Modelling of cement hydration in concrete structures with hybrid finite elements

UNIVERSIDADE DE LISBOA INSTITUTO SUPERIOR TÉCNICO Modelling of Cement Hydration in Concrete Structures with Hybrid Finite Elements Pham Tien Cuong Supervisor: Doctor Jỗo António Teixeira de Freitas Co-supervisor: Doctor Rui Manuel Carvalho Marques de Faria Thesis approved in public session to obtain the PhD Degree in Civil Engineering Jury final classification: Pass with Merit Jury Chairperson: Members of the Committee: Chairman of the IST Scientific Board Doctor Jỗo António Teixeira de Freitas Doctor Eduardo Nuno Brito Santos Júlio Doctor Mário Jorge de Seixas Pimentel Doctor Miguel Ângelo Dias Azenha Doctor Carlos Manuel Tiago Tavares Fernandes 2014 UNIVERSIDADE DE LISBOA INSTITUTO SUPERIOR TÉCNICO Modelling of Cement Hydration in Concrete Structures with Hybrid Finite Elements Pham Tien Cuong Supervisor: Doctor Jỗo António Teixeira de Freitas Co-supervisor: Doctor Rui Manuel Carvalho Marques de Faria Thesis approved in public session to obtain the PhD Degree in Civil Engineering Jury final classification: Pass with Merit Jury Chairperson: Chairman of the IST Scientific Board Members of the Committee: Doctor Jỗo António Teixeira de Freitas Professor Catedrático Instituto Superior Técnico, da Universidade de Lisboa Doctor Eduardo Nuno Brito Santos Júlio Professor Catedrático Instituto Superior Técnico, da Universidade de Lisboa Doctor Mário Jorge de Seixas Pimentel Professor Auxiliar da Faculdade de Engenharia, da Universidade Porto Doctor Miguel Ângelo Dias Azenha Professor Auxiliar da Escola de Engenharia, da Universidade Minho Doctor Carlos Manuel Tiago Tavares Fernandes Professor Auxiliar Instituto Superior Técnico, da Universidade de Lisboa Funding institutions Fundaỗóo para a Ciờncia e a Tecnologia (FCT), Grant reference number: SFRH/BD/76632/2011 2014 Abstract The main objective of this work is to develop a hybrid finite element formulation to model cement hydration in concrete structures First, a hybrid formulation of the finite element method is developed for the solution of heat transfer problems It is based on the independent approximation of the temperature and heat flux fields in the domain and on the boundary of the element, respectively The formulation is assessed in terms of convergence and robustness, and is validated using linear and nonlinear steady-state and transient test problems Second, the hybrid model is used to simulate the thermo-chemical response of ordinary Portland cement concrete using the hydration degree concept The hybrid formulation is extended to include the simulation of the effect of the heat source and to approximate directly the hydration degree field Third, the hybrid model is extended to simulate the coupled processes of heat transfer and moisture transport in concrete Two fields are approximated in the domain of the element (temperature and relative humidity) and on the boundaries (heat and moisture fluxes) The model can be used to simulate the hygro-thermo-chemical response of both hardening and hardened concrete and is valid for normal strength concrete and high-performance concrete Therefore, and besides the hydration reaction, the supporting mathematical model is extended to simulate the effects of the silica fume reaction and silicate polymerization in heat generation and in moisture consumption The corresponding degrees of reactions are also approximated in the hybrid finite element model Particular attention is paid to the simulation of the boundary conditions that occur in practical applications, which are typically nonlinear and vary in time and in space This is essential to assess the performance of the formulation using laboratory experiments and in situ tests reported in the literature These tests include one- and two-dimensional problems, as well as axisymmetric and three-dimensional applications Special attention is also paid to preserving the computational advantages offered by hybrid finite element formulations The approximation bases are orthogonal and naturally hierarchical, to induce high sparsity indices and enable adaptive p-refinement The solving system is not condensed at element level to enhance parallel processing The approximations of the state variables and of the geometry are uncoupled to enable the use of unstructured, coarse meshes of high-degree elements Approximation of the state variables is also uncoupled directionally in space to implement elements with high aspect ratios, frequently needed in the simulation of staged construction of concrete structures Keywords: Cement hydration; Heat transfer; Moisture transport; Normal strength concrete; High-performance concrete; Hybrid finite elements iii Resumo O objectivo central deste trabalho ộ desenvolver uma formulaỗóo hớbrida mộtodo dos elementos finitos para simular a hidrataỗóo cimento em estruturas de betóo Comeỗa-se