Proper Time Step Research Articles

CFD Study on Hydrodynamic Performances of a Planing Hull

The scope of the present study is to investigate the effects of various geometrical hull features, such as tunnels, spray rails and whiskers on the hydrodynamic performance of a high-speed planing hull. The criteria being tested to emphasize the boat performance are the total drag, sinkage and trim angle. In addition, the decomposition of the resistance into viscous and wave-making resistance are taken into consideration. The study starts with a validation test against experimental data in order to accentuate the capability of the Computational Fluid Dynamics CFD simulation to accurately predict the total drag and trim angle of the initial form. This is later followed by a verification study based on the Richardson Extrapolation method with a grid- and time-step-convergence test in order to predict the numerical errors during the simulation. After establishing the simulation parameters regarding the proper grid size and time step, the comparative study takes place for five hull shapes and two whisker configurations while the boat is sailing at eight different speeds. The assessment of the hydrodynamic flow parameters is evaluated compared to the initial form in order to investigate the influence of the geometry change on the hydrodynamic performances of the boat. Validation of the numerical results showed the reliability of the CFD simulation to accurately predict the drag and trim angle of the boat, while the comparative study revealed that the total drag can be reduced by up to 9%, especially at higher speeds.

Unsteady Non-Uniform Flow of Molten Metals in Rectangular Open Channels

In the metallurgical industries, it is very important to characterize the flow of molten metals in open channels given that they are transported through these devices to different plant sections. However, unlike the flow of water which has been studied since ancient times, the flow of molten metals in open channels has received little attention. The unsteady non-uniform flow of blast furnace molten pig iron in a rectangular open channel is analyzed in this work by numerical solution of the Saint-Venant equations. The influence of mesh size on the convergence of molten metal height is studied to determine the proper mesh and time step sizes. A sinusoidal inflow pulse is imposed at the entrance of the channel in order to analyze the propagation of the resulting wave. The influence of the angle of inclination of the channel and the roughness coefficient of the walls on the amplitude and the dynamic behavior of the height of the molten metal are analyzed. Phase portraits of the channel state variables are constructed and interpreted. Numerical simulations show that as the angle of inclination of the channel increases, the amplitude of the formed wave decreases. From 10 degrees onwards, the peak of the wave descends even below the initial height. On the other hand, the roughness coefficient affects the molten pig iron height profiles in an inverse way than the angle of inclination. The amplitude of the formed wave increases as the roughness coefficient increases.

Comparisons of two representative methods classified as immersed boundary and domain methods

Immersed methods have proven to be powerful numerical measures to simulate intricate fluid-structure interactions using non-conforming meshes. And they can be classified into immersed boundary methods and immersed domain methods based on the utilization of fictitious fluid around the interface or over the entire solid domain. Their performances are compared in light of several common problems for immersed methods, and two representative methods are employed by immersed boundary-lattice Boltzmann method with smoothed point interpolation method (IBLBM-SPIM) and immersed smoothed point interpolation method (IS-PIM). Numerical tests show that IS-PIM allows a fairly independent mesh size for each domain of fluid and solid, and achieves a smooth solid response; IBLBM-SPIM can ensure numerical stability in a large range of density ratio between solid and fluid by setting a proper time step to alleviate the added mass effect, and has a direct influence on the deformation and motion of solids. The present study also gives some enlightening thoughts on developing novel methods to deal with different FSI problems.

Accurately simulating nine-dimensional phase space of relativistic particles in strong fields

