arXiv subjects

Kaushik Bhattacharya

Publications and source records attributed to Kaushik Bhattacharya.

At least 37 records · Page 2Linked to original sources

arXiv2025

Spherically Symmetric, Static Solutions in Presence of Matter-Curvature Coupling

In this work we have proposed some spherically symmetric, static spacetimes in a theory of gravity which permits non-minimal coupling (NMC) between curvature of spacetime and fluid variables. It is shown that these non-minimally coupled theories may admit of new class of metric solutions. Known metric solutions from GR can also be solutions of the non-minimally coupled theories, for these cases the NMC affects the nature of the fluid which sources the spacetime. The paper presents multiple ways in which the modified field equations appearing in non-minimally coupled theories can be solved. The NMC produces multiple definitions of the stress-energy tensor. The paper discusses the complexity related to these sources of curvature as, unlike in minimally coupled general relativity, in the present theory the Ricci curvature itself can affect the stress-energy tensor of the effective fluid which seeds spacetime curvature. The various energy conditions related to various forms of possible stress-energy tensors are presented in the paper.

gr-qc↗

arXiv2025

Learning a potential formulation for rate-and-state friction

Empirical rate-and-state friction laws are widely used in geophysics and engineering to simulate interface slip. They postulate that the friction coefficient depends on the local slip rate and a state variable that reflects the history of slip. Depending on the parameters, rate-and-state friction can be either rate-strengthening, leading to steady slip, or rate-weakening, leading to unsteady stick-slip behavior modeling earthquakes. Rate-and-state friction does not have a potential or variational formulation, making implicit solution approaches difficult and implementation numerically expensive. In this work, we propose a potential formulation for the rate-and-state friction. We formulate the potentials as neural networks and train them so that the resulting behavior emulates the empirical rate-and-state friction. We show that this potential formulation enables implicit time discretization leading to efficient numerical implementation.

cond-mat.mes-hall↗

arXiv2025

Gravitational collapse of Matter in the presence of Scalar field Dark energy

This study examines the gravitational collapse of an overdense dark matter region in a coupled scalar field dark energy scenario within a flat FLRW background. It finds that, depending on the initial conditions, some overdense regions avoid collapse and expand eternally with the background. The interior overdense region follows a closed FLRW metric, while its boundary is described by generalized Vaidya spacetime, which allows flux across the boundary while preserving the homogeneity of dark energy inside. Dark matter evolves as cold dark matter, but in non-minimal coupling, the modified Klein-Gordon equation alters dark energy evolution. The results highlight the impact of coupled dark energy on dark matter virialization and cosmic structure formation.

gr-qc↗

arXiv2025

Learning Memory and Material Dependent Constitutive Laws

We propose and study a neural operator framework for learning memory- and material microstructure-dependent constitutive laws for heterogeneous materials. We work in the two-scale setting where homogenization theory provides a systematic approach to deriving macroscale constitutive laws, obviating the need to resolve complex microstructure repeatedly. However, the unit cell problems defining these constitutive models are typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Our proposed framework models homogenized constitutive laws with both memory- and microstructure-dependence through the use of Markovian recurrent and Fourier neural operators. The homogenization problem for Kelvin-Voigt viscoelastic materials is studied to provide firm theoretical foundations for our model. The theoretical properties of the cell problem in this Kelvin-Voigt setting motivate the proposed learning framework; and are also used to prove a universal approximation theorem for the learned macroscale constitutive model. Numerical experiments show that the proposed learning framework accurately learns memory- and microstructure-dependent viscoelastic and elasto-viscoplastic constitutive models, beyond the setting of the theory. Furthermore, we show that the learned constitutive models can be successfully deployed in macroscale simulation of material deformation for different microstructures without retraining.

math.NA↗

arXiv2025

Non-minimal coupling of scalar fields in the dark sector and generalization of the top-hat collapse

In this article, we propose a new way to handle interactions between two scalar fields in the cosmological backdrop where one scalar field oscillates rapidly in the cosmological time scale while the other does not show any periodic behavior in the same time scale. We have interpreted the rapidly oscillating scalar field as the dark matter candidate while the other scalar field is supposed to be the canonical quintessence field or the non-canonical phantom field. A model of a generalized top-hat-like collapse is developed where the dark sector is composed of the aforementioned scalar fields. We show how the non-minimal coupling in the dark sector affects the gravitational collapse of a slightly overdense spherical patch of the universe. The results show that one can have both unclustered and clustered dark energy in such collapses, the result depends upon the magnitude of the non-minimal coupling strength.

