In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.
Download full-text PDF |
Source |
|---|---|
| http://dx.doi.org/10.1098/rsta.2011.0340 | DOI Listing |
Publication Analysis
Top Keywords
Similar Publications
Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks.
Neural Netw
November 2024
Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, Inria, BP 7023954506 Vandœuvre-lès-Nancy Cedex, France; Institut Universitaire de France (IUF), France. Electronic address:
In this work, we explore the numerical solution of geometric shape optimization problems using neural network-based approaches. This involves minimizing a numerical criterion that includes solving a partial differential equation with respect to a domain, often under geometric constraints like a constant volume. We successfully develop a proof of concept using a flexible and parallelizable methodology to tackle these problems.
View Article and Find Full Text PDFOptimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network.
Sci Rep
October 2024
School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics, 116025, Dalian, China.
The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied.
View Article and Find Full Text PDFAn Approximate Bayesian Computation Approach for Embryonic Astrocyte Migration Model Reduction.
Bull Math Biol
September 2024
Department of Mathematics, University of Florida, Gainesville, FL, USA.
During embryonic development of the retina of the eye, astrocytes, a type of glial cell, migrate over the retinal surface and form a dynamic mesh. This mesh then serves as scaffolding for blood vessels to form the retinal vasculature network that supplies oxygen and nutrients to the inner portion of the retina. Astrocyte spreading proceeds in a radially symmetric manner over the retinal surface.
View Article and Find Full Text PDFRegularity of the Optimal Sets for a Class of Integral Shape Functionals.
Arch Ration Mech Anal
May 2024
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy.
We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain is obtained as the integral of a cost function (, ) depending on the solution of a certain PDE problem on . The main feature of these functionals is that the minimality of a domain cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions , where is the solution of the PDE with Dirichlet boundary conditions.
View Article and Find Full Text PDFQuantitative Homogenization for the Obstacle Problem and Its Free Boundary.
Arch Ration Mech Anal
August 2024
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.
In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.
View Article and Find Full Text PDFWant AI Summaries of new PubMed Abstracts delivered to your In-box?
Enter search terms and have AI summaries delivered each week - change queries or unsubscribe any time!