Runge-Kutta Time Integration Method Research Articles

Design optimisation of piezoelectric energy harvesters for bridge infrastructure

Vibrational energy harvested from the bridge excitation due to the traffic flow or the wind load can be supplied to sensors in a Structural Health Monitoring (SHM) system and help prolong its service life, reduce chemical battery waste and enable its use in remote locations. A common approach to designing a Piezoelectric Energy Harvester (PEH) consists in tuning its fundamental frequency to some target value (e.g., the fundamental frequency of a bridge). However, such approach does not answer the question if the chosen target frequency is optimal, and does not take into account the possibility to have multiple design configurations with the same fundamental frequency. In this work, we approach the problem of a PEH design optimisation in a rigorous way, using a PEH model based on the Kirchhoff–Love plate theory and Isogeometric Analysis (IGA), coupled with the Modal Order Reduction (MOR) approach and Runge–Kutta time integration method. The model is further equipped with the Particle Swarm Optimisation (PSO) algorithm that allows finding geometry which maximises energy output from a given base acceleration signal. A comprehensive study is conducted to infer the impact of a PEH geometry on its energy harvesting performance in a real-world setting by considering field health monitoring data of a large-scale cable-stayed bridge in NSW, Australia. A shape optimisation framework is developed based on acceleration events, i.e., where the level of response exceeds a certain threshold. Designs obtained in the event-based optimisation are then clustered to propose several best candidates for continuous energy generation. This work represents the first study on PEH design optimisation for real operational conditions.

A bedload transport in shallow water model applied for sediment transport in open channel flows

This paper presents the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modelled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a non-conservative hyperbolic system. To discretize it, first we consider a Non-Homogeneous Riemann Solver scheme as well as a variant based on the use of flux limiters. In order to develop second-order scheme, we use a MUSCL method incorporating slope limiters in the spatial approximation and a two-step Runge-Kutta method for time integration. The comparison between results based on the proposed scheme and analytical results shows good agreement..

Development and verification of a high-speed compressible reactive flow solver in OpenFOAM

This paper proposes a newly developed density-based compressible solver called DCRFoam on a basis of OpenFOAM platform for simulating detonation phenomena with detailed chemical kinetics. It employs AUSM+-up scheme to determine the convective fluxes and a four steps low storage Runge-Kutta time integration method for time integration. A total variation diminishing (TVD) and symmetrical flux limiter sgva is implemented for interpolation process. Also, the dynamic mesh adaptation method is applied to refine or coarsen the computational mesh dynamically since the resolution requirements of supersonic combustion are localized. The numerical solver has been validated systematically with many numerical and experimental results reported in the literature, including those from non-reacting cases (Sod’s shock tube problem, two-dimensional shock/cylinder interaction) to reacting flow cases (the ignition sequence in a shock tube, oblique detonation wave (ODW) initiating process and detonation propagating in a bifurcated channel). The results obtained by current schemes are compared with original central upwind Kurganov-Noelle-Petrova (KNP) scheme in official OpenFOAM. The corresponding result demonstrates the robustness and accuracy of the current solver.

A graphics processing unit-based robust numerical model for solute transport driven by torrential flow condition

Solute transport simulations are important in water pollution events. This paper introduces a finite volume Godunov-type model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations. The model adopts the Harten-Lax-van Leer-contact (HLLC)-approximate Riemann solution to calculate the cell interface fluxes. It can deal well with the changes in the dry and wet interfaces in an actual complex terrain, and it has a strong shock-wave capturing ability. Using monotonic upstream-centred scheme for conservation laws (MUSCL) linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy. At the same time, the introduction of graphics processing unit (GPU)-accelerated computing technology greatly increases the computing speed. The model is validated against multiple benchmarks, and the results are in good agreement with analytical solutions and other published numerical predictions. The third test case uses the GPU and central processing unit (CPU) calculation models which take 3.865 s and 13.865 s, respectively, indicating that the GPU calculation model can increase the calculation speed by 3.6 times. In the fourth test case, comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU, the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU. Therefore, it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents. It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.

