Full Time Step Research Articles

Accurate modelling of the bifacial gain potential of rooftop solar photovoltaic systems

Bifacial solar modules have emerged as a promising technology in utility-scale photovoltaic systems, experiencing significant growth and capturing a substantial market share worldwide, as reported by the International Technology Roadmap for Photovoltaic (ITRPV) 2023. Despite this progress, the potential of bifacial modules in rooftop applications remains largely unexplored. This paper aims to address this knowledge gap by conducting a comprehensive study utilizing Monte Carlo Ray Tracing techniques coupled with detailed electrical modelling. The primary objective of this study is to investigate the viability of implementing bifacial solar modules on rooftops by examining the potential energy yield gains. By conducting a detailed analysis on a representative rooftop in Canberra, Australia, real-world conditions, and variations are incorporated, providing a more accurate assessment of the energy yield gains achievable in such settings. The simulation results reveal that the implementation of bifacial solar modules on rooftops within Australia can result in energy yield gains of up to 22.6%. These findings demonstrate the considerable potential of bifacial technology in maximizing solar energy production in rooftop applications. The analysis shows significant implications of module and system design on the potential gain. For example, electrical optimisation of individual modules in a system accounted for 1.4% of the bifacial gain. The analysis considers full annual time-step simulation, typical mechanical mounting components, installation orientations and module characteristics, ensuring practical relevance and reliability of the results.

On the numerical solution of Burgers-Fisher equation by the Strang Splitting Method

In this paper, we apply the Strang splitting method to approximate the solution of Burgers-Fisher equation. This method offers a second-order accurate approximation to evolution equation by combining two non-commuting operators. This accuracy is achieved through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. The stability criteria is derived using the invariant region. The numerical results obtained are compared with the exact solution. The numerical error shows that the exact and the numerical solution are agrees with each other with a good accuracy.

High order nonlinear filter methods for subsonic turbulence simulation with stochastic forcing

Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high accuracy before the simulation reaches the statistically stationary stage is computationally intensive. The goal of this work is to employ an efficient class of high order finite difference nonlinear filter methods for subsonic turbulence simulation with stochastic forcing. The 3D Euler equations for subsonic turbulence with temporally varying stochastic forcing at rms Mach numbers up to 0.6 are numerically solved using the Strang operator splitting of the homogeneous part of the Euler equations and the forcing terms. It was shown in Yee et al. (2013) that the Strang operator splitting is more stable than solving the full Euler equation with the source term included. The spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen (2007, 2011) are used to solve the homogeneous system and an ODE solver is used to solve the forcing source terms. The nonlinear filter method includes a full time step of a spatially eighth-order central base method, using a third-order TVD Runge-Kutta time integration. The solution computed with the central base method is then nonlinearly filtered by an adaptive flow sensor and the dissipative portion of a seventh-order WENO with the Roe Riemann solver. In order to improve nonlinear stability of the base method without added numerical dissipation, the central base method discretizes the skew-symmetric split form of the inviscid flux derivatives. Both Ducros et al. and Kennedy-Gruber skew-symmetric split forms are tested. Numerical stability, computational efficiency, and effective spectral bandwidth of the nonlinear filter schemes are compared with those of second-order TVD and fifth- and seventh-order WENO methods. It is shown that the nonlinear filter method for this application is substantially more efficient, accurate and yields a superior spectral bandwidth compared to the standard TVD and WENO methods. The nonlinear filter method also demonstrates robust long-time integration for moderately compressible, statistically stationary turbulence with large-scale solenoidal forcing, including small-scale quantities such as enstrophy and mean-square dilatation.

An explicit large time step particle-in-cell scheme for nonlinear gyrokinetic simulations in the electromagnetic regime

A new algorithm for electromagnetic gyrokinetic simulations, the so called “pullback transformation scheme” proposed by Mishchenko et al. [Phys. Plasmas 21, 092110 (2014)] is motivated as an explicit time integrator reset after each full timestep and investigated in detail. Using a numerical dispersion relation valid in slab geometry, it is shown that the linear properties of the scheme are comparable to those of an implicit v∥-scheme. A nonlinear extension of the mixed variable formulation, derived consistently from a field Lagrangian, is proposed. The scheme shows excellent numerical properties with a low statistical noise level and a large time step especially for MHD modes. The example of a nonlinear slab tearing mode simulation is used to illustrate the properties of different formulations of the physical model equations.

