Figure: A wordcloud from the title of the journal papers
Unlike the 'forward problem' the class of inverse problem is generally illposed problem and consequently considered as more difficult. Professor Adhikari's research works in this field contribute to the following sub areas:
A brief description of the research topics and some key journal publications are given below.
D.1. Nanomechanical sensors
Labelfree mass sensing using cantileverlike resonators is an active field of multidisciplinary research. Depending on the size on the cantilever, the mass to be detected can vary from few atoms to relatively larger size like DNA and other biological molecules. With the advancement of manufacturing technology, micrometresize cantilever sensors are expected to be used in practice in the near future for a variety of applications including biosensors, environmental sensor, gas sensors and chemical sensors. For any mechanical sensors to work in a reliable manner, it is necessary that the object to be sensed makes a detectable change in a quantity which can be measured easily and accurately. The measurement of the resonance frequency is one of the most fundamental properties of an oscillator and can be measured reliably across a wide range of mechanical systems. Resonance based sensors offer the significant potential of achieving the highfidelity requirement of many sensing applications. This is essentially an inverse problem where the mass needs to be identified from a detectable change. Our research developed novel analytical formulation to solve the inverse problem using energy principles. This not only reduces the computational time but also improves physical understandings. We also proposed a completely new way for mass sensing using a shift in the damping factors of the system, enhancing the only existing possibility of using the frequency shift.
Bending mode of a CNT with a bio molecule 
Local mode of a CNT with a bio molecule 
Graphene based nanomechanical sensors 
Damping factor shift in cantilever sensors 
Key Journal Publications
 Adhikari, S., “Inertial mass sensing with low Qfactor vibrating microcantilevers,” Journal of Applied Physics, Vol. 52, No. 12, 2017, pp. 512526.
 Adhikari, S., “Nonlocal mechanics based computational methods for nanomechanical sensors,” Procedia Technology, Vol. 23, No. 2, 2016, pp. 719.
 Karlicic, D., Kozic, P., Adhikari, S., Cajic, M., Murmu, T., and Lazarevic, M., “Nonlocal biosensor based on the damped vibration of singlelayer graphene influenced by inplane magnetic field,” International Journal of Mechanical Sciences, Vol. 9697, No. 6, 2015, pp. 132142.
 Kam, K., Scarpa, F., Adhikari, S., and Chowdhury, R., “Graphene nanofilm as pressure and force sensor: A mechanical analysis,” Physica Status Solidi B, Vol. 250, No. 10, 2013, pp. 20852089.
 Adhikari, S. and Murmu, T., “Nonlocal mass nanosensors based on vibrating monolayer graphene sheets,” Sensors & Actuators: B. Chemical, Vol. 188, No. 11, 2013, pp. 13191327.
 Chowdhury, R., Scarpa, F., and Adhikari, S., “Molecularscale biosensing using armchair graphene,” Journal of Applied Physics, Vol. 112, No. 1, 2012, pp. 014905:16.
 Adhikari, S. and Chowdhury, R., “Zeptogram sensing from gigahertz vibration: Graphene based nanosensor,” Physica E: Lowdimensional Systems and Nanostructures,Vol. 44, No. 78, 2012, pp. 15281534.
 Chowdhury, R., Adhikari, S., and Rees, P., “Zigzag graphene nanoribbon singleelectron transistors,” Physica B: Condensed Matter, Vol. 407, No. 5, 2012, pp. 855858.
 Murmu, T. and Adhikari, S., “Nonlocal frequency analysis of nanoscale biosensors,” Sensors & Actuators: A. Physical, Vol. 173, No. 1, 2012, pp. 4148.
 Chowdhury, R. and Adhikari, S., “Boron nitride nanotubes as zeptogramscale bionano sensors: Theoretical investigations,” IEEE Transactions on Nanotechnology, Vol. 10, No. 4, 2011, pp. 659667.
 Chowdhury, R., Adhikari, S., Rees, P., Scarpa, F., and Wilks, S. P., “Graphene based biosensor using transport properties,” Physical Review B, Vol. 83, No. 4, 2011, pp. 045401:18.
 Adhikari, S. and Chowdhury, R., “The calibration of carbon nanotube based bionano sensors,” Journal of Applied Physics, Vol. 107, No. 12, 2010, pp. 124322:18.
 Chowdhury, R., Adhikari, S., and Mitchell, J., “Vibrating carbon nanotube based biosensors,” Physica E: LowDimensional Systems and Nanostructures, Vol. 42, No. 2, December 2009, pp. 104109.
