98 research outputs found
An approach to simulate interstitial processes in river beds to meet biological requirements
River engineeringRiver habitat management and restoratio
Finite Projected Entangled Pair States for the Hubbard model
We adapt and optimize the projected-pair-entangled-state (PEPS) algorithm on
finite lattices (fPEPS) for two-dimensional Hubbard models and apply the
algorithm to the Hubbard model with nearest-neighbor hopping on a square
lattice. In particular, we formulate the PEPS algorithm using projected
entangled pair operators, incorporate SU(2) symmetry in all tensor indices, and
optimize the PEPS using both iterative-diagonalization-based local bond
optimization and gradient-based optimization of the PEPS. We discuss the
performance and convergence of the algorithm for the Hubbard model on lattice
sizes of up to 8x8 for PEPS states with U(1) symmetric bond dimensions of up to
D = 8 and SU(2) symmetric bond dimensions of up to D = 6. Finally, we comment
on the relative and overall efficiency of schemes for optimizing fPEPS
Cluster-based reduced-order modelling of a mixing layer
We propose a novel cluster-based reduced-order modelling (CROM) strategy of
unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's
group (Burkardt et al. 2006) and and transition matrix models introduced in
fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a
potential alternative to POD models and generalises the Ulam-Galerkin method
classically used in dynamical systems to determine a finite-rank approximation
of the Perron-Frobenius operator. The proposed strategy processes a
time-resolved sequence of flow snapshots in two steps. First, the snapshot data
are clustered into a small number of representative states, called centroids,
in the state space. These centroids partition the state space in complementary
non-overlapping regions (centroidal Voronoi cells). Departing from the standard
algorithm, the probabilities of the clusters are determined, and the states are
sorted by analysis of the transition matrix. Secondly, the transitions between
the states are dynamically modelled using a Markov process. Physical mechanisms
are then distilled by a refined analysis of the Markov process, e.g. using
finite-time Lyapunov exponent and entropic methods. This CROM framework is
applied to the Lorenz attractor (as illustrative example), to velocity fields
of the spatially evolving incompressible mixing layer and the three-dimensional
turbulent wake of a bluff body. For these examples, CROM is shown to identify
non-trivial quasi-attractors and transition processes in an unsupervised
manner. CROM has numerous potential applications for the systematic
identification of physical mechanisms of complex dynamics, for comparison of
flow evolution models, for the identification of precursors to desirable and
undesirable events, and for flow control applications exploiting nonlinear
actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material.
Accepted for publication in Journal of Fluid Mechanic
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