Showing papers on "Finite element method published in 2008"
Book•
18 Dec 2008
TL;DR: Discontinuous Galerkin methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists and engineers who work in fluid dynamics and solid mechanics and want to use DG methods for their numerical results.
Abstract: Discontinuous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. This book covers both theory and computation as it focuses on three primal DG methods--the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation is divided into three parts: Part I focuses on the application of DG methods to second order elliptic problems in one dimension and in higher dimensions. Part II presents the time-dependent parabolic problems without and with convection. Part III contains applications of DG methods to solid mechanics (linear elasticity), fluid dynamics (Stokes and Navier Stokes), and porous media flow (two-phase and miscible displacement). Appendices contain proofs and MATLAB code for one-dimensional problems for elliptic equations and routines written in C that correspond to algorithms for the implementation of DG methods in two or three dimensions. Audience: This book is intended for numerical analysts, computational and applied mathematicians interested in numerical methods for partial differential equations or who study the applications discussed in the book, and engineers who work in fluid dynamics and solid mechanics and want to use DG methods for their numerical results. The book is appropriate for graduate courses in finite element methods, numerical methods for partial differential equations, numerical analysis, and scientific computing. Chapter 1 is suitable for a senior undergraduate class in scientific computing. Contents: List of Figures; List of Tables; List of Algorithms; Preface; Part I: Elliptic Problems; Chapter 1: One-dimensional problem; Chapter 2: Higher dimensional problem; Part II: Parabolic Problems; Chaper 3: Purely parabolic problems; Chapter 4: Parabolic problems with convection; Part III: Applications; Chapter 5: Linear elasticity; Chapter 6: Stokes flow; Chapter 7: Navier-Stokes flow; Chapter 8: Flow in porous media; Appendix A: Quadrature rules; Appendix B: DG codes; Appendix C: An approximation result; Bibliography; Index.
813 citations
TL;DR: In this article, a new material constitutive law is implemented in a 2D finite element model to analyse the chip formation and shear localisation when machining titanium alloys.
Abstract: A new material constitutive law is implemented in a 2D finite element model to analyse the chip formation and shear localisation when machining titanium alloys. The numerical simulations use a commercial finite element software (FORGE 2005®) able to solve complex thermo-mechanical problems. One of the main machining characteristics of titanium alloys is to produce segmented chips for a wide range of cutting speeds and feeds. The present study assumes that the chip segmentation is only induced by adiabatic shear banding, without material failure in the primary shear zone. The new developed model takes into account the influence of strain, strain rate and temperature on the flow stress and also introduces a strain softening effect. The tool chip friction is managed by a combined Coulomb–Tresca friction law. The influence of two different strain softening levels and machining parameters on the cutting forces and chip morphology has been studied. Chip morphology, cutting and feed forces predicted by numerical simulations are compared with experimental results.
533 citations
TL;DR: The finite element method is developed for the numerical resolution of the space and time fractional Fokker-Planck equation, which is an effective tool for describing a process with both traps and flights.
Abstract: We develop the finite element method for the numerical resolution of the space and time fractional Fokker-Planck equation, which is an effective tool for describing a process with both traps and flights; the time fractional derivative of the equation is used to characterize the traps, and the flights are depicted by the space fractional derivative. The stability and error estimates are rigorously established, and we prove that the convergent order is $O(k^{2-\alpha}+h^\mu)$, where $k$ is the time step size and $h$ the space step size. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims.
515 citations
TL;DR: This paper introduces SELFE as an open-source code available for community use and enhancement, a new finite-element model for cross-scale ocean modeling that retains key benefits of existing semi-implicit Eulerian–Lagrangian finite-volume models, but relaxation on grids, uses higher-order shape functions for elevation, and enables superior flexibility in representing the bathymetry.
514 citations
TL;DR: It is shown that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Abstract: We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
507 citations
TL;DR: In this article, the authors highlight the limitations of existing formulae used to predict numerical cohesive zone length and demonstrate modifications necessary for improved accuracy, and provide a minimum number of interface elements within the cohesive zone.
446 citations
TL;DR: In this article, a generalization of the finite cell method to three-dimensional problems of linear elasticity is presented, which combines ideas from embedding or fictitious domain methods with the p-based finite element method.
