RESEARCH ACTIVITIES

My research interests span several areas of Applied Mathematics: Numerical analysis, Scientific Computing, Ordinary and Partial Differential Equations, Mathematical Modeling, Computational Fluid Dynamics, Vibrations, Inverse Eigenvalue Problems, Control theory, Mechanical and Civil engineering. I am interested in applications of numerical analysis with the general goal of designing a robust numerical method. My recent research work focuses on designing and implementing higher-order Discontinuous Galerkin methods (DGM) for hyperbolic problems. My master thesis was devoted to solving inverse eigenvalue problems. My research program for the immediate future involves several areas in applied mathematics. Next I present my areas of expertise and describe my experience. Research projects inspired on past work are also briefly described.

Areas of Interest and Current Research Projects

1 Doctoral Research: Superconvergence Error Analysis and A posteriori Error estimation of the Discontinuous Galerkin Method for Two-dimensional Hyperbolic Problems

Many physical phenomena are represented by systems of partial differential equations having solutions with local and nonuniform multi-scale behavior such as moving fronts and boundary layers observed in fluid dynamics and materials science processes. Since these nonuniformities may form, evolve, and disappear with time, standard numerical methods become either very inefficient or impossible to use. However, Adaptive methods which variable mesh sizes and orders of approximation in both time and space are very efficient in resolving local nonuniform behaviors. Towards this, we are developing dynamic, efficient, robust, and accurate methods that handle nonuniform behaviors. These algorithms use information available during the computations to improve the quality of the solution and are guided by error estimates which indicate that elemental discretization error decrease by either decreasing the element size h or h-refinement, by increasing the polynomial degree p or p-refinement, or by repositioning the finite element vertices to equilibrate the mesh. Our software combines these enrichment techniques to obtain optimal algorithms for various time dependent systems. As mentioned above, in the heart of these algorithms lie some error estimators or indicators. Usually, error estimators or indicators are obtained by solving local problems using an approximation of degree higher that the finite element solution.

My primary research topic is the development of innovative, but reliable solution algorithms for the numerical approximation of solutions to the linear and nonlinear partial differential equations that arise in CFD. I am particular interested in conservation laws which constitute very powerful and important models for physical phenomena. They arise in many different topics like fluid mechanics, astrophysics, reactive flows, traffic modeling and several related areas. From the mathematical point of view, conservation laws are particularly interesting. They tend to develop discontinuous solutions even from smooth initial values because of the nonlinear nature of the equations.

I have devoted my doctoral work to the construction of high order DG method for hyperbolic problems in two dimensions. The numerical solution of hyperbolic systems is confronted with a list of major difficulties and questions that have been under study for many years. These include classical problems of numerically resolving shocks and discontinuities, characteristic of solutions of hyperbolic problems, while simultaneously producing high-order, non-oscillatory results near shocks and elsewhere in the solution domain. Moreover, the basic issue of quality of numerical solutions is fundamentally important: how accurate are the numerical simulations and how does one obtain the most accurate results for a fixed computational resource? These questions lie at the core of modern adaptive methods that aim to control the error in the computed solution and to optimize the computational process.

2- Selection of Physical and Geometrical Properties for the Confinement of Vibrations in Nonhomogeneous Beams

