research

Postgraduate (Master by Research / PhD) under Swinburne-National Road Safety Action Grants Program (NRSAGP) funded by Australia’s Department of Infrastructure, Transport, Regional Development, Communications and the Arts

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants

N. Sukumar              Amit Acharya

Many partial differential equations (PDEs) such as Navier–Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the L2 norm and H1 seminorm for the steady-state convection-diffusion problem.

Ponkrshnan Thiagarajan, Susanta Ghosh, "Jensen–Shannon divergence based novel loss functions for Bayesian neural networks", Neurocomputing, 2024, 129115, ISSN 0925-2312.  https://doi.org/10.1016/j.neucom.2024.129115  

Highlights

Equations not arising from a variational principle, as well as ones that arise from functionals that are not bounded above or below, can be endowed with one with favorable properties. The concept, and some examples are listed below:

The Fracture Mechanics and Structural Integrity Research Laboratory (NAMEF) of the Polytechnic School of Engineering at the University of São Paulo (EPUSP) in Brazil has an opening for a 2-year postdoctoral fellow (which may be extended to an additional year depending on funding availability) with a strong background in fracture mechanics and computational modeling of materials starting from February/2025.

Our latest paper, "Developing Mode I Cohesive Traction Laws for Crystalline UHMWPE Interphases Using Molecular Dynamics Simulations," is now freely accessible for the next 50 days from this link: https://authors.elsevier.com/a/1k9Pk3In-v14Go

I am pleased to share our latest open-access article, just published in Extreme Mechanics Letters.

A common, yet hazardous, method of transporting cylindrical tanks used to carry compressed gas involves rolling both tanks at opposite angles of inclination to the vertical. By propelling one of the tanks while maintaining point contact between the tanks, both tanks can be moved such that their centers of mass move in a straight line as demonstrated in the video below:

https://www.youtube.com/watch?v=Vgn5fv__LAk

In a paper that has just been published

Carbone and Pierro (Meccanica 48, 1819--1833 (2013)) discuss that punch shaped ends give less strong adhesion than mushrooms-ended ones because of the stress singularity at the corner. In particular, they assume large friction at the interface, which leads close to the classical Linear Elastic Fracture Mechanics singularity, corresponding to the classical scaling with the flat punch shaped pillar radius for what they call mode I of failure, ruled by the so-called Kendall formula. Their calculations are valid assuming perfect bonding for both punch or mushroom shaped ends.

dear Friends

I am happy to share our recent paper entitled ‘Nonlinear aerothermoelastic analysis of deployable control fin with actuator stiffness subjected to high-speed compressible flows’ published in the journal Acta Mechanica.

We are hiring 2 post-docs soon in Italy at the Department of Mechanics DMMM of the Politecnico di BARI.  The subject is here described.

Universal displacements are those displacements that can be maintained for any member of a specific class of linear elastic materials in the absence of body forces, solely by applying boundary tractions. For linear hyperelastic (Green elastic) solids, it is known that the space of universal displacements explicitly depends on the symmetry group of the material, and moreover, the larger the symmetry group the larger the set of universal displacements.

Gabriel Dante Lima-Chavez,        Amit Acharya,          Manas V. Upadhyay

A geometrically nonlinear theory for field dislocation thermomechanics based entirely on measurable state variables is proposed. Instead of starting from an ordering-dependent multiplicative decomposition of the total deformation gradient tensor, the additive decomposition of the velocity gradient into elastic, plastic and thermal distortion rates is obtained as a natural consequence of the conservation of the Burgers vector. Based on this equation, the theory consistently captures the contribution of transient heterogeneous temperature fields on the evolution of the (polar) dislocation density. The governing equations of the model are obtained from the conservation of Burgers vector, mass, linear and angular momenta, and the First Law. The Second Law is used to deduce the thermodynamical driving forces for dislocation velocity. An evolution equation for temperature is obtained from the First Law and the Helmholtz free energy density, which is taken as a function of the following measurable quantities: elastic distortion, temperature and the dislocation density (the theory allows prescribing additional measurable quantities as internal state variables if needed). Furthermore, the theory allows one to compute the Taylor-Quinney factor, which is material and strain rate dependent. Accounting for the polar dislocation density as a state variable in the Helmholtz free energy of the system allows for temperature solutions in the form of dispersive waves with finite propagation speed, despite using Fourier’s law of heat conduction as the constitutive assumption for the heat flux vector.

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan's problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains.

Pathrudkar, S., Thiagarajan, P., Agarwal, S. et al. Electronic structure prediction of multi-million atom systems through uncertainty quantification enabled transfer learning. npj Comput Mater 10, 175 (2024). https://doi.org/10.1038/s41524-024-01305-7 

Ponkrshnan Thiagarajan, Trisha Sain, Susanta Ghosh, "Bayesian calibration and uncertainty quantification of a rate-dependent cohesive zone model for polymer interfaces", Engineering Fracture Mechanics,

Volume 309, 2024, 110374, ISSN 0013-7944, https://doi.org/10.1016/j.engfracmech.2024.110374.

Advancements in smart soft materials are enhancing the capabilities of robotic manipulators in object interactions and complex tasks. Mechanochromic materials, acting as lightweight sensors, offer easily interpretable visual feedback for localized stress detection, structural health monitoring, and energy-efficient robotic skins.

Dear Colleagues,

Fluid Mechanics:

Variational dual solutions for incompressible fluids

Amit Acharya, Bianca Stroffolini, Arghir Zarnescu

Nonlinear Elasticity:

A hidden convexity of nonlinear elasticity

Siddharth Singh, Janusz Ginster, Amit Acharya