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1

Dual probabilistic analysis of the transient heat transfer by the stochastic finite element method with optimized polynomial basis

Rabenda M., Kamiński M.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 2/II

211--225

EN

The main aim of this work is to contrast three various probabilistic computational techniques, namely analytical, simulation and perturbation-based, in a solution of the transient heat transfer problem in specific axisymmetric problem with Gaussian uncertainty in physical parameters. It is done thanks to a common application of the Finite Element Method program ABAQUS (for the deterministic part) and symbolic algebra system MAPLE, where all probabilistic procedures have been programmed. We determine up to the fourth order probabilistic characteristics of the resulting temperatures, i.e. expectations, coefficients of variation, skewness and kurtosis together with the histograms – all as the functions of the input coefficient of variation of random heat conductivity coefficient. Stochastic perturbation technique is implemented here using the tenth order Taylor series expansion and traditional Least Squares Method released with polynomial basis whose final order is a subject of the separate statistical optimization. Probabilistic results computed show almost perfect agreement of all the probabilistic characteristics under consideration, which means that the traditional simulation method may be replaced due to the time and computer scale savings with the stochastic perturbation method.

PL

Głównym celem niniejszej pracy jest porównanie trzech różnych probabilistycznych metod numerycznych, tj. metody analitycznej, symulacyjnej, a także metody perturbacji, podczas rozwiązywania pewnego zagadnienia osiowo-symetrycznego, w którym współczynnik przewodnictwa ciepła jest Gaussowskim parametrem losowym. Porównanie takie jest przeprowadzone przy użyciu systemu Metody Elementów Skończonych ABAQUS (dla części deterministycznej rozwiązania), a także pakietu algebry komputerowej MAPLE, w którym zaimplementowano wszystkie procedury losowe. W pracy wyznacza się centralne momenty probabilistyczne do rzędu czwartego włącznie, tj. wartości oczekiwane, współczynniki wariancji, skośność i kurtozę, jak również odpowiednie histogramy – wszystkie one są wyznaczone w funkcji wejściowego współczynnika wariancji. Metoda perturbacji stochastycznej jest zaimplementowana przy użyciu rozwinięcia w szereg Taylora rzędu dziesiątego, a także z wykorzystaniem tradycyjnej Metody Najmniejszych Kwadratów. Metoda ta umożliwia wyznaczenie wielomianowej funkcji odpowiedzi, której rząd jest przedmiotem oddzielnej optymalizacji statystycznej. Otrzymane wyniki probabilistyczne pokazują bardzo dobrą zgodność wszystkich wyznaczanych charakterystyk losowych, co oznacza w praktyce, iż tradycyjna metoda symulacji może zostać zastąpiona przez metodę perturbacji stochastycznej w celu wielokrotnego zmniejszenia czasu oraz mocy obliczeniowej.

2

On the least squares stochastic finite element analysis of the steel skeletal towers exposed to fire

Kamiński M. M., Strąkowski M. T.

Archives of Civil and Mechanical Engineering

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2013

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Vol. 13, no. 2

242--253

EN

The main issue in this paper is an application of the generalized stochastic perturbation technique to thermal stresses and deformations analysis for the spatial steel tower structure exposed to a fire. This approach is based on the given order Taylor expansion of all random parameters and state functions around their mean values as well as on the Least Squares Technique to determine the analytical functions in-between design parameters and structural responses. A distribution of the temperature in the model is considered for a simplicity as the set of constant values for all structural members. The temperature equivalent to the fire exposure in the analyzed structure is taken as the input Gaussian random variable, where basic material parameters are considered as temperature-dependent and this is the basis to calculate up to the fourth probabilistic moments and characteristics of the stresses and deformations. This study is an example of a hybrid usage of the FEM engineering system ROBOT and the computer algebra system MAPLE in stochastic analysis, where thermal stresses and strains may be straightforwardly used in fire reliability analysis of the civil engineering structures with both temperature-independent and dependent material characteristics.

3

Comparison of the aluminium versus steel telecommunication towers in stochastic finite element method eigenvibrations analysis

Solecka M., Kamiński M., Szafran J.

