Introduction
Motivation and scope
This study is part of a continuous research program focused on applying hybrid finite element formulations to engineering problems Initiated at IST in the 1980s, the primary goal was to enhance the traditional finite element method, which typically uses compatible or conforming formulations, by introducing a method aimed at achieving equilibrated solutions.
The research progressed in two key directions: first, by relaxing all constraints on equilibrium and compatibility, which resulted in the creation of hybrid-mixed variants of the finite element method; second, by imposing strict conditions on the approximation bases to ensure local compliance with equilibrium, compatibility, and constitutive relations, leading to the development of the hybrid-Trefftz variant.
Initial applications of two- and three-dimensional elastostatic and elastoplastic methods focused on building foundational skills and gaining experience Subsequent phases of the study expanded to include physically and geometrically nonlinear problems, as well as the application of these concepts to structural dynamics As expertise grew, the research progressed to modeling multiphase issues, particularly in relation to incompressible soft tissues and both saturated and unsaturated soils.
Recent efforts have focused on solving multiphysics problems, specifically through the simulation of the hygro-thermo-chemo-mechanical behavior of early-age concrete structures This research, conducted with the support of LABEST – the Laboratory for Concrete Research and Structural Behavior at the University of Porto, highlights the importance of expert guidance in scientific, technical, and technological aspects related to the application The findings are significantly informed by the expertise developed in the study of early-age behavior of reinforced concrete (RC) structures.
Object and objective
This research focuses on creating and validating a hybrid finite element model that simulates the nonlinear, transient, hygro-thermo-chemical processes involved in cement hydration within concrete The advancements detailed in this study contribute to the ongoing integration of this simulation with the mechanical behavior of structural components.
This research addresses three key problems: First, it explores a hybrid finite element formulation for solving heat transfer issues by independently approximating the temperature field within the discretized structure and the heat flux at its boundaries Second, it investigates the thermo-chemical processes involved in the hydration of ordinary Portland cement (OPC) in concrete structures, assuming adequate moisture levels for the chemical reaction Lastly, it examines the hygro-thermo-chemical dynamics in high-performance concrete (HPC) structures, focusing on heat transfer and moisture transport mechanisms resulting from cement hydration, silica fume reactions, and silicate polymerization.
The modeling of the thermo-chemical problem is achieved by extending the heat transfer problem to include a heat source term that represents cement hydration While not strictly required, the finite element approximation is applied to the cement hydration field, providing a clear spatial and temporal representation of the hydration process.
The modeling of the hygro-thermo-chemical problem is enhanced by extending the necessary approximations for the relative humidity within the element's domain and the moisture flux at its boundary Additionally, the optional approximation of the chemical process is broadened to incorporate the degrees of silica fume reaction and silicate polymerization.
The research aims to evaluate and confirm the effectiveness of a proposed hybrid finite element formulation in addressing specific problems The assessment and validation of heat transfer solutions rely on rigorous convergence and robustness testing.
Validation of thermo-chemical and hygro-thermo-chemical issues is systematically conducted through laboratory tests and structural monitoring results The relative assessment relies on literature findings, which are generally derived from the established formulation of the finite element method.
Modelling of cement hydration
In recent decades, various alternative mathematical models have been developed to simulate the hydration process and microstructure evolution of cementitious materials These models are categorized into three levels based on characteristic length scale: micro-level, meso-level, and macro-level.
Micro-level models illustrate the physicochemical processes of cement constituents, highlighting the complexity of cement hydration and microstructure formation These models are essential for a comprehensive understanding of the intricate interactions that occur during these processes.
The complexity of cement hydration has led to the creation of phenomenological approaches, which characterize macro-level models in structural applications These models effectively capture the key aspects of cement hydration through straightforward analytical descriptions, utilizing coefficients and functions that are optimized based on laboratory experiments conducted under tightly controlled conditions.
Meso-scale models serve to connect micro and macro-level approaches by providing practical insights into the microstructure formation of cement paste However, this transitional phase is not addressed in the applications discussed in this article.
