相关论文: Construction of bipotentials and a minimax theorem…
This is a first paper in convex analysis dedicated to the bipotential theory, based on an extension of Fenchel's inequality. Introduced by the second author, bipotentials lead to a succesful new treatment of the constitutive laws of some…
Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. G\'ery de Saxc\'e's Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class…
We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We…
This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated…
In many practical situations, incertitudes affect the mechanical behaviour that is given by a family of graphs instead of a single one. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws,…
We present a physics-based neural network framework for the discovery of constitutive models in fully coupled thermomechanics. In contrast to classical formulations based on the Helmholtz energy, we adopt the internal energy and a…
Accurately modeling the mechanical behavior of materials is crucial for numerous engineering applications. The quality of these models depends directly on the accuracy of the constitutive law that defines the stress-strain relation.…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…
This paper explores a class of non-linear constitutive relations for materials with memory in the framework of covariant macroscopic Maxwell theory. Based on earlier models for the response of hysteretic ferromagnetic materials to…
We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a…
It exists a large class of systems for which the traditional notion of extensivity breaks down. From experimental examples we induce two general hypothesis concerning such systems. In the first the existence of an internal coordinate system…
In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For…
The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately.…
A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the…
We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform,…
Constitutive equations are developed for a polymer fluid, which is treated as a permanent network of strands bridged by junctions. The junctions are assumed to slide with respect to their reference positions under loading. Governing…
We continue the development of a method to accurately and efficiently identify the constitutive behavior of complex materials through full-field observations that we started in Akerson, Rajan and Bhattacharya (2024). We formulate the…
Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…
Accurate constitutive models of soft materials are crucial for understanding their mechanical behavior and ensuring reliable predictions in the design process. To this end, scientific machine learning research has produced flexible and…
In a previous paper, we proposed a symplectic version of Brezis-Ekeland-Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We applied it to the standard plasticity. The object of this work is…