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Combinatorial problems related to sequences with repeated entriesArchibald, Margaret Lyn 15 November 2006 (has links)
Student Number : 9708525G -
PhD thesis -
School of Mathematics -
Faculty of Science / Sequences of numbers have important applications in the field of Computer Science.
As a result they have become increasingly regarded in Mathematics, since analysis
can be instrumental in investigating algorithms.
Three concepts are discussed in this thesis, all of which are concerned with ‘words’
or ‘sequences’ of natural numbers where repeated letters are allowed:
• The number of distinct values in a sequence with geometric distri-
bution
In Part I, a sample which is geometrically distributed is considered, with the
objective of counting how many different letters occur at least once in the
sample. It is concluded that the number of distinct letters grows like log n as
n → ∞. This is then generalised to the question of how many letters occur
at least b times in a word.
• The position of the maximum (and/or minimum) in a sequence
with geometric distribution
Part II involves many variations on the central theme which addresses the
question: “What is the probability that the maximum in a geometrically distributed
sample occurs in the first d letters of a word of length n?” (assuming
d ≤ n). Initially, d is considered fixed, but in later chapters d is allowed to
grow with n. It is found that for 1 ≤ d = o(n), the results are the same as
when d is fixed.
• The average depth of a key in a binary search tree formed from a
sequence with repeated entries
Lastly, in Part III, random sequences are examined where repeated letters
are allowed. First, the average left-going depth of the first one is found,
and later the right-going path to the first r if the alphabet is {1, . . . , r} is
examined. The final chapter uses a merge (or ‘shuffle’) operator to obtain
the average depth of an arbitrary node, which can be expressed in terms of
the left-going and right-going depths.
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Análise experimental e numérico-computacional da influência do jateamento com granalha na propagação de trincas. / Experimental and numerical-computational analysis of the influence of shot peening on crack propagation.Rosalie, Beugre Ouronon Marie 20 February 2019 (has links)
O Jateamento com granalha (shot peening, em inglês) é um processo de fabricação amplamente utilizado em indústrias mecânicas, automotivas, navais, aeronáuticas e outras indústrias metalúrgicas. O objetivo principal é induzir tensões residuais de compressão para melhorar a vida à fadiga das peças e estruturas. A modelagem e simulação do processo são muito difíceis, pois envolvem a consideração de muitos e complexos aspectos. Este trabalho propõe um método de modelagem tridimensional utilizando o método dos elementos finitos para a simulação numérica desse processo assim como a sua influência na propagação de trinca, considerando efeitos pouco estudados, como o amolecimento do material devido ao aumento da temperatura na peça durante os impactos, ao comportamento elasto-plástico das granalhas e a rugosidade da peça tratada nos parâmetros da trinca. A modelagem proposta para simular o processo de jateamento com granalhas inclui: (i) análise dinâmica explícita, (ii) modelagem tridimensional, (iii) cálculo da velocidade real das granalhas, (iv) modelo de contato com coeficientes de atrito estático e dinâmico entre as áreas em contato, (v) o modelo constitutivo de Johnson-Cook para o comportamento do material alvo, (vi) um comportamento elasto-plástico multilinear para as granalhas, (vii) a consideração da cobertura real de uma área dada da superfície jateada, (viii) a consideração do aumento da temperatura no material durante o processo e (ix) o relaxamento de tensões residuais devido a este aumento de temperatura. As simulações numérico-computacionais para investigar a influência do processo em fenômenos, tais como fadiga e propagação de trinca, necessitou uma nova abordagem para a incorporação das tensões residuais num modelo de elementos finitos. Além disso, foram realizados vários ensaios quase-estáticos e dinâmicos para a determinação dos parâmetros do modelo constitutivo de Johnson-Cook para o aço AISI 5160. Para a avaliação das metodologias propostas, as tensões residuais foram introduzidas e avaliadas em uma mola parabólica automotiva, sendo que as tensões residuais foram avaliadas através da técnica de difração de raios-X, a intensidade e a cobertura do processo de jateamento com granalha na mola foram avaliadas com Lupa e rugosímetro. Também, dados experimentais de ensaios de fadiga relatados na literatura, realizados em corpos de prova tipo CT foram utilizados. A modelagem numérica do processo de jateamento com granalha desenvolvida neste trabalho consegue prever com sucesso os perfis das tensões residuais, com valores bem próximos aos obtidos experimentalmente. A investigação numérica do efeito do processo na vida à fadiga e fratura das peças comprovou a sua influência nos parâmetros de propagação de trinca nas regiões afetadas pelo processo, ocasionando um amplo aumento da vida à fadiga, e confirmou o relaxamento das tensões residuais devido aos ciclos de carregamento conforme as observações experimentais na literatura. / Shot peening is a manufacturing process widely used in the mechanical, automotive, marine, aeronautical and other metallurgical industries. The main purpose is to induce residual compression stresses to improve fatigue life. The modeling and simulation of the process are very difficult as they involve the consideration of many and complex aspects. This work proposes a three-dimensional modelling method using the finite element method for the numerical simulation of this process as well as its influence on crack propagation, considering little studied effects such as softening of the material due to the increase in temperature in the part during impacts , the elastic-plastic behavior of the shot and the roughness of the treated part in the crack parameters. The proposed modelling approach to simulate the shot peening process includes: (i) explicit dynamic analysis, (ii) three-dimensional modeling, (iii) calculation of the real velocity of the shot, (iv) contact model with static and dynamic friction coefficients between the contact areas, (v) the Johnson-Cook constitutive model for target material behavior, (vi) a multilinear elasto-plastic behavior for the shot, (vii) consideration of real coverage in a given area of the peened surface, (viii) consideration of temperature increase in the material during the process and (ix) relaxation of residual stresses due to this rise in temperature. Numerical-computational simulations to investigate the influence of the shot peening process on phenomena, such as fatigue and crack propagation, required a new approach for the incorporation of residual stress in a finite element model. In addition, several quasi-static and dynamic tests were carried out to determine the parameters of the constitutive Johnson-Cook model for AISI 5160 steel. To assess the proposed methodologies, the residual stresses were introduced and evaluated in an automotive parabolic spring, with the residual stresses being evaluated by the X-ray diffraction technique, the intensity and coverage of the shot peening process for the spring were evaluated with magnifying glass and roughness gauge. Also, fatigue tests available in the literature, carried out on types of CT specimens, were used. The numerical modeling of the shot peening process developed in this work can successfully predict the residual stress profiles, with values close to those obtained experimentally. The numerical investigation of the effect of the process in the fatigue life and fracture of the pieces proved its influence in the parameters of crack propagation in the regions affected by the process, causing a wide increase of the fatigue life, and confirmed the relaxation of the residual stresses due to the cycles loading according to experimental observations in the literature.
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