Details
| Original language | English |
|---|---|
| Title of host publication | Recent Developments and Innovative Applications in Computational Mechanics |
| Pages | 311-319 |
| Number of pages | 9 |
| Publication status | Published - 2011 |
Abstract
Structural dynamical systems under random white noise excitation can be described by time-dependent stochastic differential equations. Under white noise assumption the response process possesses Markov characteristics and the transient stochastic differential equations can be transformed into evolution equations for the probability density, the so called Fokker-Planck equations. The Fokker-Planck equation has its origin in the description of the motion of tiny particles in a fluid. Its mathematical structure is comparable with coupled advection and diffusion problems which represent a broad class of problems in engineering and natural sciences. However, the numerical treatment of advection-diffusion type partial differential equations remains a challenging task due to the non self-adjoint advection operator. Semi-discretization techniques are subject to strong limits for the relation of spatial and temporal discretization (often expressed by the Courant number) for prevention of artificially oscillatory or dispersive solutions. In this contribution a finite element approach in space and time, where the time space is discretized discontinuously is presented as superior method for the numerical treatment of hyperbolic advection-diffusion type partial differential equations. This method is referred to as Time-Discontinuous Galerkin method. It will be illustrated by the response of dynamical systems under uncertain excitation, which are described by the Fokker-Planck equation.
ASJC Scopus subject areas
- Engineering(all)
- General Engineering
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Recent Developments and Innovative Applications in Computational Mechanics. 2011. p. 311-319.
Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review
}
TY - CHAP
T1 - A time-discontinuous galerkin approach for the numerical solution of the fokker-planck equation
AU - Nackenhorst, Udo
AU - Loerke, Friederike
PY - 2011
Y1 - 2011
N2 - Structural dynamical systems under random white noise excitation can be described by time-dependent stochastic differential equations. Under white noise assumption the response process possesses Markov characteristics and the transient stochastic differential equations can be transformed into evolution equations for the probability density, the so called Fokker-Planck equations. The Fokker-Planck equation has its origin in the description of the motion of tiny particles in a fluid. Its mathematical structure is comparable with coupled advection and diffusion problems which represent a broad class of problems in engineering and natural sciences. However, the numerical treatment of advection-diffusion type partial differential equations remains a challenging task due to the non self-adjoint advection operator. Semi-discretization techniques are subject to strong limits for the relation of spatial and temporal discretization (often expressed by the Courant number) for prevention of artificially oscillatory or dispersive solutions. In this contribution a finite element approach in space and time, where the time space is discretized discontinuously is presented as superior method for the numerical treatment of hyperbolic advection-diffusion type partial differential equations. This method is referred to as Time-Discontinuous Galerkin method. It will be illustrated by the response of dynamical systems under uncertain excitation, which are described by the Fokker-Planck equation.
AB - Structural dynamical systems under random white noise excitation can be described by time-dependent stochastic differential equations. Under white noise assumption the response process possesses Markov characteristics and the transient stochastic differential equations can be transformed into evolution equations for the probability density, the so called Fokker-Planck equations. The Fokker-Planck equation has its origin in the description of the motion of tiny particles in a fluid. Its mathematical structure is comparable with coupled advection and diffusion problems which represent a broad class of problems in engineering and natural sciences. However, the numerical treatment of advection-diffusion type partial differential equations remains a challenging task due to the non self-adjoint advection operator. Semi-discretization techniques are subject to strong limits for the relation of spatial and temporal discretization (often expressed by the Courant number) for prevention of artificially oscillatory or dispersive solutions. In this contribution a finite element approach in space and time, where the time space is discretized discontinuously is presented as superior method for the numerical treatment of hyperbolic advection-diffusion type partial differential equations. This method is referred to as Time-Discontinuous Galerkin method. It will be illustrated by the response of dynamical systems under uncertain excitation, which are described by the Fokker-Planck equation.
UR - http://www.scopus.com/inward/record.url?scp=84889821560&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-17484-1_35
DO - 10.1007/978-3-642-17484-1_35
M3 - Contribution to book/anthology
AN - SCOPUS:84889821560
SN - 9783642174834
SP - 311
EP - 319
BT - Recent Developments and Innovative Applications in Computational Mechanics
ER -