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
Considerable research has been conducted on shape of pillar ends for optimal adhesion. In experiments with elastomers it has been found that mushrooms-ended ones are superior to flat-ended ones, but early experiments have suggested an extremely strong scaling in strength with pillar radius (del Campo et al Langmuir;23 :10235-43, 2007). We discuss various theories and experimental results on scaling of strength, and in particular we elaborate recent experiments on single pillars with mushroom ends finding that the scaling on strength is much less surprising.
