Research

Overview

I am dedicated to research in applied mathematics, particularly focusing on numerical solutions to partial differential equations (PDEs). My research involves various numerical techniques, such as the level-set method, finite element method (FEM), finite difference method (FDM), and programming in Matlab to simulate models. I specialize in solving complex problems in engineering, material sciences, biology (cytology), and physics (nano-level particles).

During my doctoral program, I developed a stable algorithm for simulating evolution problems in material science and physics using a level-set approach, which involve highly nonlinear PDEs. After completing my doctorate, I again joined Kyoto University as a researcher as a part of big project, focusing on the deformation of elasto-plastic materials using FEM to solve PDEs describing deformation. Then after, At Ritsumeikan University in Japan, I engaged in scientific research, specifically in the development of a mathematical model for non-monetary remittances. This collaborative effort involved working closely with fellow economists and scientists to create a comprehensive and insightful model.

In the future, I aim to construct algorithms to address complex phenomena in engineering and biology, employing various numerical tools or developing new ones. Additionally, I aspire to develop analytical or semi-analytical solutions whenever applicable.

1. Background:

1.1. Doctoral course: Evolution of particle by anisotropic mean curvature flow (AMCF) :

Consider particles with anisotropic surface energy evolve on a substrate towards a shape with minimum surface energy, see Figure 1. We are interested in the evolution of particles and their equilibrium shape (steady state). Further, we want to handle possible topological changes (singularities) without additional efforts. This problem is commonly known as obstacle problem. We use a level-set approach to solve the obstacle problem. we developed a stable thresholding scheme, which can handle topological changes and approximate anisotropic mean curvature flow, and proved its stability unconditional. Before dealing with the obstacle problem, we surveyed known convolution kernels used in thresholding scheme and we perform their comparison tests to understand: The behaviour of error and convergence order, time required for computation, ability to handle sharp corners of particles and evolution of non-convex shapes.

Figure 1: Setup of the mathematical model of a particle evolving on a flat, rigid substrate.

1.2. Postdoctoral researcher position at Kyoto university :

Deformation of elasto-plastic material: solution of minimization problem by FEM

[All figures/content belong to elastoplastic compression program group and Prof. Karel Svadlenka]

The goal is to simulate the evolution of elastoplastic material with energy modelled by E and dissipation given by D. the energy is of neo-Hook type and dissipation distance is slip difference. The setup of the problem is shown in Figure 2. Fundamentally, we desire to solve the minimization problem of sum of E and D at every time step. The problem is solved using FEM and the outcome is compared with lab experiments. My contributions include optimizing and upgrading code, reducing memory usage, and significantly improving computational efficiency. We are exploring the influence of numerical parameters on simulation results and aiming to set parameter ranges based on their roles.

Figure 2: Setup of the problem in 3D

2. Future Research Plan: Methodology and Progress :

In the coming five years, I plan to extend my ongoing research while embarking on additional projects. This section provides a concise overview of my latest research initiatives.

2.1 Simulating Biological Cells/Crystals Networks by AMCF :

Collaborating with Prof. Karel Svadlenka, I am working on the development of an algorithm to simulate the motion of biological cells/crystal networks using the anisotropic mean curvature flow (AMCF) with a level-set numerical approach. These networks, found in materials science and powder metallurgy, pose unique challenges due to their complex topological changes during evolution.

Our approach addresses these challenges by adopting a level-set method capable of handling topological changes seamlessly. We have successfully developed an algorithm for networks without volume preservation. To validate its accuracy, we have theoretical results for triple junctions and rely on experimental data for junctions with four or more interfaces.

2.2 Convergence Proof of Obstacle Problem Algorithm :

I will undertake the task of proving the convergence of the algorithm developed during my doctoral research for the obstacle problem. This algorithm describes the motion of particles on obstacles by mean curvature, and we have already established its stability, with energy decreasing in each iteration.

2.3 Particle Evolution by AMCF on Heterogeneous Substrate :

In collaboration with Dr. Hou Lingling from Hong Kong University of China, we are expanding our research to encompass the evolution of particles by anisotropic mean curvature flow on a heterogeneous substrate. This substrate allows different values to be assigned to each point, adding complexity to the problem.

Our goal is to develop an algorithm that can simulate the motion while adhering to volume constraints. We are addressing various challenges, including achieving accurate contact angle motion, managing multiple indicator functions for the heterogeneous substrate, handling restrictions on the substrate’s heterogeneity function, and addressing topological changes during evolution. We have already made significant progress in tackling some of these issues, with a particular focus on function restrictions.

These research endeavors represent my commitment to advancing the field of applied mathematics and contributing valuable insights to interdisciplinary areas.

HISTORY

1. Ph.D. Research (2022) at Kyoto University (KU): Conducted research at Kyoto University focusing on a level-set numerical approach to anisotropic mean curvature flow on obstacles. Developed a stable algorithm for simulating the evolution of particles on obstacles, addressing topological changes and achieving unconditional stability.

2. M.Sc. Research at IIT Hyderabad (2016): Explored the energy method and nonlinear stability analysis at the Indian Institute of Technology, Hyderabad. Investigated stability results obtained in standard partial differential equations using the energy method.

3. Postdoc Research at KU (2022): Collaborated with Japanese material science group and European based research group on the deformation of elastoplastic materials. Used the finite element method (FEM) and Matlab to simulate material behavior under stress and strain. Optimized and upgraded code, reduced memory usage, and improved computational efficiency.

4. Assistant Researcher at Ritsumeikan University (2023): Worked at Ritsumeikan University, Japan, on a mathematical model analyzing the impact of remittances on migrant behavior and investment decisions. Contributed to the development and analysis of the model, supported by a grant from the Japan Society for the Promotion of Science (JSPS).

5. Research Assistant at KU (2021-2022): Developed and implemented a Matlab program at Kyoto University to simulate curve evolution over time and address topological changes, enhancing computational efficiency and accuracy.

6. Research Student at KU (2017-2018): Prepared and investigated anisotropic multiphase mean curvature flow problems at Kyoto University, proposing a new thresholding algorithm for obstacle problems.

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