Forward Time Step Research Articles

A condition fitting method of temperature field for tunnel structure based on test measurements

Accurate temperature distribution of the tunnel lining under a fire scenario is crucial for conducting reliable thermal–mechanical investigations, as it directly impacts thermal stress and property degradation. In this regard, achieving consistency between simulation and actual response is imperative. However, the precision of most simulated temperature fields is hindered by a phenomenon known as the 'temperature platform', which occurs around the vaporization point of water. To this end, a Condition Fitting Method (CFM) is proposed in this paper. In CFM, optimal parameters are adjusted based on local historical measurements, enabling the acquisition of a detailed temperature field for the entire tunnel structure in real-time or even forward time step. Based on a preliminary segment fire test, the implementation of CFM successfully captures the temporal characteristics of the temperature platform on certain survey points. The results of both the current and forward time steps obtained from CFM demonstrate close agreement with the reference measurements obtained from the fire test, indicating consistency throughout the entire process. The maximum average error of the calculation result on survey points is 50.4 °C while the value for prediction result is 65.3 °C. For further investigation, a parametric study is conducted to identify the most crucial parameters. Additionally, numerical verification is performed to assess the feasibility of the proposed method and to compare the efficiency of different optimal algorithms. The results indicate that the proposed method has overcome the drawback of conventional simulation method that the temperature deviate from actual values at early stage. This achievement ensures that the proposed method provides a reliable temperature field for further investigations.

A Novel Blurring and Sharpening Techniques Using Different Images Based on Heat Equations

This paper is dedicated to show the practical part of the heat equation and its influence on images from real-life. Examples of one dimension and two dimensions are given in this paper to show how applying the forward and backward time steps on an image that is inserted as a pixel matrix in the well-known heat equation with certain parameters can make the blurring, noising, and sharpening effects. In addition, we are going to make a model for a heat diffusion in two directions x and y and assemble letters from components in matrices to apply our experiment on some medical and colorful images. Finally, we conclude that undesired details in an image can be removed by using the proposed equation.

A Flow Perspective on Nonlinear Least-Squares Problems

Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

A Forward Time Stepping Heat Conduction Model for Spot Melt Additive Manufacturing

Abstract Solidification dynamics are crucial for determining microstructure development in additively manufactured parts. Multiphysics models based on finite element or finite volume methods may help gain insight for complicated phenomena such as fluid flow, keyholing, and porosity but are too computationally expensive to use for simulating actual builds. Recent analytic and semi-analytic solutions for moving heat sources in a semi-infinite three-dimensional space provide a way to accurately estimate the solidification conditions for entire builds. The downside to these methods is that, unlike finite element or finite volume methods, they cannot use the temperature distribution of the previous timesteps to march the solution forward in time. This paper provides the mathematical formulation and implementation of a forward time stepping (FTS) approach to an existing semi-analytic solution. The speed and accuracy of the two methods are then compared for various scan patterns. The result is that, for spot melts, the forward time-stepping model provides improvements in both speed and accuracy. This is especially true for longer simulations, where the simulation can be orders of magnitude faster. The longest simulation analyzed in this paper was roughly 30× faster when using the forward time-stepping model versus the straightforward implementation of the semi-analytic solution.

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge–Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge–Kutta methods were established in Isherwood et al. (SIAM J Numer Anal 56(6):3276–3307, 2018), where it was shown that it is possible to define explicit integrating factor Runge–Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge–Kutta method that has non-decreasing abscissas. However, explicit SSP Runge–Kutta methods have an order barrier of $$p=4$$, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge–Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge–Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge–Kutta methods; furthermore, our explicit SSP two-step Runge–Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge–Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge–Kutta methods designed in Isherwood et al. (2018).

An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

In this paper, we consider two-space dimensional nonlinear Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation and two-space dimensional nonlinear fourth-order extended Fisher–Kolmogorov (EFK) equation which have a lot of applications in different branches of science, especially in mathematical biology. We present a wavelet collocation method based on Chebyshev wavelets combined with two different time discretization schemes. To generate more accurate results for Fisher–KPP equation, in time variable discretization, forward Euler timestepping scheme along with Taylor series expansion is used. Resultant time-discretized scheme has second-order accuracy and includes second-order time derivative which is evaluated from governing equation. On the other hand for EFK equation forward Euler timestepping scheme with first-order accuracy is used. Then space variables situated in the semi-discrete schemes are discretized with Chebyshev wavelet series expansion. In this way a full discrete scheme is obtained. By this approach obtainment of numerical solution of considered partial differential equations is turned into an operation of finding solution of an algebraic system of equations. Actually the solution of the algebraic system of equations is wavelet coefficients in wavelet series expansion. Putting these coefficients into Chebyshev wavelet series expansion the numerical solution of considered partial differential equations can be obtained successively. The main objective of this paper is to demonstrate that Chebyshev wavelet-based method is accurate, efficient, and reliable for nonlinear two-space dimensional high-order partial differential equations. Six test problems are considered and $$L_2$$, $$L_{\infty }$$ error norms are calculated for comparison of our numerical results with exact results whenever they are available. Also numerical results are plotted for comparison with reference solutions. The obtained results certify the applicability and efficiency of the suggested method for Fisher–KPP and EFK equations.

