Search

We use a space-time discretization based on physics informed deep learning (PIDL) to approximate solutions of a class of rate-dependent strain gradient plasticity models. The differential equation governing the plastic flow, the so-called microforce balance for this class of yield-free plasticity models, is very stiff, often leading to numerical corruption and a consequent lack of accuracy or convergence by finite element (FE) methods. Indeed, setting up the discretized framework, especially with an elaborate meshing around the propagating plastic bands whose locations are often unknown a-priori, also scales up the computational effort significantly. Taking inspiration from physics informed neural networks, we modify the loss function of a PIDL model in several novel ways to account for the balance laws, either through energetics or via the resulting PDEs once a variational scheme is applied, and the constitutive equations. The initial and the boundary conditions may either be imposed strictly by encoding them within the PIDL architecture, or enforced weakly as a part of the loss function. The flexibility in the implementation of a PIDL technique often makes for its ready interface with powerful optimization schemes, and this in turn provides for many possibilities in posing the problem. We have used freely available open-source libraries that perform fast, parallel computations on GPUs. Using numerical illustrations, we demonstrate how PIDL methods could address the computational challenges posed by strain gradient plasticity models. Also, PIDL methods offer abundant potentialities, vis-\'a-vis a somewhat straitjacketed and poorer approximant of FE methods, in customizing the formulation as per the problem objective.

Digital signatures are one of the simplest cryptographic building blocks that provide appealing security characteristics such as authenticity, unforgeability, and undeniability. In 1984, Shamir developed the first Identity-based signature (IBS) to simplify public key infrastructure and circumvent the need for certificates. It makes the process uncomplicated by enabling users to verify digital signatures using only the identifiers of signers, such as email, phone number, etc. Nearly all existing IBS protocols rely on several theoretical assumption-based hard problems. Unfortunately, these hard problems are unsafe and pose a hazard in the quantum realm. Thus, designing IBS algorithms that can withstand quantum attacks and ensure long-term security is an important direction for future research. Quantum cryptography (QC) is one such approach. In this paper, we propose an IBS based on QC. Our scheme's security is based on the laws of quantum mechanics. It thereby achieves long-term security and provides resistance against quantum attacks. We verify the proposed design's correctness and feasibility by simulating it in a prototype quantum device and the IBM Qiskit quantum simulator. The implementation code in qiskit with Jupyternotebook is provided in the Annexure. Moreover, we discuss the application of our design in secure email communication.

Stimulus-induced volumetric phase transition in gels may be potentially exploited for various bio-engineering and mechanical engineering applications. Since the discovery of the phenomenon in the 1970s, extensive experimental research has helped understand the phase transition and related critical phenomena. Yet, little insight is available on the evolving microstructure. In this article, we aim at unravelling certain geometric aspects of the micromechanics underlying discontinuous phase transition in polyacrylamide gels. Towards this, we use geometric thermodynamics and a Landau-Ginzburg type free energy functional involving a squared gradient, in conjunction with Flory-Huggins theory. We specifically exploit Ruppeiner's approach of Riemannian geometry-enriched thermodynamic fluctuation theory, which was previously employed to investigate phase transitions in van der Waals fluids and black holes. The framework equips us with a scalar curvature that is typically indicative of certain aspects of the microstructure during phase transition. Since previous studies have indicated that curvature divergence relates to correlation length divergence, we infer that the gel possesses a heterogeneous microstructure during phase transition, i.e. at critical points. Curvature also provides an insight into the universality class of phase transition and the nature of polymer-polymer interactions., Comment: 13 pages, 10 figures

