The PID controller is an elegant and versatile controller for set point tracking for a double integrator system, in particular, for engineering systems evolving on a Euclidean space. However, the ...configuration space of many mechanical systems, including interconnected ones, is a Lie group. Geometric PID control design has been proposed for mechanical systems evolving on Lie groups. In this work, we build upon this previously established framework for unconstrained mechanical systems to address systems with nonholonomic constraints. This extension encompasses many frequently encountered applications in robotics, where the constraints could either be holonomic or nonholonomic.
Understanding the switching mechanism of magnetic skyrmions is critical for realizing their potential applications in future spintronic devices. Here we study the thermodynamic stability and dynamics ...of a Néel skyrmion in an ultrathin magnetic nanodisk under biaxial in-plane strains using a combination of phase-field simulations and analytical theory. We demonstrated the switching of a circular skyrmion to a variety of magnetic configurations, including an out-of-plane monodomain or an in-plane vortex under isotropic strains and to an elliptical skyrmion or a stripe domain under anisotropic strains. We successfully formulated a Lagrangian-mechanics-based model to analytically describe the switching dynamics of a skyrmion. Both our simulations and analytical model revealed that the strain-mediated breathing dynamics of skyrmions lead to a counter-intuitive phenomenon in which a lager strain may lead to slower skyrmion-to-monodomain switching.
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•Darcy law was theoretically derived based on Lagrangian Mechanics.•Finger flow in homogenous media was characterized by the least energy loss (LEL) principle.•The LEL model concept was verified by ...the experiment in terms of finger flow area fraction.
Practical formulation for finger flow is lacking even though such a non-Darcy flow is commonly observed in homogeneous snow porous media. This study generalized physics for porous media flow based on the Lagrangian Mechanics; for instance, Darcy formula was theoretically derived minimizing the energy loss by solving the Euler-Lagrange Equation with Rayleigh dissipation function. This least energy loss (LEL) principle was then used to elucidate the finger flow formation in homogenous media. It was found that the inactive saturation (non-contributing void fraction) plays an important role in finger flow formation. This new theory also linked Darcy-Richards capillary flow and Newtonian viscous flow theories. Predicted flow area fractions by the LEL model were verified by the cold room laboratory experiment using well-controlled, homogeneous snow sample, as performed and documented by Katsushima et al. (2013).
This paper introduces a control-theoretic formulation of nonequilibrium thermodynamic systems with Gibbs states of Gaussian distributions, i.e., thermodynamic systems characterized by Gaussian ...probability densities. The distinct features of the paper are three-fold. First, the dynamics of a thermodynamic system is studied by transferring the original state variable from the non-Euclidean and nonlinear manifold state space to a Lie group, and further to the dual space of a Lie algebra, which is endowed with vector space structures. Consequently, the resulting reduced dynamics significantly reduces the high nonlinearity appearing in the original non-Euclidean state space of a thermal process. Second, the obtained equations of motion interestingly indicate that thermodynamics can be naturally viewed as a generalization of rigid body motions, and this bridges control theory, thermodynamics, information theory, and rigid body dynamics. Third, the optimality conditions for the energy-minimum optimal control problem of probability densities are derived via geometric Pontryagin’s principle by regarding the reduced dynamics as dynamical constraints, and a unified control algorithm is developed. Finally, the proposed approach is applied to three different scenarios including two benchmark examples to demonstrate the applicability and effectiveness. The purpose of the paper is to provide a deeper understanding of thermodynamics from a control perspective, and also to draw intrinsic connections between control theory, thermodynamics, information theory, and rigid body dynamics.
Eulerian-on-lagrangian cloth simulation Weidner, Nicholas J.; Piddington, Kyle; Levin, David I. W. ...
ACM transactions on graphics,
08/2018, Letnik:
37, Številka:
4
Journal Article
Recenzirano
We resolve the longstanding problem of simulating the contact-mediated interaction of cloth and sharp geometric features by introducing an Eulerian-on-Lagrangian (EOL) approach to cloth simulation. ...Unlike traditional Lagrangian approaches to cloth simulation, our EOL approach permits bending exactly at and sliding over sharp edges, avoiding parasitic locking caused by over-constraining contact constraints. Wherever the cloth is in contact with sharp features, we insert EOL vertices into the cloth, while the rest of the cloth is simulated in the standard Lagrangian fashion. Our algorithm manifests as new equations of motion for EOL vertices, a contact-conforming remesher, and a set of simple constraint assignment rules, all of which can be incorporated into existing state-of-the-art cloth simulators to enable smooth, inequality-constrained contact between cloth and objects in the world.
The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution ...from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.
•Construction of Lagrangian neural networks invariant under continuous symmetries.•Noether's Theorem guarantees exact conservation of corresponding physical quantities.•Numerically demonstrated conservation of angular momentum for three model systems.•Symmetry-enforcing layers improve stability under perturbations of initial condition.
Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing ...with time dependent problems, but essentially resting on conservation of energy and thus being not applicable to dissipative systems in a consistent way. On the other hand, there are formulations based essentially on maximizing the dissipation, working efficiently for the description of dissipative systems, but being not suitable for including inertia effects. Many attempts can be found in the literature to overcome this split into incompatible principles. However, essentially all of them possess an unnatural appearance. In this work, we suggest a solution to this dilemma resting on an additional assumption based on the thermodynamic driving forces involved. Applications to a simple dissipative structure and a material with varying mass demonstrate the capability of the proposed approach.
Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the ...warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code.
Resilient supply chains are often inherently dependent on the nature of their complex interconnected networks that are simultaneously multi-dimensional and multi-layered. This article presents a ...Supply Chain Network (SCN) model that can be used to regulate downstream relationships towards a sustainable SME using a 4-component cost function structure — Environmental (E), Demand (D), Economic (E), and Social (S). As a major generalization to the existing practice of using phenomenological interrelationships between the EDES cost kernels, we propose a complementary time varying model of a cost function, based on Lagrangian mechanics (incorporating SCN constraints through Lagrange multipliers), to analyze the time evolution of the SCN variables to interpret the competition between economic inertia and market potential. Multicriteria decision making, based on an Analytic Hierarchy Process (AHP), ranks performance quality, identifying key business decision makers. The model is first solved numerically and then validated against real data pertaining to two Small and Medium Enterprises (SMEs) from diverse domains, establishing the domain-independent nature of the model. The results quantify how increases in a production line without appropriate consideration of market volatility can lead to bankruptcy, and how high transportation cost together with increased production may lead to a break-even state. The model also predicts the time it takes a policy change to reinvigorate sales, thereby forecasting best practice operational procedure that ensures holistic sustainability on all four sustainability fronts.
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•Generalized cost function for a sustainable supply chain (cradle-to-utilization).•Lagrangian mechanics analyzing the time evolution of an Optimized Sustainable SCN (OSSCN).•Analytic Hierarchy Process ranking key econometric contributors in the OSSCN.•Case Studies of an anonymous Indian SME — two scenario model prediction & verification.•Model based policy prediction and monitoring towards a sustainable production line.
We present a three dimensional nonlinear string model based on a geometrically exact beam. The beam model is obtained by applying a variational principle using a covariant Lagrangian formulation; in ...particular, the equations of motion and the boundary conditions are treated in an unified manner. Following an analogous discrete variational principle, a Lie group variational integrator is given. The energy and momentum conservation properties of the integrator are discussed and illustrated. This geometrically exact beam serves as a basis to formulate a prestressed damped string model with coupled non trivial boundary conditions. Simulation results are discussed and validated against analytical solutions obtained in the context of a small displacement hypothesis.
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