por desenvolver uma formulaỗóo hớbrida para problemas de transferờncia de calor, a qual ộ baseada na aproximaỗóo independente da temperatura no domínio elemento e fluxo de calor nas fronteiras O desempenho da formulaỗóo ộ avaliado em termos de convergência e robustez, sendo validado recorrendo a problemas em regime estacionário e transitório, lineares e não-lineares A formulaỗóo ộ depois aplicada simulaỗóo da hidrataỗóo cimento Portland em betừes de resistờncia normal A reacỗóo e a fonte de calor associada ao processo termo-químico são descritas usando o conceito de grau de hidrataỗóo, sendo esse campo tambộm directamente aproximado na formulaỗóo por elementos finitos hớbridos Seguidamente procede-se generalizaỗóo da formulaỗóo para simular o acoplamento dos processos de transferência de calor e de transporte de humidade O modelo utilizado pode ser usado para simular o comportamento higro-termo-químico betóo nas diferentes fases de hidrataỗóo e de endurecimento, e é também válido para betões de alto desempenho, pois a formulaỗóo desenvolvida contempla as reacỗừes de fumo de sớlica e de polimerizaỗóo dos silicatos Os correspondentes graus de reacỗóo sóo tambộm directamente aproximados na formulaỗóo hớbrida desenvolvida A formulaỗóo é validada recorrendo aos resultados publicados sobre testes laboratoriais e ensaios in situ de provetes e estruturas de betão Dá-se, por isso, particular atenỗóo adequada simulaỗóo das condiỗừes de fronteira que ocorrem nos ensaios simulados numericamente, as quais são frequentemente nóo-lineares e variỏveis no espaỗo e no tempo Os testes de validaỗóo cobrem problemas unidimensionais, planos, axi-simộtricos e tridimensionais Procura-se, tambộm, conceber e implementar a formulaỗóo preservando as vantagens computacionais inerentes às variantes híbridas método dos elementos finitos As bases de aproximaỗóo sóo ortogonais e naturalmente hierỏrquicas, por forma a obter sistemas resolventes com elevados índices de esparsidade e facilitar a aplicaỗóo de processos de refinamento adaptativos O sistema resolvente não é condensado a nível elemento, a fim de potenciar a soluỗóo em processamento paralelo A aproximaỗóo das variỏveis de estado ộ desacoplada da aproximaỗóo da geometria, para permitir o uso de malhas não estruturadas envolvendo relativamente poucos elementos mas com elevado grau de aproximaỗóo A aproximaỗóo das variáveis de estado é também desacoplada direccionalmente, para permitir a utilizaỗóo de elementos com relaỗừes de dimensừes caracterớsticas muito elevadas, cuja utilizaỗóo ộ muito ỳtil na simulaỗóo processo construtivo de estruturas de betóo Palavras-chave: Hidrataỗóo cimento; Transferờncia de calor; Transporte de humidade; Betão de resistência normal; Betão de alto desempenho; Elementos finitos híbridos v Acknowledgements The work done in this thesis would not have been possible without the help and contribution of many people I would like to express my deep gratitude to all of them First, I would like to express my deepest gratitude to my supervisor, Prof Jỗo António Teixeira de Freitas for giving me the opportunity to a PhD at Departamento de Engenharia Civil, Arquitectura e Georecursos (DECivil) with his great scientific guidance, advice, and invaluable support throughout my PhD period He did his best to clarify my understanding of the non-conventional formulations of finite elements which were new to me His scientific knowledge contributed to this thesis in many ways He was always to help me solve not only problems in my work but also any difficulties during my stay in Portugal Furthermore, he always received me with sympathy for the long period of my stay I admire him for his profound knowledge, experience and his passion for research I will always be grateful for the lessons that I learned from our discussions Finally, I would like to thank him for the friendship that we have developed I express my sincerest gratitude to my co-supervisor Prof Rui Manuel Carvalho Marques de Faria from the University of Porto for all his advice, support, guidance and special contribution for this thesis He always received me warmly whenever we had meetings in Porto and he suggested many good ideas in our discussions I would like to express my thanks to the Departamento de Engenharia Civil, Arquitectura e Georecursos where I pursued my doctoral degree and to the Portuguese Foundation for Science and Technology (FCT) for their financial support through R&D project PTDC/ECM/70781/2006 launched at IST and research contract SFRH/BD/76632/2011 My thanks are also extended to O Nỳcleo de Pús-Graduaỗóo e Formaỗóo Contớnua for their kind help for my PhD’s procedure at IST I would like to thank all the staff and professors of the Structural Mechanics and Structures Section, especially Maria da Conceiỗóo Melo, Paula Cristina, Prof Moitinho de Almeida, Prof Luís Castro, Prof Orlando Pereira and Prof Eduardo Júlio for