Incomplete gamma function

Summary

In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^{\infty} t^{s-1},e^{-t}, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^{s-1},e^{-t}, dt . In both cases s is a complex parameter, su

Official source

https://en.wikipedia.org/wiki/Incomplete_gamma_function

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Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method

Sebastian Krumscheid, Fabio Nobile, Valentine Ginette Madeleine Rey

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. Then, we introduce the multilevel Monte Carlo method which is a computationally efficient sampling method for the computation of the expectation and higher statistical moments of uncertain system outputs, 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.

2019

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

Sebastian Krumscheid, Fabio Nobile, Valentine Ginette Madeleine Rey

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.

MATHICSE2018