At micro-level, all models focus on the behaviour of the cement paste as described by their constituents: tricalcium silicate (C3S), dicalcium silicate (C2S), tricalcium aluminate (C3A) and tetracalcium aluminoferrite (C4AF)
Typically, all models try to represent the long-established process of OPC hydration, e.g
Hydration in cement begins immediately upon mixing with water, with each compound reacting at different rates and generating varying amounts of heat Tricalcium aluminate (C3A) hydrates and hardens the fastest, releasing significant heat and contributing to early strength, which is why gypsum is added to retard its setting Tricalcium silicate (C3S) also hydrates quickly, producing high heat and playing a major role in initial setting and early strength, particularly in Portland cements with higher C3S content Tetracalcium aluminoferrite (C4AF) hydrates rapidly but has a minimal impact on strength; it primarily helps reduce kiln temperatures during manufacturing due to the heat it generates In contrast, dicalcium silicate (C2S) hydrates slowly, releasing less heat but is crucial for strength development beyond one week The hydration process involves complex chemical reactions and thermodynamic changes, necessitating simultaneous monitoring of cement particle dissolution and microstructure development over time Two primary modeling approaches for cement hydration at the micro-level include the discretization approach and the vector approach.
The discretization approach integrates digital image processing with the finite element method by dividing the entire volume of cement grains into smaller elements, typically using a uniform grid of volumetric pixels (voxels) Each element can be assigned unique material properties, sizes, and shapes pertinent to each phase, enabling the simulation of multiphase materials such as solids, gels, fluids, and gases, while also allowing for the definition of arbitrary grain shapes.
CEMHYD3D is a highly regarded code utilizing the discretization approach for simulating cement hydration and microstructure formation While users can adjust the size and number of voxels, the total count typically reaches around ten million, resulting in significant computational demands An illustration of the microstructure evolution of cement paste at a specific hydration degree, generated by the CEMHYD3D model, is depicted in Figure 1.1a.
Figure 1.1: Microstructure development at a certain degree of hydration
The DUCOM code, developed at the University of Tokyo's Concrete Laboratory, is a finite element program designed to evaluate the durability of concrete at early ages It also simulates the hydration process and the development of the microstructure in cement-based materials.
The vector approach eliminates the need to discretize cement particles, instead characterizing each particle by its position, size, and orientation through vectors that represent spherical shells of hydration products This method, initially proposed by Jennings and Johnson, aims to simulate the microstructure development during hydration by modeling the nucleation of spherical particles in three-dimensional space.
Recent advancements in modeling techniques, exemplified by HYMOSTRUC and IPKM, have paved the way for additional tools such as SPACE and àic These models simplify the representation of cement particles by assuming they are spherical, thereby minimizing the complexities associated with their geometric descriptions.
1.3 Modelling of cement hydration 5 of the microstructure formation of a cement paste during the hydration process obtained with models HYMOSTRUC and àic are shown in Figures 1.1b) and 1.1c), respectively
The vector-based models are specifically engineered to simulate the evolution of chemical reactions in mineral compounds The code àic methodology has emerged as a leading solution, striking an optimal balance between simulation speed and the management of extensive data related to the process Furthermore, code àic's flexible design empowers users to customize simulations by defining or altering various aspects of the process.
The macro-level simulation of cement hydration processes in engineering involves solving three interconnected equations: heat transfer, moisture transport, and chemical reactions This approach operates at a macro-scale, utilizing simplifying assumptions to create a medium that exhibits equivalent behavior.
The intervening coefficients that describe conductivity, diffusivity, specific heat, and moisture capacity are redefined to incorporate analytical expressions that capture two critical aspects of the hydration process: the heat generated from cement hydration and silica fume reactions, as well as the moisture consumption associated with these reactions, now linked to silicate polymerization Effective encoding of these phenomena is essential for accurately modeling the spatial and temporal evolution of temperature, humidity, and chemical reaction degrees Consequently, alternative macro-level models depend heavily on precise data regarding microstructure and hydration processes.
Alternative models have been introduced to explain the hardening of concrete during the hydration process, but over time, two inter-related concepts have emerged as the primary focus: equivalent age (or maturity) and degree of reaction A detailed examination of these concepts can be found in Chapter 3, while the following description highlights key milestones in their development.
