Mechanical metamaterials exhibit exotic properties that emerge from the interactions of many nearly rigid building blocks. Determining these theoretically has remained an open challenge outside a few select examples. Here, for large class periodic and planar kirigami, we provide coarse-graining rule linking design panels slits to kirigami's macroscale deformations. The procedure gives system nonlinear partial differential equations expressing geometric compatibility angle functions related...

Kirigami metamaterials dramatically change their shape through a coordinated motion of nearly rigid panels and flexible slits. Here, we study model system for mechanism-based planar kirigami featuring periodic patterns quadrilateral rhombi slits, with the goal predicting engineering scale response to broad range loads. We develop generalized continuum based on kirigami’s effective (cell-averaged) nonlinear deformation, along its slit actuation gradients thereof. The accounts three sources...

G.Parisi predicted an important variational formula for the thermodynamic limit of intensive free energy a class mean field spin glasses. In this paper, we present elementary approach to study Parisi functional using stochastic dynamic programing and semi-linear PDE. We give derivation properties PDE avoiding use Ruelle Probability Cascades Cole-Hopf transformations. As application, simple proof strict convexity functional, which was recently proved by Auffinger Chen in [2].

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|^{2}⟩≤Pe^{2} we construct steady two-dimensional flows that at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large limit. Combined with known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, conclude maximally transporting satisfy Nu∼Pe^{2/3} up to possible logarithmic corrections. bounds context Rayleigh-Bénard convection, this establishes while...

In this paper, we study the Crisanti-Sommers variational problem, which is a formula for free energy of spherical mixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spin glasses. We begin by computing dual problem using min-max argument. find that 1D obstacle type, where related...

Motivated by the search for sharp bounds on turbulent heat transfer as well design of optimal exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, exits through boundary at fixed temperature. We seek advecting flow to optimize this exchange. Previous work energy-constrained cooling with constant source has conjectured global optimizers should resemble convection rolls;...

We consider the problem of optimizing heat transport through an incompressible fluid layer. Modeling passive scalar by advection-diffusion, we maximize mean rate total a divergence-free velocity field. Subject to various boundary conditions and intensity constraints, prove that maximal scales linearly in r.m.s. kinetic energy and, up possible logarithmic corrections, as $1/3$rd power enstrophy advective regime. This makes rigorous previous prediction on near optimality convection rolls for...

Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction symmetry axis. Assuming radial symmetry, modeling suitable Dirichlet boundary conditions at center sheet, identify energy scaling law von-Kármán model. Specifically, find that three different regimes arise with increasing indentation $δ$: initially energetic cost logarithmic singularity dominates, then there...

Periodic origami patterns made with repeating unit cells of creases and panels bend twist in complex ways. In principle, such soft modes deformation admit a simplified asymptotic description the limit large number cells. Starting from bar hinge model for elastic energy generic four parallelogram panel pattern, we derive complete set geometric compatibility conditions identifying pattern’s this limit. The equations form system partial differential constraining actuation origami’s (a scalar...

We consider the axial compression of a thin elastic cylinder placed about hard cylindrical core. Treating core as an obstacle, we prove upper and lower bounds on minimum energy that depend its relative thickness magnitude compression. focus exclusively setting where radius is greater than or equal to natural cylinder. two cases: “large mandrel” case, exceeds cylinder, “neutral radii are same. In large mandrel our match in their scaling with respect thickness, compression, pre‐strain induced...

We analyze two recently proposed methods to establish a priori lower bounds on the minimum of general integral variational problems. The methods, which involve either 'occupation measures' [Korda et al., ch. 10 Numerical Control: Part A, 10.1016/bs.hna.2021.12.010] or 'pointwise dual relaxation' procedure [Chernyavsky arXiv:2110.03079], are shown produce same bound under coercivity hypothesis ensuring their strong duality. then show by minimax argument that actually evaluate for classes...