tridiagonal solver is used for concurrent processing. The middle-level parallelization solves the multiple right-hand-sides of the tridiagonal systems concurrently. The high-level parallelization is at the level of time step iterations. A PVM implementation of the parallelization has been accomplished. Different communication patterns and different PVM communication calls have been examined for best performance. The parallelized code has been tested on two parallel platforms, a cluster of workstations available at Louisiana State University and the IBM SP2 parallel computer available at Cornell Supercomputing Center. Experimental results confirm our analytical findings: the three-level parallelization is feasible and effective. A linear speedup is measured. In addition to parallelization, through profiling and performance analysis, the original sequential code is optimized as well.

Efficient material-gain models for the transmission-line laser model

Transmission-line laser models (TLLMs) are time-domain models suitable for the simulation of complex phenomena in semiconductor lasers. TLLMs include time-domain filters based on transmission-line stubs to model the spectral dependence of the material gain. In this paper, numerical simulations are presented which show that the accuracy of these gain models is dependent on the model's iteration timestep. Analytical formulae are derived that relate the accuracy of the filters to the timestep, filter centre frequency, and filter bandwindth. A new wideband stub filter which allows the material gain to be modelled using a larger timestep is presented. This is equivalent to a digital infinite-impulse-response filter, which is more computationally efficient than finite-impulse-response filters, and is unconditionally stable. The new gain model can improve the computational speed for simulating for multimode Fabry-Perot lasers by a factor of 10–100.

Low energy cluster impact simulated by molecular dynamics; angular distribution of sputtering yield and impact under various impact angles

The collision process of low energetic gold atoms and solid targets has been simulated using our molecular dynamics simulation code CLIMPACT II. The used algorithm is a third-order predictor Verlet algorithm [L. Verlet, Phys. Rev. 159 (1967) 98; W.F. van Gusteren and H.J.C. Berendsen, in: Molecular Liquids — Dynamics and Interfaces, A.J. Barnes et al., eds. (Reidel, 1984) p. 475.]. The iteration time step is continuously optimized by the program. About 50% of the total computer time is spent to integrate the motions during the first 100 fs of simulation time [B. Nitzschmann, Diploma thesis, Univ. of Oldenburg, Germany, 1992)]. When the crater formation ends and the motions in the target are slower, the step increases up to 20 times the start step size. Using this algorithm we are able to simulate a target of up to 10 5 particles. We use new nonreflecting boundary conditions. Only mechanical interactions are considered. The projectile can be chosen as a cluster with variable impact angle. Specifically the output yield under different impact angles and the distribution of the desorbed particles are presented and discussed. The temporal development of the desorption shows three distinct processes: an early explosive process, a surface ablation by an apparent surface shock wave, a final thermal evaporation.

SIMULATION OF TRANSPIRATION OF CITRUS GOVERNED BY LEAF CONDUCTANCE

A water flow model was developed which uses irradiance, leaf-to-air vapor concentration difference, and soil water potential to establish stomatal conductance. Water flow to the roots was computed using a linear approximation of radial flow through the soil toward the axis of the roots across concentric shells. Root length density and soil rooting volume within four separate layers or compartments were included in the model. The simulation was executed in small time step iterations. A small increment of transpiration was translated to a water content deficit at the root and then sequentially through the concentric shells to simulate water uptake and change of soil water potential. The change in soil water potential was used to increment changes in stomatal conductance and transpiration. The output of the model simulated the pattern of diurnal stomatal behavior observed in other types of experiments, as well as the total soil water extraction patterns of young potted citrus trees.

Hybrid symmetrical condensed node for the TLM method

An alternative symmetrical condensed node has been developed for which the iterative time step is independent of the ratio of grading on the TLM mesh. The new mode has fewer stubs than the previous node and is shown to have a reduced velocity error for axial propagation.

The imaginary time step method for Thomas-Fermi equations

The imaginary time step iteration procedure is extended to the solution of the Thomas-Fermi equations and their modifications. The convergence of the procedure is discussed and illustrated by a numerical example. The generalization to the finite temperature calculations is indicated.

A formula for the solution of the Navier-Stokes equations based on a method of Chorin

Recently, A. Chorin has found a numerical scheme for solving the Navier-Stokes equations which has the pleasing feature of not breaking down at high Reynolds numbers R . The purpose of this announcement is to present a formula which is designed to establish the convergence of Chorin's time step iteration procedure, assuming that the relevant equations (heat equation and Euler's equations) are solved exactly at each step.