Next-generation high-power laser systems that can be focused to ultra-high intensities exceeding 1023 W/cm2 are enabling new physics regimes and applications. The physics of how these lasers interact with matter is highly nonlinear, relativistic, and can involve lowest-order quantum effects. The current tool of choice for modeling these interactions is the particle-in-cell (PIC) method. In the presence of strong electromagnetic fields, the motion of charged particles and their spin is affected by radiation reaction (either the semi-classical or the quantum limit). Standard (PIC) codes usually use Boris or similar operator-splitting methods to advance the particles in standard phase space. These methods have been shown to require very small time steps in the strong-field regime in order to obtain accurate results. In addition, some problems require tracking the spin of particles, which creates a nine-dimensional (9D) particle phase space, i.e., (x,u,s). Therefore, numerical algorithms that enable high-fidelity modeling of the 9D phase space in the strong-field regime (where both the spin and momentum evolution are affected by radiation reaction) are desired. We present a new particle pusher that works in 9D and 6D phase space (i.e., with and without spin) based on analytical rather than leapfrog solutions to the momentum and spin advance from the Lorentz force, together with the semi-classical form of radiation reaction in the Landau-Lifshitz equation and spin evolution given by the Bargmann-Michel-Telegdi equation. Analytical solutions for the position advance are also obtained, but these are not amenable to the staggering of space and time in standard PIC codes. These analytical solutions are obtained by assuming a locally uniform and constant electromagnetic field during a time step. The solutions provide the 9D phase space advance in terms of a particle's proper time, and a mapping is used to determine the proper time step duration for each particle as a function of the lab frame time step. Due to the analytical integration of particle trajectory and spin orbit, the constraint on the time step needed to resolve trajectories in ultra-high fields can be greatly reduced. The time step required in a PIC code for accurately advancing the fields may provide additional constraints. We present single-particle simulations to show that the proposed particle pusher can greatly improve the accuracy of particle trajectories in 6D or 9D phase space for given laser fields. We have implemented the new pusher into the PIC code Osiris. Example simulations show that the proposed pusher provides improvement for a given time step. A discussion on the numerical efficiency of the proposed pusher is also provided.

Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes

Abstract Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the construction of explicit symplectic integrators is frequently difficult in general relativity because all variables are inseparable. Moreover, even if two analytically integrable splitting parts exist in a relativistic Hamiltonian, all analytical solutions are not explicit functions of proper time. Naturally, implicit symplectic integrators, such as the midpoint rule, are applicable to this case. In general, these integrators are numerically more expensive to solve than same-order explicit symplectic algorithms. To address this issue, we split the Hamiltonian of Schwarzschild spacetime geometry into four integrable parts with analytical solutions as explicit functions of proper time. In this manner, second- and fourth-order explicit symplectic integrators can be easily made available. The new algorithms are also useful for modeling the chaotic motion of charged particles around a black hole with an external magnetic field. They demonstrate excellent long-term performance in maintaining bounded Hamiltonian errors and saving computational cost when appropriate proper time steps are adopted.

Integrated analytical modeling of transient heat transfer inside and outside U-tube ground heat exchanger: A new angle from composite-medium method

Ground heat exchanger (GHE) is vital to geothermal energy utilization, seasonal energy storage, etc. The transient heat transfer analytical model is still a challenge for U-tube GHE due to the asymmetric structure and complex heat interaction among fluid, grout and soil. This study proposed a novel analytical model for U-tube GHE from a new angle of composite medium method. An inspiration from heat transfer of GHE in multilayer soil helps with the new modeling. Combining with the techniques of treating non-uniform pipe wall temperature/heat flux as well as effect of ground surface flux, a new analytical model of U-tube GHE is established. Both experimental data and other simulation data are used for model verifications. The investigation on the temperature evolution in longitudinal and cross section of GHE is realized through visualization method. The analysis on grout material selection and borehole depth design revealed some new results. The spatial distribution of temperature curve in different grout materials share the same shape in steady state stage. This result can offer a useful guide for quick evaluation of effect grout material. At last, the proper time step and U-tube segmentation is suggested to ensure a fast computation and a better simulation accuracy. This study offered a new paradigm for GHE simulation which may contribute to the system optimization, design and many other types modeling as well.

Assessing failure probability of coastal structures based on probabilistic representation of sea conditions at the structures’ location

In the present paper a thorough probabilistic methodology is presented, aiming at estimating the reliability of coastal structures, such as rubble mound breakwaters during their lifetime, based on the probabilistic representation of load environmental and resistance parameters. One of the innovative points and main objectives of this study is the estimation of the failure probability of a coastal structure based on the long-term wave climate at the structure's location, usually met in intermediate waters, using wave observations or measurements in deeper waters. This task is accomplished by applying a wave propagation statistical model in order that the joint probability density function of all random load parameters be estimated at the structure's location. Moreover, a relation between an event-based extreme value analysis and an analysis on sea-state conditions within storm events is derived in order that both of these two approaches could be compared estimating the same kind of failure probability; an unconditional one. The latter is properly defined here as the percentage of the structure's Lifetime that the structure will be in a failure situation. This unconditional failure probability provides direct information on the time period that the structure will be in a situation of failure (since it considers the total range of loadings and total lifetime), and thus can be incorporated more efficiently in an integrated risk analysis with consideration of social and economic costs. Besides, another specific issue of scientific originality could be considered the investigation on the proper time step denoting the sea state in the sea state analysis applied. In this manner, the actual history and shape of each storm event is taken into consideration. Furthermore, it is shown that these two approaches could be incorporated in the design of a coastal structure. Moreover, two different fully probabilistic methods, Direct Integration Method and Monte Carlo Method, were applied (and compared) by using a combination of variables with zero and non-zero hazard rate, referred here as a combined time-invariant and time-variant analysis. Finally, the effect of considering additional and different types of parameters as random variables on the assessed failure probability of the structure has been investigated. The aforementioned methodology has been applied to a sample of wave and sea level data at the structure's location generated for this purpose. The original wave data were derived from measurements in deeper waters than the structure's location, covering a period of 8 years, obtained from an oceanographic buoy, located in the western Mediterranean off Malaga, Spain. Sea level data were also obtained from a tide gauge, located in Malaga's harbor. Finally, it is shown that the methodology derived from this study could be incorporated into a coastal structure's design process to meet specific safety requirements.

