Chris Lacor Dept Mechanical Engineering
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- input uncertainties
- polynomials
- compressible navier-stokes
- uncertain input parameters
- pc decomposition
- basis functions
- gramm-schmidt procedure
- ??? ?
- intrusive approach
Intrusive Polynomial Chaos for the
Compressible Navier+Stokes
Equations
Chris Lacor
Dept. Mechanical Engineering
Vrije Universiteit Brussel
With thanks to: Dr. S. Smirnov, Ir. Ch. Vandenhaute, Ir. M.
Raes, Ir. G. Onorato
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Polynomial Chaos
Framework
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NODESIM European STREP project 1/11/06+1/03/10
WP3.3: Polynomial Chaos Methods
– TUD:non+intrusive approach
– VUB:intrusive approach
Symposium on Accuracy and Uncertainty in Flow Simulations, ONERA, Chatillon, December 3, 2010
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Polynomial Chaos
PC method relies on a probabilistic
framework.
Uncertain input parameters are modelled
as stochastic variables (or fields) with
known distributions (PDF).
For each stochastic variable a new
dimension ξ is introduced
Solution is sought in a form of PC
decomposition.
A set of orthogonal multidimensional
polynomials ψ is used as basis functions.
The polynomial type depends on the PDF
of the uncertainty, see further
The deterministic coefficients are derived
by Galerkin projection of the governing
equations.
The resulting set of equations is coupled
due to non+linear terms in the Navier+
Stokes equations.
r1r(x,t,x(q!,...,x(q!!
1n
u1u(x,t,x(q!,...,x(q!!
1n
p1p(x,t,x(q!,...,x(q!!
1n
N
∑k k1n
r(x,t,q!1r(x,t!Y(x,...,x!
k10
N
∑k k1n
u(x,t,q!1u(x,t!Y(x,...,x!
k10
N
∑k k1n
p(x,t,q!1p(x,t!Y(x,...,x!
k10
Symposium on Accuracy and Uncertainty in Flow Simulations, ONERA, Chatillon, December 3, 2010