In the early 1950s, Rastrup introduced a time-temperature function related to the heat of hydration, capturing essential phenomenological aspects It was established that for non-massive structures using standard Portland cements, approximately 50% of the total heat is released within one to three days, around 75% in one week, and nearly 90% within six months The heat of hydration is primarily influenced by the proportions of C3S and C3A in the cement, and is also affected by factors such as the water-to-cement (w/c) ratio, cement fineness, and curing temperature Rastrup's model, with ongoing adjustments, has been successfully applied to simulate cement hydration in various concrete structures, including reinforced concrete bridge decks and concrete dams.
Finite element modelling
A finite element formulation is termed mixed when it approximates two or more state variables within the element's domain, while it is classified as hybrid if at least one variable is also approximated on the boundary Hybrid finite element formulations are primarily categorized into three groups: hybrid-mixed, hybrid, and hybrid-Trefftz.
1.4 Finite element modelling 7 below, their distinguishing features are directly related to the constraints set on the approximation of the state variables
Two models exist for the formulations discussed, based on the equations chosen for direct control Originally developed in structural mechanics, particularly for linear elastostatic applications, these models are commonly referred to as the displacement model and the stress model.
The design of the displacement model in the finite element method aims to achieve solutions that meet all compatibility conditions in a strong form, both within the element's domain and along its boundary Conversely, the complementary stress model seeks to generate equilibrated solutions that satisfy equilibrium conditions in a strong form Essentially, these displacement and stress models represent extensions of the displacement and force methods used in structural analysis, with the former focusing on kinematically admissible solutions and the latter on statically admissible solutions.
1.4.1 Hybrid-mixed finite element models
The hybrid-mixed formulation utilizes the finite element method with minimal constraints on approximation bases, specifically focusing on completeness and linear independence As a result, the domain conditions of the problem—such as equilibrium, compatibility, and elasticity—are not satisfied locally, but are instead enforced in an approximate or weak form.
The development of these formulations indicates a mixed approach, where both displacement and stress fields are explicitly approximated When selecting the displacement or stress field as the primary approximation, it becomes essential to independently approximate the forces or displacements on the boundary, leading to the classification of the formulation as hybrid-mixed.
Hybrid-mixed formulations face significant challenges, primarily due to their susceptibility to spurious modes, which can result in rank-deficient solving systems, a concern highlighted by Brezzi and Fortin Additionally, these mixed formulations often require a high number of degrees-of-freedom due to the simultaneous approximation of various fields This issue can be alleviated by employing orthogonal bases, which produce sparser solving systems Furthermore, the potential to utilize any complete, linearly independent basis has led to the innovative approach of constructing approximation bases with digital functions, such as wavelets, to optimize the computational architecture utilized in their implementation.
The hybrid finite element method enhances the traditional approach by imposing additional constraints on the approximation bases within its hybrid-mixed formulation This method derives the hybrid displacement (stress) model by ensuring that the displacement (stress) basis adheres to the strong form of domain compatibility and equilibrium conditions Consequently, the hybrid finite element formulation allows for the independent approximation of the displacement (stress) field within the element's domain, while simultaneously addressing the forces (displacements) on the boundary.
The conventional formulation of the finite element method is a specific instance of the hybrid displacement model, where the displacement approximation is constrained to meet boundary and inter-element continuity conditions in a strong form, as initially proposed by Ritz in solving differential equations This continuity condition is effectively achieved through the isoparametric concept, which is a key factor in the widespread adoption of conforming finite elements in engineering applications, despite their potential to yield unsafe solutions.
Efforts to establish equilibrium elements have focused on achieving reliable solutions While it is relatively easy to create approximation bases that meet local equilibrium conditions within a domain, ensuring that both boundary and inter-element equilibrium conditions are satisfied in strong form remains challenging, often leading to spurious solutions.