An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics [arXiv:1808.04669]. to the incompressible magnetohydrodynamics (MHD). A globally divergence-free finite element space is used for both the velocity and the magnetic field. Highlights of the scheme includes global and local conservation properties, high-order accuracy, energy-stability, pressure-robustness. When forward Euler time stepping is used, we need two symmetric positive definite (SPD) hybrid-mixed Poisson solvers (one for velocity and one for magnetic field) to advance the solution to the next time level. Since we treat both viscosity in the momentum equation and resistivity in the magnetic induction equation explicitly, the method shall be best suited for inviscid or high-Reynolds number, low resistivity flows so that the CFL constraint is not too restrictive.

Numerical study on the effects of absorptivity on performance of flat plate solar collector of a water heater

The characteristics of absorber plate of a flat plate solar collector play an important role in the improvement of the performance. In this work, a numerical analysis is carried out to explore the effect of absorptivity and emissivity of absorber plate to the performance of the solar collector of a solar water heater. For a results comparison, a simple a simple solar box cooker with absorber area of 0.835 m × 0.835 m is designed and fabricated. It is employed to heat water in a container by exposing to the solar radiation in Medan city of Indonesia. The transient governing equations are developed. The governing equations are discretized and solved using the forward time step marching technique. The results reveal that the experimental and numerical results show good agreement. The absorptivity of the plate absorber and emissivity of the glass cover strongly affect the performance of the solar collector.

Numerical study on the effect of configuration of a simple box solar cooker for boiling water

In this work, a numerical study is carried out to investigate the effect of configuration of a simple box solar cooker. In order to validate the numerical results, a simple a simple solar box cooker with absorber area of 0.835 m × 0.835 m is designed and fabricated. The solar box cooker is employed to boil water by exposing to the solar radiation in Medan city of Indonesia. In the numerical method, a set of transient governing equations are developed. The governing equations are solved using forward time step marching technique. The main objective is to explore the effect of double glasses cover, dimensions of the cooking vessel, and depth of the box cooker to the performance of the solar box cooker. The results show that the experimental and numerical results show good agreement. The performance of the solar box cooker strongly affected by the distance of the double glass cover, the solar cooker depth, and the solar collector length.

Towards a Stationary Monge--Kantorovich Dynamics: The Physarum Polycephalum Experience

In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Strong stability preserving (SSP) Runge--Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of integrating factor Runge--Kutta methods have not been established. In this work we show that it is possible to define explicit integrating factor Runge--Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for explicit integrating factor Runge--Kutta methods to be SSP, namely, that they are based on explicit SSP Runge--Kutta methods with nondecreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp and that the nondecreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test cases our new explicit SSP integrating factor Runge--Kutta methods out-perform the corresponding explicit SSP Runge--Kutta methods, implicit-explicit SSP Runge--Kutta methods, and some well-known exponential time differencing methods.

Prediction of hourly ground-level PM2.5 concentrations 3 days in advance using neural networks with satellite data in eastern China

This study is an attempt to explore the effectiveness of satellite data in predicting hourly PM2.5 (Respirable particulate matter with aerodynamic diameter below 2.5 μm) concentrations for a chosen number of forward time steps over eastern China. MODIS (Moderate Resolution Imaging Spectroradiometer) aerosol optical depth (AOD), hourly forecasted meteorological variables, along with respective pollutant predictors from 2013 to 2015 were used as input to a multi-layer perceptron (MLP) type of back-propagation neural network. A novel approach, based on the correlation coefficients between surface PM2.5 and meteorological variables, was employed in selecting the averaging periods for meteorological input. Backward air mass trajectory was combined with AOD so as to explicitly measure the contribution of regional transport. Owing to the spatial variability of the AOD-PM2.5 relationship, each grid cell on the AOD retrievals was assigned to a prediction model trained by the nearest monitoring station from them. The proposed model seems to perform better in southern China. Also predictions incorporating transport predictor tend to have higher rates of detecting PM2.5 exceedance hours. We further introduced a sensitive analysis in Beijing by testing a model with surface PM2.5 input, evaluated versus the AOD input one. The two models were evaluated on the perturbed test set with input values ranging from 10 to 40%. It is found that PM model significantly outperforms the AOD one when the prediction horizon is longer than 36 h. This approach shows potential to be adapted for other regions, but will be most useful for areas without access to sophisticated deterministic models.