OpenMP parallelism in computations of three-dimensional potential numerical wave tank for fully nonlinear simulation of wave-body interaction using NURBS

The open multi-processing parallelism was implemented into a serial code to compute a high-order boundary integral equation based on the second Green's identity. on symmetric multiprocessing computer platforms. The parallel code is augmented to a three-dimensional potential numerical wave tank for fully nonlinear wave-body interaction problems in the time domain. The non-uniform rational B-Spline surface was manipulated to define the geometry of the computational boundaries and the distribution of boundary values either as a high-order iso-geometric formulation. The material node approach and fourth-order Runge-Kutta time integration method were compounded for time marching the fully nonlinear free surface boundary. The propagation of a fifth-order Stokes wave was simulated to examine the scalability of the parallel constructs implementation. Afterwards, the second-order Stokes wave interaction with a cylindrical pile was modeled as a practical case study to show the benefits of the parallel code. The results were compared with former experiments and numerical studies.

Vorticity dynamics of the three-dimensional Taylor-Green vortex problem

The three-dimensional (3D) Taylor-Green Vortex (TGV) flow problem has been used to study turbulence from genesis to eventual decay governed by the 3D Navier-Stokes equation. The evolution of the TGV shows that the solution becomes unstable at very early times and eventually becomes turbulent, but a study of this transition has not been advanced so far. The computations are performed using a high accuracy compact scheme on a uniform grid, with the fourth-order Runge-Kutta time integration method. The vector potential-vorticity (Ψ→,ω→)-formulation of the governing equations is solved in a cubic periodic domain with one complete basic unit of a TGV cell in the interior of the domain at t = 0. The TGV problem allows one to study the vorticity dynamics using highly accurate formulation because of periodic boundary conditions. Simulations performed for different Reynolds numbers and grid resolutions reveal that numerical error in computations induce a period-doubling bifurcation, which leads to new spatial symmetries maintained up to intermediate times, followed by simultaneous stretching and fragmentation of vortices resulting in a decaying turbulent flow. The compensated energy spectrum of the 3D TGV flow displays inertial subrange at t = 9, after which the generated turbulence starts decaying. The power law for turbulent kinetic energy decay is analyzed, and the decay exponent is noted to approach unity as time increases.

A numerical study of nonlinear infrasound propagation in a windy atmosphere.

Direct numerical simulations of the two-dimensional unsteady compressible Navier-Stokes equations are performed to study the acoustic field generated by an infrasonic source in a realistic atmosphere. Some of the main phenomena affecting the propagation of infrasonic waves at large distances from the source are investigated. The effects of thermal and wind-related refraction on the signals recorded at ground level are highlighted, with particular emphasis on the phase shift induced by the presence of caustics in the acoustic field. Nonlinear waveform steepening associated with harmonic generation, and period lengthening, both of which are typical of large source amplitudes, are illustrated, and the importance of thermoviscous absorption in the upper atmosphere is clearly demonstrated. The role of diffraction in the shadow zone, around caustics and at stratospheric altitudes is also pointed out. The Navier-Stokes equations are solved using high-order finite-differences and a Runge-Kutta time integration method both originally developed for aeroacoustic applications, along with an adaptive shock-capturing algorithm which allows high-intensity acoustic fields to be examined. An improvement to the shock detection procedure is also proposed in order to meet the specificities of nonlinear propagation at long range. The modeling as well as the numerical results are reported in detail and discussed.