The development of a finite element method that satisfies both strong form domain equilibrium conditions and weak form boundary equilibrium conditions is straightforward, as demonstrated by the hybrid stress element proposed by Pian This method not only ensures local equilibrium within the element's domain but also effectively emulates conforming elements, which has led to its widespread inclusion in the libraries of leading commercial finite element software.
The process involves condensing the element's solving system based on its boundary degrees of freedom, which represent displacements This results in a stiffness format characteristic of the conforming element Consequently, aside from this element-level operation, all other procedures within a commercial code can be utilized for finite element analysis, including mesh generation, system definition, and solution Additionally, post-processing must be tailored at the element level to accurately extract the approximated and equilibrated stress field.
1.4.3 Hybrid-Trefftz finite element models
The hybrid-Trefftz variant of the finite element method enhances the approximation basis by deriving it directly from the formal solution of the governing differential equations, ensuring that it meets all domain conditions—equilibrium, compatibility, and elasticity—in steady-state or elastostatic problems This method focuses on identifying the optimal combination of these bases to effectively satisfy boundary conditions, either through displacement continuity in the displacement model or force equilibrium in the alternative stress model.
This method, commonly outlined in introductory textbooks on solving systems of differential equations, involves three key steps: first, identifying a particular solution to enhance convergence in the finite element approach; second, determining the solutions of the homogeneous differential equation to establish the finite element approximation basis; and finally, combining these solutions to achieve the most accurate approximation for the boundary conditions.
In the general framework of solving partial differential equations, this was the idea that Trefftz offered [171] as the alternative proposed earlier by Ritz [150] His contribution to the
The 'textbook approach' involves enforcing boundary conditions in weak form, a concept significantly advanced in the finite element method by Jirousek, with Herrera contributing to its formalization.
The hybrid-Trefftz formulation offers significant advantages over the hybrid variant, notably its immunity to rank-deficiency and enhanced convergence rates These benefits arise from the selection of the approximation basis, as the Trefftz approach utilizes formal solutions from the mathematical model, effectively filtering out non-physical solutions and ensuring that the model accurately reflects the underlying physics of the problem.
Modelling options
This research focuses on two key aspects: the mathematical model employed to simulate the hygro-thermo-chemical behavior of both early-age and hardened concrete structures, and the finite element formulation used to solve this complex problem.
The cement hydration model developed by Ulm and Coussy, and further refined by Cervera et al., has been validated through a review of specialized literature, demonstrating its consistency and general applicability This model effectively simulates the hydration process phenomena with the precision necessary for engineering applications.
Original contributions
rather complex phenomena, the main limitation of their simulation of cement hydration is the reliance on functional parameters that may be difficult to define through laboratory testing
The integration of the chemical reaction model with macro-scale heat transfer and moisture transport models results in a complex, nonlinear parabolic boundary value problem that is transient in nature.
The assessment of a hybrid-mixed finite element formulation revealed no computational advantages, and the limited experience in solving nonlinear coupled problems with such elements raised concerns about identifying and addressing spurious modes Consequently, the nonlinear characteristics of the problem led to the decision against developing a hybrid-Trefftz formulation for hygro-thermo-chemical analysis Ultimately, the choice was made to adopt the hybrid finite element formulation as the most effective compromise.
The final decision involved selecting between a 'displacement' or 'stress' model for the hybrid finite element formulation in thermal (moisture) analysis A 'stress' model directly approximates heat (moisture) flow within the element's domain and temperature (relative humidity) at its boundary In contrast, the 'displacement' model focuses on approximating the temperature (relative humidity) field in the domain, while the boundary approximation pertains to heat (moisture) flux.
The selection of a model is primarily influenced by the level of control it offers over the quality of relevant information In this context, the 'displacement' model is particularly advantageous, as it allows the analyst to regulate relative humidity, which is crucial since environmental exposure can impact cement hydration Furthermore, direct control over the temperature field is essential, given its significant effect on the mechanical behavior of the structure.
The selection of the finite element model necessitated the creation of a new computational code, as current commercial options lack the flexibility to implement a hybrid ‘displacement’ element for hygro-thermo-chemical analysis Although the development and validation process requires significant investment, it ultimately allows for the establishment of a code specifically designed for this unique application.