424 citations
TL;DR: This work introduces a new offline basis-generation algorithm based on the derivation of rigorous a-posteriori error estimates in various norms for general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations.
Abstract: The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.
420 citations
TL;DR: In this article, the discretization behavior of classical finite element and NURBS approximations on problems of structural vibrations and wave propagation was studied, and it was shown that, on the basis of equal numbers of degrees-of-freedom and bandwidth, NURbs have superior approximation properties.
393 citations
TL;DR: In this article, different engineering materials are classified into four impact cases according to their physical and mechanical properties, i.e., soft/soft, hard/hard, soft/hard and hard/soft (particle/substrate).
TL;DR: In this article, a benchmark comparison between various numerical codes (Eulerian and Lagrangian, Finite Element and Finite Difference, with and without markers) as well as a laboratory experiment is presented.
TL;DR: In this article, a 3D constitutive model is developed to describe the finite deformation thermo-mechanical response of amorphous shape memory polymers (SMPs).
Abstract: Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature TH>Tg. It will maintain this deformed shape after subsequently lowering the temperature to TL Tg, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.
TL;DR: The performance of finite element methods for dynamic crack propagation in brittle materials is studied in this paper, where three methods are considered: the extended finite element method (XFEM), element deletion method and interelement crack method.
Abstract: The performance of finite element methods for dynamic crack propagation in brittle materials is studied. Three methods are considered: the extended finite element method (XFEM), element deletion method and interelement crack method. The extended finite element method is a method for arbitrary crack propagation without remeshing. In element deletion methods, elements that meet a fracture criterion are deleted. In interelement crack methods, the crack is limited to element edges; the separation of these edges is governed by a cohesive law. We show that XFEM and interelement method show similar crack speeds and crack paths. However, both fail to predict a benchmark experiment without adjustment of the energy release rate. The element deletion method performs very poorly for the refinements studied, and is unable to predict crack branching.
TL;DR: In this paper, a quadrilateral element with smoothed curvatures for Mindlin-Reissner plates is proposed, where the curvature at each point is obtained by a non-local approximation via a smoothing function.
TL;DR: This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties.
Abstract: This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Galerkin formulation using the generalized gradient smoothing technique and smoothing domains constructed in various ways. The numerical methods developed based on this smoothed bilinear form will be spatially stable and convergent and possess three major important properties: (1) it is variationally consistent, if the solution is sought in a Hilbert space; (2) the stiffness of the discretized model will be reduced comp...
TL;DR: In this paper, a finite element analysis (FEA) of machining of TiAl6V4 both for conventional and high speed cutting regimes is presented, where cutting force, chip morphology and segmentation are taken into account due to their predominant roles to determine machinability and tool wear.
TL;DR: A maximum principal strain criterion can be defined a suitable candidate for the in vivo risk factor assessment on long bones by correctly identifying the level of failure risk and the location of fracture onset in all the modelled specimens.
TL;DR: In this paper, micro-computed X-ray tomography is used to develop 3D images of the morphology of polyester urethane and Duocel aluminum foams with different average cell sizes.
TL;DR: In this paper, a three-dimensional, thermo-elastic-plastic, large deformation finite element method (FEM) is used to simulate welding distortion in a low carbon steel butt-welded joint with 1mm thickness.
Book•
01 Jan 2008
TL;DR: In this paper, the authors present a review of the literature on finite element fracture models and their application in the field of finite element finite element models (FEM) and fracture mechanics.
