arXiv subjects

L. Mahadevan

Publications and source records attributed to L. Mahadevan.

At least 55 records · Page 3Linked to original sources

arXiv2024

Phase transitions in rolling of irregular cylinders and spheres

When placed on an inclined plane, a perfect 2D disk or 3D sphere simply rolls down in a straight line under gravity. But how is the rolling affected if these shapes are irregular or random? Treating the terminal rolling speed as an order parameter, we show that phase transitions arise as a function of the dimension of the state space and inertia. We calculate the scaling exponents and the macroscopic lag time associated with the presence of first and second order transitions, and describe the regimes of co-existence of stable states and the accompanying hysteresis. Experiments with rolling cylinders corroborate our theoretical results on the scaling of the lag time. Experiments with spheres reveal closed orbits and their period-doubling in the overdamped and inertial limits respectively, providing visible manifestations of the hairy ball theorem and the doubly-connected nature of SO(3), the space of 3-dimensional rotations. Going beyond simple curiosity, our study might be relevant in a number of natural and artificial systems that involve the rolling of irregular objects, in systems ranging from nanoscale cellular transport to robotics.

physics.class-ph↗

arXiv2024

Data-driven quasiconformal morphodynamic flows

Temporal imaging of biological epithelial structures yields shape data at discrete time points, leading to a natural question: how can we reconstruct the most likely path of growth patterns consistent with these discrete observations? We present a physically plausible framework to solve this inverse problem by creating a framework that generalises quasiconformal maps to quasiconformal flows. By allowing for the spatio-temporal variation of the shear and dilatation fields during the growth process, subject to regulatory mechanisms, we are led to a type of generalised Ricci flow. When guided by observational data associated with surface shape as a function of time, this leads to a constrained optimization problem. Deploying our data-driven algorithmic approach to the shape of insect wings, leaves and even sculpted faces, we show how optimal quasiconformal flows allow us to characterise the morphogenesis of a range of surfaces.

cs.CG↗

arXiv2024

Approximate Lie symmetries and singular perturbation theory

Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact that all exact solutions of differential equations are consequences of (Lie) symmetries, we reformulate perturbation theory for differential equations in terms of expansions of the Lie symmetries of the solutions. This is a change in perspective from the usual method of obtaining series expansions of the solutions themselves. We show that these approximate symmetries are straightforward to calculate and are never singular; their integration is therefore an easier way of constructing uniformly convergent solutions. This geometric viewpoint naturally subsumes the RG-inspired approach of Chen, Goldenfeld and Oono, the method of multiple scales, and the Poincare-Lindstedt method, by exploiting a fundamental class of symmetries that we term ``hidden scale symmetries''. It also clarifies when and why these singular perturbation methods succeed and just as importantly, when they fail. More broadly, direct, algorithmic identification and integration of these hidden scale symmetries permits solution of problems where other methods are impractical.

math-ph↗

arXiv2023

Optimal strategies for kiiking: active pumping to invert a swing

Kiiking is an extreme sport in which athletes alternate between standing and squatting to pump a standing swing till it is inverted and completes a rotation. A minimal model of the sport may be cast in terms of the control of an actively driven pendulum of varying length to determine optimal strategies. We show that an optimal control perspective, subject to known biological constraints, yields time-optimal control strategy similar to a greedy algorithm that aims to maximize the potential energy gain at the end of every cycle. A reinforcement learning algorithms with a simple reward is consistent with the optimal control strategy. When accounting for air drag, our theoretical framework is quantitatively consistent with experimental observations while pointing to the ultimate limits of kiiking performance.

physics.pop-ph↗

arXiv2023

An additive framework for kirigami design

We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.

cond-mat.soft↗

arXiv2023

Zonal flows and reversals of cortically confined active suspensions

At sufficiently high concentrations, motile bacteria suspended in fluids exhibit a range of ordered and disordered collective motions. Here we explore the combined effects of confinement, periodicity and curvature induced by the active motion of E. coli bacteria in a thin spherical shell (cortex) of an oil-water-oil (O/B/O) double emulsion drop. Confocal microscopy of the bacterial flow fields shows that at high density and activity, they exhibit azimuthal zonal flows which oscillate between counterclockwise and clockwise circulating states. We characterize these oscillatory patterns via their Fourier spectra and the distributions of their circulation persistence times. To explain our observations, we used numerical simulations of active particles and characterize the two-dimensional phase space of bacterial packing fraction and activity associated with persistent collective motions. All together, our study shows how geometric effects lead to new types of collective dynamics.

