Evaluation of Accuracy and Efficiency of Numerical Methods for Contact Problems

Tato diplomová práce se zabývá variačními metodami, které umožňují formulovat problém kontaktu lineárně pružného tělesa bez tření jako nepodmíněnou variační rovnost, která může být posléze diskretizována a řešena metodou konečných prvků. Hlavní důraz je kladen na Nitscheho metody podle Wriggerse a Zavariseho [56] a podle Fabrého, Pousina a Renarda [15]. V současné době nejrozšířenější konečněprvkové softwarové balíky, jako jsou ANSYS, ABAQUS a COMSOL, využívají pro modelování kontaktu především standardní metody penalty a smíšené metody [57, Kapitola 1.1.1, p.7]. Ukazuje se, že právě Nitscheho metody mají potenciál překonat klasické obtíže spojené se standardními metodami penalty a smíšenými metodami. Na rozdíl od metod penalty jsou Nitscheho metody konzistentní a kontaktní okrajové podmínky jsou vynuceny přesně (na teoretické úrovni). Je také možné využít mnohem menší hodnotu parametru penalty, čímž se lze vyhnout problémům spojeným se špatným podmíněním úlohy, charakteristickým pro metody penalty. Nitscheho metoda ale současně nevyžaduje přidání žádných dalších neznámých (Lagrangeových multiplikátorů) a výsledný diskrétní systém tak není nadbytečně rozšířen, jako je tomu v případě smíšených metod. Oproti smíšeným metodám také není třeba věnovat pozornost splnění Babuškovy-Brezziho podmínky. V této diplomové práci se ukazuje, že analyzované Nitscheho metody úzce souvisejí s metodami penalty a metodou augmentovaného lagrangiánu. V práci jsou prezentovány slabé formulace těchto metod a zkoumají se rozdíly mezi formulací Nitscheho metody podle Wriggerse a podle Fabrého, Pousina a Renarda. Všechny metody jsou implementovány do prostředí FEniCS (výpočetní platforma pro řešení parciálních diferenciálních rovnic metodou konečných prvků) a jejich přesnost a výkonnost se testuje na různých dvourozměrných a trojrozměrných problémech kontaktu lineárně pružného tělesa s dokonale tuhou rovinou. Na jednoduchém dvourozměrném příkladu je ukázáno, že funkce, kterou získáme jako levou stranu diskretizované slabé formy Wriggersovy varianty Nitscheho metody, není spojitá vzhledem k neznámým stupňům volnosti. Tento poznatek vysvětluje problémy s konvergencí Newtonovy metody při řešení Wriggersovou variantou Nitscheho metodou, které jsme zaznamenali při numerických experimentech.This thesis is concerned with various methods that allow us to formulate the frictionless linear elastic contact problems as an unconstrained variational equality, which is then discretised and solved with the finite element method. The main focus is on Nitsche methods in the forms used respectively by Wriggers and Zavarise [56] and Fabré, Pousin and Renard [15]. Currently, standard penalty and mixed methods are dominant in the modern leading finite element software packages such as ANSYS, ABAQUS and COMSOL [57, Chapter 1.1.1, p.7]. Nitsche methods display a potential to overcome classic drawbacks of the penalty and mixed methods. Unlike penalty methods, Nitsche methods are consistent, and contact boundary conditions are enforced precisely (on the theoretical level). Also, a significantly smaller value of the penalty parameter is necessary and the possible ill-conditioning, so characteristic for penalty methods, is thus avoided. At the same time, no additional unknowns (Lagrange multipliers) are introduced; thus, the corresponding discrete system is not enlarged, and one does not have to worry about the Babuška-Brezzi condition. In this thesis was shown that the analysed Nitsche methods are closely related to penalty methods and the augmented Lagrangian method. The weak forms of all these methods are presented, and differences between Wriggers' version and Fabré, Pousin and Renard's version of Nitsche method are investigated. All methods are implemented in FEniCS (the computational platform for solving partial differential equations with the finite element method), and their accuracy and efficiency is tested on various two- and three-dimensional numerical examples of contact of an elastic body with a rigid plane. By means of the simple two-dimensional example it is shown that the function obtained as the left-hand side of the discretised weak form of the Nitsche-Wriggers method is not continuous with respect to the unknown displacement DOFs. This finding explains the convergence problems (of Newton's method) that the Nitsche-Wriggers method suffers from, unlike other investigated methods

Predicting the impact of water transport on carbonation-induced corrosion in variably saturated reinforced concrete