An Efficient One-Step Leapfrog WCS-FDTD Method

For solving the electromagnetic problems with fine structures in two directions, this letter presents an efficient three-dimensional (3-D) one-step leapfrog weakly conditionally stable finite-difference time-domain (WCS-FDTD) method, in which the updating manner of field quantities is the same as that of the traditional FDTD method that electric and magnetic fields are calculated at every full time-step but interlaced half a step apart in both time and space. Furthermore, by introducing an auxiliary field variable group denoted by m, the proposed algorithm is reformulated in a much simpler form with more concise right-hand sides for an efficient implementation. Compared to the original WCS scheme in Chen and Wang's work (IEEE Trans. Electromagn. Compat., vol. 49, no. 2, pp. 419-426, May 2007), the proposed method is more efficient while keeping the same numerical properties.

An interpretation and derivation of the lattice Boltzmann method using Strang splitting

The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank–Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity.

Effect of Pruning and Smoothing while Using M5 Model Tree Technique for Reservoir Inflow Prediction

This study reports the performance of an M5 model tree (MT) and the effects of pruning and smoothing applied to reservoir inflow prediction. The full year and seasonal monthly time step MT predictions were compared with conventional univariate autoregressive integrated moving average (stochastic) models. It was found that stochastic models could not predict the future inflows in a better way, because the observed series had not followed any particular distribution. However, it was found that the stochastic models showed better improvement using a logarithmic-transformed series, but the logarithmic-transformed MT results showed otherwise. The model validation was performed using the comparison of goodness of fit measures, standard statistics, time series, and scatter plots of predicted inflows with observed inflows. The effect of pruning each leaf in the MT model was also studied. Instead of pruning all the leaves, leading to lesser predictive accuracy, selective pruning was carried out based on the import...

Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

The goal of this paper is to generalize the well-balanced approach for non-equilibrium flow studied by Wang et al. (2009) [29] to a class of low dissipative high-order shock-capturing filter schemes and to explore more advantages of well-balanced schemes in reacting flows. More general 1D and 2D reacting flow models and new examples of shock turbulence interactions are provided to demonstrate the advantage of well-balanced schemes. The class of filter schemes developed by Yee et al. (1999) [33], Sjögreen and Yee (2004) [27] and Yee and Sjögreen (2007) [38] consist of two steps, a full time step of spatially high-order non-dissipative base scheme and an adaptive non-linear filter containing shock-capturing dissipation. A good property of the filter scheme is that the base scheme and the filter are stand-alone modules in designing. Therefore, the idea of designing a well-balanced filter scheme is straightforward, i.e. choosing a well-balanced base scheme with a well-balanced filter (both with high-order accuracy). A typical class of these schemes shown in this paper is the high-order central difference schemes/predictor–corrector (PC) schemes with a high-order well-balanced WENO filter. The new filter scheme with the well-balanced property will gather the features of both filter methods and well-balanced properties: it can preserve certain steady-state solutions exactly; it is able to capture small perturbations, e.g. turbulence fluctuations; and it adaptively controls numerical dissipation. Thus it shows high accuracy, efficiency and stability in shock/turbulence interactions. Numerical examples containing 1D and 2D smooth problems, 1D stationary contact discontinuity problem and 1D turbulence/shock interactions are included to verify the improved accuracy, in addition to the well-balanced behavior.

An Efficient 2D ADI-FDTD Modeling of High-Order, Frequency-Dependent Media

The alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is extended for modeling the propagation of transverse-electric waves in high-order, frequency-dependent media. Ohm's law is considered as the constitutive relation with the complex conductivity assumed to be a high-order rational function of the frequency. Contrary to previously reported approaches, the integration of the constitutive relation over a full time-step is not split into several substeps, which improves the computational efficiency of the resulting algorithm. In addition, the numerical dispersion relation is derived in a closed form. To illustrate the validity and accuracy of the formulation, we consider a muscle model that consists of a static conductivity and three Debye poles.