all their assistance and support My special thanks are conveyed to Prof Carlos Manuel Tiago Tavares Fernandes, from whom I received great help, support and encouragement as well as for the many daily talks about many aspects of life that we had, in particular, cycling and football discussions I sincerely wish to thank Prof João José Rio Tinto de Azevedo and his aunt, Ms Maria Tinto for letting me stay in her apartment without paying during my last year of my PhD I want to take this opportunity to thank Prof Miguel Ângelo Dias Azenha from the School of Engineering of the University of Minho He never hesitated to share his scientific expertise, particularly in early-age concrete behaviour Additionally, he gave me many good comments and suggestions, as well as many test data used in this work vii Acknowledgements viii I would also like to thank Prof Giovanni Di Luzio at Department of Structural Engineering, Politecnico di Milano, Italy, for his kind help with many discussions about the mathematical model of moisture as well as the data used in this thesis I gratefully acknowledge Maria Cristina (my supervisor’s wife) for her kindness, encouragement as well as her regular and kind regards to my family in Vietnam I would like to thank her for the close friendship that we have built My thanks also go to my colleagues at DECivil: Cristina Lopez, Vasco Mano, Júlio Pargana and Maria José I will always remember the fun moments we had together My particular gratitude to Luís Mendes, my office mate, who helped me a lot, namely with Matlab programing, and we also had a lot of fun during the first two years of my PhD I express many thanks to another office mate, Vladimir Ivannikov, a Russian PhD student who shared with me many types of fun for almost three years and half, such as watching together many games of European football cups, going to watch the games in Estádio da Luz, the home of the Benfica team, and travelling I have also spent many great moments with Vietnamese friends, who were studying and working in Lisbon: Hồng Văn Xiêm, Trần Nguyễn Ngun Khơi, Lê Hồng Trang, Hàn Thế Anh, Đinh Thanh Giang, Trần Thế Hùng, Nguyễn Thành Trung, Nguyễn Hà Anh Tuấn, Nguyễn Hoàng Sơn and Prof Phan Thành An Thank you all for sharing a good time with me during my stay in Lisbon My thanks also go to other friends: Diego Felix de Souza, Sanchit Singh, Mauricio Ramalho, Suresh, Dhiraj K Shah, Berner Panti and João Matos Cardoso for the great times of relaxing and travelling during my stay in Portugal, especially with all the funny things they did that I cannot forget Furthermore, I am particularly grateful to my close friend, Cao Đức Toàn, without whose help I may not have been to Lisbon I would like to thank my mother, mother-in-law, sisters, brothers, sisters-in-law and brothersin-law, who encouraged, motivated and supported me throughout my years of study They admire me and I deeply love them Finally, my most special words are for my wife, Lê Thị Tuyết Chinh and my son, Phạm Gia Hoàng, for having me in their hearts Nothing could have been possible without their love, encouragement and patience They gave me the motivation to work and to overcome my hard time during my stay in Portugal Pham Tien Cuong Lisbon, June 2014 152 Chapter 5: Closure Different alternatives are offered for the analytical definition of the cement hydration, namely piecewise linear, exponential and power laws They can be selected to best-fit the experimental data, and the supporting information is typically drawn from the temperature rise obtained under adiabatic or semi-adiabatic conditions, or based on isothermal characterization tests of cement hydration The silica fume reaction function is defined in exponential form, whereas a linear function is used to express the rate of silicate polymerization Their characterization is also based on data obtained through experimental testing The model is designed to simulate the wide variety of boundary conditions that can be encountered in practical concrete applications Their description can be complex, as environmental conditions vary in space and in time through air temperature, ambient relative humidity and wind speed Thermal conductance or resistance conditions are included in the model to simulate the contact conditions that occur in phased construction This option is easily implemented with hybrid elements, though often omitted in commercial codes as it requires the use of interface elements Users of commercial codes often choose to implement instantaneous continuity of temperature on contact, which leads to a localized instability of the solution that tends to rapidly decay in time The same technique has been extended to the moisture transport problem Furthermore, a different approach has been followed to simulate convective moisture transfer It consists in using a moisture transfer coefficient dependent on moisture capacity, proved