cond-mat.soft↗

arXiv2023

Non-planar snake gaits: from Stigmatic-starts to Sidewinding

Of the vast variety of animal gaits, one of the most striking is the non-planar undulating motion of a sidewinder. But non-planar gaits are not limited to sidewinders. Here we report a new non-planar mode used as an escape strategy in juvenile anacondas (Eunectes notaeus). In the S-start, named for its eponymous shape, transient locomotion arises when the snake writhes and bends out of the plane while rolling forward about its midsection without slippage. To quantify our observations, we present a mathematical model for an active non-planar filament that interacts anisotropically with a frictional substrate and show that locomotion is due to a propagating localized pulse of a topological quantity, the link density. A two-dimensional phase space characterized by scaled body weight and muscular torque shows that relatively light juveniles are capable of S-starts but heavy adults are not, consistent with our experiments. Finally, we show that a periodic sequence of S-starts naturally leads to a sidewinding gait. All together, our characterization of a novel escape strategy in snakes highlights the role of topology in locomotion, provides a phase diagram for mode feasibility as a function of body size, and suggests a role for the S-start in the evolution of sidewinding.

q-bio.QM↗

arXiv2023

Statics and diffusive dynamics of surfaces driven by $p$-atic topological defects

Inspired by epithelial morphogenesis, we consider a minimal model for the shaping of a surface driven by $p$-atic topological defects. We show that a positive (negative) defect can dynamically generate a (hyperbolic) cone whose shape evolves diffusively, and predict that a defect of charge $+1/p$ leads to a final semi-cone angle $β$ which satisfies the inequality $\sinβ\ge 1 - \frac{1}{p} + \frac{1}{2p^2}$. By exploiting the fact that for axisymmetric surfaces, the extrinsic geometry is tightly coupled to the intrinsic geometry, we further show that the resulting stationary shape of a membrane with negligible bending modulus and embedded polar order is a deformed lemon with two defects at antipodal points. Finally, we close by pointing out that our results may be relevant beyond epithelial morphogenesis in such contexts as shape transitions in macroscopic closed spheroidal surfaces such as pollen grains.

cond-mat.soft↗

arXiv2022

Localization in musical steelpans

The steelpan is a pitched percussion instrument that takes the form of a concave bowl with several localized dimpled regions of varying curvature. Each of these localized zones, called notes, can vibrate independently when struck, and produces a sustained tone of a well-defined pitch. While the association of the localized zones with individual notes has long been known and exploited, the relationship between the shell geometry and the strength of the mode confinement remains unclear. Here, we explore the spectral properties of the steelpan modeled as a vibrating elastic shell. To characterize the resulting eigenvalue problem, we generalize a recently developed theory of localization landscapes for scalar elliptic operators to the vector-valued case, and predict the location of confined eigenmodes by solving a Poisson problem. A finite element discretization of the shell shows that the localization strength is determined by the difference in curvature between the note and the surrounding bowl. In addition to providing an explanation for how a steelpan operates as a two-dimensional xylophone, our study provides a geometric principle for designing localized modes in elastic shells.

cond-mat.soft↗

arXiv2022

Explosive rigidity percolation in kirigami

Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of $k$ choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design and control of mechanical metamaterials.

cond-mat.soft↗

arXiv2022

Optimal intercellular competition in senescence and cancer

Effective multicellularity requires both cooperation and competition between constituent cells. Cooperation involves sacrificing individual fitness in favor of that of the community, but excessive cooperation makes the community susceptible to senescence and aging. Competition eliminates unfit senescent cells via natural selection and thus slows down aging, but excessive competition makes the community susceptible to cheaters, as exemplified by cancer and cancer-like phenomena. These observations suggest that an optimal level of intercellular competition in a multicellular organism maximizes organismal vitality by delaying the inevitability of aging. We quantify this idea using a statistical mechanical framework that leads to a generalized replicator dynamical system for the population of cells that change their vitality and cooperation due to somatic mutations that make them susceptible to aging and/or cancer. By accounting for the cost of cooperation and strength of competition in a minimal setting, we show that our model predicts an optimal value of competition that maximizes vitality and delays the inevitability of senescence or cancer.

q-bio.PE↗

arXiv2022

Quasicrystal kirigami

Kirigami, the art of introducing cuts in thin sheets to enable articulation and deployment, has become an inspiration for a novel class of mechanical metamaterials with unusual properties. Here we complement the use of periodic tiling patterns for kirigami designs by showing that quasicrystals can also serve as the basis for designing deployable kirigami structures, and analyze the geometrical, topological and mechanical properties of these aperiodic kirigami structures.