gr-qc↗

arXiv2025

Constitutive relations from images

Constitutive relations close the balance laws of continuum mechanics and serves as the surrogate for a material in the design and engineering process. The problem of obtaining the constitutive relations is an indirect inverse problem where both the relation and the quantities that define the relation have to be inferred from experimental observations. The advent of full-field observation techniques promises a new ability of learning constitutive relations in realistic operational conditions. However, this is done in two steps, first obtaining deformations from the images, and then obtaining the constitutive relation from deformations and forces. This leads to a variety of difficulties. In this paper, we propose a novel approach that enables us to obtain constitutive relation directly from the raw data consisting of images and force measurements.

cond-mat.mtrl-sci↗

arXiv2025

Comprehensive Study of $k$-essence Model: Dynamical System Analysis and Observational Constraints from Latest Type Ia Supernova and BAO Observations

We constrain the parameters of the $k$-essence scalar field model with inverse square and exponential potentials using data sets including Pantheon+SHOES and the Dark Energy Survey (DES) of Type Ia supernovae, Baryon Acoustic Oscillation (BAO) data from SDSS and DESI surveys, and direct measurements of the Hubble parameter and redshift obtained from the differential age method (CC). We also provide a brief perspective on the dynamical evolution of both models and derive stability constraints on the model parameters, which are then used to set appropriate priors. We adopt a Bayesian inference procedure to estimate the model parameters that best fit the data. A comprehensive analysis in light of observational data shows that the $k$-essence model fits well across all data combinations. However, according to the BIC criterion, the $Λ$CDM model provides a slightly better fit compared to the $k$-essence model.

astro-ph.CO↗

arXiv2025

End equilibrium state of a spherical gravitational collapse in the presence of matter and scalar field

We explore the possibilities of modeling a spherically symmetric static spacetime that can emerge as the end state of gravitational collapse, by considering it to be seeded by a composite fluid made of matter and a scalar field. In this scenario, the matter represents dark matter, while the scalar field represents dark energy. On certain scales, dark energy is believed to significantly influence the structure formation of dark matter. Various models describe the possible impacts of dark energy on structure formation under different scenarios. By investigating an inhomogeneous scalar field representing dark energy, coupled with dark matter, we demonstrate that this two-component fluid can seed spacetimes forming the final equilibrium state. We derive solutions for the scalar field and potential for Joshi-Malafarina-Narayan (JMN) spacetimes.

gr-qc↗

arXiv2025

Grain-size dependence of plastic-brittle transgranular fracture

The role of grain size in determining fracture toughness in metals is incompletely understood with apparently contradictory experimental observations. We study this grain-size dependence computationally by building a model that combines the phase-field formulation of fracture mechanics with dislocation density-based crystal plasticity. We apply the model to cleavage fracture of body-centered cubic materials in plane strain conditions, and find non-monotonic grain-size dependence of plastic-brittle transgranular fracture. We find two mechanisms at play. The first is the nucleation of failure due to cross-slip in critically located grains within transgranular band of localized deformation, and this follows the classical Hall-Petch law that predicts a higher failure stress for smaller grains. The second is the resistance to the propagation of a mode I crack, where grain boundaries can potentially pin a crack, and this follows an inverse Hall-Petch law with higher toughness for larger grains. The result of the competition between the two mechanisms gives rise to non-monotonic behavior and reconciles the apparently contradictory experimental observations.

cond-mat.mtrl-sci↗

arXiv2024

Learning constitutive relations from experiments: 1. PDE constrained optimization

We propose a method to accurately and efficiently identify the constitutive behavior of complex materials through full-field observations. We formulate the problem of inferring constitutive relations from experiments as an indirect inverse problem that is constrained by the balance laws. Specifically, we seek to find a constitutive behavior that minimizes the difference between the experimental observation and the corresponding quantities computed with the model, while enforcing the balance laws. We formulate the forward problem as a boundary value problem corresponding to the experiment, and compute the sensitivity of the objective with respect to model using the adjoint method. The resulting method is robust and can be applied to constitutive models with arbitrary complexity. We focus on elasto-viscoplasticity, but the approach can be extended to other settings. In this part one, we formulate the method and demonstrate it using synthetic data on two problems, one quasistatic and the other dynamic.

cond-mat.mtrl-sci↗

arXiv2024

Systematic design of compliant morphing structures: a phase-field approach

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this sharp interface problem, prove its convergence as the regularization length approaches 0 and present an efficient numerical implementation. We illustrate the strengths of our approach through a series of numerical examples.

math.NA↗

arXiv2024

Artificial sunflower: Light-induced deformation of photoactive shells

Photomechanically active materials undergo reversible deformation on illumination, making them ideal for remote, tether-free actuation. Much of the work on these materials has focused on one-dimensional structures, such as strips. In this paper, we explore photomechanically active two-dimensional structures such as sheets and shells. When illuminated, such structures undergo spontaneous bending due to the limited penetration of light. However, the geometry of the shell constrains possible deformation modes: changes in Gauss curvature lead to in-plane stretching, against which shells are very stiff. Therefore, there is a complex coupling between the photomechanical actuation and the mechanical behavior of a shell. We develop and implement a novel approach to study photomechananically active shells. This method is a discrete shell model which captures the interplay between actuation, stretching, bending, and geometric changes. Through a series of examples, we explore these complex interactions, demonstrating how one can design shells that deform to follow the source of illumination.

