Shue-Sum Chow, Mathematics

Academic Objectives

The project is concerned with the study of wave propagation in metamaterials. We recall the work plan described in the original proposal and provide an evaluation of the academic objectives.

1. Gathering of information about existing mathematical models for various metamaterials and related references on the mathematical properties of FDTD and finite element methods.
   This task has been completed as we have compiled a detailed list of references and currently available commercial and academic codes for studying of the propagation of EM waves and acoustic waves.
2. Evaluation, selection and design of benchmark problems. A concurrent activity is to have students investigate various simple optical effects such as the presence or absence of rainbow in droplets made up of metamaterials.
   We have a good set of benchmark problems involving only one spatial dimension as and some partial information for two or three dimensional problems. The study of rainbow e ect is completed and a paper has been completed and ready for submission. Several related codes are also made available publicly.
3. Formulate time domain computational models for three or four main metamaterial models and use FDTD method to build benchmark data bank for future reference and comparison.
   For one dimensional problems, we have generated a set benchmark problems and a FDTD code. However, much work still remains for two and three dimensional problems.
4. Investigate numerical performance and issues in using the method of lines to solve benchmark problems.
   This is ongoing as higher dimensional problems have a different level of computational challenges.
5. Implementation of the computational models on various machine architectures.
   The group of students involved are not that interested in coding on various machine architectures and so currently all the codes are written either in matlab or geogebra.
   The students involved in the project have plenty of opportunities to learn to write codes for scientific computing and to use LATEX to prepare reports and presentation slides. Many had given several presentations in local and regional conferences. Two of the students are involved in preparing academic papers for journal submission.

Mentoring Environment

Each student met with me individually for about half an hour to an hour once a week (and occasionally more often) to report on their progress. From Septem- ber 2013 onward, we also have a group meeting once a week in which students would give short presentations of the work they did and to exchange ideas. I also setup a shared google drive folder so information such as academic papers the group found and their reports and presentations may be shared and disseminated efficiently. It seems to me that students find the setup satisfactory and productive. The students also have access to the mathematics department’s undergraduate mentoring lab but most students seem to prefer using their own computers and BYU supplied software such as matlab.

Participated Students

The students involved in the projects are Nirdosh Chapagain, Ammon Washburn, Elise Cope Hardle, Steffan Larsen, Yunwoo Jang.

Nirdosh Chapagain started his research work in the summer term of 2013. Initially Nirdosh was very enthusiastic and showed a lot of potential. To get started, I asked him to look into the rainbow e ect in meatmaterials, a topics that was rst investigated by Bradford Tuck eld before his graduation in De- cember 2012. Nirdosh took the initiative to learn to use the Geogebra package and produced a code to generate rainbow effect in a sphere consists of some hypothetical metamaterial enclosing a spherical water droplet. However, around the middle of November, Nirdosh became less focused and often missed our individual and group meetings. After mid January I have not seen him again and can only communicate with him intermittently via email. By the third week of February I received an email informing me that he is having some immigrational issues and is on his way back to Provo. After that I have no success in contacting him. He did not show up for his talk at the Utah Conference on Undergraduate Research.

Ammon Washburn started his research work in the fall semester of 2013. Ammon worked on the use of the method of perfectly matched layer (PML) to handle radiation boundary conditions in wave propagation problems. Ammon focused on applying PML to acoustic metamaterials and had developed a good understanding of the technique. In consideration of Ammon’s interest in computational work, I asked him to implement a shooting method for a class of gradient dependent nonlinear boundary value problems. The method is not directly related to but has some relevance to the metamaterial project. Am- mon completed the implementation shortly after his graduation in April 2014. Unfortunately, during his move to Arizona to commence his Ph.D. study in mathematics, Ammon lost his code and has no backup. Ammon completed his recoding in early December and we are ready to nish up the last section of a paper that will be submitted to a tier two academic journal within a couple of weeks. Ammon presented his work in the College’s Student Research Conference and in an undergraduate research session at the MAA Intermountain Section Conference in March 2014.

Elise Cope started her research work in January 2014. Elise worked on the use of the so-called Bayliss-Turkel boundary condition to handle the Sommerfeld radiation boundary conditions in wave propagation problems. Elise gave presentations in the College’s Student Research Conference and in an undergrad- uate research session at the MAA Intermountain Section Conference in March 2014. She was chosen as the best speaker in her SRC session which included some graduate student talks. Elise is planning to attend a graduate school.

Steffan Larsen started his research work in the fall semester of 2013. Steffan worked on the topic of space-filling curves and their applications with metamaterials. He gave presentations in the Utah Conference on Undergraduate Research, the College’s Student Research Conference and in the MAA Inter- mountain Section Conference in 2014.

At the beginning of spring term in 2014, Steffan decided to complete the work on rainbow effect in metamaterials and has managed not only confirmed the absence of rainbow formation in spherical metamaterials, as demonstrated by some of my previous students, but also shown the presence of rainbow in prism-shaped metamaterials. Steffan took the initiative to learn to use the Geogebra package and produced several code to investigate the rainbow effect. These codes are posted on the website GeogebraTube. We are in the final stage of preparing a paper for submission sometime next week.

Yunwoo Jang started her research work in the Spring term of 2014. She worked on applying the Bayliss-Turkel radiation boundary conditions to the wave equation and Helmholtz equation and made an interesting observation for the spherically symmetric case that has useful implication not only for the project but also to an inverse source problem that I have been working on. Yunwoo has also completed much of the background work such as classi cation of metamaterials and preparing a survey of currently available software for studying wave propagatioon in metamaterials. She is currently preparing a one dimensional FDTD code to start building a set of benchmark problems.

Results/findings of the project

Due to some misunderstanding the timing of fund transfer from the College, recruitment of students did not start until mid February of 2013, with the unfortunate result of not being able to nd suitable student participants until Spring term of 2013. Despite the delayed start of the project, much has been accomplished by the students.

1. Two papers have been produced as a result of the project. One relates to the rainbow e ect in metamaterials and the other is concerned with a nonlinear shooting method for a special class of boundary value problems. Both will be submitted to academic journals in the near future.
2. Several presentations at local and regional conferences were made.
3. Several pieces of code related to rainbow effect in metamaterials have been deposited in the publicly accessible website GeogebraTube.
4. A one dimensional FDTD matlab code is constructed and used to evaluate some benchmark problems.
5. One student mentioned that his work on metamaterials has helped strength- ened his successful application to a graduate school.

Budget

Currently over half of the budget is not spent. Most of the spent money was for student wages, but only for the summer term 2013 until fall semester 2014, since I had not been able to recruit student to participate in the winter semester and spring term of 2013. Further contributed to the , as the mathematics department has decided to contribute some of its undergraduate research funds to the student wages, the amount of fund spent is

The budget allocated for traveling was used mainly to pay for conference reg- istration fees as the conferences the students attended this year are all located locally, and due to scheduling issue, we are not able to invite Professor Li from UNLV to visit here or to have our students visit him. All these contributed to a minimal travelling expense being incurred.

The largest expense in supply is the purchase of an ipad. The initial plan to purchase COMSOL software is delayed as students have found my very old version is sucient for their purpose. The purchase of graphics cards is also delayed because of a lack of interest in the students to learn to program graphics cards.

As the project su ered a major delay in its startup, it is my hope that I may be able to keep some of the MEG fund to continue to pay for undergraduate research work in the winter semester and possibly spring and summer terms in order to nish the project with the completion of the benchmark set of problems for the two and three dimensional cases.