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The finite element method in heat transfer and fluid dynamics. 3rd ed. (English) Zbl 1257.80001

CRC Series in Computational Mechanics and Applied Analysis. Boca Raton, FL: CRC Press (ISBN 978-1-4200-8598-3/hbk). xiii, 500 p. (2010).
This third edition is an updated and extended version of a bestselling monograph, written by two well-known authors, about numerical simulation of the two disciplines heat transfer and fluid dynamics, closely coupled in many engineering and scientific applications, by the finite element method.
The text is based on graduate lectures and extended courses for a wide spectrum of engineering students and on own practical experiences. Also professionals in modeling of heat transfer, incompressible viscous flow, and heat convection and corresponding codes can profit from the book. Proofs and a more theoretical presentation are avoided in the application oriented text. It is an advantage for the reader to be familiar with basic numerical analysis, linear algebra, thermodynamics and fluid mechanics but, the most important ideas of these fields are summarized in the first chapter “Equations of heat transfer and fluid mechanics”, in the introductions of the chapters, and in the appendices. For readers interested in more details of the derivation of the equations the authors refer to theoretical textbooks.
The eight chapters of the 500 pages long book are divided into sections and subsections. Additionally the monograph contains three appendices, a glossary of relevant terms, the prefaces of the three editions, and resumes of the authors. The chapters and the appendices are completed by references for additional reading. Problems to improve the skill of the reader are formulated at the end of the most chapters. Solutions of the problems are not given.
The equations of heat transfer and fluid dynamics are presented in Chapter 1 starting with classifications of thermodynamics and fluid mechanics.
Three basic forms of heat transfer are considered: conduction (transfer of heat in a medium by diffusion, Fourier’s law), convection (energy transport by the motion of fluid, Newton’s law of cooling) and radiation (emission of electromagnetic energy due to the temperature of the medium, Stefan-Boltzmann law). Based on the properties of the medium, a distinction is drawn between inviscid and viscous (laminar and turbulent, characterized by the Reynolds number) as well as incompressible and compressible fluids. The notions of ideal, Newtonian and non-Newtonian fluid are defined
Vectors and tensors including corresponding index notations and summation conventions are introduced. The gradient vector, the Del operator, the divergence and curl of a vector, and the Laplacian operator are formulated in orthogonal, cylindrical and spherical coordinate systems.
Considering two subregions which contain a moving fluid and a solid body, conservation laws are described for combined convective and conductive thermal transport. The Eulerian or spatial description, commonly used to study fluid flows, and the Lagrangian or material description, mostly applied in solid body heat conduction and the stress and deformation of solid bodies, are mentioned. The authors focus here on the Eulerian description. The conservation of mass (continuity equation) is formulated in the divergence and in the advective (non-conservation) form by means of the Eulerian derivative including the special expressions for steady state conditions and constant density. Newton’s second law describes the motion of a continuous medium and results, applied to fluid mechanics, in the Navier equation, which is the divergence form of the conservation of momenta for compressible flows. An advective form commonly used with incompressible flows is also presented. The conservation of energy is based on the first law of thermodynamics.
Additionally, equation of states are required and formulated. Constitutive equations are introduced for laminar Newtonian fluids. Explicitly, the authors point to the fact that the divergence and the advective forms of the transport operators, related by the continuity equation, could differ from each other in a discretized computational form because the continuity equation may be satisfied only approximately. Thus, they propose a weighted combination of the two forms with a view to the finite element method.
Based on the advective form of the conservation and the constitutive equations the governing equations for isotropic, Newtonian, viscous, incompressible fluids and the energy equation are expressed in terms of the so-called primitive variables (\(\vec v\) velocity, \(P\) pressure, \(T\) temperature) in vector form, Cartesian and cylindrical component form. Another variables choice is possible but, the primitive variable approach is the mostly used one with a number of advantages.
Furthermore, the fluid motion and energy transport are considered for porous media. The corresponding generalized Darcy equations are also known as Forchheimer-Brinkman or Darcy-Forchheimer equations.
