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1

Application of reanalysis methods in structural mechanics

Delyová I., Frankovský P., Bocko J., Sivák P., Kurimský R.

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 3

49--62

EN

When designing structures, it is often necessary to re-analyse a structure that is different in some parts from the original one. As real structures are often complex, their analysis is therefore very challenging. In such cases, reanalysis methods are advantageously used. The aim of this paper is to approach the problem of solving the constructions using reanalysis method in which the time taken in solving algebraic equations is reduced. In particular, the purpose of this work is to demonstrate on a chosen system the time savings and the advantages of the chosen direct efficient reanalysis method for a given design problem. A basic condition for meeting these criteria is the modernization of computational procedures in the mechanics of compliant solids.

2

Construction of algebraic and difference equations with a prescribed solution space

Moysis L., Karampetakis N. P.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 1

19--32

EN

This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ) β (k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ) . This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ) β (k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

3

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Shiri B., Shahmorad S., Hojjati G.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 2

341--355

EN

In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

4

Supernodal algorithm versus object oriented technique in sparse linear equations solver

Stabrowski M.

Przegląd Elektrotechniczny

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2005

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R. 81, nr 2

44-47

EN

Supernodal technique, used in the field of sparse linear equation solvers, introduces dense kernel for speeding-up the computations. SuperLU solver is the most sophisticated and efficient example of this technique. The research reported here shows that supernodal solver is really fast, but this spectacular speed is obtained through abandoning of numerical stability. It has been shown that SuperLU solver fails in whole class of practical real-world problems. An attempt to characterize and diagnose this problems in terms of matrix parameters has been madę. Pointer solver, being stable numerically and fairly fast alternative for supernodal solver is presented quite comprehensively.

PL

Metoda superwęzłów stosowana przy rozwiązywaniu rzadkich układów równań algebraicznych polega na specjalnym traktowaniu przydiagonalnych fragmentów macierzy rzadkich. Fragmenty te traktowane są jako submacierze pełne, przez co uzyskuje się przyspieszenie obliczeń kosztem, jak podają niektórzy badacze, utraty stabilności numerycznej. Praca jest próbą analizy tego problemu w kategoriach parametrów macierzy rozwiązywanego układu równań. Przedstawia stabilną numerycznie i dość szybką alternatywę - solver ,,wskaźnikowy" (pointer solver).

5

Positive solutions to polynomial matrix equations

Kaczorek T.

Applied Mathematics and Computer Science

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1998

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Vol. 8, no 2

403-415

EN

Necessary and sufficient conditions are established for the existence of positive solutions to polynomial matrix equations. Two methods are proposed for determination of such solutions. As an example of applications it is shown that the determination of the transfer matrix of a positive controller for a closed-loop system with a given transfer matrix can be reduced to finding solutions to two suitable polynomial matrix equations.