Deviation analysis of loosely coupled quasi-static method for fluid-thermal interaction in hypersonic flows

Abstract Fluid-thermal interaction plays an important role in the design and development of hypersonic vehicles. Among the different ways to couple the flow field with the material thermal response, the loosely coupled quasi-static method is widely used. In this paper, deviation formulas based on one-dimensional model are derived for uncoupled method and loosely coupled quasi-static method to estimate the structural temperature deviations at the stagnation point in different forward time steps. To verify the formulas, fluid-thermal coupling numerical simulations are conducted for 2-D and 3-D hypersonic fluid-thermal interaction problems. The results show that the temperature deviation formulas agree quite well with the deviations from numerical simulation. The derived deviation formulas can serve as some guidance for selecting an appropriate forward time step for the loosely coupled quasi-static method.

A novel hybrid strategy for PM2.5 concentration analysis and prediction

The analysis and prediction of air pollutants are of great significance in environmental research today since airborne pollution is a substantial threat, especially in urban agglomerations of China. To develop more effective warning systems and management advice, the authorities and city dwellers need more accurate forecasts of the air pollution. Most previous analysis systems were based on costly observation apparatus at fixed sites, forecasting models were usually built on observations within a certain range, and some observations contained biases. In this paper, a novel and effective framework, termed HML-AFNN, was successfully developed to analyse and forecast the concentration of particular matter (PM2.5) for a selected number of forward time steps. In a simulation of the trajectory of air pollutants, the high-dimension association rules (HDAR) approach considered the tempo-spatial relations, as well as the meteorological and geographical factors of the ambient regions, as parameters. In addition, the learning vector quantization (LVQ) network was adopted to select the appropriate inputs to improve the efficiency of the training process. Moreover, an adaptive fuzzy neural network (AFNN), a combination of neural and fuzzy logic, was utilized to analyse and predict the PM2.5 concentration. The experiment results of our study on two major urban agglomerations of China, the Jing-Jin-Ji area and Pearl River Delta, over a period of more than one year demonstrated that the developed hybrid HML-AFNN model outperforms a plain AFNN, an HM-AFNN model without LVQ and the least squares support vector machines (LS-SVM); this superior performance can be determined from the values of several error indexes, including MAE, MAPE and band errors. This hybrid model, which has robust and accurate results, shows the potential to be a political and administrative method to issue effective early warnings and to design suitable abatement strategies.

Numerical Study on Natural Vacuum Solar Desalination System with Varying Heat Source Temperature

A natural vacuum desalination unit with varying low grade heat source temperature is investigated numerically. The objective is to explore the effects of the variable temperature of the low grade heat source on performances and characteristics of the desalination unit. The specifications of the desalination unit are naturally vacuumed with surface area of seawater in evaporator and heating coil are 0.2 m2 and 0.188 m2, respectively. Temperature of the heating coil is simulated based on the solar radiation in the Medan city. A program to solve the governing equations in forward time step marching technique is developed. Temperature of the evaporator, fresh water production rate, and thermal efficiency of the desalination unit are analysed. Simulation is performed for 9 hours, it starts from 8.00 and finishes at 17.00 of local time. The results show that, the desalination unit with operation time of 9 hours can produce 5.705 L of freshwater and thermal efficiency is 81.8 %. This reveals that varying temperature of the heat source of natural vacuum desalination unit shows better performance in comparison with constant temperature of the heat source.