to be essential for recovering the results registered in experimental testing In what regards numerical modelling, the hybrid formulation used here is based on independent approximations of the temperature and relative humidity fields in the domain of the element, and of the heat and moisture flux fields on the boundaries To facilitate postprocessing, the degrees of cement hydration, silica fume reaction and silicate polymerization are also directly approximated in the domain of the element The approximation bases are used to enforce on average, in the sense of Galerkin methodology, the domain and boundary conditions of the problem The resulting weak forms of the thermo-chemical and hygro-thermo-chemical applications addressed in this research define symmetric, first-order, nonlinear problems They are integrated in time using the thetamethod and solved with the Newton-Raphson method Analytical expressions are used to calculate all gradient terms, thus avoiding the loss of accuracy caused by finite difference approximations The domain and boundary approximation bases are implemented with orthogonal and naturally hierarchical functions They lead to highly sparse solving systems and a naturally padaptive implementation The systems are also well-suited to parallelization, as the domain degrees-of-freedom are strictly element dependent, and the boundary degrees-of-freedom are shared by at most two connecting elements 5.3 Results 153 Modelling is flexible because these approximations are independent and, for each field, different degrees can be used in each spatial direction This feature is very useful when implementing elements with high aspect-ratios In addition, the approximation bases are uncoupled from the geometric mapping of the element This opens the possibility of implementing the formulation on unstructured, coarse meshes of high-degree elements and obtain accurate and numerically stable solutions 5.3 Results The objectives initially set on the development, validation and assessment of a hybrid variant of the finite element method designed to simulate the hygro-thermo-chemical behaviour of concrete structures since early-ages have been successfully met The mathematical model used to simulate cement hydration is powerful and open to further refinement, and the numerical solution method – the centre of this research – proved well in both absolute and relative terms It has been shown that the hybrid finite element provides good levels of accuracy, yields relatively high rates of convergence, is robust in terms of element shape distortion and performs well under transient and strongly nonlinear conditions The formulation is very flexible, as it allows the user to set different levels of refinement in the definition of the geometry and of the level of approximation in each of the supporting problems of heat transfer and moisture control It has also been shown that this flexibility substantially simplifies the simulation of singular and boundary layer fields, and it is instrumental to simulate the phased construction of concrete structures However, and because they were not set as objectives for this research, two features essential to the analysis of real engineering problems have not been demonstrated: suitability to adaptive p-refinement and to parallel processing The results reported in this thesis confirm that the hybrid formulation of the finite element method can recover the levels of accuracy obtained with the conform formulation using simpler meshes and, according to the published results, requiring considerably less degreesof-freedom In addition, the numerical tests illustrate the gains that can be attained by uncoupling the approximations of the geometric and state variables to use unstructured meshes with uncoupled finite element bases (in the sense of using different degrees of approximation in each direction, and independently of the geometric mapping) It has been shown that this technique performs well when implementing elements with high aspect-ratios The price paid for this gain in flexibility and suitability to adaptive refinement and parallelization is the increase in the number of degrees-of-freedom per element caused by the additional approximation of the flux fields This may be relevant in solid mechanics problems, but has a marginal effect in the scope of the applications presented here, where convective boundary conditions dominate, and temperature or relative humidity are rarely prescribed Moreover, this relative increase in the number of degrees-of-freedom per element 154 Chapter 5: Closure does not imply a necessarily higher number of degrees-of-freedom of the assembled mesh, as shown by the extensive testing reported in this thesis Setting apart flexibility in modelling, where the advantage of the hybrid element is quite clear, it is difficult, and even unfair in some aspects, to establish a