The focus of my current research is on the non-classical continuum mechanics of solids, wherein the aim is to describe the defect structure (micro-cracks, dislocations, disclinations to wit) using such homogenized tensorial quantities as curvature, torsion and non-metricity. Within this framework, I wish to understand the consequences of defect motion and other constraints – geometrical (e.g. gauge symmetries) or thermodynamic (e.g. the entropy inequality) – in modifying the balance laws and material constitution that describe a component’s journey to failure through brittle or ductile damage. The question to which I seek an answer is if such a non-classical model could be a macro-continuum extension to what molecular dynamic simulations accomplish in the nanoscale. A related curiosity is to explore if defects could be so engineered as to control damage and failure in components according to certain design criteria. In order to realize this objective, the basic issues that I, along with my collaborators and students, have been trying to address are summarized below.
An effort at understanding and characterizing the non-local (non-Euclidean), non-equilibrium macro-mechanics of solids, such as metals, ceramics, foams and polymers, under varying ambient temperature and strain rate is currently under progress. The original motivation for this work came from a recently concluded and DRDO-funded inter-institutional initiative in developing computationally efficient, yet accurate and physically based, thermo-visco-plastic and damage models that can realistically predict the response of armour materials under ballistic impacts. The two essential aspects of such powerful predictive models are: 1) the geometric description (using torsion, curvature and non-metricity tensors) of moving defects such as dislocations, voids and micro-cracks responsible for plasticity and damage and 2) the non-equilibrium thermodynamic aspects that govern dissipation. The following related developments, some of which are currently being reviewed, should contribute to the realization of a physics-based predictive model in simulating the macro-continuum signatures of the microscopic defects.
A new global optimization method, belonging to the broad class of evolutionary stochastic search algorithms, has recently been developed. Starting with an initial scatter of the design variables to be optimized against a single or a set of cost functions, the optimality condition is provided by a novel stochastic characterization of the cost functions as well as the constraint equalities in that their diffusive variations around the mean optimal solution must behave as zero-mean martingales. In other words, the notion of optimization is now related to a martingale problem. An absolutely continuous change of measure is then effected on the design variables to iteratively force the cost functions and constraints behave according to the above characterization, which in turn yield the optimal design variables. Implemented using an appropriate expansion consistent with the stochastic calculus, this change of measure yields an additive gain-type correction term, which could be interpreted as providing a strictly non-Newton direction, to update the design variables. The stochastic search, directed this way, is typically seen to converge to the global optimum significantly faster than most competing schemes, e.g. variants of the genetic algorithm, differential evolution, particle swarm optimization etc. Indeed, it is also demonstrated that many basic search ideas, encoded within some of the existing global optimization schemes, may also be modified as martingale problems, thereby improving the performance of these methods (see journal publication 6). This body of work encompasses a very broad class of Np-hard search/optimization and inverse problems. Noteworthy applications in the last category of problems include medical imaging of vibro-acoustic or photo-acoustic types (see journal publications 120, 124) and recovery of earthquake source parameters in the 2004 Andaman-Sumatra tsunami event (see journal publication 116).
An important precursor to the work on optimization has been a family Monte Carlo filters, of the non-iterative and iterative types. With a view to applications to dynamical system identification and structural health assessment problems, these filters have been developed by rigorously extracting, through manipulations on the Kushner-Startonovich equation, a gain-type additive update term to correct the particle positions within a Monte Carlo setup. It is conclusively shown that the non-iterative variant, which is computationally efficient, is accurate enough for most state/system identification problems of practical interest. These filters, unlike most existing ones, work with very low process noise, relative low ensemble size and can be applied to large dimensional problems. In general, they also yield highly accurate model parameter estimates with lower sampling fluctuations, even when the ensemble size is smaller by an order.
Based upon a new stochastic characterization of the functional discretization errors in finite element or mesh-free methods and the powerful notion of a change of measure, a methodology to correct for this error and thus arrive at solutions with higher space/time resolution, without recourse to h or p refinement, is currently under development.
Our earlier research on transversal linearization has led to a new linearization paradigm that bypasses the derivation of tangent system matrices in a class of applications involving the evolutions of nonlinear dynamical systems. For stochastically excited nonlinear oscillators of engineering interest, the transversal and Girsanov linearizations provide twin simulation strategies that achieve, in principle, very high numerical accuracy in both direct simulation and nonlinear system identification.
One of our contributions in computational solid mechanics is a more recent method, the smooth DMS-FEM, which endows a given finite element mesh (in 2D or 3D) with strictly C1 or still higher order continuous shape functions and thus potentially achieves an unparalleled numerical accuracy in the solutions of a class of highly ill-conditioned problems of great relevance in engineering science. The DMS-FEM offers a computationally cheaper alternative to the mixed finite element method (FEM) in obtaining smooth stress/strain distributions whilst being extremely tolerant to mesh distortions. This method is the culmination of my earlier efforts at reconciling the mesh-based FEM with a class of mesh-free methods with globally smooth, polynomial reproducing shape functions with NURBS (non-uniform rational B-splines) as the basic building block. Other than the convexity it affords in the numerical approximation, the use of NURBS has the added advantage of being a vehicle for a seamless interface between the solid modeler and the solver, thereby avoiding the costly data transfer between the two software modules. Yet another application of such methods, of relevance in aerospace engineering, has been in numerically obtaining globally smooth deformed shapes of wrinkled/slack membranes and their experimental validations. The research interest also includes the numerically stabilized mesh-free computations of impact dynamic systems, strain gradient plasticity problems and shallow water equations modeling the propagation of tsunami waves.
Our research on inverse problems, as reflected in some of the recent publications on stochastic filters, provide a bridge across the apparently disparate disciplines of stochastic dynamics and computational mechanics. It is also here that our work assumes a cross-disciplinary hue. Along with a few collaborators, we have been applying such inverse algorithms in developing low-cost scanning mechanisms for early detection of inhomogeneous inclusions (cancers) in soft-tissue organs. In collaboration with physicists, we have recently proposed a new imaging modality that combines ultrasound probing with optical sensing to image soft tissue organs based on their vibration characteristics. Efforts are underway to convert this idea into a practical imaging system.