Abstract: Dedication. Preface . Nomenclature . Chapter 1 Introduction. 1.1 ANALYSIS OF STRUCTURES. 1.2 ANALYSIS OF DISCONTINUITIES. 1.3 FRACTURE MECHANICS. 1.4 CRACK MODELLING. 1.4.1 Local and non-local models. 1.4.2 Smeared crack model. 1.4.3 Discrete inter-element crack. 1.4.4 Discrete cracked element. 1.4.5 Singular elements. 1.4.6 Enriched elements. 1.5 ALTERNATIVE TECHNIQUES. 1.6 A REVIEW OF XFEM APPLICATIONS. 1.6.1 General aspects of XFEM. 1.6.2 Localisation and fracture. 1.6.3 Composites. 1.6.4 Contact. 1.6.5 Dynamics. 1.6.6 Large deformation/shells. 1.6.7 Multiscale. 1.6.8 Multiphase/solidification. 1.7 SCOPE OF THE BOOK. Chapter 2 Fracture Mechanics, a Review. 2.1 INTRODUCTION. 2.2 BASICS OF ELASTICITY. 2.2.1 Stress-strain relations. 2.2.2 Airy stress function. 2.2.3 Complex stress functions. 2.3 BASICS OF LEFM. 2.3.1 Fracture mechanics. 2.3.2 Circular hole. 2.3.3 Elliptical hole. 2.3.4 Westergaard analysis of a sharp crack. 2.4 STRESS INTENSITY FACTOR, K . 2.4.1 Definition of the stress intensity factor. 2.4.2 Examples of stress intensity factors for LEFM. 2.4.3 Griffith theories of strength and energy. 2.4.4 Brittle material. 2.4.5 Quasi-brittle material. 2.4.6 Crack stability. 2.4.7 Fixed grip versus fixed load. 2.4.8 Mixed mode crack propagation. 2.5 SOLUTION PROCEDURES FOR K AND G . 2.5.1 Displacement extrapolation/correlation method. 2.5.2 Mode I energy release rate. 2.5.3 Mode I stiffness derivative/virtual crack model. 2.5.4 Two virtual crack extensions for mixed mode cases. 2.5.5 Single virtual crack extension based on displacement decomposition. 2.5.6 Quarter point singular elements. 2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM). 2.6.1 Plastic zone. 2.6.2 Crack tip opening displacements (CTOD). 2.6.3 J integral. 2.6.4 Plastic crack tip fields. 2.6.5 Generalisation of J . 2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL. 2.7.1 Nodal solution. 2.7.2 General finite element solution. 2.7.3 Equivalent domain integral (EDI) method. 2.7.4 Interaction integral method. Chapter 3 Extended Finite Element Method for Isotropic Problems. 3.1 INTRODUCTION. 3.2 A REVIEW OF XFEM DEVELOPMENT. 3.3 BASICS OF FEM. 3.3.1 Isoparametric finite elements, a short review. 3.3.2 Finite element solutions for fracture mechanics. 3.4 PARTITION OF UNITY. 3.5 ENRICHMENT. 3.5.1 Intrinsic enrichment. 3.5.2 Extrinsic enrichment. 3.5.3 Partition of unity finite element method. 3.5.4 Generalised finite element method. 3.5.5 Extended finite element method. 3.5.6 Hp-clouds enrichment. 3.5.7 Generalisation of the PU enrichment. 3.5.8 Transition from standard to enriched approximation. 3.6 ISOTROPIC XFEM. 3.6.1 Basic XFEM approximation. 3.6.2 Signed distance function. 3.6.3 Modelling strong discontinuous fields. 3.6.4 Modelling weak discontinuous fields. 3.6.5 Plastic enrichment. 3.6.6 Selection of nodes for discontinuity enrichment. 3.6.7 Modelling the crack. 3.7 DISCRETIZATION AND INTEGRATION. 3.7.1 Governing equation. 3.7.2 XFEM discretization. 3.7.3 Element partitioning and numerical integration. 3.7.4 Crack intersection. 3.8 TRACKING MOVING BOUNDARIES. 3.8.1 Level set method. 3.8.2 Fast marching method. 3.8.3 Ordered upwind method. 3.9 NUMERICAL SIMULATIONS. 3.9.1 A tensile plate with a central crack. 3.9.2 Double edge cracks. 3.9.3 Double internal collinear cracks. 3.9.4 A central crack in an infinite plate. 3.9.5 An edge crack in a finite plate. Chapter 4 XFEM for Orthotropic Problems. 