A Lumped-Parameter Modeling Methodology for One-Dimensional Hyperbolic Partial Differential Equations Describing Nonlinear Wave Propagation in Fluids

Most physical phenomena and processes encountered in engineering problems are governed by partial differential equations, PDEs. Disciplines that use partial differential equations to describe the phenomena of interest include fluid mechanics, where one is interested in predicting the flow of gases and liquids around objects such as cars and airplanes, flow in long distance pipelines, blood flow, ocean currents, atmospheric dynamics, air pollution, underground dispersion of contaminants, plasma reactors for semiconductor equipments, and flow in gas turbine and internal combustion engines. In solid mechanics, problems encountered in vibrations, elasticity, plasticity, fracture mechanics, and structure loadings are governed by partial differential equations. The propagation of acoustic and electromagnetic waves, and problems in heat and mass transfer are also governed by partial differential equations. Numerical simulation of partial differential equations is far more demanding than that of ordinary differential equations. Also the diversity of types of partial differential equations precludes the availability of general purpose “canned” computer programs for their solutions. Although commercial codes are available in different disciplines, the user must be aware of the workings of these codes and/or perform some complementary computer programming and have a basic understanding of the numerical issues involved. However, with the advent of faster computers, numerical simulation of physical phenomena is becoming more practical and more common. Computational prototyping is becoming a significant part of the design process for engineering systems. With ever increasing computer performance the outlook is even brighter, and computer simulations are expected to replace expensive physical testing of design prototypes.

For the quite extensively developed partial differential equations (PDE) backstepping methodology for coupled linear hyperbolic PDEs, we provide a generalization from finite collections of such PDEs, whose states at each location in space are vector-valued, to previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs, whose states are function-valued. The motivation for studying such systems comes from traffic applications (where driver and vehicle characteristics are continuously parametrized), fluid and structural applications, and future applications in population dynamics, including epidemiology. Our design is of an exponentially stabilizing scalar-valued control law for a PDE system in two independent dimensions, one spatial dimension and one ensemble dimension. In the process of generalizing PDE backstepping from finite to infinite collections of PDE systems, we generalize the results for PDE backstepping kernels to the continuously parametrized Goursat-form PDEs that govern such continuously parametrized kernels. The theory is illustrated with a simulation example, which is selected so that the kernels are explicitly solvable, to lend clarity and interpretability to the simulation results.

In recent years, due to the increased fossil fuel costs and environmental concerns, there has been a renewed interest in absorption cooling (using low-grade heat source) systems for refrigeration and space cooling applications. Although, the stand-alone coefficient of performance (COP) is a concern with such systems, absorption cooling can be a useful add-on that improves the overall efficiency of conventional vapor compression cooling cycle. A local company based in Las Vegas which is involved in the development of advanced HVAC technologies, has developed a natural gas fueled internal combustion (IC) engine driven heat pump. This system recovers the rejected heat from the IC engine during the heating cycle, thus, increasing the heat delivered and improving the system’s overall efficiency. However, during the cooling cycle the rejected heat is dissipated to the ambient air through radiators. The overall efficiency of the system can be improved if the heat rejected during the cooling cycle can be recovered and used for space cooling or refrigeration applications. In this study, a vapor compression refrigeration system coupled with an absorption cooling system is simulated using MATLAB. The vapor compression system is driven by a natural gas fueled IC engine and the waste heat from the engine is used to drive the absorption cooling system. The waste heat is recovered both from gas exhaust and engine cooling systems. The developed simulation model is used to find the transients of both the vapor absorption and compressions systems for varied cooling demands. Important parameters such as coolant temperature and exhaust gas temperature are obtained from experimental data. This paper presents the most efficient load distribution between the vapor compression and absorption cooling systems.

The combustion of natural gas reduces fuel cost and generates less emissions of carbon dioxide and particulate matter (PM) than diesel and gasoline. Replacing diesel by natural gas in internal combustion engines is of great interest for transportation and stationary power generation. Dual fuel combustion is an efficient way to burn natural gas in internal combustion engines. In natural gas–diesel dual fuel engines, unburned hydrocarbon emissions increase with increasing natural gas fraction. Many studies have been conducted to improve the performance of natural gas–diesel dual fuel engines and reported the performance of combustion and emissions of regulated pollutants and total unburned hydrocarbon at various engine operating strategies. However, little has been reported on the emissions of different unburned hydrocarbon components. In this paper, an experimental investigation was conducted to investigate the combustion performance and emissions of various unburned hydrocarbon components, including methane, ethane, ethylene, acetylene, propylene, formaldehyde, acetaldehyde, and benzaldehyde, at a low engine load condition. The operating conditions, such as engine speed, load, intake temperature, and pressure, were well controlled during the experiment. The combustion and emissions performance of pure diesel and natural gas–diesel dual fuel combustion were compared. The effect of diesel injection timing was analyzed. The results show that appropriately advancing diesel injection timing to form a homogeneous charge compression ignition (HCCI)-like combustion is beneficial to natural gas–diesel dual fuel combustion at low load conditions. The emissions of different unburned hydrocarbon components changed in dual fuel combustion, with emissions of some unburned hydrocarbon components being primarily due to the combustion of natural gas, while those of others being more related to diesel combustion.