The phenomenological mathematical models utilized to simulate the early-age response of Ordinary Portland Cement (OPC) and High-Performance Concrete (HPC) structures are derived from specialized literature The primary aim of this research is to evaluate a hybrid variant of the finite element method for simulating cement hydration in concrete structures, focusing on the contributions of this study within that framework.
Insufficient flexibility in commercial finite element codes often results in unnecessary modeling simplifications To address this issue, significant effort has been dedicated to developing comprehensive coupled, nonlinear, and transient conceptual models, leveraging the capability of specialized codes for more effective analysis.
Users of the codes developed in this research can opt for either the comprehensive hygro-thermo-chemical model or employ common engineering simplifications for analyzing early-age concrete structures They have the flexibility to choose or customize alternative analytical or experimental definitions for key coefficients that influence heat transfer, moisture transport, and cement hydration processes Additionally, the supporting codes are designed to accommodate a diverse range of boundary conditions encountered in real-world applications, taking into account the variability of environmental factors such as air temperature, relative humidity, and wind speed over time and space.
In the early exploration of specialized literature, it was noted that authors often preferred the conventional formulation of the finite element method for heat transfer problems due to its robustness Their focus primarily lay on refining and expanding the mathematical modeling of their specific applications However, it became apparent that the implementation of conventional finite elements was becoming increasingly computationally intensive as the complexity of the mathematical models grew This challenge is not an inevitable result of enhancing the modeling of physical or chemical processes; rather, it stems from surpassing the limitations inherent in the design of the conforming finite element numerical solution tool.
The finite element method is fundamentally based on isoparametric transformations, where the same transformation is applied to both geometry and state variables To maintain conformity, which is essential for monotonic convergence, structured meshes are required; however, this necessity often leads to excessive mesh refinement.
Heat transfer modeling often involves challenges that arise from combining domains with significantly different geometric and thermal properties Additionally, coupled problems frequently feature state variables that necessitate varying degrees of approximation to achieve satisfactory accuracy These complexities are particularly evident in the simulation of cement hydration within early-age concrete structures.
This study focuses on decoupling the representation of geometry from the approximation of state variables The design of finite elements is primarily influenced by the geometry and hygro-thermal properties of structural components, while the approximation of state variables depends on their spatial and temporal evolution Notably, the moisture field is highly sensitive to boundary layer effects resulting from drying on exposed surfaces.
Outline of the thesis
The uncoupling of geometry and field variables enhances the approximation of temperature and relative humidity within each element, while also refining the heat and moisture fluxes at the element's boundary This approach can significantly increase the degrees of freedom in the algebraic solving system, necessitating careful formulation design to maintain competitive performance Consequently, the strategy involves applying established concepts from solid mechanics to hygro-thermo-chemical analysis.
The article discusses three key ideas for improving computational efficiency in large-scale problems First, it highlights the use of complete orthogonal approximation bases, resulting in a highly sparse solving system where less than 1% of coefficients need to be computed and stored Second, it emphasizes the advantage of naturally hierarchical approximation bases to facilitate adaptive p-refinement procedures Lastly, it advocates for solving the full system, including both domain and boundary degrees-of-freedom, rather than condensing the system on boundary variables, which is common in hybrid finite element formulations This approach enhances solutions through parallel processing, as domain variables like temperature and relative humidity are element-dependent, while boundary variables such as heat and moisture fluxes are shared between connecting elements.
This research explores three innovative options for modeling the staged construction of concrete structures Firstly, it utilizes unstructured meshes, allowing connections between the sides of multiple elements, enhancing flexibility in design Secondly, it employs varying degrees of approximation in different directions of a typical element, which proves effective without sacrificing numerical stability or accuracy, even with high aspect-ratio elements Lastly, it directly approximates the degrees of cement hydration, silica fume reaction, and silicate polymerization While these approximations are not strictly necessary, they provide a comprehensive understanding of the hygro-thermo-chemical processes involved in cement hydration, including the spatial and temporal variations of temperature, relative humidity, and chemical reaction rates.
