Coupling of adaptive refinement with variational multiscale element free Galerkin method for high gradient problems

In this thesis, a new adaptive refinement coupled with variational multiscale element free Galerkin method (EFGM) is developed for solving high gradient problems. The aim of this thesis is to propose a new framework of moving least squares (MLS) approximation with coupling method based on the variational multiscale concept. Additional new nodes will be inserted automatically at high gradient regions by adaptive algorithm based on refinement criteria. An enrichment function is embedded in the MLS approximation for the fine scale part of the problem. Besides, this new technique will be parallelized by using OpenMP which is based on shared memory architecture. The proposed new approach is first applied in two-dimensional large localized gradient problem, transient heat conduction problem as well as Burgers' equation in order to analyze the accuracy of the proposed method and validated with an available analytic solutions. The obtained numerical results show a very good agreement with the analytic solutions and is able to obtain more accurate results than the standard EFGM. It is found that the average relative error of this new method is reduced in the range of 15% to 70%. Besides, this new method is also extended to solve two-dimensional sine-Gordon solitons. The results obtained show good agreement with the published results. Moreover, the parallelization of adaptive variational multiscale EFGM can improve the computational efficiency by reducing the execution time without loss of accuracy. Therefore, the capability and robustness of this new method has the potential to investigate more complicated problems in order to produce higher precision solutions with shorter computational time

Multiscale Mathematical Modelling of Nonlinear Nanowire Resonators for Biological Applications

Nanoscale systems fabricated with low-dimensional nanostructures such as carbon nanotubes, nanowires, quantum dots, and more recently graphene sheets, have fascinated researchers from different fields due to their extraordinary and unique physical properties. For example, the remarkable mechanical properties of nanoresonators empower them to have a very high resonant frequency up to the order of giga to terahertz. The ultra-high frequency of these systems attracted the attention of researchers in the area of bio-sensing with the idea to implement them for detection of tiny bio-objects. In this thesis, we originally propose and analyze a mathematical model for nonlinear vibrations of nanowire resonators with their applications to tiny mass sensing, taking into account thermal, piezoelectric, electromagnetic, surface, and external excitations.~The mathematical models for such nanowires are formulated using the Euler-Bernoulli beam theory in conjunction with the nonlocal differential constitutive relations of Eringen type. In order to analyze the obtained nonlinear partial differential equation (PDE), we first use the Galerkin method in combination with a perturbation technique to obtain the primary resonance.~After finding the primary resonance, a parametric sensitivity analysis is carried out to investigate the effects of key parameters on the sensitivity of the nanowire resonators in mass sensing.~Our main hypothesis is that bio-particles attached to the surface of the nanowire resonator would result in a detectable shift in the value of the jump frequency.~Therefore, a mathematical formula is developed based on the jump frequency to scrutinize the sensitivity of the considered nanowire resonators. Our mass sensitivity analysis aims at the improved capability of the nanowire resonators in detection of tiny bio-particles such as DNA, RNA, proteins, viruses, and bacteria.~Numerical solutions, obtained for the general nonlinear mathematical model of nanowire resonators, using the Finite Difference Method, are compared with the results obtained with a simplified approach described above. Finally, we investigate the sensitivity of the nanowire resonator for mass sensing using molecular dynamics simulations to provide a validation for our results from the obtained continuum models. It is expected that the results of this research may assist in our better understanding of key characteristics of nanowire resonators for their applications in detection of bio-particles, ultimately impacting the development of advanced approaches to disease diagnostics and treatments