• Geometrically inspired and gauge theories for continuum mechanics of solids (emphasizing non-Euclidean defect kinematics)
  • Non-equilibrium thermodynamics of solids and fluctuation relations valid far from equilibrium
  • Defect engineering and metamaterials with acoustic band gaps
  • Optimization based on stochastic search on Riemannian manifolds
  • Computational Mechanics: Mesh-free and Finite Element Methods; Peridynamics; SPH-based simulations in computational impact dynamics
  • Mechanics of structured continua; the role of local symmetry in the continuum physics of solids; applications to modelling of plasticity, damage, piezo-electricity etc.
  • Stochastic Filtering Techniques for Inverse Problems; nonlinear structural system identification and medical diagnostic imaging in soft materials
  • Bayesian updates on Riemannian manifolds

Research Statement

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.

  • Since the presence of defects implies that the material body is non-Euclidean, the notion of a geodesic is affected and the kinematic description (e.g. the metric, volume, connection, strain etc.), including the concept of a derivative or a differential, needed to transport a vector from a point to another, is altered so as to reflect the non-trivial connection on the manifold.
  • Defect motion may simultaneously occur over a wide range of time rates and hence one of my objectives is to exploit a space-time connection, which may be non-metric, to explicate on the so called pseudo-forces that may occur during impacts, rapid crack propagation etc.
  • In the perspective of gauge theory of solids, we look for certain local symmetries of the Lagrangian (e.g. invariances to local translation, local rotation and conformal scaling) to extract the Euler-Lagrange equations of motion for the defective solid body. The demand on local time translation symmetry means that these equations could be dissipative.
  • I am applying these tools mostly to develop non-classical models for plasticity and damage and for the determination of residual stresses.
  • I plan to develop a family of stochastic search techniques based on the processes evolving on a Riemannian manifold and exploit the faster convergence possible with these methods to solve a range of inverse problems – mostly for material profiling exercises.
  • We also intend to pose and solve the inverse problem of engineered defects: to find an optimal defect configuration for maximal peak load, ductility etc.
  • On the experimental side, I wish to set up an additive manufacturing facility (as part of our newly established Centre of Excellence in Advanced Mechanics of Materials; see below) to manufacture such defective components and experimentally verify their predicted response.

More Details on Recent Research: Accomplished and Ongoing

Geometrically inspired continuum models for Plasticity and Damage

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.

  • New peridynamics approaches:
    In view of an inherently non-smooth nature of defect-induced flows, a new family of peridynamic formulations as a generic route to capturing possible non-locality in the nonlinear mechanics of solids has recently been developed. This approach obtains the governing, non-local equations for momentum balance in an integro-differential form, wherein the basic response field need not be differentiable. Of specific current interest is the development of a family of non-polar peridynamic models that can treat a whole range of complex response scenarios – from progressive damage (or material degradation) to fragmentation (material disintegration) – in metals, composites, foams and ceramics.
  • A dynamic flow rule in plasticity and damage:
    It incorporates a fictive micro- or configurational inertia contribution from moving defects, has been introduced to model plastic deformation under high strain rates and damage. The configurational inertia or the pseudomomentum is shown to slow down the plastic deformation rate or delay the damage onset and propagation.
  • Dynamic plasticity and damage based on exterior calculus:
    Beginning with the geometrical representations of deformation incompatibilities using the tools of exterior calculus, the final aim here is to arrive at continuum models for dynamic plasticity and damage (in crystalline and amorphous solids) based on the gauge field theory. Of specific interest under very large strain rates is the notion of defect current, which is defined in terms of the incompatibility of the velocity gradient with the rate of elastic distortion, expressed using appropriate differential 1-forms. The defect current, other than the torsion 2-form (defect density tensor in the classical parlance), appears to be the ideal quantity to describe the dynamics of defects such as dislocations. A set of closed field equations evolving these quantities may be derived using appropriate gauge invariance principles. It is believed that a pseudo-momentum (or configurational momentum) term in the field equation for defect motion could play a significant role in an accurate prediction of the dynamic response under high strain rates.
  • A two-temperature thermodynamic model for metal plasticity and fluctuation relations:
    Modeling of plasticity in polycrystalline solids, precipitated by dislocation motion, is facilitated by the introduction of a configuration subsystem characterized by the dislocation density along with a kinetic-vibrational subsystem defined through the vibration of atomic lattices. Though coupled, the two sub-systems have their own thermodynamics and hence separately defined temperatures. Replacing the second law of thermodynamics by an appropriate form of non-equilibrium integral fluctuation relation, this setup enables development of sharper material constitutive models consistent with the inherently non-equilibrium aspects of plasticity (see journal publication 117).
  • Martingale approach and a generalized fluctuation relation:
    Starting with a macroscopic (continuum-level) ensemble of material configurations defined over a probability space, physically admissible future changes in the material must conform to a valid sequence of probability measures in the sense of Kolmogorov's extension theorem. This means that any physical and observable material change can be described by a Radon-Nikodym derivative, the ratio of the incremental measures at the current and initial times. Since the Radon-Nikodym derivative is itself a martingale as a stochastic process, the problem of describing the evolving constitution of a material, including possible phase changes, essentially reduces to a martingale problem whose solution provides for a generalized fluctuation relation valid within a strictly non-equilibrium setup. This work, currently under progress, may yield an entirely new viewpoint for what is loosely described today as the 'non-equilibrium entropy'. In addition, by Doob's h-transform (or entropy-transform), one can also derive the necessary modifications in the governing equations of motion associated with the Radon-Nikodym derivative.

Stochastic Filtering, Global Optimization and Applications

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.

Previous Research Accomplishments (1998-2011)

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.