This research is divided into three key chapters: Chapter 2 focuses on solving nonlinear transient heat transfer problems through a hybrid finite element formulation Chapter 3 narrows this formulation for the thermo-chemical analysis of early-age concrete structures Finally, Chapter 4 expands the formulation to incorporate the influence of moisture content in the concrete mix on the cement hydration process.
Chapter 2 is used to establish the assumptions and the notation adopted in the definition of heat transfer problems However, its main role is to present the criteria adopted to approximate the geometry of the structure and to approximate the state variables Thus, it is in Chapter 2 that the basic concepts of the hybrid finite element formulation used throughout this research are established and clarified The formulation is so developed as to ensure that the relative strengths of the hybrid version of the finite element method are preserved in the present context, namely high sparsity and suitability to adaptive p-refinement and parallel processing Particular attention is given to the concepts that are shared with the conventional formulation of the finite element method and to those that distinguish the two approaches
Chapter 2 closes with the presentation of a comprehensive set of tests on numerical performance They address the basic issues of convergence and sensitivity to shape distortion, and include all the benchmarks typically used in the assessment of conventional finite elements, covering one-, two-, and three-dimensional problems, as well as steady-state and transient applications Moreover, they are extended to illustrate the modelling of singular heat flow fields in cracked plates and the solution of a strongly nonlinear problem, the simulation of the thermal response of a RC beam strengthened with carbon fibre reinforced polymer laminates subjected to fire
Chapter 3 focuses on modeling cement hydration in early-age concrete structures, assuming the use of ordinary Portland cement with sufficient free water for the hydration reaction The chapter provides a detailed simulation of the cement hydration heat source, emphasizing the concept of the degree of cement hydration and its connection to the widely used equivalent age concept Additionally, it presents the models utilized for simulating the cement hydration reaction and thoroughly addresses the characterization of relevant thermo-chemical properties.
The hybrid finite element formulation introduced in Chapter 2 is enhanced to model the coupled thermo-chemical process, aiming to maintain the essential computational features that are crucial for simulating the casting of concrete structures.
This article presents a first test demonstrating the application of a hybrid formulation under various boundary and initial conditions related to thermal conductance mechanisms in staged construction Three specific tests are selected for evaluation: the one-dimensional modeling of reinforced concrete (RC) slab casting, the axisymmetric simulation of a wind tower RC foundation casting, and the two-dimensional modeling of phased construction for a roller-compacted concrete (RCC) gravity dam The final test showcases the three-dimensional extension of the proposed formulation.
Chapter 4 addresses the simulation of cement hydration in high-performance concrete structures Three problems are now strongly coupled, namely the heat transfer caused by the chemical reactions, the moisture transport associated with the same process and the
1.7 Outline of the thesis 15 dependence of the chemical reactions on both temperature and availability of water The notation used in Chapters 2 and 3 is redefined to establish a compact description of the hygro- thermo-chemical problem
The mathematical modeling of heat transfer now incorporates a heat source that accounts for both cement hydration and silica fume reactions, while typically overlooking the effects of silicate polymerization Before introducing the moisture transport model, it is crucial to define key concepts related to water content, specifically evaporable and non-evaporable (chemically bound) water These definitions are vital for establishing the sink term (water consumption) in the moisture transport model, which is elaborated upon with a focus on moisture capacity and diffusivity definitions.
In terms of numerical implementation, the hygro-thermo-chemical extension of the formulation is designed to preserve the main features secured for the thermo-chemical model
This article emphasizes the importance of numerical validation, featuring two distinct sets of tests The first set simulates laboratory experiments focused on temperature and moisture variations under adiabatic, self-desiccation, and drying conditions The second set demonstrates the practical application of the formulation in engineering scenarios Notably, only one relevant test, the drying of a concrete wall, is documented in existing literature To enhance this analysis, the study includes a reexamination of the wind tower RC foundation casting discussed in Chapter 3, specifically evaluating the impact of moisture transport modeling.
The thesis concludes by evaluating the key findings and exploring potential avenues for further research In addition to the ongoing development of hygro-thermo-chemo-mechanical analysis, Chapter 5 highlights specific aspects of the current research that require enhancement or warrant additional investigation.
