JADHAV RESEARCH
Web pages of Ajit Jadhav
Home Training Consultancy Work Experience Academics Research Publications Miscellaneous Links	Research -- Computational Mechanics -- Fundamentals of Physics -- Fundamental Algorithmics   Computational Mechanics: This field is broad enough to cover computational field theory, and then also certain other topics such as kinematics (or motion analysis) of machine elements, theory of structures and their design, etc. However, my own interest is mostly limited to the computational modeling of continua--how to use computers to predict quantities like temperatures, velocity, electric field, stresses, etc., that develop within the real (3D) components and systems during service. In computational mechanics, I have invented a new method that I call FAQ (short for Fields As Quanta). I have so far published four peer-reviewed papers at two international meets covering various aspects of this new approach. This new approach is the topic of my on-going Ph D. For my current Ph D work, I am only interested in applying the FAQ method to what I call "Helmholtzian" equations--i.e. the linear Wave, Heat/Diffusion, and Poisson/Laplace equations taken together as a single class. Mathematical literature, unfortunately, does not separately define such a class. So, the name given it--"Helmholtzian"--is new, coined by me. The new class should not be confused with the well-known Helmholtz equation which describes the space part of waves. Of course, the Helmholtz equation does fall in the Helmholtzian class. However, the class I mean above is much broader in scope. Thus, the class also includes the time-dependent aspects of waves apart from the diffusion and Poisson/Laplace equations. From a mathematical point of view, comparatively speaking, the Helmholtzian class is a simpler one. As of today, for the purpose of the Ph D thesis, I am only applying FAQ to the Helmholtzian class. However, it is obvious that the FAQ method should also be applicable to many other field problems such as: viscous (or Navier-Stokes kind of) flow; stress fields in solids; fracture-mechanical analysis (involving singular stress fields near cracks); etc. I hope to be able to apply the FAQ method to these challenging problems after the Ph. D. thesis is complete. (These latter problems sure are challenging. Just by way of an indication: Navier-Stokes equations are at the heart of a million dollar prize declared by the Clay Mathematics Institute!) But when it comes to applications, even the equations from the simpler Helmholtzian class are applicable across a wide spectrum of physical phenomena in both science and engineering. Just consider this (partial) list of applications: ideal fluid flow; viscous flow in ducts; underground seepage of fluids; steady-state and transient heat conduction in solids; radiative heat transfer; conjugate-mode heat transfer; diffusion in solids; torsion of thin plates; static and dynamic electromagnetic fields; electronic drift in metals; waves like: elastic waves in earth-crust, ultrasonic waves in nondestructive testing, radar waves, ocean waves at ports; illumination for architectural design of spaces; acoustic characteristics of concert halls; etc. So, the range of even the "simpler" Helmholtzian equations is quite mind-boggling! Another thing. The extension of the FAQ method cannot be simply mathematical. An identification of some sort of fundamental physical/engineering principles will have to go hand in hand with the next developments I mention above. BTW, in this research, to compare the results given out by the FAQ method with FEM, I implemented my own FEM solver and post-processor. (See the comparative results of FEM and FAQ in the 2003 paper, which is available for download following the link on the Publications page.)   Fundamentals of Physics: Actually, I have only little formal training in quantum physics. (A few graduate courses on solid-state physics, x-ray and electron diffraction, structural metallurgy, and so on, but no separate course-sequence on QM proper.) So, initially, my interest in the topic was not much different from what a layman would keep. But then, I found that I could resolve the fundamental issue of wave-particle duality using this new approach (i.e. the FAQ method). Once I got admitted for the current Ph. D. program, within a year, I published two conference papers on these ideas. (See the Publications page.) The papers I just mentioned were only limited to photons (bosons), and the computer simulation for them was limited to a 2D (or planar) interference chamber. This was mostly because of the lack of space in the paper (and time available to prepare them!) Further, in these papers, I also could not cover a treatment of phases, i.e. from the perspective of the new approach (the FAQ method). I now intend to write a journal paper on this somewhat extended topic. As a related development, equally interesting it was to correct a misconception from classical physics (diffraction theory). This misconception seems to have been propagating intact right since the year 1818! So, I also published a separate paper to correct it. Sometime later on, it will be interesting to pursue a broader application of the FAQ method for propagation of electrons (i.e. fermions) too. BTW, you might want to start reading more about this "Most Beautiful Experiment in Physics".   Fundamental Algorithmics: That is, the P-vs.-NP problem. (But, now a days, hardly anything else from the field of algorithmics itself really interests me.) Like everyone else, I too am aware of the Clay Institute’s million dollar prize for this problem! But the prize (really) has not been the motivation. In fact, the first time I found the P-vs.-NP problem mentioned very succinctly was in Dennis Shasha’s book: "Out of Their Minds." I "discovered" the book for myself only recently, in 2004! Soon after reading the description of the traveling salesman problem in that book, for a while, I thought I could see a way to crack the problem! But, I have not been able to find a mathematician or computer scientist who would collaborate with me, to translate the solution I have in mind, into the kind of language which will be acceptable to the mathematical community, i.e., in the terms compatible to the Clay Institute’s official statement of the problem. Now, unless one puts one’s thinking in precise mathematical terms, even a "solution" that seems fool-proof is only so much so. So, since I can't even have the solution informally verified, I have stopped thinking about it! Then, why do I mention it here? Because, I believe I am thinking of something that people don't seem to, and feel, they should! So, overall, though this research is now on the back burner, I remain open for any possible collaboration of this kind with a mathematician or a computer scientist.
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