The equations so far treated for non-isothermal, viscous, incompressible media can be used for many heat transfer models, but they are limited to flows with small variations in density and pressure relative to the static state and consequently also to small temperature differences. Thus, the restriction on small density variations must be avoided to allow for large temperature differences in convective heat transfer. Because the corresponding fully compressible flow equations also contain acoustic waves for the slow speed flow, which are not important for the heat transfer, two versions of acoustically filtered equations are presented.
Another topic are auxiliary transport equations for flow problems that cannot be described by the state equations and have to be coupled to the non-isothermal flow problem.
In the case of chemically reactive fluid, the incompressible fluid is a mixture of various chemical media. Thus, conservation principles have to be formulated for each species and for the mixture as a whole. Reaction kinetics, the stoichiometry and the properties of the species have to be taken into account in the equations.
The described equations have to be completed by initial and/or boundary conditions. Dirichlet or essential boundary conditions and Neumann or natural boundary conditions are formulated in the component form for the Navier-Stokes equations. They are also generalized to periodic conditions in a flow. Interface conditions between two immiscible fluids are presented for several problem types. The interface location and shape is usually unknown a priori and one has a free boundary problem. Corresponding conditions for the time-dependent and the time-independent free surface flows are presented.
Generally, the same initial and boundary conditions of the Navier-Stokes equations can be applied to the Forchheimer-Brinkman equations for flows in porous media. But, if the Brinkman term is excluded, fewer boundary conditions are needed.
The boundary conditions of the thermal part of the fluid region generally contain temperature specifications in the case of Dirichlet conditions and heat flux specifications for the natural conditions. A set of special cases are outlined.
For time-dependent problems initial conditions for the dependent variable are required. They must satisfy the conservation equations.
Phase transitions result in free or moving boundary problems in which the unknown location of the moving boundary or interface becomes a part of the solution. A limited class of melting/solidification phase changes is treated.
The treatment of radiation is restricted to problems that only takes into account the surface temperature distribution.
The chapter concludes with a summary of the treated governing equations, constitutive relations, and boundary conditions.
Chapter 2 is devoted to the finite element method (FEM). The authors start with the characterization of the FEM as a generalization of the classical variational and of the weighted-residual methods, in which, roughly seaking, the solution of the differential equation is assumed to be a linear combination of appropriate approximation functions multiplied by unknown expansion parameters in the whole domain. The unknown parameters are determined by requiring that the linear combination satisfy the differential equation, selectively in a weighted-integral manner. The approximation functions have to fulfill the boundary conditions. Real-world problems are often defined on regions with a complex geometry, and it is a disadvantage of the variational and weighted residual methods to find appropriate approximation functions that satisfy the corresponding boundary conditions. These difficulties are overcame by the FEM, in which the region is divided into a set of small, non-overlapping subdomains, the finite elements, over which functions are approximated by interpolation functions, generally polynomials. The variational and weighted residual methods are element-wise applied resulting in a set of algebraic relations among the unknown parameters. (The phrase “finite element” is used twofold in the book referring to both the geometry of the element and the degree of approximation for the dependent unknown over the element or to the geometry only.) For readers, that are interested in details about the outlined characterization of the FEM, the authors mainly refer to a book of the first author [An introduction to the finite element method. New York etc.: McGraw-Hill Book Company, 3rd ed. (2006; Zbl 0561.65079)].
In the present monograph, the major steps of the FEM are demonstrated considering the equation for the heat distribution (the time derivative term is set to zero), that reduces to the Poisson equation for an isotropic medium, with Dirichlet and Neumann boundary conditions in a two-dimensional orthotropic medium in a rectangular Cartesian system, as model equation.