The research objectives outlined in this chapter have been successfully achieved, as evidenced by the results discussed in Chapters 2 to 4 The hybrid finite element formulation demonstrated effective performance in addressing hygro-thermo-chemical problems, achieving satisfactory accuracy with coarse and potentially unstructured high-degree element meshes This approach requires fewer degrees of freedom compared to traditional conforming finite elements, highlighting its efficiency in solving complex issues.
Hybrid elements for heat transfer
Introduction
Heat transfer problems are usually solved using the conventional formulation of the finite element method, very much in consequence of the original work of Zienkiewicz and Cheung
The conventional formulation in finite element analysis relies on isoparametric transformations, where the same functions are utilized to map the geometry of the element and approximate the temperature field A crucial aspect of this approach is the use of master elements, which support transformations defined by nodal polynomials This ensures that the geometric mapping remains conforming and the temperature field is continuous, achieved by maintaining geometric and thermal continuity at the element nodes.
The established performance of heat transfer conforming elements highlights their convergence and robustness, yet they face notable challenges due to the isoparametric concept The sensitivity of the temperature field solution to shape distortion is a significant weakness, and achieving convergence under high temperature gradients necessitates extensive local h-refinement, which can exacerbate shape distortion Additionally, adaptive p-refinement poses computational challenges, as it requires the recalculation of all submatrices impacted by the refinement Furthermore, parallelization efforts are complicated by the shared nodal variables among all elements connected to a common node.
The limitations in temperature approximation stem from the coupling with the geometry mapping of the master element, leading to the development of hybrid finite element formulations Despite a wealth of literature on alternative hybrid formulations across various applications, their implementation in heat transfer problems has been scarce since the foundational work of Fraeijs de Veubeke and Hogge on steady-state issues While there have been extensions to transient analyses using the Trefftz variant, applications of hybrid formulations in thermal problems, such as thermoelastic fracture and inverse heat conduction, remain limited.
This chapter focuses on the application of hybrid finite element formulations to solve nonlinear, transient heat transfer problems, highlighting the modeling flexibility that these hybrid elements have demonstrated in various applications.
This chapter begins by defining the mathematical problem, establishing essential terminology and notation It clarifies the formulation of the hybrid finite element model by independently defining the basic equations of heat flow equilibrium, conduction, and the thermal gradient The domain conditions are supported by appropriate initial and boundary conditions, with initial conditions expressed in terms of temperature or temperature time rates Boundary conditions are categorized under generalized Neumann conditions—prescribing heat flux, convection, radiation, and thermal conductance—and Dirichlet conditions, which involve prescribed temperature and thermal resistance.
The second part of the chapter focuses on the spatial and temporal discretization of the problem Temporal discretization is briefly outlined, utilizing a well-established trapezoidal time integration rule For spatial discretization, the analysis employs the master element concept from conventional finite element methods, with a notable modification that involves decomposing the sides of two-dimensional elements or the faces of solid elements to facilitate the creation of unstructured meshes.
The hybrid finite element formulation for heat transfer problems is explored in detail, highlighting its distinct concepts compared to the conventional finite element method This formulation is structured into four key stages: first, identifying the state variables for independent approximation, specifically temperature within the element and heat flux at its boundary; second, utilizing orthogonal and naturally hierarchical polynomials for these approximations; third, formulating the finite element equations through the weak form of the heat transfer problem; and fourth, assembling and solving the elementary finite element equations Emphasis is placed on the unique properties of the resulting finite element system, particularly regarding sparsity, adaptive p-refinement, and parallelization The discussion on numerical implementation focuses on the differences between programming hybrid and traditional finite element formulations.
Chapter 2 closes with the presentation of a comprehensive set of tests on numerical performance They address the basic robustness issues, namely convergence and sensitivity to shape distortion, and include all the benchmarks typically used in the assessment of conventional finite elements, covering one-, two-, and three-dimensional problems, as well as steady-state and transient applications Moreover, they are extended to illustrate the modelling of singular heat flow fields and the solution of strongly nonlinear problems.
