Publication

MATHICSE Technical Report : Quantifying uncertainties in contact mechanics of rough surfaces using the Multilevel Monte Carlo method

Abstract

We quantify the effect of uncertainties on quantities of interest related to contact mechanics of rough surfaces. Specifically, we consider the problem of frictionless non adhesive normal contact between two semi infinite linear elastic solids subject to uncertainties. These uncertainties may for example originate from an incomplete surface description. To account for surface uncertainties, we model a rough surface as a suitable Gaussian random field whose covariance function encodes the surface's roughness, which is experimentally measurable. Within this stochastic framework, we first introduce the complete random contact model, which includes the precise definition of the considered class of rough random surfaces as well as the study of a practical random surface generator. Then, we introduce the multilevel Monte Carlo method which is a computationally efficient sampling method for the computation of statistical moments of uncertain system output's, such as those derived from contact simulations. In particular, we consider two different quantities of interest, namely the contact area and the number of contact clusters, and show via numerical experiments that the multilevel Monte Carlo method offers significant computational gains compared to an approximation via a classic Monte Carlo sampling.

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Related concepts (32)
Contact mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces (known as normal stress) and frictional stresses acting tangentially between the surfaces (shear stress). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied normal forces and by the adhesion present on surfaces in close contact, even if they are clean and dry.
Surface roughness
Surface roughness can be regarded as the quality of a surface of not being smooth and it is hence linked to human (haptic) perception of the surface texture. From a mathematical perspective it is related to the spatial variability structure of surfaces, and inherently it is a multiscale property. It has different interpretations and definitions depending from the disciplines considered. Surface roughness, often shortened to roughness, is a component of surface finish (surface texture).
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
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