Audit logs are one of the most important tools for transparently tracking system events and maintaining continuous oversight in corporate organizations and enterprise business systems. There are many cases where the audit logs contain sensitive data, or the audit logs are enormous. In these situations, dealing with a subset of the data is more practical than the entire data set. To provide a secure solution to handle these issues, a sanitizable signature scheme (SSS) is a viable cryptographic primitive. Herein, we first present the first post-quantum secure multivariate-based SSS, namely Mul-SAN. Our proposed design provides unforgeability, privacy, immutability, signer accountability, and sanitizer accountability under the assumption that the MQ problem is NP-hard. Mul-SAN is very efficient and only requires computing field multiplications and additions over a finite field for its implementation. Mul-SAN presents itself as a practical method to partially delegate control of the authenticated data in avenues like the healthcare industry and government organizations. We also explore using Blockchain to provide a tamper-proof and robust audit log mechanism.

Going beyond the classical Gaussian approximation of Einstein's fluctuation theory, Ruppeiner gave it a Riemannian geometric structure with an entropic metric. This yielded a fundamental quantity - the Riemannian curvature, which was used to extract information on the nature of interactions between molecules in fluids, ideal gases and other open systems. In this article, we examine the implications of this curvature in a non-equilibrium thermodynamic system where relaxation is sufficiently slow so as not to invalidate the local equilibrium hypothesis. The non-equilibrium system comprises of a rubbery polymer undergoing strain induced crystallization. The curvature is found to impart information on a spurious isochoric energy arising from the conformational stretching of already crystallized segments. This unphysical component perhaps arises as the crystallized manifold is considered Euclidean with the stretch measures defined via the Euclidean metric. The thermodynamic state associated with curvature is the key to determine the isochoric stretch and hence the spurious energy. We determine this stretch and propose a form for the spurious free energy that must be removed from the total energy in order that the correct stresses are recovered., Comment: 19 pages, 17 figures

A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with a $C^0$ mixed FE approximations. The core idea behind the approach is to introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role of section force). Well known frames like the Frenet-Serret are defined only using the centerline, which demands higher-order smoothness for the centerline approximation. Decoupling the frame from the position vector of the base curve leads to a description of torsion and curvature that is independent of the position information, thus allowing for simpler interpolations. This approach is then cast in the language of exterior calculus. In this framework, the strain energy may be interpreted as a differential form leading to the natural force-displacement (velocity) pairing. The four-field mixed variational principle we propose has frame, differential forms, and position vector as input arguments. While the position vector is interpolated linearly, the frames are interpolated as piecewise geodesics on the rotation group. Similarly, consistent interpolation schemes are also adopted to obtain finite dimensional approximations for other differential forms aswell. Using these discrete approximations, a discrete mixed variational principle is laid out which is then numerically extremized. The discrete approximation is then applied to benchmark problems, our numerical studies reveal an impressive performance of the proposed method without numerical instabilities or locking.

Markov Chain Monte Carlo (MCMC) is one of the most powerful methods to sample from a given probability distribution, of which the Metropolis Adjusted Langevin Algorithm (MALA) is a variant wherein the gradient of the distribution is used towards faster convergence. However, being set up in the Euclidean framework, MALA might perform poorly in higher dimensional problems or in those involving anisotropic densities as the underlying non-Euclidean aspects of the geometry of the sample space remain unaccounted for. We make use of concepts from differential geometry and stochastic calculus on Riemannian manifolds to geometrically adapt a stochastic differential equation with a non-trivial drift term. This adaptation is also referred to as a stochastic development. We apply this method specifically to the Langevin diffusion equation and arrive at a geometrically adapted Langevin dynamics. This new approach far outperforms MALA, certain manifold variants of MALA, and other approaches such as Hamiltonian Monte Carlo (HMC), its adaptive variant the no-U-turn sampler (NUTS) implemented in Stan, especially as the dimension of the problem increases where often GALA is actually the only successful method. This is evidenced through several numerical examples that include parameter estimation of a broad class of probability distributions and a logistic regression problem., Comment: 43 pages, 9 figures