On the Data-Driven Generalized Cell Mapping Method

Global analysis is often necessary for exploiting various applications or understanding the mechanisms of many dynamical phenomena in engineering practice where the underlying system model is too complex to analyze or even unavailable. Without a mathematical model, however, it is very difficult to apply cell mapping for global analysis. This paper for the first time proposes a data-driven generalized cell mapping to investigate the global properties of nonlinear systems from a sequence of measurement data, without prior knowledge of the underlying system. The proposed method includes the estimation of the state dimension of the system and time step for creating a mapping from the data. With the knowledge of the estimated state dimension and proper mapping time step, the one-step transition probability matrix can be computed from a statistical approach. The global properties of the underlying system can be uncovered with the one-step transition probability matrix. Three examples from applications are presented to illustrate a quality global analysis with the proposed data-driven generalized cell mapping method.

L1 adaptive output‐feedback control of multivariable nonlinear systems subject to constraints using online optimization

SummaryIn this paper, an L1 adaptive output‐feedback controller is developed for multivariable nonlinear systems subject to constraints using online optimization. In the L1 adaptive architecture, an adaptive law will update the adaptive parameters that represent the nonlinear uncertainties such that the estimation error between the predicted state and the real state is driven to zero at every integration time step. Of course, neglection of the unknowns for solving the error dynamic equations will introduce an estimation error in the adaptive parameters. The magnitude of this error can be lessened by choosing a proper sampling time step. A control law is designed to compensate the nonlinear uncertainties and deliver a good tracking performance with guaranteed robustness. Model predictive control is introduced to solve a receding horizon optimization problem with various constraints maintained. Numerical examples are given to illustrate the design procedures, and the simulation results demonstrate the availability and feasibility of the developed framework.

A High Order Positivity Preserving DG Method for Coagulation-Fragmentation Equations

We design, analyze, and numerically validate a novel discontinuous Galerkin (DG) method for solving the coagulation-fragmentation equations. The DG discretization is applied to the conservative form of the model, with flux terms evaluated by Gaussian quadrature with $Q=k+1$ quadrature points for polynomials of degree $k$. The first moment (total mass) is naturally conserved by the scheme construction, and the positivity of the mass density is enforced by the use of a scaling limiter based on positive cell averages. The positivity of cell averages is shown to propagate by the time discretization, provided a proper time step restriction is imposed.

Numerical and Physical Investigation of the Performance of Turbulence Modeling Schemes around a Scour Hole Downstream of a Fixed Bed Protection

Local scour occurs around hydraulic structures such as piers, bed protections, and dikes. In this study, the turbulent flow around a scour hole downstream of a fixed bed protection was investigated. Numerical modeling with OpenFOAM was applied to compute the flow velocity and turbulent kinetic energy with respect to flow conditions by changing water depth. A proper computational grid size and time step for simulations are suggested. Three typical turbulent models, k − ε , k − ω , and k − ω S S T , were considered for simulating the flow around a scour hole. The performances of the three models were evaluated by comparing them with numerical and laboratory experimental results. Mean flow velocity profiles computed by the three turbulent schemes are generally in good agreement with laboratory measurements. However, k − ω has a limitation in simulating reversal flow in the scour hole, and the k − ε model does not predict turbulent kinetic energy well near the bottom. Thus, this study found that the most suitable turbulent model for simulating flow around a scour hole downstream of a fixed bed protection is the k − ω S S T model.