Numerical simulation of shallow-water flooding using a two-dimensional finite volume model

A 2-D Finite Volume Model (FVM) is developed for shallow water flows over a complex topography with wetting and drying processes. The numerical fluxes are computed using the Harten, Lax, and van Leer (HLL) approximate Riemann solver. Second-order accuracy is achieved by employing the MUSCL reconstruction method with a slope limiter in space and an explicit two-stage Runge-Kutta method for time integration. A simple and efficient method is introduced to deal with the wetting and drying processes without any correction of the numerical flux term or the source term. In this new method, a switch of alternative schemes is used to compute the water depths at the cell interface to obtain the numerical flux. The model is verified against benchmark tests with analytical solutions and laboratory experimental data. The numerical results show that the model can simulate different types of flood waves from the ideal flood wave to cases over complex terrains. The satisfactory performance indicates an extensive application prospect of the present model in view of its simplicity and effectiveness.

Pulsatile Poiseuille flows in microfluidic channels with back-and-forth mode

The numerical solver for the velocity field equation describing laminar pulsatile flows driven by a timedependent pressure drop in the straight microfluidic channel of square cross-section is developed. In the computational algorithm, an orthogonal collocation on finite element scheme for spatial discretizations is combined with an adaptive Runge-Kutta method for time integration. The algorithm with the 1,521 computational nodes and the accuracy up to O(10 –5 ) is applied to the flow in the back-and-forth standing mode with the channel hydraulic diameter (Dh) in the range 10 – 500 µm and the oscillating frequency (f) of 1 to 100 Hz. As a result, a periodic steady state is defined as the flow condition where there would be no net movement after long time elapses. Following by the retardation phenomena in a cycle, reversal of the axial velocity is observed at the channel center. Major attention is focused on the influences of the size of channel cross-section and the oscillating frequency. Increasing Dh and f results in the decrease in the amplitude of mean velocity but the increase in the start-up time. Larger time delay occurs by low-frequency pulsation.

Flood Simulation Using a Well-Balanced Shallow Flow Model

This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against...

A relaxation scheme for a multi-class Lighthill-Whitham-Richards traffic flow model

We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-explicit Runge-Kutta method for time integration. The resulting method retains the simplicity of the relaxation schemes. There is no need to involve Riemann solvers and characteristic decomposition. Even the computation of the eigenvalues is not required. This makes the scheme particularly well suited for the MCLWR model in which the analytical expressions of the eigenvalues are difficult to obtain for more than four classes of road users. The numerical results illustrate the effectiveness of the presented method.

Large-Eddy Simulation of a Round Jet in Crossflow

A numerical simulation of a round turbulent jet issuing perpendicularly into a laminar M = 0.2 boundary-layer crossflow is performed at a jet-to-crossflow momentum ratio of 3.3 and a jet Reynolds number of 6930. The mixing behavior of the jet in crossflow is investigated by computing the evolution of a passive scalar. The spatial discretization of the computational domain consists of a block-structured multiblock grid with 58 blocks and a total of 8.9 million cells. The NSMB (Navier-Stokes Multiblock) simulation code uses a finite-volume discretization with a skew-symmetric fourth-order central scheme and a four-stage Runge-Kutta method for time integration. To account for the subgrid scales, the approximate-deconvolution model is employed in a multiblock formulation. Of special interest is the mixing behavior of the jet with the crossflow connected with the complex vortex systems in the mixing region. Results are compared with incompressible large-eddy simulation data and experimental measurements at similar flow parameters. Significant findings comprise the study of the jet development along its trajectory, as well as the upstream influence of the crossflow into the nozzle. Furthermore, the three-dimensional temporal oscillation of the ejected jet body is investigated and an attempt is made to correlate it to the formation of vortices in the mixing region.

