2,591 research outputs found
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
Finite element methods for deterministic simulation of polymeric fluids
In this work we consider a finite element method for solving the coupled Navier-Stokes (NS) and Fokker-Planck (FP) multiscale model that describes the dynamics of dilute polymeric fluids. Deterministic approaches such as ours have not received much attention in the literature because they present a formidable computational challenge, due to the fact that the analytical solution to the Fokker-Planck equation may be a function of a large number of independent variables. For instance, to simulate a non-homogeneous flow one must solve the coupled NS-FP system in which (for a 3-dimensional flow, using the dumbbell model for polymers) the Fokker-Planck equation is posed in a 6-dimensional domain. In this work we seek to demonstrate the feasibility of our deterministic approach. We begin by discussing the physical and mathematical foundations of the NS-FP model. We then present a literature review of relevant developments in computational rheology and develop our deterministic finite element based method in detail. Numerical results demonstrating the efficiency of our approach are then given, including some novel results for the simulation of a fully 3-dimensional flow. We utilise parallel computation to perform the large-scale numerical simulations
Reduced-order modeling of transonic flows around an airfoil submitted to small deformations
A reduced-order model (ROM) is developed for the prediction of unsteady transonic flows past an airfoil submitted to small deformations, at moderate Reynolds number. Considering a suitable state formulation as well as a consistent inner product, the Galerkin projection of the compressible flow Navier–Stokes equations, the high-fidelity (HF) model, onto a low-dimensional basis determined by Proper Orthogonal Decomposition (POD), leads to a polynomial quadratic ODE system relevant to the prediction of main flow features. A fictitious domain deformation technique is yielded by the Hadamard formulation of HF model and validated at HF level. This approach captures airfoil profile deformation by a modification of the boundary conditions whereas the spatial domain remains unchanged. A mixed POD gathering information from snapshot series associated with several airfoil profiles can be defined. The temporal coefficients in POD expansion are shape-dependent while spatial POD modes are not. In the ROM, airfoil deformation is introduced by a steady forcing term. ROM reliability towards airfoil deformation is demonstrated for the prediction of HF-resolved as well as unknown intermediate configurations
A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation
A numerical method for the two-dimensional, incompressible Navier--Stokes
equations in vorticity--streamfunction form is proposed, which employs
semi-Lagrangian discretizations for both the advection and diffusion terms,
thus achieving unconditional stability without the need to solve linear systems
beyond that required by the Poisson solver for the reconstruction of the
streamfunction. A description of the discretization of Dirichlet boundary
conditions for the semi-Lagrangian approach to diffusion terms is also
presented. Numerical experiments on classical benchmarks for incompressible
flow in simple geometries validate the proposed method
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