A heterogeneous multiscale method for stiff combustion chemistry integration in reactive flows

The concept of heterogeneous multiscale method (HMM) is applied into combustion chemistry integration in reactive flows in this study. It adopts a macro solver to integrate the slow variables with global time steps and a micro solver to integrate the fast variables with smaller inner time steps. Explicit or implicit schemes with different orders of accuracy could be applied in the macro and micro solvers. In this study, we formulated HMM with a first-order Euler forward scheme in the micro solver and a second-order trapezoidal rule in the macro solver. An adaptive time step method was proposed to better evaluate a proper time step and control the error in this HMM. This formulation of HMM with the adaptive time step method has been implemented and tested in homogeneous constant volume auto-ignition in hydrogen, methane and butane mixture under various initial temperatures, pressures and equivalence ratios. The speed-up factor of 100–700 has been achieved compared with the implicit solver DASAC. In addition, the numerical accuracy and stability of this HMM formulation have been discussed. It was also applied into multi-dimensional Reynolds-averaged Navier–Stokes (RANS) engine simulations and good efficiency and accuracy was accomplished. In order to better optimize its efficiency performance in multi-dimensional reactive flow simulations, a hybrid scheme with VODE and HMM was proposed and tested in non-homogeneous charged engine simulations. In sum, the concept of HMM in stiff combustion chemistry integration is numerically stable, accurate and efficient.

Optimal evaluation of time step size in numerical simulation for two-dimensional flow sensing

This paper proposes an optimal evaluation of time step size for numerical computations based on the 180 numerical simulations of the two-dimensional unsteady flow at low Reynolds number around a circular cylinder. A proper time step size is very important to obtain the right magnitude and frequency for the numerical computation to simulating the real flow. The optimal evaluation is found out after analyzing the influences of the time step sizes on the Strouhal numbers, vortex-shedding periods, lift and drag coefficients. The time step size is found out to be a function of the dividend of vortex-shedding period, the velocity, the feature size and the Strouhal number. The optimal dividend point of the vortex-shedding periods for the optimal time step size is in the interval $$\left[ {50,90} \right) $$ , and the average partition counts of the interval is an optimum approximation. So, the proposed optimal evaluation of time step size is calculated by $$D/\left( {14U} \right) $$ , of which D is the feature size and U is the velocity. The calculation to decide time step size is efficient and consequently the numerical simulations are more stable. Numerical results based on the provided time step size also bring on better agreements of the model parameters to reflect the real flow theoretically.

Advancement in simulating moisture diffusion in electronic packages under dynamic thermal loading conditions

The normalization approach has been accepted as a routine to overcome the concentration discontinuity at multi-material interfaces in the moisture diffusion simulation by the finite element method. However, applying the normalization approach directly in the commercial finite element software under a dynamic thermal loading condition, such as reflow process, may lead to erroneous results. Special techniques are needed to obtain correct results. In this study, different approaches that can deal with the moisture diffusion under dynamic thermal loading conditions were reviewed and compared with case studies. Advantages and disadvantages of each approach were analyzed and discussed. Theoretical derivation was developed to show the direct concentration approach (DCA) violates the law of mass conservation. The effect of Fourier number on the accuracy of the DCA was also examined. The internal source approach was shown to be a universal method in modeling the moisture diffusion in a multi-material system under dynamic thermal loading conditions. The newest version of ANSYS also showed good performance in solving the diffusion problem under transient thermal loadings if a proper time step size was used.

Application of 2D numerical model to unsteady performance evaluation of vertical-axis tidal current turbine

Tidal current energy is renewable and sustainable, which is a promising alternative energy resource for the future electricity supply. The straight-bladed vertical-axis turbine is regarded as a useful tool to capture the tidal current energy especially under low-speed conditions. A 2D unsteady numerical model based on Ansys-Fluent 12.0 is established to conduct the numerical simulation, which is validated by the corresponding experimental data. For the unsteady calculations, the SST model, 2×105 and 0.01 s are selected as the proper turbulence model, mesh number, and time step, respectively. Detailed contours of the velocity distributions around the rotor blade foils have been provided for a flow field analysis. The tip speed ratio (TSR) determines the azimuth angle of the appearance of the torque peak, which occurs once for a blade in a single revolution. It is also found that simply increasing the incident flow velocity could not improve the turbine performance accordingly. The peaks of the averaged power and torque coefficients appear at TSRs of 2.1 and 1.8, respectively. Furthermore, several shapes of the duct augmentation are proposed to improve the turbine performance by contracting the flow path gradually from the open mouth of the duct to the rotor. The duct augmentation can significantly enhance the power and torque output. Furthermore, the elliptic shape enables the best performance of the turbine. The numerical results prove the capability of the present 2D model for the unsteady hydrodynamics and an operating performance analysis of the vertical tidal stream turbine.

