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Nowadays, the condition-based maintenance (CBM), in which repairs are triggered by the heuristic symptoms of the component faults, is finding increasing applications in the industrial fields. However, for the high-precision machine tools, the conventional CBM might not be the optimal option, which is uneconomic and incapable of ensuring their machining accuracy. In order to overcome these shortcomings, this paper propose the manufacturing-error-based maintenance (MEBM), where the repairs are initiated based on the manufacturing errors instead of the heuristic symptoms. In MEBM, repairs are taken properly at the occurrence of the excessive machining errors, and therefore, the premature and redundant maintenance can be avoided and the maintenance cost can be minimized; what is more, the machining errors are controlled in the closed loops, and therefore, the machining accuracy can be guaranteed. Based on the principles of the MEBM, a prototype maintenance system--the transient backlash error (TBE)-based maintenance system--is established. To achieve this aim, first, the width of the backlash in the mechanical chain is measured by utilizing the built-in encoders and the analytical mapping relationship between the backlash width and the TBE is derived. Relying on these foundations, the TBE can be indirectly estimated. Then, the warning threshold of the TBE is customized according to the permissible roundness error of the workpiece. Thus, the maintenance actions can be precisely implemented: when the monitored TBE exceeds its warning threshold, maintenance workers will be notified to lessen the backlash width, and meanwhile, the permissible maximal size for the backlash will also be informed. [ABSTRACT FROM AUTHOR]

Divergence- L q and divergence-measure tensor fields are introduced and their properties are analyzed. These measure-theoretic tools are then used to study the gradient flow of linear growth functionals of maps into the unit sphere. A BV-solution concept is introduced and the existence of such a solution is established for the flow using the energy method together with a regularization and a penalization technique. Based on the analytical results, practical fully discrete finite element methods are then proposed for computing weak solutions of the gradient flow, and the convergence of the proposed numerical method is also proved. [ABSTRACT FROM AUTHOR]

Abstract  This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included. [ABSTRACT FROM AUTHOR]

Summary We derive rates of convergence for regularization procedures (characterized by a parameter ?) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods. [ABSTRACT FROM AUTHOR]

Summary. We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + ?( ?? u- ? -1 f( u)) = 0, where ? > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on only in some lower polynomial order for small ?. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ? ? 0 in [29]. [ABSTRACT FROM AUTHOR]

Summary. We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation ut-Δu+ε-2f(u)=0 arising from phase transition in materials science, where ε is a small parameter known as an ``interaction length''. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ε. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u0. In particular, all our error bounds depend on ${{\frac{{1}}{{\varepsilon}}}}$ only in some lower polynomial order for small ε. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. [ABSTRACT FROM AUTHOR]