In-depth Exploration of the Multigrid Method to Simulate Elastohydrodynamic Line Lubrications with Smooth, Wavy, and Rough Surfaces

Due to high efficiency, multigrid (MG) algorithms developed by Lubrecht and Venner or others have been widely applied to solve the Reynolds equation in lubrication simulations. However, such algorithms are complex in nature and in-depth understandings and further development are of interest. This work proposes a new restriction operator of pressure to simplify the relaxation of the load balance equation and constructs several new relaxation processes based on key options of relaxations when either pressures or changes of pressure are evaluated from the Reynolds equation. In addition, effects of cycle types, treatments of cavitation boundary, line solvers, relaxation factors, and differential schemes are revealed. This paper further implements a mass conservation algorithm into the MG code in order to deal with micro-Cavitations. Characteristics of film thickness, pressure, flow continuity, and residuals, resulting from smooth, wavy, or rough surfaces are discussed. Finally, the results from the last correction cycles at various levels are recommended to be used for better accuracy.

How to Handle Missing Values in Multi-Criteria Decision Aiding?

It is often the case in the applications of Multi-Criteria Decision Making that the values of alternatives are unknown on some attributes. An interesting situation arises when the attributes having missing values are actually not relevant and shall thus be removed from the model. Given a model that has been elicited on the complete set of attributes, we are looking thus for a way -- called restriction operator -- to automatically remove the missing attributes from this model. Axiomatic characterizations are proposed for three classes of models. For general quantitative models, the restriction operator is characterized by linearity, recursivity and decomposition on variables. The second class is the set of monotone quantitative models satisfying normalization conditions. The linearity axiom is changed to fit with these conditions. Adding recursivity and symmetry, the restriction operator takes the form of a normalized average. For the last class of models -- namely the Choquet integral, we obtain a simpler expression. Finally, a very intuitive interpretation is provided.

A Trilinear Restriction Estimate for the Hyperbolic Paraboloid with Sharp Dependence on Transversality

The problem of restriction in harmonic analysis revolves around the operator $\mathcal{R}:f\mapsto \hat{f}|_{S}$, where $S$ is a manifold in $\mathbb{R}^n$. A fundamental inequality of the multilinear theory of the restriction operator is $\big \lVert{\prod _{k=1}^{n}\mathcal{R}^{*} f_{k}}\big \rVert _{L^{2/(n-1)}(\mathbb{R}^{n})}\le C\prod _{k=1}^{n}\lVert{f_k}\rVert _{L^{2}(S_k)}$, where $\mathcal{R}^{*}$ is the dual operator and the hypersurfaces $S_k$ satisfy a condition of transversality. This estimate was proven by Bennett, Carbery, and Tao, up to logarithmic losses, but without optimal dependence on transversality in the constant $C$. For subsets of the paraboloid in $\mathbb{R}^{3}$, Ramos got the sharp dependence on transversality. In this paper we extend this result to subsets of the hyperbolic paraboloid.

A note on properties of the restriction operator on Sobolev spaces

In our companion paper [

Lower bounds for quasianalytic functions, I. How to control smooth functions

Let $\mathscr{F}$ be a class of functions with the uniqueness property: if $f\in \mathscr{F}$ vanishes on a set $E$ of positive measure, then $f$ is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. the estimate from below for the norm of the restriction operator $f\mapsto f\big|_E$ or, equivalently, a lower bound for $|f|$ outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part, we consider quasianalytically smooth functions. This part relies upon Bang's approach and includes the proofs of relevant results of Bang. In the second part, which is to be published separately, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence.The proofs are based on the elementary calculus technique.