Numerical study of ice-induced loads and responses of a monopile-type offshore wind turbine in parked and operating conditions

Offshore wind is an attractive source of renewable energy. In regions with cold climates, such as Canada, the Baltic Sea, and Bohai Bay, a good understanding of ice loads is essential to design a reliable and cost effective support structure for an offshore wind turbine. This paper presents a study of the dynamic ice–structure interaction of a commonly used monopile-type offshore wind turbine in drifting level ice in both parked and operating conditions. A semi-empirical numerical model for ice–structure interaction was coupled to the aero-hydro-servo-elastic simulation tool HAWC2. A convergence study was performed to determine the proper time step and simulation length to obtain reliable response prediction. The simulated ice load was compared with the formulations in international standards. The coupling between the ice loads and the structural response of the monopile-type wind turbine was investigated and found to be important. The effects of the ice characteristics (e.g., ice thickness and drifting speed) were examined with the turbine being in parked and operating conditions. Compared with current numerical model, standards from International Electrotechnical Commission (IEC) and International Organization for Standardization (ISO) predict lower ice loads. 3-D effect of ice–monopile interaction and failure of local ice sheet are considered. More stochastic dynamic phenomena are captured for ice–structure interaction in our numerical model. The effect of the ice thickness on the response was found to be significant. A negligible drifting speed effect is found on the bending moment response in the fore-aft direction. There is a large increase in the fore-aft response as the inclination angle of the cone increases. Further sensitivity studies and validation against model tests will be performed for the conical waterline of the wind turbine.

Flood Analysis Using Hydrological Modeling. Case Study – The Flood In The Upper Catchment Of River Geru, Galaţi County, Romania

Abstract This paper analyzes the flood that occurred between 11th and 13th of September 2013 in the upper catchment of the river Geru. The flood was simulated using the program Mike by DHI with the Unitary Hydrograph Method. As input data, we used the precipitation measured at the Automated Hydrological Sensor Station Cudalbi and radar precipitations. We analyzed the importance of accuracy for input data on the simulation results and the direct influence of setting the proper time steps in achieving the simulated discharge hydrograph. It appears that radar precipitations used as input data lead to a discharge hydrograph with low errors for amplitude and phase of the runoff peak. The model can be used in the future to reproduce the floods produced in the analyzed catchment and to study the influence of physical and geographical characteristics of the hillslopes.

Issues on the Choice of a Proper Time Step in Molecular Dynamics

Abstract We will discuss herein issues on the proper time step in molecular dynamics simulations. Determination of the proper time step is very important in the simulations, however, still it is determined rather empirically by trial and error and also kept fixed throughout the simulation. These cause severe problems sometimes. We raise these issues and propose a new method, which is quite different from the conventional one.

Algorithms for Solving Darcian Flow in Structured Porous Media

This paper presents several algorithms that were implemented in DRUtES [1], a new open source project. DRUtES is a finite element solver for coupled nonlinear parabolic problems, namely the Richards equation with the dual porosity approach (modeling the flow of liquids in a porous medium). Mass balance consistency is crucial in any hydrological balance and contaminant transportation evaluations. An incorrect approximation of the mass term greatly depreciates the results that are obtained. An algorithm for automatic time step selection is presented, as the proper time step length is crucial for achieving accuracy of the Euler time integration method. Various problems arise with poor conditioning of the Richards equation: the computational domain is clustered into subregions separated by a wetting front, and the nonlinear constitutive functions cover a high range of values, while a very simple diagonal preconditioning method greatly improves the matrix properties. The results are presented here, together with an analysis.

COMPUTATIONAL FLUID DYNAMIC ANALYSIS OF HIGHWAY BRIDGES EXPOSED TO HURRICANE WAVES

In present paper, numerical code STAR CCM+ by CD-adapco which works based on compressible two-phase Navier Stokes equations is used to evaluate hydrodynamic forces exerted on prototype of I10 Bridge over Escambia Bay which was extensively damaged during Hurricane Ivan. Volume of Fluid (VOF) is used to capture dynamic free surface which is well suited for simulating complex discontinuous free surface associated with wave-deck interactions. 2D and 3D models were setup and properly configured. Simulations were conducted on High performance Computing and Communication Center (HPCC) at University of Southern California. Simulation results are compared to experimental data available from Hinsdale Wave Laboratory at Oregon State University. Comparison of experimental data to simulation results show the importance of proper mesh size and time step choice on accuracy of horizontal and vertical hydrodynamic force predictions applied to bridge superstructure.