Fine tuning of numerical algorithms

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Dr. Pataricza András
Department of Measurement and Information Systems

Nowadays in the design of products and industrial processes computational modelling are more and more important because with computer a few different settings can be tried out without the performance of the more expensive experiments. However, in the study of complex systems the available computational capacity might determine the finesse and the reality of the applied model. The efficient usage of computational capacity determines the utility of the simulation. Hence, in my work I examine such an approach increasing the efficiency of modelling if on the important spot of the domain fine resolution, on the less interesting locations rough resolution are applied, following the dynamic of simulation process.

The general problem was studied on the example of a particularly important chemical analytical method where the qualitative or quantitative composition of a sample is deduced from the signal (e.g. electric current) measured by the detector. Based on the previous investigations a cheap and sensitive iondetector (e. g. suitable for potassium or sodium concentration determination from blood or water sample) could be developed from the so-called chemical system acid-base diode.

The simulation program solves the partial differential equations describing the diode using finite element method (FEM). In this approximation the geometric domain, where the solution is looked for, is divided into smaller parts called finite elements. This is called mesh generation.

Since the mathematical model of the diode is one dimensional, in this case subdivision means line elements (beams). The number of applied elements on a section determines the accuracy of the approximation. The numerical solution of the differential equation system is reproduced by polynomial approximation.

In case of acid-base diode a chemical reaction proceeds. The location of it is a critical zone where extremely high resolution must be applied than onto other parts of the diode to obtain a realistic simulation result. The difficulty of transient simulation is the time-dependence of this zone’s position (it “moves”) thus the high resolution zone must be put always elsewhere (adaptive mesh refinement), hence fine resolution on the whole domain causes unreal computational complexity.

Based on the time-dependent modelling of acid-base diodes I studied the acceleration opportunities of adaptive meshing techniques.

The impact of the applied mesh on the simulation time and on the error of numerical approximation were studied, because the first one is failed, the second one is improved in case of finer mesh.

Couple of potential monitoring functions (the space derivatives of the equation’s variables, chemical reaction factor) are selected with which the mesh density can be optimized.


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