An Alternating-Direction Implicit FDTD Modeling of Dispersive Media Without Constitutive Relation Splitting

In recent years, the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method has been extended to dispersive media. To this end, a constitutive relation is appended to Maxwell's curl equations. A common feature of all of the existing formulations is that the integration over a full time step, of Maxwell's equations and the constitutive relation, is split into two sub-steps.

On the calculation of velocity-dependent properties in molecular dynamics simulations using the leapfrog integration algorithm

Widely used programs for molecular dynamics simulation of (bio)molecular systems are the Verlet and leapfrog algorithms. In these algorithms, the particle velocities are less accurately propagated than the positions. Important quantities for the simulation such as the temperature and the pressure involve the squared velocities at full time steps. Here, we derive an expression for the squared particle velocity at full time step in the leapfrog scheme, which is more accurate than the standardly used one. In particular, this allows us to show that the full time step kinetic energy of a particle is more accurately computed as the average of the kinetic energies at previous and following half steps than as the square of the average velocity as implemented in various molecular dynamics codes. Use of the square of the average velocity introduces a systematic bias in the calculation of the instantaneous temperature and pressure of a molecular dynamics system. We show the consequences when the system is coupled to a thermostat and a barostat.

An incompressible multi-phase SPH method

An incompressible multi-phase SPH method is proposed. In this method, a fractional time-step method is introduced to enforce both the zero-density-variation condition and the velocity-divergence-free condition at each full time-step. To obtain sharp density and viscosity discontinuities in an incompressible multi-phase flow a new multi-phase projection formulation, in which the discretized gradient and divergence operators do not require a differentiable density or viscosity field is proposed. Numerical examples for Taylor–Green flow, capillary waves, drop deformation in shear flows and for Rayleigh–Taylor instability are presented and compared to theoretical solutions or references from literature. The results suggest good accuracy and convergence properties of the proposed method.

A second order volume of fluid (VOF) scheme for numerical simulation of 2-D breaking waves

Among all environmental forces acting on ocean structures and marine vessels, those resulting from wave impacts are likely to yield the highest loads. Being highly nonlinear, transient and complex, a theoretical analysis of their impact would be impossible without numerical simulations. In this paper, a pressure-split two-stage numerical algorithm is proposed based on Volume Of Fluid (VOF) methodology. The algorithm is characterized by introduction of two pressures at each half and full cycle time step, and thus it is a second-order accurate algorithm in time. A simplified second-order Godunov-type solver is used for the continuity equations. The method is applied to simulation of breaking waves in a 2-D water tank, and a qualitative comparison with experimental photo observations is made. Quite consistent results are observed between simulations and experiments. Commercially available software and Boundary Integral Method (BIM) have also been used to simulate the same problem. The results from present code and BIM are in good agreement with respect to breaking location and timing, while the results obtained from the commercial software which is only first-order accurate in time has clearly showed a temporal and spatial lag, verifying the need to use a higher order numerical scheme.

Symmetric source implementation for the ADI-FDTD method

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method has been introduced to overcome the Courant limit present in the standard FDTD scheme. However, the ADI-FDTD scheme may produce small asymmetries in the field distributions according to the source implementation. In this communication, we compare different source implementation strategies and show that ADI-FDTD scheme has no asymmetry errors up to the numerical noise level, and can be implemented most accurately if the excitation is directly incorporated within the tridiagonal matrix and if the time discretization of the source is done appropriately within each full time step.

Sub-timing in fluid flow and transport simulations

A sub-time stepping method is described for computational fluid dynamics problems that utilize implicit-type time marching procedures to resolve transients. In this method, small time-step sizes are used in portions of a domain where interest and activity is high, with larger time-step sizes being applied in other locations. The sub-time step is an integral portion of a larger time step––i.e., multiple sub-time steps over a sub-timed part of a domain add up to the time interval of the full time step used over the remainder of the domain. The technique is particularly suitable for extensive simulations where large portions of a domain are temporally over-discretized. The principles underlying implementation of the implicit sub-timing procedure, the computational effort in relation to conventional implicit time-stepping methods, and an analysis of the effects of sub-timing on the matrix structure are presented. Feasibility and applicability of the implicit sub-timing method is demonstrated through three proof-of-the-concept example problems. These examples include implementation of implicit sub-timing to one-dimensional as well as two-dimensional flow and solute transport problems.