D.2. Identification of nonlinear systems
The inverse problem of estimating timeinvariant (static) parameters of nonlinear systems is considered in this research. The state and parameter estimates of systems exhibiting oscillation obtained from noisy observational data provide valuable information for assessing the safety and reliability of the engineering system in its operational state. The parameter estimation problem falls in the general category of system identification. We explore the feasibility of joint state and parameter estimation of noisedriven chaotic systems using the extended Kalman filter (EKF), ensemble Kalman filter (EnKF), and particle filter (PF). We consider the chaotic vibration of a noisy Duffing oscillator perturbed by combined harmonic and random inputs ensuring a transition probability density function (pdf) of motion which displays strongly nonGaussian features. Our system offers computational simplicity while exhibiting a kaleidoscope of dynamical behaviour with a slight change of input and system parameters. An extensive numerical study is undertaken to contrast the performance of various nonlinear filtering algorithms with respect to the sparsity of observational data and strength of model and measurement noise.
Phasespace diagram of a chaotic Duffing oscillator 
Parameter estimates with sparse observational data of a Duffing oscillator 
Key Journal Publications
 Khalil, M., Sarkar, A., Adhikari, S., and Poirel, D., “The estimation of timeinvariant parameters of noisy nonlinear oscillatory systems,” Journal of Sound and Vibration,Vol. 344, No. 5, 2015, pp. 81100.
 Khalil, M., Sarkar, A., and Adhikari, S., “Tracking noisy limit cycle oscillation with nonlinear filters,” Journal of Sound and Vibration, Vol. 329, No. 2, January 2010, pp. 150170.
 Khalil, M., Sarkar, A., and Adhikari, S., “Nonlinear filters for chaotic oscillatory systems,” Nonlinear Dynamics, Vol. 55, No. 12, January 2009, pp. 113137.
D.3. Model updating and damage detection
System identification plays a crucial role in the modelbased prediction of dynamical systems. A finite representation of a continuous media described by a partial differential equation usually leads to a discrete dynamical system. In practical applications involving linear continuous operators, the discrete system is fully characterised by the socalled mass, stiffness and damping matrices. Once these matrices are estimated with reasonable confidence, such discrete model can predict the response of the underlying dynamical system to external disturbances. In the context of structural dynamics, the identification of these system matrices is achieved by traditional modal analysis techniques. In the lowfrequency domain, only a few modes contribute to the total response of the system. Experimental modal analysis permits the extraction of such modal parameters through experimental measurements. However, there are uncertainties in the measured data and also in the modelling approaches. Our vision is to develop probabilistic model updating to address this issue. We also use spectral methods for system identification and damage detection for highfrequency applications.
AIRMOD (a) the physical structure and (b) the finite element model 
Comparison between an initial value, updated and measured frequency response function of a dynamic system 
Book
Key Journal Publications
 Machado, M. R., Adhikari, S., DosSantos, J. M. C., and Arruda, J. R. F., “Estimation of beam material random field properties via sensitivitybased model updating using experimental frequency response functions,” Mechanical Systems and Signal Processing, Vol. 102, No. 3, 2018, pp. 180197.
 Khodaparast, H. H., Govers, Y., , Dayyani, I., Adhikari, S., Link, M., and Friswell, M. I., “Fuzzy finite element model updating of the DLR AIRMOD test structure,” Applied Mathematical Modelling, Vol. 52, No. 12, 2017, pp. 512526.
 Machado, M. R., Adhikari, S., and DosSantos, J. M. C., “A spectral approach for damage quantification in stochastic dynamic systems,” Mechanical Systems and Signal Processing, Vol. 88, No. 5, 2017, pp. 253273.
 Boulkaibet, I., Mthembua, L., Marwala, T., Friswell, M. I., and Adhikari, S., “Finite element model updating using Hamiltonian Monte Carlo technique,” Inverse Problems in Science & Engineering, Vol. 25, No. 7, 2017, pp. 10421070.
 Boulkaibet, I., Mthembua, L., Marwala, T., Friswell, M. I., and Adhikari, S., “Finite element model updating using the shadow hybrid Monte Carlo technique,” Mechanical Systems and Signal Processing, Vol. 5253, No. 1, 2015, pp. 115132.
 Jacquelin, E., Adhikari, S., and Friswell, M. I., “A secondmoment approach for direct probabilistic model updating in structural dynamics,” Mechanical Systems and Signal Processing, Vol. 29, No. 5, 2012, pp. 262283.
 Mthembu, L., Marwala, T., Friswell, M. I., and Adhikari, S., “Model selection in finite element model updating using the Bayesian evidence statistic,” Mechanical Systems and Signal Processing, Vol. 25, No. 7, 2011, pp. 23992412.
 Friswell, M. I. and Adhikari, S., “Structural health monitoring using shaped sensors,” Mechanical System and Signal Processing, Vol. 24, No. 3, March 2010, pp. 623635.
 Adhikari, S. and Friswell, M. I., “Distributed parameter model updating using the KarhunenLoeve expansion,” Mechanical System and Signal Processing, Vol. 24, No. 2, February 2010, pp. 326339.
 du Bois, J. L., Adhikari, S., and Lieven, N. A. J., “Error analysis in trifilar inertia measurements,” Experimental Mechanics, Vol. 49, No. 4, August 2009, pp. 533540.
