27,564 research outputs found
A Pseudospectral Approach to High Index DAE Optimal Control Problems
Historically, solving optimal control problems with high index differential
algebraic equations (DAEs) has been considered extremely hard. Computational
experience with Runge-Kutta (RK) methods confirms the difficulties. High index
DAE problems occur quite naturally in many practical engineering applications.
Over the last two decades, a vast number of real-world problems have been
solved routinely using pseudospectral (PS) optimal control techniques. In view
of this, we solve a "provably hard," index-three problem using the PS method
implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In
contrast to RK-type solution techniques, no laborious index-reduction process
was used to generate the PS solution. The PS solution is independently verified
and validated using standard industry practices. It turns out that proper PS
methods can indeed be used to "directly" solve high index DAE optimal control
problems. In view of this, it is proposed that a new theory of difficulty for
DAEs be put forth.Comment: 14 pages, 9 figure
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Parallel-In-Time Simulation of Eddy Current Problems Using Parareal
In this contribution the usage of the Parareal method is proposed for the
time-parallel solution of the eddy current problem. The method is adapted to
the particular challenges of the problem that are related to the differential
algebraic character due to non-conducting regions. It is shown how the
necessary modification can be automatically incorporated by using a suitable
time stepping method. The paper closes with a first demonstration of a
simulation of a realistic four-pole induction machine model using Parareal
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
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