The elements are uniquely defined by a set of points in the element, the so-called element nodes. Various geometric shapes (triangle, quadrilateral) and approximation orders (linear, quadratic, and so on) are described. The sum of all elements, the assembly or the finite element mesh of the domain, represents the whole region. The dependent unknown function \(T(x,y)\) over a finite element is expressed by \[ T(x,y) \approx T^e(x,y) = \sum_{j=1}^n T_j^e \psi_j^e(x,y), \] where \(T^e(x,y) \) is an approximation of \(T(x,y)\) over the finite element and the \(\psi_j^e(x,y)\) are the approximation functions associated with the element. The nodal values \(T_j^e \) are the function values of \(T^e(x,y) \) at the element nodes, which have to be determined such that the governing equation is fulfilled in a weighted integral sense taking into account the boundary conditions. This results in a system of algebraic equations for the nodal values \(T_j^e\), denoted as “finite element model”. The authors prefer weak-form finite element models in which the weight and approximation functions coincide. The matrix notation of the algebraic equations with the coefficient or conductivity matrix is derived.
Especially, the lowest-order polynomial, that must be linear in both \(x\) and \(y\), is shown to be associated with the linear triangular element, where the vertices of the triangle are the three element nodes. The Lagrange interpolation functions of this three-node triangle are presented, their use approximates the curved surface of the solution by a piecewise linear function. Local and global coordinates are introduced. The corresponding integrals are evaluated and the matrix notation is presented. For details about the derivation, the authors often refer to the book of the first author [loc. cit.].
Other topics of Chapter 2 are linear rectangular elements, the evaluation of corresponding boundary integrals, the assembly of the elements, and the treatment of time-dependent problems by a two-stage approximation, i.e., a spatial discretionary approximation and a temporal approximation by a finite difference method resulting in a semidiscrete finite element method. A library of two-dimensional triangular and rectangular master finite elements is represented. The elements are well-presented by figures in this chapter and the following ones. Irregular shaped elements (e.g., triangles with curved primitives, quadrilateral elements, here restricted to isoparametric elements) can be taken into account by coordinate transformations with the calculation of the corresponding Jacobian matrix. The authors treat the Gauss-Legendre quadrature formula for the integration needed for the transformation of shaped elements to the master elements. For in some applications effective rectangular elements without interior nodes (so-called serendipity elements), i.e., with fewer nodes compared to higher-order Lagrange elements, one refers to the first author’s book [loc. cit.] but the interpolation functions for a 2-D quadratic serendipity element are formulated.
Furthermore, mesh generation instructions, guidelines for boundary flux representations, and the treatment of boundary conditions are outlined. The last section is devoted to examples, e.g., the treatment of heat conduction in a square plate using meshes of linear triangular and rectangular elements. The accuracy of the computations are compared with each other, with the analytical solution and a finite difference one.
Chapter 3 is devoted to the finite element method for three-dimensional heat conduction problems, i.e., the elements introduced in Chapter 2 have to be replaced by corresponding three-dimensional elements, such as tetrahedral, hexahedral (brick) and prism elements including the shaped forms. The corresponding interpolation functions are presented but not derived in detail. The energy equation, already introduced in Chapter 1, is considered with boundary conditions which include conduction, convection and radiation. Taking into account a temperature dependency of the material coefficients, the problems may be nonlinear. The time-depended heat equation is treated in two steps, the spatial discretization by finite elements, resulting into a set of ordinary differential equations, and the temporal approximation of the ordinary differential system for instance by a finite difference method, which leads to a system of algebraic equations with the nodal values (temperature values) at a time \(t_{n+1}\) that has to be computed from the previous time step \(t_n\). The matrix form of the \(i\)th ordinary differential equation of the semidiscrete, weak-form Galerkin finite element model is deduced. Computational details for the consideration of the surface flux are given. Methods for the solution of nonlinear equations (Picard scheme, Newton iteration) and linear equations (direct and iterative methods) are outlined. For the finite difference integration of the transient problems the Crank-Nicolson method, the Euler backward and forward method, and predictor-corrector methods are sketched including time step control, convergence and stability considerations.
A special section is devoted to the conjunction of the finite element heat conduction equations with an enclosure radiation problem. The coupled conduction-radiation problem is generally highly nonlinear due to the fourth power surface temperature dependence in the boundary condition (Stefan-Boltzmann law). The advantages and drawbacks of coupled and decoupled algorithms for the numerical solution are discussed. The authors present the matrix forms of the methods and refer to original literature for a detailed description of the procedures.
