A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

Autoignition in nonpremixed flow

The objective of this investigation has been to improve understanding of autoignition processes in nonpremixed flow fields of the types encountered in Diesel-engine ignition, through theoretical analyses that employ asymptotic methods of applied mathematics. The work was intended to develop formulas and equations that can be used in activities of applied research, such as code development, aimed at providing tools useful for the design of Diesel engines. The formulas may also be used directly for ignition estimates.Characteristic time scales were identified for these ignition problems. Their relative magnitudes were employed to define different regimes of ignition and to obtain simplified partial differential equations that describe ignition in these regimes. Effects of turbulence on ignition were addressed. Special attention was devoted to unsteady mixing layers, involving both variable strain and variable pressure, for which ignition-time formulas were derived. In addition, ignition analyses were completed for variable-volume chambers with arbitrary initial spatial variations of temperature and composition, to determine pressure histories produced by ignition-front propagation. These studies were based on one-step, Arrhenius approximations for the chemical kinetics and were restricted to ignition stages that precede ordinary flame propagation. Additional work considered triple-flame propagation that can odcur in mixing layers after ignition, with this same chemical-kinetic description, and asymptotic analysis of n-heptane ignition on the basis of a four-step, semi-empirical model for the chemical kinetics. In this latter study, the region of negative effective overall activation energy, between 800 K and 1100 K, was identified as exhibiting unusual ignition dynamics, and the asymptotic ignition-time formulas were shown to give good agreement with predictions of numerical integrations. This research has helped to strengthen the foundations of ignition theory for nonuniform media. It provided simplified descriptions of ignition processes that can be employed in studies of Diesel combustion that are oriented more towards development than are the present investigations. The asymptotic methods employed in this work thus appear capable of providing quite useful results

Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs

Development of a Systems Engineering Model of the Chemical Separations Process: Final Report

The whole chemical separation process is complex to the point that definitely requires certain level of systematic coordination. To perform smoothly and meet the target extraction rates among those processes, this research proposed a general-purpose systems engineering model. A general purposed systems engineering model, Transmutation Research Program System Engineering Model Project (TRPSEMPro), was developed based on the above design concept. The system model includes four main parts: System Manager, Model Integration, Study Plan, and Solution Viewer. TRPSEMPro can apply not only to chemical separation process, but also a general system model. Software engineering and Object Oriented Analysis and Design (OOA&D) play a critical role during our software development. Through the application of OOA&D, the user can define objects and concepts from our problem domain that is quantitatively described by Unified Modeling Language (UML). The logical software objects were created from the previous definition. Meanwhile, different design patterns were also applied during the detailed design phase. Finally, those designed components were implemented by using MicrosoftTM.Net, the most up-to-date object-oriented programming language framework from Microsoft. Currently, only the UREX process module is available and ready to be implemented. Since extraction modules can be developed from various agencies with different development concepts and programming conventions, an intermediate bridge or interpreter is generally required. The system connects the only available process, UREX and with the TRPSEMPro system model from the AMUSESimulator interface. The AMUSESimulator communicates with the calculation engine AMUSE macros designed for the UREX process. A user-friendly GUI in AMUSESimulator allows the user to efficiently define the UREX process – flowsheet, input streams, sections, and stages

Continuous Biochemical Processing: Investigating Novel Strategies to Produce Sustainable Fuels and Pharmaceuticals

Biochemical processing methods have been targeted as one of the potential renewable strategies for producing commodities currently dominated by the petrochemical industry. To design biochemical systems with the ability to compete with petrochemical facilities, inroads are needed to transition from traditional batch methods to continuous methods. Recent advancements in the areas of process systems and biochemical engineering have provided the tools necessary to study and design these continuous biochemical systems to maximize productivity and substrate utilization while reducing capital and operating costs. The first goal of this thesis is to propose a novel strategy for the continuous biochemical production of pharmaceuticals. The structural complexity of most pharmaceutical compounds makes chemical synthesis a difficult option, facilitating the need for their biological production. To this end, a continuous, multi-feed bioreactor system composed of multiple independently controlled feeds for substrate(s) and media is proposed to freely manipulate the bioreactor dilution rate and substrate concentrations. The optimal feed flow rates are determined through the solution to an optimal control problem where the kinetic models describing the time-variant system states are used as constraints. This new bioreactor paradigm is exemplified through the batch and continuous cultivation of β-carotene, a representative product of the mevalonate pathway, using Saccharomyces cerevisiae strain mutant SM14. The second goal of this thesis is to design continuous, biochemical processes capable of economically producing alternative liquid fuels. The large-scale, continuous production of ethanol via consolidated bioprocessing (CBP) is examined. Optimal process topologies for the CBP technology selected from a superstructure considering multiple biomass feeds, chosen from those available across the United States, and multiple prospective pretreatment technologies. Similarly, the production of butanol via acetone-butanol-ethanol (ABE) fermentation is explored using process intensification to improve process productivity and profitability. To overcome the inhibitory nature of the butanol product, the multi-feed bioreactor paradigm developed for pharmaceutical production is utilized with in situ gas stripping to simultaneously provide dilution effects and selectively remove the volatile ABE components. Optimal control and process synthesis techniques are utilized to determine the benefits of gas stripping and design a butanol production process guaranteed to be profitable

