Bo Dong, Master - US grants

This Small Business Innovation Research Phase I project aims to develop a distributed acoustic emission monitoring technology. The new technology will provide real-time acoustic monitoring using a single fiber optic cable, distributed temperature information will also be generated simultaneously. The low physical profile renders it embeddable and minimally intrusive. The technology is designed to be low-cost, which ensures its wide-scale commercialization potential. In this Phase I research, a prototype sensor system will be built and applied to demonstrate highly sensitive acoustic detection. The focus will be on feasibility proof of the technology.

The broader impact/commercial potential of this project includes the development of a sensor technology that is highly desired in multiple industrial sectors. The potentially low-cost technology will enable large-scale deployment of the sensor to revolutionize the sensing methodology used in a number of applications. Accurate locating and quantifying of acoustic signatures will directly benefit seismic oil & gas exploration. Distributed acoustic monitoring will seek immediate implementation in multi-zone production in gas wells, heavy oil steam-assisted gravity drainage operation surveillance and flowline hydrates, scale and corrosion monitoring. Long-span, real-time acoustic monitoring also provides a viable means for pipeline health monitoring. Another beneficiary of the technology will be distributed partial discharge (PD) detection within power transformers. It is hoped that the new sensing technology will dramatically improve the operation safety while reduce the overall cost of such industries. Successful demonstration of the proposed sensing principle will also shed light on a number of other sensing applications requiring high-speed, densely-multiplexed operations.

The project focuses on developing novel numerical methods for simulating the Korteweg-de Vries (KdV) type equations, that model phenomena in areas such as fluid mechanics, nonlinear optics, acoustics, and plasma physics. For example, the KdV equation has been used in the modeling of shallow water waves and the study of Tsunami waves. The new numerical tools developed under this project will provide scientists with a better understanding of theoretically unresolved issues on the mathematical properties of solutions to KdV type equations. Furthermore, the proposed project will provide accurate and efficient numerical algorithms for the simulation of nonlinear dispersive wave propagation in various applications. These proposed research topics will have a positive impact across the mathematical sciences and have significant applications in many scientific areas that rely on the study of non-linear phenomena. This project will involve undergraduate and graduate students and focus on involving student from groups traditionally underrepresented in the sciences. By working on the project, the students will benefit from novel ideas for new algorithm design, approaches for rigorous mathematical analysis, and advanced skills in implementation.

The objective of the project is to devise and analyze the first superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized mixed methods for solving the KdV equations and their multidimensional generalizations. The proposed project includes a comprehensive coverage of new algorithm design that is backed up by solid analysis and made practical by efficient implementation. The P.I. proposes to carry out a detailed study of superconvergent HDG methods and hybridized mixed methods for KdV type problems in the following steps: First, the P.I. will develop novel HDG methods and hybridized mixed methods for stationary third-order linear equations, focusing on the discretization of the third-order differential operator. Superconvergence properties of the approximations will be computationally and analytically investigated. Second, the P.I. would like to solve the third-order KdV equations by using implicit schemes for time discretization to avoid extremely small time steps and developing new HDG methods and hybridized mixed methods for spatial discretization. Error analysis will be carried out, and superconvergence and conservativity properties will be studied. Third, the P.I. plans to extend these superconvergent methods to multidimensional KdV type equations such as the Kadomtsev-Petviashvili equation, and the hybridization technique will make the methods efficiently implementable in multiple dimensions.

The international conference entitled "Recent Advances and Challenges in Discontinuous Galerkin Methods and Related Approaches" will be held at the Institute for Mathematics and its Applications at the University of Minnesota from June 29 to July 1, 2017. This award supports junior participants' travel. The event brings together a variety of researchers from at least 14 countries/regions. They range from internationally renowned experts to early career mathematicians and PhD students. The event will summarize recent advances made both in the theory and implementation of the Discontinuous Galerkin and related numerical approaches, and to identify new challenges and opportunities in these areas. The conference will also have a significant educational component, with each talk required to feature introductory parts at a level accessible to graduate students, and ample discussion sessions throughout the conference. To further promote cross-pollination and mentoring, there will be moderated panel sessions where participants explore the frontiers of different research areas, possibilities of new connections between areas and new applications, and exciting opportunities of new collaborations. It will help junior researchers broaden their perspective and create research ties with more senior members in these fields.

Discontinuous Galerkin and related approaches have been adopted in areas ranging from mechanical engineering to the simulation of muscles. In recent decades, deep theoretical advances have been made and wide-ranging applications discovered for these approaches. They often lead to design of novel methods (e.g. Hybridizable discontinuous Galerkin methods, Virtual Element Methods, etc.) with superior properties in terms of accuracy, versatility, robustness and computational efficiency. They also leave open many exciting problems. This conference presents a rare but timely opportunity to summarize recent advances both in the theory and implementation of these methods, identify new challenges, and map out future research directions in related areas. The bringing together of people from different fields such as engineers, applied mathematicians, national lab researchers, will lead to cross-fertilization of ideas that normally does not happen in a conference of this size.