For the treatment of different properties (phase changes, anisotropic conductivities, inhomogeneous material) variable coefficients are introduced into the matrix formulation of the FEM with the aim to reduce quadrature computation times at the expense of some collateral storage requirements.
Special elements, such as beams, wires, cables, shells and so on, are considered for specific solid geometries, besides the hitherto described workhorse elements. Another topics are the treatment of special contact boundaries, such as partially covered surfaces, splitting methods for the regard of kinetic equations in chemical reacting systems, and post processing techniques.
The finite element analysis described in this chapter is illustrated by a number of two- and three-dimensional heat conduction and radiation examples (Stefan problem, nonlinear material properties, welding, brazing, anisotropic conductivity, drag bit analysis, and so on) using the FEM code COYOTE [D. K. Gartling, R. E. Hogan and M. W. Glass, COYOTE – a finite element computer program for nonlinear heat conduction problems. Sandia National Laboratories Report, 2009–4926, Albuquerque, New Mexico (2009), http://prod.sandia.gov/techlib/access-control.cgi/2009/094926.pdf].
Chapter 4 is the largest of all. The motion of viscous incompressible fluids (gas or liquid) under isothermal conditions is governed by the conservation laws of mass, momenta and energy. The corresponding system of equations consists of coupled, nonlinear partial differential equations in terms of velocity components, pressure and temperature. When the temperature effect can be neglected, the energy equation is uncoupled from the momentum equation. Thus, for isothermal flows only the Navier-Stokes equations (conservation of momenta and constitutive relations) have to be solved together with the continuity equation (conservation of mass). The authors use a formulation of these equations which contain only the terms of the velocity components and the pressure. Two weak form of Galerkin finite element models are considered. (I) The velocity-pressure or mixed model contains both variables in a single formulation. (II) The penalty function model considers the continuity equation as a constraint among the velocity components which has to be satisfied by a least-squares approximation. Two implementations of the penalty method, the so-called reduced integration penalty FEM and the consistent penalty model, are described.
The matrix forms of the weak form of Galerkin FEM’s are presented referring again to the first author’s book [loc. cit.] for some details. Especially, the mixed FEM is illustrated for porous flow. The numerical solution of the matrix equations depends on the features of the involved matrices which represent the mass, the convective transport term, the viscous diffusion term and the divergence operator. A number of literature advices are given for the choice of the solution methods.
The authors point out that the pressure plays a special role in incompressible fluids: The interpolation function used for the pressure should be selected one order less than that used for the velocity field, and the Ladyzhenskaya-Babuska-Brezzi (LBB) condition must be satisfied to get stability, accuracy and convergence for the mixed and the penalty function model, otherwise incorrect highly oscillations in the pressure solutions have to be due. Because in the case of equal order elements the velocity solution will be in general accurate also some algorithms are known which stabilize the pressure oscillations. For a deeper understanding of this problems it is referred to additional literature. Quadrilateral (nine-node rectangular) and triangular elements for the velocity interpolation and continuous and discontinuous approximations for the pressure interpolation are presented, both satisfying the LBB condition or not. For three-dimensional viscous flow problems some octahedral, e.g., the eight-node tetrahedron (brick), and tetrahedral elements are analogically discussed.
The initial boundary value problems of fluid dynamics are mathematically demanding due to the high variations in the time and length scales. Subsequently, the next sections of this chapter are devoted to the solution of the nonlinear equations, the time approximation, stabilized methods, least squares FEM’s, free surface problems, the simulation of the highly complex turbulence problem from an engineering point of view (averaged Navier-Stokes equations because of a high spectrum of length, large eddy simulation, direct numerical simulation, and so on) including numerical examples solved using different computer codes.
