Mathematical modelling of spark-ignition engines

The flow field, scavenging efficiency, power output, heat transfer losses, and unburned hydrocarbon emissions have been numerically studied by means of a two-equation model of turbulence in a four-stroke, homogeneous-charge, spark-ignition engine. The engine is equipped with an intake valve, an exhaust valve, and a constant rate heat source which simulates the spark plug. Combustion has been modelled by means of a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression. The numerical results indicate that the intake stroke is characterized by the formation of two eddies which persist in the compression stroke. Turbulence is generated at the shear layers of the air jet drawn into the cylinder, but its level decreases in the compression stroke. Due to the heat released by the spark plug and the chemical reaction, a spherical flame kernel is formed. This kernel evolves into a cylindrical flame when the flame front reaches the piston. Fuel remains unburnt at the corner between the cylinder head and the cylinder wall due to heat transfer losses. The numerical results also indicate that despite uncertainties about the turbulence and heat transfer models, an engine model such as the one studied here can be used to understand the flow field, heat transfer losses, scavenging efficiency, and power output in conventional spark-ignition engines. Such capabilities are very helpful in the development and optimization stages of engines. For example, here the engine model thermal and scavenging efficiencies are 15.69% and 94%, respectively. The peak pressure is 33 atm and occurs at 6° ATDC. The unburnt hydrocarbon emissions are 7.41% of the total fuel admitted into the cylinder.     

Air path identification of diesel engines by LPV techniques for gain scheduled control

Modelling is a key element to improve the performance of engine control systems, but many factors like non-linearity and complexity complicate the derivation of sufficiently precise physical models. This motivates an increasing interest in data based models. Linear models can successfully represent the engine operation in some reduced regions, but tend to fail when large operating regions must be considered. This motivates the interest in deriving and using gain scheduling models or their natural extension, the linear parameter varying (LPV) models. In this article we propose to model the air path of diesel engines using input–output LPV models with a physically motivated structure and parameters estimated from data. These models are shown to combine good precision with simplicity and allow the systematic design of optimal and robust control systems, and can be determined in a very short time if sufficient data are available.     

Mathematical modeling and validation of mass transfer phenomenon in homogeneous charge compression ignition engines based on a thermodynamic multi zone model

The main purpose of the current study is mathematical modelling and validation of mass transfer phenomenon in homogeneous charge compression ignition engines. A validated multi-zone model coupled to a semi-detailed chemical kinetics is used to predict homogeneous charge compression ignition combustion and emissions. Heat and Mass transfer submodels are linked to the multi-zone model. Bulk flow and diffusion mass transfer between zones are considered. The results indicate that the diffusion mass transfer is negligible in homogeneous charge compression ignition engines. Bulk flow mass transfer plays a critical role in homogeneous charge compression ignition simulation and applying it in the multi-zone model leads to accurate prediction of the start of combustion, peak pressure and exhaust emissions. The results show that the maximum error changes from 90% to 5% in carbon monoxide prediction and from 98% to 14% in unburned hydrocarbons prediction, using the mass transfer submodel.     

水泥生产的质量——成本控制与优化

对影响水泥的 8个质量指标的 5个因素进行对称一单纯形设计与试验 ,用混料规范多项式拟合响应曲面 ,建立质量一成本控制优化的数学模型 ,对理论最优解的利益区域进行极端顶点设计并按排小磨试验 ,确定质量较好成本较低的水泥生产方案 .     

BLUP estimation of linear mixed-effects models with measurement errors and its applications to the estimation of small areas

The linear mixed-effects model （LMM） is a very useful tool for analyzing cluster data. In practice, however, the exact values of the variables are often difficult to observe. In this paper, we consider the LMM with measurement errors in the covariates. The empirical BLUP estimator of the linear combination of the fixed and random effects and its approximate conditional MSE are derived. The application to the estimation of small area is provided. Simulation study shows good performance of the proposed estimators.     

Analysis and prediction of hazard risks caused by tropical cyclones in Southern China with fuzzy mathematical and grey models

A hazard-risk assessment model and a grey hazard-year prediction model (GHYPM) are constructed by integrating recent advances in the fuzzy mathematics, grey theory and information spread technique, and then applied to 17-year tropical cyclones (TCs) hazards in Southern China. In constructing the models, a genetic fuzzy mathematical algorithm is first developed to calculate the categorical and ranking weights of TC hazard impact and cause indicators, from which their combined weights are obtained after optimization. The hazard impact and cause index series are then found by coupling the combined weights with their corresponding down-scaled indicators. A two-dimensional normal-spread technique is employed to create a primitive information matrix and a fuzzy relation matrix in order to make fuzzy rough inference of hazard risks with the factorial space theory. An exceeded probability model is developed to assess the possibility of exceeding any given hazard-year category. Results from the GHYPM show that the simulated hazard risk values are more or less consistent with the hazard-impact index series, with more than 60% probability of exceeding a moderate hazard year in Southern China. Results also show small relative errors of the GHYPM, indicating its applicability to the prediction of TC hazard-years up to 20 years.     

Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps

It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.     

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump‐diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises. Copyright © 2008 John Wiley & Sons, Ltd.