My research interests lie in the field of Numerical analysis, Scientific Computing, optimal control problems, mathematical physics (solid mechanics), and modelling of dynamical systems. More specifically, my research focuses on design, analysis and implementation of numerical methods including Finite difference method, Adaptive finite elements, and Discrete Element Method for the efficient numerical solutions of ordinary or partial differential equations which model physical systems. Mainly, I study hyperbolic and elliptic problems for elastic solid mechanics, parabolic problems for biological sciences and singular boundary value problems arising in physiology. Moreover, my research focuses on the modeling, simulation and optimization of the process of crystallization.

**1. Research on Numerical Methods for Singular boundary value problems:**

Currently, the main focus of my research is to develop new numerical techniques based on a uniform mesh and/or a non-uniform mesh for the numerical approximation of a class of singular boundary value problem arising in Physiology. In particular, the proposed scheme will be applied to a nonlinear singular boundary value problem representing a heat conduction model of the human head. This will give us the opportunity to obtain better numerical results of the distribution of heat sources in the human head. The method is based on the combination of three-point finite difference method and decomposition technique. Besides the numerical design I will establish the order of convergence of the proposed scheme theoretically and observe numerically with nonlinear examples. The development of the new algorithm will allow us not only to approximate the numerical solution of Physiological problems but also to solve other similar singular boundary value problems. Furthermore, I will compare the results obtained using the proposed method with those obtained by other numerical methods existing in the literature in order to see the advantages that the new method would have. Moreover, the behaviour of the solution in the vicinity of the singular points of the singular boundary value problems will be analyzed.

In the study of nonlinear phenomena in engineering and sciences, many mathematical models lead to singular two-point boundary value problems. For instance, this problem arises in the study of electro hydrodynamics, in the theory of thermal explosions, in the study of generalized axially symmetric potentials in a rectangle and in the study of transport process and so on. These problems also arise in Physiology, for example, in the study of various tumour growth problems, in the study of steady-state oxygen diffusion in a cell with Michaelis-Menten update kinetics and in the study of heat sources in the human head.

**Objective**

• Development of decomposition finite difference method based on a uniform or non-uniform mesh

• Development of decomposition spline method based on a uniform mesh or non-uniform mesh

• Convergence analysis of the methods

• Derivation of the rate of convergence of the methods

• Illustration of the methods and verification of theoretical order of convergence

• Comparison with the exact solution and other existing numerical methods

• Implementations of proposed scheme to Physiological problems

**2. Research on DEM simulation for granular flows:**

Currently, a large amount of money is spent on the transportation and processing associated with the storage and containment of granular materials. However, about 50% of the money is unnecessarily spent because of problems related to the transport of the material from one part of the factory to another part of the factory. Now, to have a look from another angle, it is often assumed that the side wall of a material container receives a constant force from the granular material inside. The common example of this issue is a model of a silo which is of great concern to various industries such as agricultural, pharmaceutical and mining industries, and all construction-based industries. However, this assumption is wrong, and in the general case, forces are non-uniformly propagated within the material, so they are also non-uniformly distributed at the wall of the silo. In some cases, if the force is much larger in some parts of the container than in other parts, the silo might collapse. For in order to avoid problems such as the collapse or breach of a silo, one can simply increase the thickness of the walls by a generously chosen safety margin, which would be unnecessary if we had the knowledge how to design the silo in a proper way, especially taking into account the expected distribution of forces inside the silo. Therefore, the understanding of the basic physical principles behind the stress distribution in static granular materials is clearly important.

A simple example out of a collection of granular arrangements is the static sand pile. The formation of a sand pile is related to the fundamental behaviour of granular materials, including particle packing, avalanches, segregation and pressure distribution. The sandpile is of significant scientific interest, in practical applications, such as silos, dams, and embankments, the effects of retaining walls are likely to be very important. The practice of storing granular materials in the form of sand piles occurs in many industrial situations dealing with particulate materials. Examples include the pharmaceutical industry relying on the processing of powders and tablets, the agricultural industry, coal industry and the food processing industries where seeds, coal (grain) and foods are transported and manipulated. Moreover, the storing of the material in a pile may be useful in fertilizer and mining industries. Thus, the flow of granular materials through a funnel (to form a pile) is an important problem for many industrial processes.

**Goal:**

The goal of this study is to contribute to the understanding of mechanical properties, physical properties and effective material behaviour of non-cohesive granular materials, especially in the static limit. To achieve this goal, an efficient numerical technique, so called the discrete element method (DEM) is adopted to simulate the dynamics of granulates made up from differently shaped particles. Numerical simulations is performed on two-dimensional (2D) systems, in which symmetric sand pile, asymmetric sand pile or rectangular system of assemblies is constructed from several thousands of convex polygonal particles with varying shapes, sizes and edge numbers. The particles are poured either from a point source or a line source.

I focus on computing the macroscopic continuum quantities and elastic constants of the resulting sand piles and show how the construction history of the sand piles, shape and size distribution of the particles and inter-particle friction affects their mechanical properties including stress, strain, fabric and density, and elastic material behaviour as well. Simulation results of stress distributions will be qualitatively compared with the available analytical predictions for the stress tensor and with that measured experimentally. I wish to develop constitutive relations for the sandpile model using not only the stress strain tensor, but also the density and or the fabric tensor and establish the relation between the continuums quantities for the sandpiles obtained from the simulation. Moreover, I focus on investigating numerically the mechanical properties of a static granular assembly in the form of rectangular system of granular material with different amounts of disorder.