686 research outputs found
Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems
We are interested in high-order linear multistep schemes for time
discretization of adjoint equations arising within optimal control problems.
First we consider optimal control problems for ordinary differential equations
and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas
BDF methods preserve high--order accuracy. Subsequently we extend these results
to semi--lagrangian discretizations of hyperbolic relaxation systems.
Computational results illustrate theoretical findings
Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems.
Gradients of the objective function are most efficiently computed by
approximate adjoint variables. High accuracy with moderate computing time can
be achieved by such time integration methods that satisfy a sufficiently large
number of adjoint order conditions and supply gradients with higher orders of
consistency. In this paper, we upgrade our former implicit two-step Peer
triplets constructed in [Algorithms, 15:310, 2022] to meet those new
requirements. Since Peer methods use several stages of the same high stage
order, a decisive advantage is their lack of order reduction as for
semi-discretized PDE problems with boundary control. Additional order
conditions for the control and certain positivity requirements now intensify
the demands on the Peer triplet. We discuss the construction of 4-stage methods
with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of
practical interest. We prove convergence for s-stage methods, for instance,
order s for the state variables even if the adjoint method and the control
satisfy the conditions for order s-1, only. Numerical tests show the expected
order of convergence for the new Peer triplets.Comment: 47 pages, 5 figure
A Review of Time Relaxation Methods
The time relaxation model has proven to be effective in regularization of Navier–Stokes Equations. This article reviews several published works discussing the development and implementations of time relaxation and time relaxation models (TRMs), and how such techniques are used to improve the accuracy and stability of fluid flow problems with higher Reynolds numbers. Several analyses and computational settings of TRMs are surveyed, along with parameter sensitivity studies and hybrid implementations of time relaxation operators with different regularization techniques
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.3°)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84°)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability
Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method
This paper addresses the problem of the design optimization of turbomachinery components
under thermo-mechanical constraints, with focus on a radial turbine impeller for turbocharger
applications. Typically, turbine components operate at high temperatures and are exposed to
important thermal gradients, leading to thermal stresses. Dealing with such structural requirements
necessitates the optimization algorithms to operate a coupling between fluid and structural solvers
that is computationally intensive. To reduce the cost during the optimization, a novel multiphysics
gradient-based approach is developed in this work, integrating a Conjugate Heat Transfer procedure
by means of a partitioned coupling technique. The discrete adjoint framework allows for the ecient
computation of the gradients of the thermo-mechanical constraint with respect to a large number
of design variables. The contribution of the thermal strains to the sensitivities of the cost function
extends the multidisciplinary outlook of the optimization and the accuracy of its predictions, with
the aim of reducing the empirical safety factors applied to the design process. Finally, a turbine
impeller is analyzed in a demanding operative condition and the gradient information results in a
perturbation of the grid coordinates, reducing the stresses at the rotor back-plate, as a demonstration
of the suitability of the presented method
On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics
This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit
Nonlinear Evolution Equations: Analysis and Numerics
The workshop was devoted to the analytical and numerical investigation of nonlinear evolution equations. The main aim was to stimulate a closer interaction between experts in analytical and numerical methods for areas such as wave and Schrödinger equations or the Navier–Stokes equations and fluid dynamics
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