Modeling of directional uncertainty using moments of the angular central Gaussian

Abstract: The Gaussian distribution is commonly used to model uncertainty for all kind of problems. However, for directional data like fiber orientations in injection molding simulations the canonical choice is the so called angular central Gaussian (ACG) distribution, which arises as analytical solution to Jeffrey's equation which is used to model the orientation of a elliptical fiber suspended in a flow field. Computations are favorably performed using moments of the density instead of the density itself, leading to the so called Folgar‐Tucker equation. In this differential equation for the second order moment also the fourth order moment arises, which has to be expressed in terms of the second order moment in order to close the equation. This is called the “closure problem”, which has been addressed in many publications with various proposals for the solution, among which the fourth order moment of the ACG distribution represents the exact solution. The ACG is obtained by normalizing a centered multivariate Gaussian and therefore the individual components of the ACG distribution are no longer independent and the moments of the ACG distribution are coupled with its covariance parameter in a more complicated way in the form of elliptic integrals. This beautiful distribution is not very well studied and hardly used for directional statistics, where more empirical or wrapped distributions are employed instead. In this article we compare the ACG to the Bingham distribution and discuss the current status of computing moments as well as the analytical and numerical solution of the closure problem.

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch

References and Relationships

• Bibliographic citation: Modeling of directional uncertainty using moments of the angular central Gaussian ; volume:21 ; number:1 ; year:2021 ; extent:2 ; Proceedings in applied mathematics and mechanics ; 21, Heft 1 (2021) (gesamt 2)

Contributors, Places and Time

• Creator: Ospald, Felix ; Herzog, Roland

Further information

• DOI: 10.1002/pamm.202100159
• URN: urn:nbn:de:101:1-2021121514183733285547
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 15.08.2025, 7:32 AM CEST