Large-Eddy Simulation of Separated Flow in a Channel with Streamwise-Periodic Constrictions

Large-eddy simulations using the approximate-deconvolution subgrid-scale model of compressible flow in a channel with streamwise-periodic constrictions are performed at two Reynolds numbers of 2800 and 10,595. The goal of the study is the evaluation of this subgrid-scale model for massively separated flows. The Mach number was chosen as 0.2 to facilitate comparison with incompressible flow data. A relatively coarse structured grid and a finite-volume discretization employing a skew-symmetric fourth-order central scheme are used with a four-stage Runge-Kutta method for time integration. The results are compared to data from the literature obtained with incompressible direct numerical simulations, as well as highly resolved large-eddy simulations. Despite the chosen coarse resolution, good agreement of the statistical quantities and important flow features, such as separation and reattachment locations, is achieved. In contrast to previous works we also consider the flow properties at the straight upper wall. We introduce a new measure quantifying the time fraction of reverse flow along the wall. Furthermore, we analyze the energy density spectra of the velocity fluctuations and study the turbulence structure with Lumley's flatness parameter and realizability map. We also illustrate the vortex systems associated with the shedding and other instantaneous flow structures.

Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations

Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.

Design study of a small scale soft recovery system

A soft recovery system (SRS) is a device that stops a high speed projectile without damaging the projectile. The SRS is necessary to verify the shock resistant requirements of microelectronics and electro-optic sensors in smart munitions, where the projectiles experience over 20,000 g acceleration inside the barrel. In this study, a computer code for the performance evaluation of a SRS based on ballistic compression decelerator concept has been developed. It consists of a time accurate compressible one-dimensional Euler code with use of deforming grid and a projectile motion analysis code. The Euler code employs Roe’s approximate Riemann solver with a total variation diminishing (TVD) method. A fully implicit dual time stepping method is used to advance the solution in time. In addition, the geometric conservation law (GCL) is applied to predict the solutions accurately on the deforming mesh. The equation of motion for the projectile is solved with the four-stage Runge-Kutta time integration method. A small scale SRS to catch a 20 ㎜ bullet fired at 500 m/s within 1,600 g-limit has been designed with the proposed method.

SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME

A numerical method for solving the two-dimensional shallow water equations was presented. This method was based on the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving (SSP) Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.

Direct numerical simulation of the turbulent flow in an elliptical pipe

The turbulent flow in a pipe with an elliptical cross section is directly simulated at Re = 4000 (where the Reynolds number Re is calculated in terms of the mean velocity and the hydraulic diameter). The incompressible Navier-Stokes equations are solved in curvilinear orthogonal coordinates by using a central-difference approximation in space and a third-order accurate semi-implicit Runge-Kutta method for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and of the pressure gradient in the kinetic energy production. The distributions of the mean and fluctuation characteristics of the turbulent motion over the pipe’s cross section are computed.

Effective cell-centred time-domain Maxwell’s equations numerical solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell’s equations on multi-dimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge–Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.

Interpolated Differential Operator (IDO) Scheme for Direct Numerical Simulation of Two-Dimensional Homogeneous Isotropic Turbulence

Direct Numerical Simulation (DNS) of two-dimensional homogeneous isotropic turbulence is conducted using the Interpolated Differential Operator (IDO) scheme, which is highly accurate numerical scheme based on spatial discretization. Fifth-order Hermite interpolation and four stage Runge-Kutta time integration methods are employed. The incompressible Navier-Stokes equation is solved by the SMAC algoritim under the same conditions (initial and boundary conditions, comparable grid number, and time step) as the spectral method. It is found that the present IDO scheme has almost comparable accuracy with that of the spectral method in all the statistical quantities and the energy spectrum.

Comparison of time domain numerical solvers for the propagation of a Gaussian pulse inside a rectangular waveguide

We analyse techniques for increasing the accuracy and efficiency of a finite-volume time-domain (FV-TD) cell-centred computational methodology. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell equations on multi-dimensional structured grid networks are investigated and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed by the use of Gaussian microwave pulse simulations. The implementation of typical electromagnetic boundary conditions is also deliberated. Finally, a comparison of the classical finite-difference time-domain (FD-TD) method and FV-TD numerical results for a standard case study in rectangular waveguides allows the accuracy of the developed methods to be assessed.