Mechanics and Mechanical Engineering

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2011

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Vol. 15, nr 2

101-116

EN

The main aim of this paper is to make a comparison of the eigenfrequencies of the high telecommunication towers alternatively manufactured using the stainless steel and the aluminium components. It is provided each time assuming that the Young modulus of the applied materialź is the Gaussian input random variable and using the generalized stochastic perturbation method using the global version of the Response Function Method. Up to the fourth order probabilistic moments and characteristics are computed in the three dimensional Finite Element Method model of the tower composed from the continuous linear elastic edge beams spanned by the large number of the linear elastic bars. A computational part of the work is made using the hybrid usage of the computer algebra system MAPLE and the FEM engineering package ROBOT used widely in the civil engineering practice.

4

The stochastic perturbation - based finite volume method for 1 & 2D fluid and heat flow problems

Kamiński M., Ossowski R. L.

Mechanics and Mechanical Engineering

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2010

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Vol. 14, nr 1

151-173

EN

This article is devoted to the mathematical formulation and computational implementation of the Stochastic Finite Volume Method for 1 and 2D fluid and heat flow problems. It is based on the stochastic generalized perturbation technique, which allows for a determination of the probabilistic moments of the state variables or functions for the general stationary transport equations with random parameters. Both numerical case studies contain a comparison of the stochastic perturbation approach of different orders, their relations to the Monte-Carlo simulation results as well as the effect of the perturbation parameter and input coefficient of variation on the output state functions.

5

A generalized version of the perturbation-based stochastic finite difference method for elastic beams

Kamiński M.

Journal of Theoretical and Applied Mechanics

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2009

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Vol. 47 nr 4

957-975

EN

The main idea of this paper is to demonstrate a stochastic computational technique consisting of the generalized stochastic perturbation method using the Taylor expansions of random variables and the classical Finite Difference Method based on regular grids. As it is documented by computational illustrations, it is possible to determine, using this approach, also higher probabilistic moments for any random dispersion of input variables unlike in the second order second moment technique worked out before. A numerical algorithm is implemented here using straightforward partial differentiation of hierarchical equations with respect to the random input quantity and further symbolic computations of probabilistic moments and characteristics by the system MAPLE.

PL

Głównym celem niniejszej pracy jest zastosowanie probabilistycznej analizy numerycznej składającej się z uogólnionej metody perturbacji opartej na szeregu Taylora ze współczynnikami losowymi oraz z Metody Różnic Skończonych dla siatek regularnych. Jak wykazano w przykładach numerycznych, używając tego podejścia można również wyznaczać momenty probabilistyczne wyższych rzędów dla dowolnych funkcji losowych, co było niemożliwe dla zastosowań metod drugiego rzędu znanych wcześniej. Zastosowany tutaj algorytm numeryczny jest oparty na różniczkowaniu bezpośrednim hierarchicznych równań równowagi względem przyjętych zmiennych i na symbolicznym wyznaczaniu momentów i charakterystyk losowych przy pomocy programu MAPLE.

6

The stochastic perturbation - based finite volume method for the flow problems

Kamiński M., Ossowski R. L.

Journal of Technical Physics

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2009

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Vol. 50, no 4

297-315

EN

This article is devoted to the mathematical formulation and computational implementation of the Stochastic Finite Volume Method for heat and fluid flow problems. It is based on the stochastic generalized perturbation technique, which allows for determination of the probabilistic moments of the state variables or functions of the general stationary transport equations with random parameters. Both numerical case studies contain a comparison of the stochastic perturbation approach of different orders, their relations to the Monte-Carlo simulation results as well as the effect of the perturbation parameter and input coefficient of variation on the output state functions.

7

Eigenvalue analysis for high telecommunication towers with lognormal stiffness by the response function method and SFEM

Kamiński M. M., Szafran J.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

279-290

EN

The main aim of this paper is to demonstrate the application of the generalized stochastic perturbation techniąue to model the lognormal random variables in structural mechanics. This is done to study probabilistic characteristics of the eigenvibrations for the high telecommunication towers with random stiffness, which are modeled as the linear elastic 3D trusses. The generalized perturbation technique based on the Taylor expansion is implemented using the Stochastic Finite Element Method in its Response Function version. The main difficulty here, in a comparison to this techniąue previous applications, is a necessity of both odd and even order terms inclusion in all the Taylor expansions. The hybrid numerical approach combines the traditional FEM advantages with the symbolic computing and its visualization power and it enables for a verification of probabilistic convergence of the entire computational procedure.