4.1 INTRODUCTION. 4.2 ANISOTROPIC ELASTICITY. 4.2.1 Elasticity solution. 4.2.2 Anisotropic stress functions. 4.2.3 Orthotropic mixed mode problems. 4.2.4 Energy release rate and stress intensity factor for anisotropic. materials. 4.2.5 Anisotropic singular elements. 4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP. 4.3.1 Near crack tip displacement field (class I). 4.3.2 Near crack tip displacement field (class II). 4.3.3 Unified near crack tip displacement field (both classes). 4.4 ANISOTROPIC XFEM. 4.4.1 Governing equation. 4.4.2 XFEM discretization. 4.4.3 SIF calculations. 4.5 NUMERICAL SIMULATIONS. 4.5.1 Plate with a crack parallel to material axis of orthotropy. 4.5.2 Edge crack with several orientations of the axes of orthotropy. 4.5.3 Single edge notched tensile specimen with crack inclination. 4.5.4 Central slanted crack. 4.5.5 An inclined centre crack in a disk subjected to point loads. 4.5.6 A crack between orthotropic and isotropic materials subjected to. tensile tractions. Chapter 5 XFEM for Cohesive Cracks. 5.1 INTRODUCTION. 5.2 COHESIVE CRACKS. 5.2.1 Cohesive crack models. 5.2.2 Numerical models for cohesive cracks. 5.2.3 Crack propagation criteria. 5.2.4 Snap-back behaviour. 5.2.5 Griffith criterion for cohesive crack. 5.2.6 Cohesive crack model. 5.3 XFEM FOR COHESIVE CRACKS. 5.3.1 Enrichment functions. 5.3.2 Governing equations. 5.3.3 XFEM discretization. 5.4 NUMERICAL SIMULATIONS. 5.4.1 Mixed mode bending beam. 5.4.2 Four point bending beam. 5.4.3 Double cantilever beam. Chapter 6 New Frontiers. 6.1 INTRODUCTION. 6.2 INTERFACE CRACKS. 6.2.1 Elasticity solution for isotropic bimaterial interface. 6.2.2 Stability of interface cracks. 6.2.3 XFEM approximation for interface cracks. 6.3 CONTACT. 6.3.1 Numerical models for a contact problem. 6.3.2 XFEM modelling of a contact problem. 6.4 DYNAMIC FRACTURE. 6.4.1 Dynamic crack propagation by XFEM. 6.4.2 Dynamic LEFM. 6.4.3 Dynamic orthotropic LEFM. 6.4.4 Basic formulation of dynamic XFEM. 6.4.5 XFEM discretization. 6.4.6 Time integration. 6.4.7 Time finite element method. 6.4.8 Time extended finite element method. 6.5 MULTISCALE XFEM. 6.5.1 Basic formulation. 6.5.2 The zoom technique. 6.5.3 Homogenisation based techniques. 6.5.4 XFEM discretization. 6.6 MULTIPHASE XFEM. 6.6.1 Basic formulation. 6.6.2 XFEM approximation. 6.6.3 Two-phase fluid flow. 6.6.4 XFEM approximation. Chapter 7 XFEM Flow. 7.1 INTRODUCTION. 7.2 AVAILABLE OPEN-SOURCE XFEM. 7.3. FINITE ELEMENT ANALYSIS. 7.3.1 Defining the model. 7.3.2 Creating the finite element mesh. 7.3.3 Linear elastic analysis. 7.3.4 Large deformation. 7.3.5 Nonlinear (elastoplastic) analysis. 7.3.6 Material constitutive matrix. 7.4 XFEM. 7.4.1 Front tracking. 7.4.2 Enrichment detection. 7.4.3 Enrichment functions. 7.4.4 Ramp (transition) functions. 7.4.5 Evaluation of the B matrix. 7.5 NUMERICAL INTEGRATION. 7.5.1 Sub-quads. 7.5.2 Sub-triangles. 7.6 SOLVER. 7.6.1 XFEM degrees of freedom. 7.6.2 Time integration. 7.6.3 Simultaneous equations solver. 7.6.4 Crack length control. 7.7 POST-PROCESSING. 7.7.1 Stress intensity factor. 7.7.2 Crack growth. 7.7.3 Other applications. 7.8 CONFIGURATION UPDATE. References . Index
TL;DR: In this article, a combined fracture-plastic model for concrete is presented, which combines the fracture model and the crack band approach. And the model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions.