 Khalil, M., Adhikari, S., and Sarkar, A., “Linear system identification using proper orthogonal decomposition,” Mechanical System and Signal Processing, Vol. 21, No. 8, November 2007, pp. 31233145.
D.4. Identification of damping
Characterisation of damping forces in a vibrating structure has long been an active area of research in structural dynamics. The demands of modern engineering have led to a steady increase of interest in recent years. However, in spite of a large amount of research, understanding of damping mechanisms is still quite primitive. A major reason for this is that, by contrast with inertia and stiffness forces, it is not in general clear which state variables are relevant to determine the damping forces. By far the most common approach is to assume socalled 'viscous damping', a linear model in which it is supposed that the instantaneous generalised velocities are the only relevant state variables that determine damping. This approach was first introduced by Rayleigh in 1877 via his famous 'dissipation function', a quadratic expression for the energy dissipation rate with a symmetric matrix of coefficients, the 'damping matrix'. A further idealisation, also pointed out by Rayleigh, is to assume the damping matrix to be a linear combination of the mass and stiffness matrices. Rayleigh was quite clear that this idea was suggested for mathematical convenience only, because it allows the damping matrix to be diagonalised simultaneously with the mass and stiffness matrices, preserving the simplicity of uncoupled, real normal modes as in the undamped case. Since its introduction, this model has been used extensively and is now usually known as 'Rayleigh damping', 'proportional damping' or 'classical damping'.
Unfortunately, there is no physical reason why a general system should behave like this. In fact, practical experience in modal testing shows that most reallife structures do not do so, as they possess complex modes instead of real normal modes. As Sestieri and Ibrahim (1994) have put it '... it is ironic that the real modes are in fact not real at all, in that in practice they do not exist, while complex modes are those practically identifiable from experimental tests. This implies that real modes are a pure abstraction, in contrast with complex modes that are, therefore, the only reality!' However, consideration of complex modes in the experimental modal analysis has not been very popular among researchers. Our work uses complex modes to identify a realistic model of damping. Both viscous and nonviscous damping modes are considered. New identification methods have been proposed and independently validated.
An experimental system used in out study 
Fitting the modal damping matrix 
Book
Key Journal Publications
 Papai, F., Adhikari, S., and Wang, B., “Estimation of modal dampings for unmeasured modes,” Slovak Journal of Civil Engineering, Vol. 20, No. 4, 2012, pp. 1727.
 Adhikari, S. and Phani, A., “Experimental identification of generalized proportional damping,” Transactions of ASME, Journal of Vibration and Acoustics, Vol. 131, No. 1, January 2009, pp. 011008:112.
 Adhikari, S. and Woodhouse, J., “Identification of damping: Part 3, symmetrypreserving method,” Journal of Sound and Vibration, Vol. 251, No. 3, March 2002, pp. 477490.
 Adhikari, S. and Woodhouse, J., “Identification of damping: Part 4, error analysis,” Journal of Sound and Vibration, Vol. 251, No. 3, March 2002, pp. 491504.
 Adhikari, S., “Lancaster’s method of damping identification revisited,” Transactions of ASME, Journal of Vibration and Acoustics, Vol. 124, No. 4, October 2002, pp. 617627.
 Adhikari, S. and Woodhouse, J., “Identification of damping: Part 1, viscous damping,” Journal of Sound and Vibration, Vol. 243, No. 1, May 2001, pp. 4361.
 Adhikari, S. and Woodhouse, J., “Identification of damping: Part 2, nonviscous damping,” Journal of Sound and Vibration, Vol. 243, No. 1, May 2001, pp. 6388.
D.5. Digital twin
A digital twin is an avatar of a real physical system which exists in the computer. While a computer model of a physical system attempts to closely match the behaviour of a physical system, the digital twin also tracks the temporal evolution of the physical system. The recent rise of digital twins motivated academic and industrial researchers to formalise and standardise underlying procedures and methodologies. We clearly see the benefit of such an approach due to the possibility of wider adoption of digital twins across several industrial sectors. However, a major drawback of such an approach is jargondominated literature, which many early adopters of digital twins may find it difficult to tailor for their specific application sector. Keeping this background in mind, we propose a specific but original approach to develop digital twins for structural dynamic systems. Our main scientific proposition is that the digital twin evolves at a timescale which is much slower than the dynamics of the system. This makes it possible to identify crucial system parameters as a function of the `slowtime' from continuously measured data.
The overview of constructing a digital twin for a singledegreeoffreedom dynamic system 
Digital twin constructed with different slowtime sampling 
Key Journal Publications
 Ganguli, R. and Adhikari, S. “The digital twin of discrete dynamic systems: Initial approaches and future challenges,” Applied Mathematical Modelling, in press.
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