Chapter 5 treats the coupling of fluid mechanics and heat transfer. Firstly, the convection problems are classified in two categories. Depending on forces, it is differentiated between free (or natural) convection and forced convection. Another classification can be made taking into account the compressibility of the fluid. The Boussinesq model assumes an incompressible flow with small density variations and relative small temperature differences. The non-Boussinesq models or low-speed compressible flows have been already mentioned above. The FEM, based on the weak formulation of the coupled partial differential equations in terms of the velocity field, pressure and temperature, is considered both for viscous incompressible fluids with free convection and for low-speed compressible and nonisothermal porous flows applying analogically the elements and methods developed already in Chapters 3 and 4. The methods are illustrated again by a number of numerical examples using different known FEM codes.
Fluids that are used in the chemical, petroleum, food or polymer industries can often not be described by the Newtonian (linear) constitutive relations. These non-Newtonian fluids are the subject of Chapter 6. A distinction is drawn between inelastic fluids (also referred to as fluids without memory) and viscoelastic fluids. The discrimination is of importance both from the physical and computational point of view. Inelastic fluids can be considered as a generalization of Newtonian fluids and are treated by algorithms which differ only a little from the methods used for the Newtonian case. In contrast to it, the treatment of viscoelastic fluids requires new methods.
Finite element simulations for both inelastic and viscoelastic fluids are treated starting with the complete representation of the involved equations (conservation of mass, momentum, and energy and several non-Newtonian constitutive relations) of inelastic fluids including boundary conditions. The mixed and the Penalty model are adapted for the FEM of inelastic flows.
The power law model including variations is introduced for viscoelastic fluids. The description of the governing equations is limited, for simplicity, to two-dimensional incompressible viscous isotropic fluids. Furthermore, it is assumed that the fluid motion is laminar and isothermal, the latter avoiding the inclusion of the energy equation. For the various constitutive equations (Maxwell fluids, Johnson-Segalman and Phan Thien-Tunner models) the authors prefer the differential rather than the integral model. A matrix formulation of the finite element model is developed, accompanied by numerous references to additional formulations, to unresolved problems, existence and uniqueness, the choice of the constitutive equations, and numerical problems.
The two categories of non-Newtonian fluids are demonstrated by a number of examples.
Chapter 7 is devoted to multiphysics problems. Coupled problems have been treated already in Chapters 3 and 5. Now the coupling is extended to finite element solutions of solid mechanics and electromagnetic problems. Thus, the reader will be additionally confronted with the kinematics of deformations, equilibrium statements of a solid, Maxwell equations, the implementation of the coupled algorithms and numerical examples.
The storage and computational requirements of the treated problems are very high. Thus, the question of parallel processing considered in the last chapter exhibits a reasonable supplement. The reader will find short sections about the classification of parallel computers, the topology of the interconnections of the processors, languages and communication utilities for parallel computing, about algorithmic efficiency and scalability. Besides these general statements about parallel computing, the treatment of the generic steps of the FEM (preprocessing, element matrix building, matrix solvers, postprocessing) adapted to parallel processing are outlined, oriented mainly towards domain composition methods on MIMD machines (multiple instruction multiple data), accompanied by references to corresponding computer codes.
The chapters of the book are completed by three appendices, describing the use of the FEM code FEM2DHT, an extension of the code from the book of first author [loc. cit.], direct and iterative methods for the solution of linear equations, and the solution of nonlinear equations.

MSC:

80-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to classical thermodynamics
76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
76M10 Finite element methods applied to problems in fluid mechanics
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
76M20 Finite difference methods applied to problems in fluid mechanics
65Z05 Applications to the sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
80A20 Heat and mass transfer, heat flow (MSC2010)
80A32 Chemically reacting flows
80A22 Stefan problems, phase changes, etc.
76D05 Navier-Stokes equations for incompressible viscous fluids
76Rxx Diffusion and convection
76S05 Flows in porous media; filtration; seepage
76A10 Viscoelastic fluids
76F65 Direct numerical and large eddy simulation of turbulence
76A05 Non-Newtonian fluids
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
74A05 Kinematics of deformation
74S05 Finite element methods applied to problems in solid mechanics
97M50 Physics, astronomy, technology, engineering (aspects of mathematics education)

Citations:

Zbl 0561.65079

Software:

COYOTE
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