This article discusses a mixed FE technique for 3D nonlinear elasticity using a Hu-Washizu (HW) type variational principle. Here, the deformed configuration and sections from its cotangent bundle are taken as additional input arguments. The critical points of the HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. The present FE approximation distinguishes a vector from a 1-from, a feature not commonly found in FE approximations. This point of view permits us to construct finite elements with vastly superior performance. Discrete approximations for the differential forms appearing in the variational principle are constructed with ideas borrowed from finite element exterior calculus. The discrete equations describing mechanical equilibrium, compatibility and constitutive rule, are obtained by extemizing the discrete functional with respect to appropriate DoF, which are then solved using the Newton's method. This mixed FE technique is then applied to benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding.

This article offers a new perspective for the mechanics of solids using moving Cartan's frame, specifically discussing a mixed variational principle in non-linear elasticity. We treat quantities defined on the co-tangent bundles of reference and deformed configurations as additional unknowns. Such a treatment invites compatibility of the fields with base-space (configurations of the body), so that the configuration can be realised as a subset of the Euclidean space. We first rewrite the metric and connection using differential forms, which are further utilised to write the deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We show that, for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of stress. Using these, we write a mixed energy functional in terms of differential forms, whose extremum leads to the compatibility of deformation, constitutive rules and equations of equilibrium. Finite element exterior calculus is then utilised to construct a finite dimensional approximation for the differential forms appearing in the variational principle. These approximations are then used to construct a discrete functional which is then numerically extremised. This discertization leads to a mixed method as it uses independent approximations for differential forms related to stress and deformation gradient. The mixed variational principle is then specialized for 2D case, whose discrete approximation is applied to problems in nonlinear elasticity. An important feature of our FE technique is the lack of additional stabilization. From the numerical study, it is found that the present discretization also does not suffer form locking and related convergence issues., Comment: arXiv admin note: text overlap with arXiv:2009.00275

We discuss the Hu-Washizu (HW) variational principle from a geometric standpoint. The mainstay of the present approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as primal. Such a treatment invites compatibility equations so that the base space (configurations of the solid body) could be realised as a subset of an Euclidean space. Cartan's method of moving frames and the associated structure equations establish this compatibility. Moreover, they permit us to write the metric and connection using 1-forms. With the mathematical machinery provided by differentiable manifolds, we rewrite the deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the HW functional in terms of frames and differential forms. Finally, we show that the compatibility of deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional when varied with respect to traction 1-forms, deformation 1-forms and the deformation. This new perspective that involves the notion of kinematic closure precisely explicates the necessary geometrical restrictions on the variational principle, without which the deformed body may not be realized as a subset of the Euclidean space. It also provides a pointer to how these restrictions could be adjusted within a non-Euclidean setting.

We propose a method for developing the flows of stochastic dynamical systems, posed as Ito's stochastic differential equations, on a Riemannian manifold identified through a suitably constructed metric. The framework used for the stochastic development, viz. an orthonormal frame bundle that relates a vector on the tangent space of the manifold to its counterpart in the Euclidean space of the same dimension, is the same as that used for developing a standard Brownian motion on the manifold. Mainly drawing upon some aspects of the energetics so as to constrain the flow according to any known or prescribed conditions, we show how to expediently arrive at a suitable metric, thus briefly demonstrating the application of the method to a broad range of problems of general scientific interest. These include simulations of Brownian dynamics trapped in a potential well, a numerical integration scheme that reproduces the linear increase in the mean energy of conservative dynamical systems under additive noise and non-convex optimization. The simplicity of the method and the sharp contrast in its performance vis-a-vis the correspondent Euclidean schemes in our numerical work provide a compelling evidence to its potential., Comment: 22 pages, 4 figures

