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Analysis of involute-curve geometry using position gradients: Application to gear-tooth profile

MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 2024年第12期52卷 10723-10745页

作者： Shabana, Ahmed A. Univ IL Chicago Dept Mech & Ind Engn Chicago IL USA

This paper examines using non-rational polynomials to approximate the involute-curve geometry, which is fundamental in the design of gear systems. The parametric form of the involute curve in terms of an involute-angle parameter is first presented. To be able to compare the analytical geometry results and numerical results obtained using the finite-element (FE) method for describing the geometry, the relationship between the involute-angle parameter and the FE spatial coordinate is defined. A new relationship between the involute-curve arc length and the coordinate used in the FE parameterization is also developed. To this end, the involute-curve equation is used to determine the involute-angle parameter at the intersection with circles with arbitrary radii including the gear pitch, addendum, and base circles. The parameter relationship allows defining the nodal position-gradient vectors defined by differentiation with respect to the FE spatial coordinate from involute-curve tangent vector. Two procedures for determining the left involute curve from the right involute curve of the gear tooth are described. In the first procedure, a coordinate transformation based on Rodrigues transformation with axis of rotation defined by the tooth-center line is used;while in the second procedure, the analytical expression of the left involute curve is defined using the gear-tooth thickness at the base circle. A general procedure for determining the position gradients for a given geometry obtained from analytical curves or using imaging techniques is introduced. Such a procedure can be used in case of worn gear teeth that cannot be described by simple functions. A planar rectangular element based on the absolute nodal coordinate formulation (ANCF) is used to approximate the gear-tooth geometry. While the involute geometry cannot be described exactly using non-rational polynomials, the analysis and results of this investigation demonstrate that the FE solution leads to accurate geo

关键词： Gear geometry curve parameterization involute curve position gradients finite-element approximation absolute nodal coordinate formulation

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Analysis of a fully discrete approximation for the classical Keller-Segel model: Lower and a priori bounds

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2021年 85卷 69-81页

作者： Vicente Gutierrez-Santacreu, Juan Rafael Rodriguez-Galvan, Jose Univ Seville Dept Matemat Aplicada ETSI Informat Avda Reina Mercedes S-N E-41012 Seville Spain Univ Cadiz Dept Matemat Fac Ciencias Campus Univ Puerto Real E-11510 Cadiz Spain

This paper is devoted to constructing approximate solutions for the classical Keller-Segel model governing chemotaxis. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds - positivity for the cell density and nonnegativity for the chemoattractant density-, are bounded in the L-1(Omega)-norm, satisfy a discrete energy law, and have a priori energy estimates. The latter is achieved by means of a discrete Moser-Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.

关键词： Keller-Segel equations Non-linear parabolic equations finite-element approximation Lower bounds A priori bounds

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Cooperative Field Prediction and Smoothing via Covariance Intersection

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 797-808页

作者： Li, Zhuo You, Keyou Song, Shiji Tsinghua Univ Dept Automat Beijing 100084 Peoples R China Tsinghua Univ BNRist Beijing 100084 Peoples R China

This work studies the field prediction and smoothing problems, where the spatio-temporal field in 2-D is described by a stochastic dynamical system and observed by a number of spatially deployed sensors. We adopt a finite-element technique to approximate the field dynamics with piece-wise Gaussian functions, leading to a high-dimensional linear stochastic system. By exploiting its sparsity, a local covariance intersection-based filter and smoother are developed in each sensor only for a moderate number of state variables via communications with nearby sensors. Such a cooperative scheme is both communication and computation efficient. We prove the uniform stability of the local filter and smoother under mild conditions, and validate their effectiveness on two application examples: the temperature prediction of a metal rod and the source localization of a PM2.5 field with a real dataset in a city of China.

关键词： Sensors Smoothing methods Temperature sensors Task analysis Stochastic systems Metals Sensor systems Field prediction and smoothing finite-element approximation covariance intersection cooperative estimation

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Numerical solution for an aggregation equation with degenerate diffusion

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APPLIED MATHEMATICS AND COMPUTATION 2020年 377卷 125145-000页

作者： Carlos Cabrales, Roberto Gutierrez-Santacreu, Juan Vicente Rafael Rodriguez-Galvan, Jose Univ La Serena Inst Invest Multidisciplinaria Ciencia & Tecnol La Serena Chile Univ Seville Dept Matemat Aplicada I ETSI Informat Avda Reina Mercedes S-N E-41012 Seville Spain Univ Cadiz Dept Matemat Fac Ciencias Campus Univ Puerto Real E-11510 Cadiz Spain

A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a finite element method together with a mass lumping technique and an extra stabilizing term plus a semi-implicit Euler time integration. Then we carry out a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that the sequence of finite element approximations converges toward the unique weak solution of the model at hands. In doing so, nonnegativity is attained due to the stabilizing term and the acuteness on partitions of the computational domain, and hence a priori energy estimates of finite element approximations are established. As we deal with a nonlinear problem, some form of strong convergence is required. The key compactness result is obtained via an adaptation of a Riesz-Frechet-Kolmogorov criterion by perturbation. A numerical example is also presented. (C) 2020 Elsevier Inc. All rights reserved.