conclusive comparative analysis of the relative performances of hybrid and conventional elements A direct comparison can only be fairly set in the context of the solution of transient heat transfer problems, addressed in Chapter The results there presented show that the hybrid element performs at least as well as the conventional element Comparisons in the contexts of thermo-chemical and hygrothermo-chemical analyses, addressed in Chapters and 4, would be biased because it would imply assessing special purpose codes against extensions of commercial codes with built-in design limitations No mention is made to storage requirements and execution times in the reports from where the comparative tests presented here have been taken Even if they did, that information is of limited interest because commercial codes are written to be run in different computing platforms The number of degrees-of-freedom is commonly used to assess relative performance in broad terms Measured in this way, the tests reported here indicate that the hybrid element has a higher potential It is recalled that the more demanding tests presented here are run in out-of-the-shelf laptop computers in a time scale of minutes Real life problems would justify the use of parallel processing to attain running times adequate for professional practice 5.4 Improvements In computational terms, special attention has been paid to minimization of storage requirements by exploiting the high sparsity of the hybrid finite element solving systems However, there is room for improvement in this topic, as the simplest sparse system storage scheme has been implemented (the value of the non-null coefficient and its row and column positions are stored) Although a quite efficient direct solver was used, based on the multifrontal method, no assessment tests have been made on the use of the eventually faster preconditioned versions of the biconjugate gradient algorithm They may prove to be better suited to the time stepping, iterative solution of nonlinear systems Another area of immediate improvement is user-friendliness, particularly in what concerns input and output interfaces New users would need specific training to efficiently handle the codes produced in the framework of this research This is certainly true in what concerns meshing and choice of approximation bases, as the concepts used are quite different from those that experienced users of commercial finite element codes are familiar with 5.5 Extensions 155 5.5 Extensions The most obvious extension of the research reported here is its coupling with the modelling of the mechanical response of early-age concrete structures That extension has been set as the ultimate objective of the research project that supported this thesis, and is already under development under a different doctoral project As the mechanical response is usually assumed to be geometrically linear, the finite element modelling consists is coupling the hybrid finite element formulation of the hygro-thermochemical problem with a hybrid-Trefftz formulation of the mechanical problem It is based on the stress model of that variant of the finite element method, as the main priority is to have an effective control of the stress field during the hardening process Within the framework of the research reported here, one-dimensional, two-dimensional, axisymmetric and three-dimensional models have been developed for heat transfer and thermo-chemical problems All those models, with the exception of the latter, were developed for the hygro-thermo-chemical problems Thus, their extension to three-dimensional applications is a natural development of the research reported here This extension is necessary to model correctly most engineering applications to sensitive structures, namely bridges and dams The three-dimensional extension has implications on both modelling and computer implementation Computing with reasonable execution times would certainly imply the full use of the p-refinement and parallel processing features inherent to the hybrid formulation used here It terms of modelling it would not affect concepts or techniques, but would have to centre instead in the generalization of the ambientdependent boundary conditions It would imply investing in geometry control, to model correctly solar positioning and shading in convective and 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the finite element method 1.3 Modelling of cement hydration 1.3 Modelling of cement hydration Over the... Abstract The main objective of this work is to develop a hybrid finite element formulation to model cement hydration in concrete structures First, a hybrid formulation of the finite element method... degree of cement hydration α ∞ , α c∞ ultimate degree of cement hydration α ad degree of cement hydration under adiabatic condition α c0 threshold value of degree of cement hydration α0 , αc0 initial