A conformal gauge theory is used to describe and unify myriad electromechanical and magnetomechanical coupling effects observed in solid continua. Using a space-time pseudo-Riemannian metric in a finite-deformation setup and exploiting the local conformal symmetry of the Lagrangian, we derive Cauchy-Maxwell (CM) equations that seamlessly combine, for the first time, Cauchy's elasto-dynamic equations with Maxwell's equations for electromagnetism. Maxwell's equations for vacuum are recoverable from our model, which in itself also constitutes a new derivation of these equations. With deformation gradient and material velocity coupled in the Lagrange density, various pseudo-forces appear in the Euler-Lagrange equations. These forces, not identifiable through classical continuum mechanics, may have significance under specific geometric or loading conditions. As a limited illustration on how the CM equations work, we carry out semi-analytical studies, viz. on an infinite body subject to isochoric deformation and a finite membrane under both tensile and transverse loading, considering piezoelectricity and piezomagnetism. Our results show that under specific loading frequencies and tension, electric and magnetic potentials may increase rapidly in some regions of the membrane. This may have significance in future studies on efficient energy harvesting., Comment: 46 pages, 7 figures, 3 tables

We seek a rational route to large-deformation, thermo-mechanical modeling of solids with metrical defects. It assumes the reference and deformed geometries to be of the Weyl type and introduces the Weyl one-form -- an additional set of degrees of freedom that determine ratios of lengths in different tangent spaces. The Weyl one-form prevents the metric from being compatible with the connection and enables exploitation of the incompatibility for characterizing metrical defects in the body. When such a body undergoes temperature changes, additional incompatibilities appear and interact with the defects. This interaction is modeled using the Weyl transform, which keeps the Weyl connection invariant whilst changing the non-metricity of the configuration. An immediate consequence of the Weyl connection is that the critical points of the stored energy are shifted. We exploit this feature to represent the residual stresses. In order to relate stress and strain in our non-Euclidean setting, use is made of the Doyle-Ericksen formula, which is interpreted as a relation between the intrinsic geometry of the body and the stresses developed. Thus the Cauchy stress is conjugate to the Weyl transformed metric tensor of the deformed configuration. The evolution equation for the Weyl one-form is consistent with the two laws of thermodynamics. Our temperature evolution equation, which couples temperature, deformation and Weyl one-form, follows from the first law of thermodynamics. Using the model, the self-stress generated by a point defect is calculated and compared with the linear elastic solutions. We also obtain conditions on the defect distribution (Weyl one-form) that render a thermo-mechanical deformation stress-free. Using this condition, we compute specific stress-free deformation profiles for a class of prescribed temperature changes., Comment: 26 pages; 3 figures

A gauge theory of solids with conformal symmetry is formulated to model various electromechanical and magnetomechanical coupling phenomena. If the pulled back metric of the current configuration (the right Cauchy-Green tensor) is scaled with a constant, the volumetric part of the Lagrange density changes while the isochoric part remains invariant. However, upon a position dependent scaling, the isochoric part also loses invariance. In order to restore the invariance of the isochoric part, a 1-form compensating field is introduced and the notion of a gauge covariant derivative is utilized to minimally replace the Lagrangian. In view of obvious similarities with the Weyl geometry, the Weyl condition is imposed through the Lagrangian and a minimal coupling is employed so the 1-form could evolve. On deriving the Euler-Lagrange equations based on the action functional, we observe a close similarity with the governing equations for flexoelectricity under isochoric deformation if the exact part of 1-form is interpreted as the electric field and the anti-exact part as the polarization vector. Next, we model piezoelectricity and electrostriction phenomena by contracting the Weyl condition in various ways. Applying the Hodge decomposition theorem on the 1-form which leads to the curl of a pseudo-vector field and a vector field, we also model magnetomechanical phenomena. Identifying the pseudo-vector field with magnetic potential and the vector part with magnetization, flexomagnetism, piezomagnetism and magnetostriction phenomena under isochoric deformation are also modeled. Finally, we consider an analytical solution of the equations for piezoelectricity to provide an illustration on the insightful information that the present approach potentially provides., Comment: 46 pages, 1 figure