Research Projects

Thermodiffusion in an elastic solid results from the coupling of strain, temperature, and mass diffusion fields, and it plays a critical role in various engineering applications, particularly in satellite and aircraft operations. There is considerable interest in diffusion processes for the manufacturing of integrated circuits, integrated resistors, semiconductor substrates, and transistors. It is also a key component in the heat and mass transfer processes involved in enhancing oil extraction conditions from deposits. Understanding diffusion properties in thin thermoelastic plates is critical in the study of advanced materials. The first goal of the project is to study the mathematical and numerical analysis of coupled linear and nonlinear systems of Partial Differential Equations (PDEs) that govern the behaviour of thin structures under loads. Besides outcomes relevant to the theoretical and numerical aspects of thin structures, the results of this research will have potential applications for the design and analysis of thin structures; for example in microelectronics, biomedical engineering, and energy storage. The project involves study of deep fundamental research of thin and very thin plates with an application potential. It aims to study the theoretical aspects of existence of weak solutions, derive regularity results on the exact solution, propose effective numerical schemes that take into account the compatible nonstandard finite element methods for approximating second and fourth-order spatial operators and first and second order time derivative terms, try to provide a unified analysis the schemes for stability, convergence, develop a priori error estimates, propose novel algorithms, study the implementation aspects, and validate the theoretical estimates. The second goal is to develop deep neural network (DNN) solvers for fourth-order partial differential equations (PDEs), specifically, the biharmonic equation arising in plate bending and related models. DNN-based solvers are mesh-free, can be trained with scattered data, and are well-suited for challenging linear and nonlinear higher-order PDEs where traditional numerical methods are challenging with high computational costs when the domain is high-dimensional or has complex geometry. The proposed research aims to advance the state-of-the-art in solving fourth-order PDEs using deep neural network–based methods and by conducting a rigorous error analysis. We expect to deliver robust, efficient, and accurate solvers for biharmonic problems and possibly nonlinear models. The outcomes of this research will be significant for applications in structural mechanics, fluid dynamics, and materials science.