A novel forecasting model based on a hybrid processing strategy and an optimized local linear fuzzy neural network to make wind power forecasting: A case study of wind farms in China

As a crucial issue in the wind power industry, it is a tough and challenging task to predict the wind power accurately because of its nonlinearity, non-stationary and chaos. In this paper, we propose a novel hybrid model, which combines an integrated processing strategy and an optimized local linear fuzzy neural network, to forecast the wind power. First, discrete wavelet transform and singular spectrum analysis are used to filter out the noises and extract the trends from original wind power series, respectively. Then, the novel no-negative-constraint-combination theory together with the CS algorithm are adopted to integrate these two subseries obtained from the first step to retain strengths of each method. Based on the phase space reconstruction model, we could determine the most proper structure of the input sets and the output sets. Next, the local linear fuzzy neural network, with the initial rule consequent parameters optimized by the seeker optimization algorithm, is utilized to make wind power forecasts for a selected number of forward time steps. The numerical results from two experiments demonstrate that the proposed hybrid model is an effective approach to predict wind power, and the accuracy of prediction is highly improved compared with conventional forecasting models.

Study on the performance of natural vacuum desalination system using low grade heat source

In the present work characteristics and performance of a natural vacuum desalination system using low grade heat source is studied numerically and experimentally. In the numerical work, a program to solve the governing equations in forward time step marching technique is developed. In the experimental work a lab scale natural vacuum system using electric heater as a heat source is designed and tested. The surface area of seawater in the evaporator is 0.5m. The numerical and experimental results show a good agreement. By using numerical simulations effect operating conditions such as heat source temperature, condenser temperature, heating coil surface area, and the present of heat recovery unit are explored. The simulation showed that, the present system with evaporator surface area of 0.2m2, operation time of 9h and source temperature of 80°C will produce 6.63L of fresh water ant thermal efficiency is 70.4%. The main conclusion can be drawn here is that the performance of the natural vacuum desalination system is mainly affected by maximum temperature in evaporator and minimum temperature in condenser. On the other hand, the surface area of heating coil and the heat recovery unit showed only a small effect.

Ugotavljanje števila vnaprejšnjih časovnih korakov za sekvenčni Beckov približek pri reševanju problemov inverzne toplotne prevodnosti

Ugotavljanje števila vnaprejšnjih časovnih korakov za sekvenčni Beckov približek pri reševanju problemov inverzne toplotne prevodnosti

The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes

High-order temporal discretizations for hyperbolic conservation laws have historically been formulated as either a method of lines (MOL) or a Lax--Wendroff method. In the MOL viewpoint, the partial differential equation is treated as a large system of ordinary differential equations (ODEs), where an ODE tailored time integrator is applied. In contrast, Lax--Wendroff discretizations immediately convert Taylor series in time to discrete spatial derivatives. In this work, we propose the Picard integral formulation (PIF), which is based on the method of modified fluxes, and is used to derive new Taylor and Runge--Kutta (RK) methods. In particular, we construct a new class of conservative finite difference methods by applying weighted essentially nonoscillatory (WENO) reconstructions to the so-called time-averaged fluxes. Our schemes are automatically conservative under any modification of the fluxes, which is attributed to the fact that classical WENO reconstructions conserve mass when coupled with forward Euler time steps. The proposed Lax--Wendroff discretization is constructed by taking Taylor series of the flux function as opposed to Taylor series of the conserved variables. The RK discretization differs from classical MOL formulations because we apply WENO reconstructions to time-averaged fluxes rather than taking linear combinations of spatial derivatives of the flux. In both cases, we only need one projection onto the characteristic variables per time step. The PIF is generic and lends itself to a multitude of options for further investigation. At present, we present two canonical examples: one based on Taylor, and the other based on the classical RK method. Stability analyses are presented for each method. The proposed schemes are applied to hyperbolic conservation laws in one and two dimensions and the results are in good agreement with current state-of-the-art methods.

Numerical modeling of the 2D time-domain transient electromagnetic secondary field of the line source of the current excitation

To effectively minimize the electromagnetic field response in the total field solution, we propose a numerical modeling method for the two-dimensional (2D) timedomain transient electromagnetic secondary field of the line source based on the DuFort-Frankel finite-difference method. In the proposed method, we included the treatment of the earth-air boundary conductivity, calculated the normalized partial derivative of the induced electromotive force (Emf), and determined the forward time step. By extending upward the earth-air interface to the air grid nodes and the zero-value boundary conditions, not only we have a method that is more efficient but also simpler than the total field solution. We computed and analyzed the homogeneous half-space model and the flat layered model with high precision—the maximum relative error is less than 0.01% between our method and the analytical method—and the solution speed is roughly three times faster than the total-field solution. Lastly, we used the model of a thin body embedded in a homogeneous half-space at different delay times to depict the downward and upward spreading characteristics of the induced eddy current, and the physical interaction processes between the electromagnetic field and the underground low-resistivity body.