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Realtime Motion Planning for Manipulator Robots under Dynamic Environments: An Optimal Control Approach

This report presents optimal control methods integrated with hierarchical control framework to realize real-time collision-free optimal trajectories for motion control in kinematic chain manipulator (KCM) robot systems under dynamic environments. Recently, they have been increasingly used in applications where manipulators are required to interact with random objects and humans. As a result, more complex trajectory planning schemes are required. The main objective of this research is to develop new motion control strategies that can enable such robots to operate efficiently and optimally in such unknown and dynamic environments. Two direct optimal control methods: The direct collocation method and discrete mechanics for optimal control methods are investigated for solving the related constrained optimal control problem and the results are compared. Using the receding horizon control structure, open-loop sub-optimal trajectories are generated as real-time input to the controller as opposed to the predefined trajectory over the entire time duration. This, in essence, captures the dynamic nature of the obstacles. The closed-loop position controller is then engaged to span the robot end-effector along this desired optimal path by computing appropriate torque commands for the joint actuators. Employing a two-degree of freedom technique, collision-free trajectories and robot environment information are transmitted in real-time by the aid of a bidirectional connectionless datagram transfer. A hierarchical network control platform is designed to condition triggering of precedent activities between a dedicated machine computing the optimal trajectory and the real-time computer running a low-level controller. Experimental results on a 2-link planar robot are presented to validate the main ideas. Real-time implementation of collision-free workspace trajectory control is achieved for cases where obstacles are arbitrarily changing in the robot workspace

Model Order Reduction in Porous Media Flow Simulation and Optimization

Subsurface flow modeling and simulation is ubiquitous in many energy related processes, including oil and gas production. These models are usually large scale and simulating them can be very computationally demanding, particularly in work-flows that require hundreds, if not thousands, runs of a model to achieve the optimal production solution. The primary objective of this study is to reduce the complexity of reservoir simulation, and to accelerate production optimization via model order reduction (MOR) by proposing two novel strategies, Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM), and Quadratic Bilinear Formulation (QBLF). While the former is a training-based approach whereby one runs several reservoir models for different input strategies before reducing the model, the latter is a training-free approach. Model order reduction by POD has been shown to be a viable way to reduce the computational cost of flow simulation. However, in the case of porous media flow models, this type of MOR scheme does not immediately yield a computationally efficient reduced system. The main difficulty arises in evaluating nonlinear terms on a reduced subspace. One way to overcome this difficulty is to apply DEIM onto the nonlinear functions (fractional flow, for instance) and to select a small set of grid blocks based on a greedy algorithm. The nonlinear terms are evaluated at these few grid blocks and interpolation based on projection is used for the rest of them. Furthermore, to reduce the number of POD-DEIM basis and the error, a new approach is integrated in this study to update the basis online. In the regular POD-DEIM work flow all the snapshots are used to find one single reduced subspace, whereas in the new technique, namely the localized POD-DEIM, the snapshots are clustered into different groups by means of clustering techniques (k-means), and the reduced subspaces are computed for each cluster in the online (pre-processing) phase. In the online phase, at each time step, the reduced states are used in a classifier to find the most representative basis and to update the reduced subspace. In the second approach in order to overcome the issue of nonlinearity, the QBLF of the original nonlinear porous media flow system is introduced, yielding a system that is linear in the input and linear in the state, but not in both input and state jointly. Primarily, a new set of variables is used to change the problem into QBLF. To highlight the superiority of this approach, the new formulation is compared with a Taylor's series expansion of the system. At this initial phase of development, a POD-based model reduction is integrated with the QBLF in this study in order to reduce the computational costs. This new reduced model has the same form as the original high fidelity model and thus preserves the properties such as stability and passivity. This new form also facilitates the investigation of systematic MOR, where no training or snapshot is required. We test these MOR algorithms on the SPE10 and the results suggest twofold runtime speedups for a case study with more than 60,000 grid blocks. In the case of the QBLF, the results suggests moderate speedups, but more investigation is needed to accommodate an efficient implementation. Finally, MOR is integrated in the optimization work flow for accelerating it. The gradient based optimization framework is used due to its efficiency and fast convergence. This work flow is modified to include the reduced order model and consequently to reduce the computational cost. The water flooding optimization is applied to an offshore reservoir benchmark model, UNISIM-I-D, which has around 38,000 active grid blocks and 25 wells. The numerical solutions demonstrate that the POD-based model order reduction can reproduce accurate optimization results while providing reasonable speedups