cond-mat.soft↗

arXiv2022

How to grow a flat leaf

Growing a flat lamina such as a leaf is almost impossible without some feedback to stabilize long wavelength modes that are easy to trigger since they are energetically cheap. Here we combine the physics of thin elastic plates with feedback control theory to explore how a leaf can remain flat while growing. We investigate both in-plane (metric) and out-of-plane (curvature) growth variation and account for both local and nonlocal feedback laws. We show that a linearized feedback theory that accounts for both spatially nonlocal and temporally delayed effects suffices to suppress long wavelength fluctuations effectively and explains recently observed statistical features of growth in tobacco leaves. Our work provides a framework for understanding the regulation of the shape of leaves and other laminar objects.

cond-mat.soft↗

arXiv2022

Active entanglement enables stochastic, topological grasping

Grasping, in both biological and engineered mechanisms, can be highly sensitive to the gripper and object morphology, as well as perception, and motion planning. Here we circumvent the need for feedback or precise planning by using an array of fluidically-actuated slender hollow elastomeric filaments to actively entangle with objects that vary in geometric and topological complexity. The resulting stochastic interactions enable a unique soft and conformable grasping strategy across a range of target objects that vary in size, weight, and shape. We experimentally evaluate the grasping performance of our strategy, and use a computational framework for the collective mechanics of flexible filaments in contact with complex objects to explain our findings. Overall, our study highlights how active collective entanglement of a filament array via an uncontrolled, spatially distributed scheme provides new options for soft, adaptable grasping.

cond-mat.soft↗

arXiv2022

Modular representation and control of floppy networks

Geometric graph models of systems as diverse as proteins, robots, and mechanical structures from DNA assemblies to architected materials point towards a unified way to represent and control them in space and time. While much work has been done in the context of characterizing the behavior of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, under-constrained networks that are far more common in nature and technology. Here we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero-energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network, and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.

cs.RO↗

arXiv2021

Statistics and Topology of Fluctuating Ribbons

Ribbons are a class of slender structures whose length, width, and thickness are widely separated from each other. This scale separation gives a ribbon unusual mechanical properties in athermal macroscopic settings, e.g. it can bend without twisting, but cannot twist without bending. Given the ubiquity of ribbon-like biopolymers in biology and chemistry, here we study the statistical mechanics of microscopic inextensible, fluctuating ribbons loaded by forces and torques. We show that these ribbons exhibit a range of topologically and geometrically complex morphologies exemplified by three phases - a twist-dominated helical phase (HT), a writhe-dominated helical phase (HW), and an entangled phase - that arise as the applied torque and force is varied. Furthermore, the transition from HW to HT phases is characterized by the spontaneous breaking of parity symmetry and the disappearance of perversions that characterize chirality reversals. This leads to a universal response curve of a topological quantity, the link, as a function of the applied torque that is similar to magnetization curves in second-order phase transitions.

cond-mat.stat-mech↗

arXiv2021

Optimal transport and control of active drops

Understanding the complex patterns in space-time exhibited by active systems has been the subject of much interest in recent times. Complementing this forward problem is the inverse problem of controlling active matter. Here we use optimal control theory to pose the problem of transporting a slender drop of an active fluid and determine the dynamical profile of the active stresses to move it with minimal viscous dissipation. By parametrizing the position and size of the drop using a low-order description based on lubrication theory, we uncover a natural ''gather-move-spread'' strategy that leads to an optimal bound on the maximum achievable displacement of the drop relative to its size. In the continuum setting, the competition between passive surface tension, and active controls generates richer behaviour with futile oscillations and complex drop morphologies that trade internal dissipation against the transport cost to select optimal strategies. Our work combines active hydrodynamics and optimal control in a tractable and interpretable framework, and begins to pave the way for the spatiotemporal manipulation of active matter.

cond-mat.soft↗

arXiv2021

Prestrain-induced contraction in 1D random elastic chains

Prestrained elastic networks arise in a number of biological and technological systems ranging from the cytoskeleton of cells to tensegrity structures. To understand the response of such a network as a function of the prestrain, we consider a minimal model in one dimension. We do this by considering a chain (1D network) of elastic springs upon which a random, zero mean, finite variance prestrain is imposed. Numerical simulations and analytical predictions quantify the magnitude of the contraction as a function of the variance of the prestrain, and show that the chain always shrinks. To test these predictions, we vary the topology of the chain and consider more complex connectivity and show that our results are relatively robust to these changes.

cond-mat.stat-mech↗