关键词： finite-element approximation Aggregation equation Nonlinear diffusion

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Design of interval observers and controls for PDEs using finite-element approximations

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AUTOMATICA 2018年 93卷 302-310页

作者： Kharkovskaya, Tatiana Efimov, Denis Polyakov, Andrey Richard, Jean-Pierre Ecole Cent Lille CRIStAL UMR CNRS 9189 BP 48 F-59651 Villeneuve Dascq France ITMO Univ Dept Control Syst & Informat 49 Kronverkskiy Av St Petersburg 197101 Russia Non A Team Inria Parc Sci Haute Borne40 Av Halley F-59650 Villeneuve Dascq France

Synthesis of interval state estimators is investigated for the systems described by a class of parabolic Partial Differential Equations (PDEs). First, a finite-element approximation of a PDE is constructed and the design of an interval observer for the derived ordinary differential equation is given. Second, the interval inclusion of the state function of the PDE is calculated using the error estimates of the finite-element approximation. Finally, the obtained interval estimates are used to design a dynamic output stabilizing control. The results are illustrated by numerical experiments with an academic example and the Black-Scholes model of financial market. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Interval observers PDE finite-element approximation

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Convergence of an implicit Euler Galerkin scheme for Poisson-Maxwell-Stefan systems

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2019年第3期45卷 1469-1498页

作者： Juengel, Ansgar Leingang, Oliver Vienna Univ Technol Inst Anal & Sci Comp Wiedner Hauptstr 8-10 A-1040 Vienna Austria

A fully discrete Galerkin scheme for a thermodynamically consistent transient Maxwell-Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the diffusive dynamics of an isothermal ionized fluid mixture with vanishing barycentric velocity. The equations are studied in a bounded domain, and different molar masses are allowed. The Galerkin scheme preserves the total mass, the nonnegativity of the particle densities, their boundedness and satisfies the second law of thermodynamics in the sense that the discrete entropy production is nonnegative. The existence of solutions to the Galerkin scheme and the convergence of a subsequence to a solution to the continuous system is proved. Compared to previous works, the novelty consists in the treatment of the drift terms involving the electric field. Numerical experiments show the sensitive dependence of the particle densities and the equilibration rate on the molar masses.

关键词： Maxwell-Stefan systems Cross diffusion Ionized fluid mixtures Entropy method finite-element approximation Galerkin method Numerical convergence

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Interval observers for PDEs: approximation approach

Interval observers for PDEs: approximation approach

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10th IFAC Symposium on Nonlinear Control Systems (NOLCOS)

作者： Kharkovskaia, Tatiana Efimov, Denis Polyakov, Andrey Richard, Jean-Pierre Ecole Cent Lille CRIStAL UMR CNRS 9189 BP 48 F-59651 Villeneuve Dascq France ITMO Univ Dept Control Syst & Informat 49 Kronverkskiy Av St Petersburg 197101 Russia Inria Non A Team Parc Sci Haute Borne40 Av F-59650 Villeneuve Dascq France

The problem of interval state estimation is studied for systems described by parabolic Partial Differential Equations (PDEs). The proposed solution is based on a finite-element approximation of PDE, with posterior design of an interval observer for the obtained ordinary differential equation. The interval inclusion of the state function of PDE is obtained using the estimates on the error of discretization. The results are illustrated by numerical experiments with an academic example. (C) 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Interval observers PDE finite-element approximation

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Rate-independent elastoplasticity at finite strains and its numerical approximation

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MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 2016年第12期26卷 2203-2236页

作者： Mielke, Alexander Roubicek, Tomas Weierstr Inst Angew Anal & Stochast Mohrenstr 39 D-10117 Berlin Germany Humboldt Univ Inst Math Rudower Chaussee 25 D-12489 Berlin Germany Charles Univ Prague Math Inst Sokolovska 83 CZ-18675 Prague 8 Czech Republic Czech Acad Sci Inst Thermomech Dolejskova 5 CZ-18200 Prague 8 Czech Republic

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

关键词： Plasticity quasistatic evolution energetic solutions dissipation distance hardening polyconvexity Ciarlet-Necas condition Signorini contact finite-element approximation Gamma-convergence Lavrentiev phenomenon second-grade nonsimple materials

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Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2012年第18期89卷 2460-2478页

作者： Cox, Sonja Hausenblas, Erika Delft Univ Technol Delft Inst Appl Math NL-2600 GA Delft Netherlands Univ Leoben Dept Math & Informat Technol A-8700 Leoben Austria

We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is almost sure uniform convergence, i.e. pathwise convergence. The proofs are based on a recent perturbation result for such equations.

关键词： stochastic differential equations stochastic partial differential equations perturbation spectral Galerkin method finite-element approximation

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Energy transport in semiconductor devices

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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 2010年第1期16卷 1-22页

作者： Juengel, Ansgar Vienna Univ Technol Inst Anal & Sci Comp A-1040 Vienna Austria

The modelling, analysis and numerical approximation of energy-transport models for semiconductor devices are reviewed. The derivation of the partial differential equations from the semiconductor Boltzmann equation is sketched. Furthermore, the main ideas for the analytical treatment of the equations, employing thermodynamic principles, are given. A new result is the proof of the weak sequential stability of approximate solutions to some time-dependent energy-transport equations with physical transport coefficients. The discretization of the stationary model using mixed finite elements is explained, and some numerical results in two and three space dimensions are presented. Finally, energy-transport models with lattice heating or quantum corrections are reviewed.

关键词： energy-transport equations semiconductors existence analysis weak sequential stability finite-element approximation

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