A modelling framework for predicting carbonation-induced corrosion in reinforced concrete is presented. The framework constituents include a new model for water transport in cracked concrete, a link between corrosion current density and water saturation, and a theory for characterising concrete carbonation. The theoretical framework is numerically implemented using the finite element method and model predictions are extensively benchmarked against experimental data. The results show that the model is capable of accurately predicting carbonation progress, as well as wetting and drying of cracked and uncracked concrete, revealing a very good agreement with independent experiments from a set of consistent parameters. In addition, insight is gained into the evolution of carbonation penetration and corrosion current density under periodic wetting and drying conditions. Among others, we find that cyclic wetting periods significantly speed up the carbonation progress and that the induced corrosion current density is very sensitive to concrete saturation

Longevity of Cane Corso Italiano dog breed and its relationship with hair colour

The Cane Corso Italiano belongs among the new dog breeds that were fully recognised by Federation Cynologique Internationale (FCI) in 2007. For the first time, this study describes a median lifespan using the data of 232 dogs of the Cane Corso Italiano breed collected from kennels and individual owners from 25 countries. The median lifespan of the whole examined group is 9.29 years (IQR 6.98-11.12, IQR = Interquartile Range). This paper is the first to describe the possible relationship between median lifespan and hair colour within one breed. The longest living group is formed by black brindle coloured dogs, with a median of 10.30 years (IQR 8.33-13.00), and brindle coloured dogs, with a median of 10.13 years (IQR 7.12-11.25). The median lifespan of black brindle dogs exceeded the overall median lifespan of all dogs by 1.01 year and the median lifespan of other colour dogs by 2.21 years. Our results suggest a possible way for a prolongation of age at death of the Cane Corso Italiano breed using appropriate breeding. The median lifespan of male Cane Corso Italiano dogs is 9.25 years (IQR 6.97-11.00) and female Cane Corso Italiano dogs 9.33 years (IQR 7.00-11.31). The statistical analysis using the Independent Samples Student’s t test confirmed that the lifespan of female dogs did not exceed the median lifespan of male dogs (P>0.01)

Inheritance of coat colour in the cane Corso Italiano dog

Abstract Background The inheritance of different coat colours in the Cane Corso Italiano dog has not been described thus far. We analysed data from 23,271 dogs and bitches using the Cane Corso Italiano Pedigree Database. We are describing for the first time the coat colour segregation ratios in Cane Corso Italiano offspring arising from crosses between parents of all possible coat colour combinations. Results Segregation ratios that do not follow a Mendelian pattern suggest that additional genes are active in the determination of coat colour. Segregation ratios of offspring produced by parental crossing (male colour A x female colour B) were compared with the ratios of offspring produced by reciprocal crossing (male colour B x female colour A) in all possible coat colour combinations. Most of the segregation ratios were the same, but some segregation ratios in reciprocal crosses differed. This result suggests that at least one gene responsible for coat colour is located on a sex chromosome. The sex ratio was analysed in the offspring of all colour groups. A ratio of 1:1 was not confirmed in 8 colour groups by the chi-square test. Conclusions We described for the first time coat colour segregation ratios in Cane Corso Italiano dogs. Furthermore, we present the hypothesis that at least one gene responsible for coat colour is located on a sex chromosome

Longevity of Cane Corso Italiano dog breed and its relationship with hair colour

The Cane Corso Italiano belongs among the new dog breeds that were fully recognised by Federation Cynologique Internationale (FCI) in 2007. For the first time, this study describes a median lifespan using the data of 232 dogs of the Cane Corso Italiano breed collected from kennels and individual owners from 25 countries. The median lifespan of the whole examined group is 9.29 years (IQR 6.98-11.12, IQR = Interquartile Range). This paper is the first to describe the possible relationship between median lifespan and hair colour within one breed. The longest living group is formed by black brindle coloured dogs, with a median of 10.30 years (IQR 8.33-13.00), and brindle coloured dogs, with a median of 10.13 years (IQR 7.12-11.25). The median lifespan of black brindle dogs exceeded the overall median lifespan of all dogs by 1.01 year and the median lifespan of other colour dogs by 2.21 years. Our results suggest a possible way for a prolongation of age at death of the Cane Corso Italiano breed using appropriate breeding. The median lifespan of male Cane Corso Italiano dogs is 9.25 years (IQR 6.97-11.00) and female Cane Corso Italiano dogs 9.33 years (IQR 7.00-11.31). The statistical analysis using the Independent Samples Student’s t test confirmed that the lifespan of female dogs did not exceed the median lifespan of male dogs (P>0.01).Keywords: Cane Corso Italiano, Dog Breed, Kennel, Life Prolongation, Lifespa