796 research outputs found
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
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Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter
This paper investigates the state estimation of a high-fidelity spatially
resolved thermal- electrochemical lithium-ion battery model commonly referred
to as the pseudo two-dimensional model. The partial-differential algebraic
equations (PDAEs) constituting the model are spatially discretised using
Chebyshev orthogonal collocation enabling fast and accurate simulations up to
high C-rates. This implementation of the pseudo-2D model is then used in
combination with an extended Kalman filter algorithm for differential-algebraic
equations to estimate the states of the model. The state estimation algorithm
is able to rapidly recover the model states from current, voltage and
temperature measurements. Results show that the error on the state estimate
falls below 1 % in less than 200 s despite a 30 % error on battery initial
state-of-charge and additive measurement noise with 10 mV and 0.5 K standard
deviations.Comment: Submitted to the Journal of Power Source
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
Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).In this thesis we study the initialization of multistep methods and parametrize some well-known classesof multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)multistep methods and parametric formulation of blocked multistep methods.Depending on the number of steps, a multistep method requires adequate number of initial values tostart the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.The proposed starters estimate the error by embedded methods.The second part concerns the variable step-size blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize blocked multistep methods forthe solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments. For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulationallows time adaptivity by construction
Optimization of a fed-batch bioreactor for 1,3-propanediol production using hybrid nonlinear optimal control
A nonlinear hybrid system was proposed to describe the fed-batch bioconversion of glycerol to 1,3-propanediol with substrate open loop inputs and pH logic control in previous work [47]. The current work concerns the optimal control of this fed-batch process. We slightly modify the hybrid system to provide a more convenient mathematical description for the optimal control of the fed-batch culture. Taking the feeding instants and the terminal time as decision variables, we formulate an optimal control model with the productivity of 1,3-propanediol as the performance index. Inequality path constraints involved in the optimal control problem are transformed into a group of end-point constraints by introducing an auxiliary hybrid system. The original optimal control problem is associated with a family of approximation problems. The gradients of the cost functional and the end-point constraint functions are derived from the parametric sensitivity system. On this basis, we construct a gradient-based algorithm to solve the approximation problems. Numerical results show that the productivity of 1,3-propanediol can be increased considerably by employing our optimal control policy
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