cond-mat.mtrl-sci↗

arXiv2024

Reassessing constant-roll Warm Inflation

Departing from standard slow-roll conditions is one way of putting the inflationary paradigm to test, and constraining the dynamics of the inflaton field with a constant-rate of roll of the inflaton field, a.k.a. the constant-roll scenario, is one way of exploring such deviation from the standard slow-roll dynamics. In this manuscript we explore such a possibility in a variant inflationary scenario, known as Warm Inflation. We construct and derive the conditions for having constant-roll WI models where inflation lasts at least for 60 $e-$folds, gracefully exits the constant-roll inflation phase, and maintains near thermal equilibrium of the system which is an essential feature of WI in the slow-roll regime. We show that while certain models of WI (the ones with dissipative coefficient as a function of temperature alone) can accommodate constant-roll dynamics, others (with dissipative coefficient as a function of temperature and the inflaton field both) fail to maintain thermal equilibrium once the constant-roll condition is imposed and hence cannot produce a constant-roll WI phase.

astro-ph.CO↗

arXiv2024

Adhesion of a nematic elastomer cylinder

Liquid crystal elastomers are cross-linked elastomer networks with liquid crystal mesogens incorporated into the main or side chain. Polydomain liquid crystalline (nematic) elastomers exhibit unusual mechanical properties like soft elasticity, where the material deforms at nearly constant stress, due to the reorientation of mesogens. In this paper, we use numerical simulation to study the implication of the remarkable elastic softness on a classical problem of adhesion. This study reveals that the soft elasticity of nematic elastomers dramatically affects the interfacial stress distribution at the interface of a nematic elastomer cylinder adhered to a rigid substrate. The stress near the edge of the nematic cylinder under tensile load deviates from the singular behavior predicted for linear elastic materials, and the maximum normal stress reduces dramatically. Moreover, the location of maximum interfacial stress shifts from the edge to the center of the nematic cylinder when the applied tensile force goes beyond a critical value. We discuss the implications for adhesion. The results are consistent with the available experimental data.

cond-mat.soft↗

arXiv2024

Neural Operator: Learning Maps Between Function Spaces

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

cs.LG↗

arXiv2024

Iterated learning and multiscale modeling of history-dependent architectured metamaterials

Neural network based models have emerged as a powerful tool in multiscale modeling of materials. One promising approach is to use a neural network based model, trained using data generated from repeated solution of an expensive small scale model, as a surrogate for the small scale model in application scale simulations. Such approaches have been shown to have the potential accuracy of concurrent multiscale methods like FE2, but at the cost comparable to empirical methods like classical constitutive models or parameter passing. A key question is to understand how much and what kind of data is necessary to obtain an accurate surrogate. This paper examines this question for history dependent elastic-plastic behavior of an architected metamaterial modeled as a truss. We introduce an iterative approach where we use the rich arbitrary class of trajectories to train an initial model, but then iteratively update the class of trajectories with those that arise in large scale simulation and use transfer learning to update the model. We show that such an approach converges to a highly accurate surrogate, and one that is transferable.

physics.app-ph↗

arXiv2024

Gravitational collapse of matter in the presence of non-minimally coupled Quintessence and Phantom-like scalar fields

This paper explores the evolution of the over-dense region of dark matter in the presence of a non-minimally coupled scalar field which is used to model quintessence and phantom-like dark energy. We focus on algebraic coupling, where the interaction Lagrangian is independent of the derivatives of the scalar field. To make our model more relativistic, like the minimal coupling scenario we studied earlier, we consider a spacetime structure that is internally closed Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime and externally the generalized Vaidya spacetime. This structure allows non-zero matter flux at the boundary of the over-dense region. Our investigation reveals that an increment of the coupling strength causes dark energy to cluster with dark matter at a certain cosmological scale where the influence of dark energy cannot be ignored. This phenomenon arises from the specific nature of the non-minimal coupling considered in this paper. While the evolution of matter's energy density remains unchanged, the scalar field's Klein-Gordon equation is modified, causing dark energy to deviate from its homogeneous state and cluster with dark matter. Similar to minimal coupling scenarios, closed spherical regions do not collapse within certain parameter ranges, exhibiting eternal expansion within the spatially flat FLRW spacetime acting as voids with decreasing matter density. The study extends our understanding of the cosmological scenarios where the virialization of the over-dense regions of dark matter is influenced by the non-minimally coupled dark energy.

gr-qc↗

arXiv2024

Learning Homogenization for Elliptic Operators

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context; in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.

math.NA↗