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  • Opus Repository ZIB  (16)
  • 1990-1994  (16)
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  • 1
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    The Treatment of Macromolecular Processes with Chain-Length-Dependent Reaction Coefficients - An Examplefrom Soot Formation. (1991)
    Details
    Publication Date: 2014-02-26
    Description: The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 2
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    Quadratic 0/1 Optimization and a Decomposition Approach for the Placement of Electronic Circuits. (1992)
    Details
    Publication Date: 2020-10-05
    Description: The placement in the layout design of electronic circiuts consists of finding a non- overlapping assignment of rectangular cells to positions on the chip so what wireability is guaranteed and certain technical constraints are met.This problem can be modelled as a quadratic 0/1- program subject to linear constraints. We will present a decomposition approach to the placement problem and give results about $NP$-hardness and the existence of $\varepsilon$-approximative algorithms for the involved optimization problems. A graphtheoretic formulation of these problems will enable us to develop approximative algorithms. Finally we will present details of the implementation of our approach and compare it to industrial state of the art placement routines. {\bf Keywords:} Quadratic 0/1 optimization, Computational Complexity, VLSI-Design.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
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    Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy. (1992)
    Details
    Publication Date: 2014-02-26
    Description: Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 4
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    Numerical Treatment of Countable Systems of Ordinary Differential Equations. (1990)
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    Publication Date: 2014-02-26
    Description: Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and medicine. They can be considered as ordinary differential equations in sequence spaces. In this work, a full adaptive algorithm for the computational treatment of such systems is developed. The method combines time discretization with extrapolation in Hilbert spaces with a discrete Galerkin approach as discretization of the stationary subproblems. The Galerkin method is based on orthogonal functions of a discrete variable , which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations can be assembled either by use of analytical properties of the orthogonal functions or numerically by a multilevel summation algorithm. The resulting algorithm CODEX is applied to many examples of technological interest, in particular from polymer chemistry.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 5
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    Recent Developments in Chemical Computing (1990)
    Details
    Publication Date: 2020-12-14
    Language: English
    Type: article , doc-type:article
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  • 6
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    Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions (1992)
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    Publication Date: 2020-03-20
    Language: English
    Type: bookpart , doc-type:bookPart
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  • 7
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    Optimal Control of Plotting and Drilling Machines (1991)
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    Publication Date: 2020-08-05
    Language: English
    Type: article , doc-type:article
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  • 8
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    Recent Developments in Chemical Computing. (1990)
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    Publication Date: 2014-02-26
    Description: The paper surveys three aspects of chemical computing, which seem to play a role in recent developments. First, extrapolation methods for the numerical treatment of differential- algebraic equations are introduced. The associated extrapolation code LIMEX has reached a certain level of sophistication, which makes it a real competitor to the elsewhere widely used multi-step code DASSL of Petzold. Second, adaptive methods of lines for partial differential equations such as those arising in combustion problems are treated. Both static and dynamic regridding techniques are discussed in some detail. Finally, some new ideas about the treatment of the kinetic equations arising from polymer reactions are presented. The new feature of the suggested approach is the application of a Galerkin procedure using sets of orthogonal polynomials over a discrete variable (which, of course, in the case of polymer reactions is the polymer degree). The new approach may open the door to a new reliable low dimensional treatment of complex polymer reactions.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 9
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    Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions. (1990)
    Details
    Publication Date: 2014-02-26
    Description: The discrete Galerkin method developed by the authors has turned out to be an efficient tool for the computational treatment of very large scale ODE systems arising in polyreaction kinetics. Up to now, this approach has been worked out in detail for homogeneous polymer reactions. The present paper deals with one line of possible extensions of the method to the case of so-called heterogeneous processes, which may appear e. g. in smog reactions. The associated mathematical models involve reaction coefficients depending on the chain length of the reacting polymer. The herein suggested extension is worked out in some detail on the basis of the earlier paper. In addition, a numerical example describing polymer degradation is included.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 10
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    Symmetry Breaking Bifurcations of Chaotic Attractors (1994)
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    Publication Date: 2014-02-26
    Description: In an array of coupled oscillators {\em synchronous chaos} may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries and this leads to the idea of {\em symmetry breaking bifurcations of chaotic attractors}. Finally the results and ideas are illustrated for several systems of coupled oscillators.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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