591 research outputs found
Probability density adjoint for sensitivity analysis of the Mean of Chaos
Sensitivity analysis, especially adjoint based sensitivity analysis, is a
powerful tool for engineering design which allows for the efficient computation
of sensitivities with respect to many parameters. However, these methods break
down when used to compute sensitivities of long-time averaged quantities in
chaotic dynamical systems.
The following paper presents a new method for sensitivity analysis of {\em
ergodic} chaotic dynamical systems, the density adjoint method. The method
involves solving the governing equations for the system's invariant measure and
its adjoint on the system's attractor manifold rather than in phase-space. This
new approach is derived for and demonstrated on one-dimensional chaotic maps
and the three-dimensional Lorenz system. It is found that the density adjoint
computes very finely detailed adjoint distributions and accurate sensitivities,
but suffers from large computational costs.Comment: 29 pages, 27 figure
The prospect of using LES and DES in engineering design, and the research required to get there
In this paper we try to look into the future to divine how large eddy and
detached eddy simulations (LES and DES, respectively) will be used in the
engineering design process about 20-30 years from now. Some key challenges
specific to the engineering design process are identified, and some of the
critical outstanding problems and promising research directions are discussed.Comment: accepted for publication in the Royal Society Philosophical
Transactions
Optimization of Gaussian Random Fields
Many engineering systems are subject to spatially distributed uncertainty,
i.e. uncertainty that can be modeled as a random field. Altering the mean or
covariance of this uncertainty will in general change the statistical
distribution of the system outputs. We present an approach for computing the
sensitivity of the statistics of system outputs with respect to the parameters
describing the mean and covariance of the distributed uncertainty. This
sensitivity information is then incorporated into a gradient-based optimizer to
optimize the structure of the distributed uncertainty to achieve desired output
statistics. This framework is applied to perform variance optimization for a
model problem and to optimize the manufacturing tolerances of a gas turbine
compressor blade
Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations
The adjoint method, among other sensitivity analysis methods, can fail in
chaotic dynamical systems. The result from these methods can be too large,
often by orders of magnitude, when the result is the derivative of a long time
averaged quantity. This failure is known to be caused by ill-conditioned
initial value problems. This paper overcomes this failure by replacing the
initial value problem with the well-conditioned "least squares shadowing (LSS)
problem". The LSS problem is then linearized in our sensitivity analysis
algorithm, which computes a derivative that converges to the derivative of the
infinitely long time average. We demonstrate our algorithm in several dynamical
systems exhibiting both periodic and chaotic oscillations.Comment: submitted to JCP in revised for
- …