iMechanica

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.

Job Description:

There is some rumour on the Nobel prize for physics by Hopfield and AI.  I found this post by the well known German AI scientist Juergen Schmidhuber. 

 However, the publication output of this Jurgen is also strange... not many top journals....

https://scholar.google.com/citations?user=gLnCTgIAAAAJ&hl=en&oi=ao

Hello iMechanica Community, We are organizing a mini symposium (MS) titled "226 - Machine Learning-Based Modeling, Prediction, and Optimization in Advanced Manufacturing and Multiphysical Properties of Materials" at the 18th U.S. National Congress on Computational Mechanics (USNCCM18), to be held in Chicago, Illinois, from July 20-24, 2025.

https://apply.interfolio.com/158038

There will be a mini-symposium titled “Mechanics and Physics of Granular Materials – 25th Anniversary of the Technical Committee" as part of Engineering Mechanics Institute Conference (EMI 2025) in Anaheim, California (May 27-30, 2025). This mini-symposium focuses on the mechanics of granular systems over a broad range of scales and phase regimes.
 
The Abstract submission deadline is December 20, 2024. To submit an abstract, please visit:
https://lnkd.in/gURTW8at
 

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

We have two new PhD positions to work on Mechanics and Manufacturing/Characterization of nanomaterials and nanocomposites with starting dates of Spring 2025 or Fall 2025. A stipend, full tuition and health insurance will be provided. If interested, please email your CV/resume and all your college/postgraduate transcrips to cke AT binghamton.edu.

This is a reminder that our next monthly webinar is December 5, 3-4pm EST. The speaker will be Associate Professor Chao Hu from University of Connecticut. We are hoping the format will promote a lively interactive discussion and engage both junior and senior members of our community. Look forward to seeing you there.

Monthly Webinar by USACM, TTA-Uncertainty Quantification and Probabilistic Modeling

December 5; 3pm EST

Speaker: Associate Professor Chao Hu, University of Connecticut

I am happy to announce that a postdoctoral fellowship is available in the Dolbow Research Group at Duke University, working in the area of computational fracture mechanics.  The fellowship provides the opportunity to work on an emerging class of complete fracture models that incorporate the three ingredients that are necessary to be predictive with elastic brittle materials: their elasticity, their fracture toughness, and their strength.  

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:

Author: Mahendra Gattu, Department of Civil Engineering, NIT-Rourkela-769008.

Abstract

The problem of fracture in quasi-brittle materials is strongly linked with a zone of micro-cracking known as the fracture process zone (FPZ). For monotonic loading, the material length scale parameter D0 is used in strength scaling law to describe the transition from strength criteria to linear elastic fracture mechanics criteria.

The Department of Engineering Science at the University of Oxford intends to appoint an Associate Professor of Engineering Science (Solid Mechanics) with effect from 1 October 2025 (or as soon as possible thereafter). The successful candidate will work at the Department of Engineering Science (Central Oxford) and will be offered a Tutorial Fellowship at Pembroke College. The appointment will be initially for five years at which point, upon completion of a successful review, the post-holder will be eligible for reappointment to the retiring age.

Monthly Webinar: USACM, TTA-Nanotechnology and Lower Scale Phenomena

This is a reminder that our next monthly webinar is November 27, 2-3pm CST. The speaker will be Professor Gábor Csányi from the University of Cambridge. We are hoping the format will promote a lively interactive discussion and engage both junior and senior members of our community. Look forward to seeing you there. Talk and discussions will be archived and available on our website if accepted by the speaker.

November 27; 2pm CST

Speaker: Prof. Gábor Csányi, University of Cambridge

The Mechanics of Soft Materials Lab (https://www.msm.seas.ucla.edu/) led by Prof Lihua Jin in the Mechanical and Aerospace Engineering at University of California, Los Angeles (UCLA) has a PhD opening to be filled in 2025 fall. Successful candidates are expected to work on modeling, simulations, and experiments of soft materials, soft tissue and soft robots. The ideal candidates will be highly motivated with a strong background in solid mechanics and computation, and be fluent in spoken and written English.

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

Dielectric rods have been employed in various electromechanical applications, including energy harvesters and sensors. This paper develops a general framework to model large deformations in dielectric rods, considering both direct and converse flexoelectric effects. Initially, we derive the governing differential equations for a three-dimensional dielectric continuum solid to model large deformations, incorporating converse flexoelectricity. Then, we derive the equilibrium equations for the flexoelectric strain-gradient special Cosserat rod.

In unidirectional sliding of rubber contacts on smooth hard surface it has been found that contact shrinks largely in longitudinal directions, and generally much less in the transverse direction, and two explanations have been suggested to explain this: one is the effect of mixed mode fracture mechanics in the presence of adhesion (with mode II reducing adhesion and mode III less clear), and another uniquely based on finite strain effects even for a simple material model as neo-Hookean hyperelastic material.

A fully supported Ph.D. position is open immediately in the Advanced Manufacturing Research Group within the Department of Aerospace Engineering at Embry-Riddle Aeronautical University (ERAU), starting in Fall 2025. Our interdisciplinary research focuses on advancing additive manufacturing technologies for high-performance composite materials, with applications spanning aerospace, automotive, electronics, and biomedical industries.

Two PhD/PostDoc positions open for Fall 2025 at the University of Wisconsin-Madison on: 

(1) Multiphysics modeling of biological systems and/or neuronal mechanics (NSF and ONR funded)

(2) Microstructure modeling in metal Additive Manufacturing (ONR funded).

Both positions focus on extensive computational mechanics and multiphysics modeling, and involve modeling phenomena across length scales. Prospective students with interest in numerical modeling, solid mechanics, multiphysics and/or biomechanics may email CV's to shiva dot rudraraju at wisc dot edu