ADER: Arbitrary High Order Godunov Approach

This paper concerns the construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws. The schemes result from extending the ADER approach, which is related to the ENO/WENO methodology. Our schemes are conservative, one-step, explicit and fully discrete, requiring only the computation of the inter-cell fluxes to advance the solution by a full time step; the schemes have optimal stability condition. To compute the intercell flux in one space dimension we solve a generalised Riemann problem by reducing it to the solution a sequence of m conventional Riemann problems for the kth spatial derivatives of the solution, with ke0, 1,m, m−1, where m is arbitrary and is the order of the accuracy of the resulting scheme. We provide numerical examples using schemes of up to fifth order of accuracy in both time and space.

Uncertainty analysis of decomposing polyurethane foam

Sensitivity/uncertainty analyses are necessary to determine where to allocate resources for improved predictions in support of our nation’s nuclear safety mission. Yet, sensitivity/uncertainty analyses are not commonly performed on complex combustion models because the calculations are time consuming, CPU intensive, nontrivial exercises that can lead to deceptive results. To illustrate these ideas, a variety of sensitivity/uncertainty analyses were used to determine the uncertainty associated with thermal decomposition of polyurethane foam exposed to high radiative flux boundary conditions. The polyurethane used in this study is a rigid closed-cell foam used as an encapsulant. The response variable was chosen as the steady-state decomposition front velocity. Four different analyses are presented, including (1) an analytical mean value (MV) analysis, (2) a linear surrogate response surface (LIN) using a constrained latin hypercube sampling (LHS) technique, (3) a quadratic surrogate response surface (QUAD) using LHS, and (4) a direct LHS (DLHS) analysis using the full grid and time step resolved finite element model. To minimize the numerical noise, 50 μm elements and approximately 1 ms time steps were required to obtain stable uncertainty results. The complex, finite element foam decomposition model used in this study has 25 input parameters that include chemistry, polymer structure, and thermophysical properties. The surrogate response models (LIN and QUAD) are shown to give acceptable values of the mean and standard deviation when compared to the fully converged DLHS model.

A computation of a capacitor motor with the first harmonic method

A coupled electromagnetic-circuit model that allows any time dependence of the terminal voltage is described in the context of the capacitor motor. This model, called the first harmonic method, is an approximation of the full time step model. Practical results and comparisons with experiments are provided.

A multi-step computation scheme: Decoupling kinetic processes from physical transport in water quality models

This paper presents a computation scheme that considers accuracy and efficiency in the numerical treatment of the kinetic processes in water quality models. The proposed scheme involves decoupling of physical transport and kinetic processes in the governing mass-balance equations. The decoupled equations are solved by employing multi-step computation, in which the kinetic processes are applied for the first half time step, followed by the physical transport for a full time step, and the kinetic processes are applied again for the remaining second half time step. The kinetic equations are solved numerically or analytically, which may require prior linearization of the equations. The numerical solution scheme of the kinetic equations alternates between explicit and implicit schemes, and is equivalent to the second-order accurate Crank-Nicolson solution. A hypothetical model, which is simple enough to have an analytical solution, is used to explain the concept, and to demonstrate the accuracy and computational efficiency of the proposed scheme. The proposed scheme and three other traditional solution schemes are compared to the analytical solution. The proposed scheme shows better accuracy and computational efficiency compared to other schemes.

Calculating short range forces on a massively parallel computer: SPH on the Connection Machine

Three implementations of SPH on a Connection Machine are presented. The first one is a highly parallel direct summation implementation which maximally utilizes CM computational power but suffers from N 2 scaling. The second one uses cells to group particles in space and hence reduce the number of inter-particle forces to be computed. Very high execution speeds are obtained for regular particle distributions by processing these cells in parallel, however, for uneven particle distributions, the approach degrades again into N 2 scaling behavior. The third approach uses multiple grids and the duplication of particles to combine linear scaling with good load-balancing. A full SPH time step takes 3.1 seconds for a regular particle distribution and 46 seconds for an irregular one for 128k particles on a 64k CM200.