The combustion of natural gas reduces fuel cost and generates less emissions of carbon dioxide and particulate matter than diesel and gasoline. Replacing diesel by natural gas in internal combustion engines is of great interest for transportation and stationary power generation. Dual fuel combustion is an efficient way to burn natural gas in internal combustion engines. In natural gas–diesel dual fuel engines, unburned hydrocarbon emissions increase with increasing natural gas fraction. Many studies have been conducted to improve the performance of natural gas–diesel dual fuel engines and reported the performance of combustion and emissions of regulated pollutants and total unburned hydrocarbon at various engine operating strategies. However, little has been reported on the emissions of different unburned hydrocarbon components. In this paper, an experimental investigation was conducted to investigate the combustion performance and emissions of various unburned hydrocarbon components, including methane, ethane, ethylene, acetylene, propylene, formaldehyde, acetaldehyde and benzaldehyde, at a low engine load condition. The operating conditions, such as engine speed, load, intake temperature and pressure, were well controlled during the experiment. The combustion and emissions performance of pure diesel and natural gas–diesel dual fuel combustion were compared. The effect of diesel injection timing was analyzed. The results show that appropriately advancing diesel injection timing to form a homogeneous charge compression ignition-like combustion is beneficial to natural gas–diesel dual fuel combustion at low load conditions. The emissions of different unburned hydrocarbon components changed in dual fuel combustion, with emissions of some unburned hydrocarbon components being primarily due to the combustion of natural gas, while those of others being more related to diesel combustion.

Solving parabolic and hyperbolic equations by the generalized finite difference method

This paper study the problem of designing a fuzzy adaptive controller based on H ∞ observer for a class of uncertain nonlinear system modeled by first-order hyperbolic partial differential equation (PDE). The purpose of this paper is to design a new adaptive fuzzy law based on the principle of observer such that the first-order hyperbolic PDE system is exponentially stable with a desired H ∞ performance for disturbance attenuation, while the control constraint is respected. Initially, a Takagi-Sugeno (T-S) hyperbolic PDE model is proposed to represent the nonlinear PDE systems. Then the fuzzy adaptive observer with H ∞ controller is designed through seeking the feasible solutions of existing linear matrix inequality (LMI) optimization techniques, so using adjusting observer parameters, the control effectiveness can be estimated by state observer. For uncertain system, by designing the estimate of state observer, and introduced approximately error compensation, then the requirement of square integrability for optimal approximation error is inessential. Finally, the proposed design method is proved that the controller based on fuzzy adaptive H ∞ observer will stabilize the first-order hyperbolic PDE system.

The work for the subcontract was situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to diffusion-dominated PDEs, with some recent success in scalar linear hyperbolic PDEs. Given the ubiquity of numerical simulations of hyperbolic PDEs throughout the sciences, in particular, nonlinear hyperbolic systems, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems beyond simple scalar and linear cases, which is the main topic of this subcontract. The main focus of the work was to further develop and perfect coarse-grid operators for the Multigrid Reduction-inTime (MGRIT) method applied to hyperbolic PDEs that were recently proposed in PhD thesis, based on a modified semi-Lagrangian approach.

Ecological life cycle analysis-based installation capacity determination of building energy-supply systems

Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system

Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs

Real Time Neutron Radiography (RTNR) is rapidly becoming a valuable tool for nondestructive testing and basic research with a wide variety of applications in the field of engine technology. The Phoenix Memorial Laboratory (PML) at the University of Michigan has developed a RTNR facility and has been using this facility to study several phenomena that have direct application to internal combustion and gas turbine engines. These phenomena include; 1) the study of coking and debris deposition in several gas turbine nozzles (including the JT8D), 2) the study of lubrication problems in operating standard internal combustion engines and in operating automatic transmissions (1, 2, 3), 3) the location of lubrication blockage and subsequent imaging of the improvement obtained from design changes, 4) the imaging of sprays inside metallic structures in both a two-dimensional, standard radiographic manner (4, 5) and in a computer reconstructed, three-dimensional, tomographic manner (2, 3), and 5) the imaging of the fuel spray from an injector in a single cylinder diesel engine while the engine is operating. This paper will show via slides and real time video, the above applications of RTNR as well as other applications not directly related to gas turbine engines.

Control of hyperbolic partial differential equations (PDEs) is a truly interdisciplinary area of research in applied mathematics nurtured by challenging problems arising in most modern applications ranging from road traffic, gas pipeline management, blood circulation, to opinion dynamics and socio-economical models, as well as in environmental and biological issues, or more recently in the analysis of deep learning and machine learning methods. The topic has gained an increasing attraction of researchers in the last twenty years due to fundamental theoretical as well as numerical advances achieved in the field of nonlinear hyperbolic PDEs. The hyperbolic and the control of PDEs communities, while pursuing separate interests in their respective range of action with a different focus and, often, with a different array of technical tools, do share a substantial body of common knowledge and background. We think the time is right and the momentum is propitious to bring those communities together at a joint workshop, to mutually stimulate each other and inter- act with one another, for a marked advancement of this area of research on a broad spectrum of control ranging from theoretical to numerical problems and covering also the emerging challenges involving the interplay between (topics of) control and learning. In order to also attract young scientists to this striving field we focus on selected lectures, in-depth discussions, transfer of information from senior to young researchers, and vice versa, and invited plenary talks.

In Chs. 5 and 8 we discussed the implementation of continuous Galerkin (CG) methods in one dimension for hyperbolic and elliptic equations, respectively. In Ch. 12 we introduced the extension of the CG method to two dimensions by describing its implementation for elliptic partial differential equations (PDEs). In this chapter we extend the CG method for the application of hyperbolic equations in two dimensions. We also discuss the addition of diffusion operators.

In this contribution we present a new practice to deal with the state estimation problem of the process governed by hyperbolic partial differential equation (PDE) model. In this paper, an approach based on the spacial discretization is adopted to provide an equivalent uncertain state space which describes accurately the original PDE systems. Subsequently, based on the uncertain model a high gain observer is synthesized. The proposed observer allows us to achieve the desired performances with comprehensive and simple convergence analysis. Finally, an assessment of the proposed observer is carried out through simulation by using it to estimate the oil temperature for a parabolic solar collector.