334 research outputs found
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks
Nonlinear model predictive control (NMPC) often requires real-time solution
to optimization problems. However, in cases where the mathematical model is of
high dimension in the solution space, e.g. for solution of partial differential
equations (PDEs), black-box optimizers are rarely sufficient to get the
required online computational speed. In such cases one must resort to
customized solvers. This paper present a new solver for nonlinear
time-dependent PDE-constrained optimization problems. It is composed of a
sequential quadratic programming (SQP) scheme to solve the PDE-constrained
problem in an offline phase, a proper orthogonal decomposition (POD) approach
to identify a lower dimensional solution space, and a neural network (NN) for
fast online evaluations. The proposed method is showcased on a regularized
least-square optimal control problem for the viscous Burgers' equation. It is
concluded that significant online speed-up is achieved, compared to
conventional methods using SQP and finite elements, at a cost of a prolonged
offline phase and reduced accuracy.Comment: Accepted for publishing at the 58th IEEE Conference on Decision and
Control, Nice, France, 11-13 December, https://cdc2019.ieeecss.org
Solving 2D Time-Fractional Diffusion Equations by Preconditioned Fractional EDG Method
Fractional differential equations play a significant role in science and technology given that several scientific problems in mathematics, physics, engineering and chemistry can be resolved using fractional partial differential equations in terms of space and/or time fractional derivative. Because of new developments in the analysis and understanding of many complex systems in engineering and sciences, it has been observed that several phenomena are more realistically and accurately described by differential equations of fractional order. Fast computational methods for solving fractional partial differential equations using finite difference schemes derived from skewed (rotated) difference operators have been extensively investigated over the years. The main aim of this paper is to examine a new fractional group iterative method which is called Preconditioned Fractional Explicit Decoupled Group (PFEDG) method in solving 2D time-fractional diffusion equations. Numerical experiments and comparison with other existing methods are given to confirm the superiority of our proposed method
Status of research at the Institute for Computer Applications in Science and Engineering (ICASE)
Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science is summarized
Coupled/combined compact IRBF schemes for fluid flow and FSI problems
The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)
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