Domain and boundary definitions
The analysis is conducted within a Cartesian coordinate system (x, y, z) that remains constant over time (t) and is divided into finite elements Each typical element is defined by its domain V and boundary Γ, as illustrated in Figure 2.1.
Figure 2.1: Domain and boundary of a typical finite element
For simplicity of presentation, it is convenient to separate the boundary in two complementary parts, the generalized Neumann boundary, Γ N , and the generalized Dirichlet boundary Γ D :
The generalized Neumann boundary combines the portions where the heat flux is prescribed, Γ q , and where convection, Γ c , radiation, Γ r , convective-radiation, Γ cr , and conductance, Γ cd , conditions are applied:
The generalized Dirichlet boundary combines the portions where the temperature is prescribed, Γ T , the inter-element boundaries of the element, Γ i , and the boundaries whereon thermal resistance conditions may apply, Γ rs :
The conditions that are enforced in each of the boundaries identified above are defined in Section 2.5
Domain equations of thermal analysis
The distribution of temperature T in a control volume V is described by the transient heat conduction equation,
The equation T + c Ṫ = ∇ · σ + Q̇ describes the relationship between temperature, heat flow, and heat sources in a given volume Here, ∇ T represents the gradient vector, indicating how temperature changes in space, while σ denotes the heat flow vector The term c refers to the volumetric specific heat, Ṫ signifies the rate of temperature change over time, and Q̇ represents the heat source within the system Each component of this equation has dimensions of watts per cubic meter (W/m³), highlighting its application in thermal analysis.
The temperature gradient is a vector quantity defined as,
T in V ε ∇= (2.5) or ε = ∂ ∂ { T x / ∂ ∂ T y / ∂ ∂ T z / } T in explicit form Thus, its unit is Kelvin per meter ( / ).K m
The constitutive relation, defined by Fourier's law of conduction, links heat flow to the temperature gradient, expressed as V σ = − k ε, where k represents the symmetric thermal conductivity matrix Thermal conductivity coefficients (W/m/K) indicate the heat transfer rate through a unit thickness of material per unit area and temperature difference In this context, isotropic conductivity is assumed, represented as k i j = k δ i j, with δ i j being the Kronecker symbol It is important to note that thermal conductivity can vary with time, space, and temperature.
The heat conduction domain conditions (2.4) to (2.6) are complemented by appropriate boundary and initial conditions.
Initial conditions
Heat conduction equation (2.4) is a first-order differential equation in time, for which one out of the two initial conditions needs to be provided for the entire domain, namely,
T T at t t in V= = (2.8) where T 0 and T0 are values of temperature and its time rate at initial time, t 0 Condition (2.7) is used in most applications.
Boundary conditions
For convenience, the boundary conditions are written as follows for the generalized Neumann and Dirichlet boundaries,
In the equation T T= onΓ (2.10), the unit outward normal vector is denoted by n, with q N representing the prescribed heat flux and T D indicating the prescribed temperature The boundary functions have been designed to account for variations over time and space, and their definitions are provided in accordance with decompositions (2.2) and (2.3).
By definition, the Neumann equation (2.9) is written for a prescribed heat flux q t( , ) x as:
They are extended as follows to include convection conditions,
T σ =h T T c − a onΓ c n (2.12) radiation and convective-radiation conditions,
T σ =h T T cr − a onΓ cr n (2.14) and conductance conditions:
Convection is the method of heat transfer occurring between a surface and a fluid, typically air, in contact with it This process is governed by Newton's law of cooling, which incorporates the convective heat transfer coefficient (h c) measured in W/m²/K, along with the surface temperature (T).
The air temperature, denoted as Ta, significantly affects the convective heat transfer coefficient (hc), which is influenced by various factors including fluid properties like density, viscosity, specific heat, and conductivity, as well as fluid velocity and the texture of the external surface According to McAdams, the convective coefficient can be accurately determined by considering both wind speed and the texture of the external surface, as detailed in Table 2.1.
Nature of surface Wind speed ν w ( / ) m s w 4.88 ν ≤ 4.88