TL;DR: In this article, a wave/finite element (WFE) method is described by which the dispersion relations for a two-dimensional structural component can be predicted from a finite element (FE) model.
TL;DR: In this paper, the authors investigate the band structure of a phononic crystal of finite thickness constituted of a periodical array of cylindrical dots deposited on a thin plate of a homogeneous material and show that this structure can display a low-frequency gap, as compared to the acoustic wavelengths in the constituent materials.
Abstract: We investigate theoretically the band structure of a phononic crystal of finite thickness constituted of a periodical array of cylindrical dots deposited on a thin plate of a homogeneous material. We show that this structure can display a low-frequency gap, as compared to the acoustic wavelengths in the constituent materials, similarly to the case of locally resonant structures. The opening of this gap requires an appropriate choice of the geometrical parameters, and in particular the thickness of the homogeneous plate and the height of the dots. However, the gap persists for various combinations of the materials constituting the plate and the dots. Besides, the band structure can exhibit one or more higher gaps whose number increases with the height of the cylinders. We discuss the condition to realize waveguiding through a linear defect inside the phononic crystal dots. The numerical simulations are performed by using the finite difference time domain and the finite element methods.
TL;DR: A correction of the densitometric calibration should be used when evaluating bone ash-density from clinical CT scans, to avoid ash- density underestimation and overestimation for low- and high-density bone tissue, respectively; the ash/apparent-density ratio can be assumed constant in human femurs and the correction improves significantly the model accuracy.
01 Apr 2008-Precision Engineering-journal of The International Societies for Precision Engineering and Nanotechnology
TL;DR: In this paper, a comparison of various compliance/stiffness equations of circular flexure hinges with FEA results is presented, based on the limitations of these design equations, a guideline for selecting the most accurate equations for hinge design calculations is presented.
Abstract: This article presents the comparison of various compliance/stiffness equations of circular flexure hinges with FEA results. The limitation of these equations at different t / R (R is the radius and t is the neck thickness) ratios are revealed. Based on the limitations of these design equations, a guideline for selecting the most accurate equations for hinge design calculations are presented. In addition to the review and comparisons, general empirical stiffness equations in the x- and y-direction were formulated in this study (with errors less than 3% when compared to FEA results) for a wide range of t / R ratios ( 0.05 ≤ t / R ≤ 0.8 ).
Book•
13 Jun 2008
TL;DR: The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients and presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions.
Abstract: Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.
TL;DR: In this article, a continuous adjoint formulation of the cost function of ducted flows is derived and the adjoint equations and boundary conditions for typical cost functions are derived for two and three-dimensional geometries.
Abstract: Topology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two- and three-dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities. Copyright © 2008 John Wiley & Sons, Ltd.
TL;DR: In this paper, a new model for the generation of the transversal cross-section of long-fibre reinforced composites with high fiber volume fraction, characterised by a random distribution of the reinforcements, is presented.
Book•
04 Dec 2008
TL;DR: In this article, a unified treatment of fluid mechanics, analysis, and numerical analysis is intended for graduate students in mathematics, engineering, physics, and the sciences who are interested in understanding the foundations of methods commonly used for flow simulations.
Abstract: This book treats the numerical analysis of finite element computational fluid dynamics. Assuming minimal background, the text covers finite element methods; the derivation, behavior, analysis, and numerical analysis of Navier Stokes equations; and turbulence and turbulence models used in simulations. Each chapter on theory is followed by a numerical analysis chapter that expands on the theory. The chapters contain numerous exercises. Introduction to the Numerical Analysis of Incompressible Viscous Flows provides the foundation for understanding the interconnection of the physics, mathematics, and numerics of the incompressible case, which is essential for progressing to the more complex flows not addressed in this book (e.g., viscoelasticity, plasmas, compressible flows, coating flows, flows of mixtures of fluids, and bubbly flows). With mathematical rigor and physical clarity, the book progresses from the mathematical preliminaries of energy and stress to finite element computational fluid dynamics in a format manageable in one semester. Audience: This unified treatment of fluid mechanics, analysis, and numerical analysis is intended for graduate students in mathematics, engineering, physics, and the sciences who are interested in understanding the foundations of methods commonly used for flow simulations.