This project is conducted in collaboration with Prof. Ali Nayfeh of the Department of Engineering Science and Mechanics at Virginia Tech. It is part of the NSF funded project between Virginia Tech and EPT. The aim of the work was to employ the inverse eigenvalue problem to estimate the physical and geometrical properties of the beams. Therefore, the objective was to determine such properties, namely the cross-section area, Young’s modulus of elasticity, mass density per unit length, and second moment area, for a given set of frequencies and mode shapes. In this work, the confinement of flexural vibrations in nonhomogeneous beams was formulated as one of two types of an inverse eigenvalue problem. The first one determines the beam’s geometrical and physical parameters and natural frequencies for a prescribed set of confined mode shapes. The second approximates these parameters for a given set of confined mode shapes and frequencies. In both problems, a set of mode shapes, which satisfy all of the boundary conditions and yield vibration confinement in prespecified spatial subdomains of the beam, were selected. Because closed-form solutions are not available in both methods, the differential quadrature method was employed to discretize the spatial domain, and thus replace the eigenvalue problem as a system of algebraic equations that incorporates the parameters evaluated at all grid points. In both confinement problems, the unknown physical and geometrical properties must be positive and were approximated using functions constructed from polynomials. These functions are specified at the beam’s left end, right end, or both. It is shown that the physical and geometrical properties can be reconstructed from a few mode shapes. The approximate parameters were finally substituted in the eigenvalue problem to confirm the confined mode shapes of the beam. We predict that the outcome of this study on the confinement of vibrations in beams can be extended to the class of two-dimensional and three-dimensional linear flexible structures, such as plates and cylinders. This work generated my second journal paper. This is a still open project. This area has certainly caught my attention. In the near future, I would like to extend this work for two and three dimensional problems.

3- Modal identification and model updating of a reinforced concrete bridge

This project was my engineering training during the entire summer of 2003 and was funded by the Center for Testing and Construction Techniques. The project deals with the development of a rational methodology for the assessment of older reinforced concrete bridges. This methodology is based on the following steps: (a) response-only ambient vibration measurement of the bridge; (b) output-only modal signature identification of the bridge using the Enhanced Frequency Domain Decomposition technique (EFDDT); (c) finite element model updating which yields a linear elastic finite element model that reproduces as much as possible the real experimental behavior of the bridge; and (d) estimation of maximum bridge capacity and prediction of its failure modes based on detailed nonlinear finite element analyses. The bridge is made of a continuous four-span simply supported reinforced concrete slab without girders resting on elastomeric bearings at each support. The EFDDT was applied to extract the dynamic characteristics of the bridge. The finite element model was updated in order to obtain a reasonable correlation between experimental and numerical modal properties. For the model updating part of the study, the parameters selected for the updating process include the concrete modulus of elasticity, the elastic bearing stiffness and the foundation spring stiffnesses. The application of the proposed methodology led to a relatively faithful linear elastic model of the bridge in its present condition. The modal properties for the first three vibration modes were successfully identified using the EFDDT. The maximum error between the model and test frequencies before updating reached a value of 24% which indicated the need to update the finite element model. These errors became less than 6.8% after updating. Furthermore, a reasonable correlation between the experimental and finite element mode shapes was obtained at least for the first three vibration modes. The dynamic response of the bridge was shown to be more sensitive to changes in the concrete modulus of elasticity. Updating results revealed that the bridge has suffered relatively minor damage. This work generated my first international journal paper. Although this type of work was new to me, I was able to understand the details of the work in extremely short period.

4 Plans for Future Research

I have highlighted a few of the initial research directions I am interested in. These are starting points for exploration, and many interesting and unexpected research challenges are sure to arise during the process. There are a whole lot of long term goals. In my future research I would like to (i) Develop a working code of the DGM applied to systems of convection-diffusion problems using general meshes, (ii) Study the error for the DGM including the rates of convergence and hope to derive error estimates, (iii) Extend the error estimates and superconvergence results of the DGM to locally refined meshes with hanging nodes and unstructured tetrahedral meshes for hyperbolic systems and (iv) Continue the work in this area and develop more efficient and reliable error estimators that can be used to control the accuracy of the solution. I will also continue to work on stable higher order finite elements methods for singularly perturbed elliptic and parabolic systems in two and three dimensions. With a strong background in applied and computational mathematics, as well as a good background in computational mechanics, I would also like to participate in multidisciplinary research in computational sciences and engineering. I am also interested in studying numerical solutions of PDEs using other higher order methods such as finite differences, finite volume schemes and spectral methods with broader application. I believe that collaboration, especially across disciplines, exposes one to new perspectives, allowing experiences and lessons previously isolated to their fields to be leveraged to produce high-quality research. To this end, I have collaborated and co-authored papers with researchers in different areas of Mathematics and Mechanics from different Universities, hoping to grow this aspect of my research even further in the future. It would be very exciting to be involved in such an active research group.