Encyclopedia of Computational Mechanics

VOLUME 1: FUNDAMENTALS. List of Contributors. Preface. 1. Fundamentals, Introduction and Survey (Erwin Stein). Motivation and Scope. 2. Finite Difference Methods (Owe Axelsson). Introduction. Two-point Boundary Value Problems. Finite Difference Methods for Elliptic Problems. Finite Difference Methods for Parabolic Problems. Finite Difference Methods for Hyperbolic Problems. Convection--Diffusion Problems. A Summary of Difference Schemes. References. Further Reading. 3. Interpolation in h-version Finite Element Spaces (Thomas Apel).. Introduction. Finite Elements. Definition of Interpolation Operators. The Deny--Lions Lemma. Local Error Estimates for the Nodal Interpolant. Local Error Estimates for Quasi-Interpolants. Example for a Global Interpolation Error Estimate. Related Chapters. References. 4. Finite Element Methods (Susanne C. Brenner and Carsten Carstensen). Introduction. Ritz--Galerkin Methods for Linear Elliptic Boundary Value Problems. Finite Element Spaces. A Priori Error Estimates for Finite Element Methods. A Posteriori Error Estimates and Analysis. Local Mesh Refinement. Other Aspects. Acknowledgments. References. Further Reading. 5. The p-version of the Finite Element Method (Ernst Rank, Barna Szabo and Alexander Duster). Introduction. Implementation. Convergence Characteristics. Performance Characteristics. Applications to Nonlinear Problems. Outlook. Acknowledgements. Notes. References. Further Reading. 6. Spectral Methods (Claudio Canuto and Alfio Quarteroni). Introduction. Fourier Methods. Algebraic Polynomial Expansion. Algebraic Expansions on Triangles. Stokes and Navier--Stokes Equations. Advection Equations and Conservation Laws. The Spectral Element Method. The Mortar Method. References. 7. Adaptive Wavelet Techniques in Numerical Simulation (Wolfgang Dahmen, Ronald DeVore and Albert Cohen). Introduction. Wavelets. Evolution Problems---Compression of Flow Fields. Boundary Integral Equations---Matrix Compression. A New Adaptive Paradigm. Construction of Residual Approximations and Complexity Analysis. Acknowledgment. Notes. References. Further Reading. 8. Plates and Shells: Asymptotic Expansions and Hierarchic Models (Zohar Yosibash, Monique Dauge and Erwan Faou). Introduction. Multiscale Expansions for Plates. Hierarchical Models for Plates. Multiscale Expansions and Limiting Models for Shells. Hierarchical Models for Shells. Finite Element Methods in thin Domains. Acknowledgments. References. Further Reading. 9. Mixed Finite Elements Methods (Franco Brezzi, Ferdinando Auricchio and Carlo Lovadina). Introduction. Formulations. Stability of Saddle-Points in Finite Dimensions. Applications. Techniques for Proving the inf--sup Condition. References. 10. Meshfree Methods (Timon Rabczuk, Ted Belytschko, Sonia Fernandez-Mendez and Antonio Huerta). Introduction. Approximation in Meshfree Methods. Discretization of Partial Differential Equations. Radial Basis Functions. Discontinuities. Blending Meshfree Methods and Finite Elements. References. 11. Discrete Element Method (Nenad Bicanic). Introduction. Basic Discrete Element Framework and Regularization of Nonsmooth Contact Conditions. Characterization of Interacting Bodies and Contact Detection. Imposition of Contact Constraints and Boundary Conditions. Modeling of Block Deformability. Transition Continuum/Discontinuum, Fragmentation in Discrete Element Methods. Time Integration---Temporal Discretization, Energy Balance, and Discrete Element Implementation. Associated Frameworks and Developments. References. Further Reading. 12. Boundary Element Methods: Foundation and Error Analysis (W. L. Wendland and G. C. Hsiao). Introduction. Boundary Integral Equations. Variational Formulations. The Galerkin-BEM. The Role of Sobolev Index. Concluding Remarks. Acknowledgments. References. Further Reading. 13. Coupling of Boundary Element Methods and Finite Element Methods (Ernst P. Stephan). Introduction. Symmetric Coupling of Standard Finite Elements and Boundary Elements. Fast Solvers for the hp-version of FE/BE Coupling. Least Squares FE/BE Coupling Method. FE/BE Coupling for Interface Problems with Signorini Contact. Applications. Concluding Remarks. References. 14. Arbitrary Lagrangian--Eulerian Methods (J. Donea, J.-Ph. Ponthot, A. Rodriguez-Ferran and A. Huerta). Introduction. Descriptions of Motion. The Fundamental ALE Equation. ALE Form of Conservation Equations. Mesh-update Procedures. ALE Methods in Fluid Dynamics. ALE Methods in Nonlinear Solid Mechanics. References. 15. Finite Volume Methods: Foundation and Analysis (Timothy Barth and Mario Ohlberger). Introduction: Scalar Nonlinear Conservation Laws. Finite Volume (FV) Methods for Nonlinear Conservation Laws. Higher-order Accurate FV Generalizations. Further Advanced Topics. Concluding Remarks. References. Further Reading. 16. Geometrical Modeling of Technical Objects (F.-E. Wolter, M. Reuter and N. Peinecke). Architecture of Modeling Systems. Voxel Representation. Surface Patches. Boundary Representation. Constructive Solid Geometry. Medial Modeling. Attributes. Outlook and Concluding Remarks. References. 17. Mesh Generation and Mesh Adaptivity (P. Laug, P. L. George, P. J. Frey, H. Borouchaki and E. Saltel). Introduction. A Brief History. Mesh-Generation Methods. Quality Meshing and Adaptivity. Adaptive FEM Computations. Large-Size Problem, Parallelism and Adaptivity. Meshing for Moving Boundary Problems. Application Examples. Conclusions. References. 18. Computational Visualization (William J. Schroeder and Mark S. Shephard). Introduction. Data Forms. Visualization Algorithms. Volume Rendering. Methods in Large Data Visualization. Taxonomy for Data Visualization Systems. Interfacing the Computational System with the Visualization System. References. 19. Linear Algebraic Solvers and Eigenvalue Analysis (Henk A. van der Vorst). Introduction. Mathematical Preliminaries. Direct Methods for Linear Systems. Preconditioning. Incomplete LU Factorizations. Methods for the Complete Eigenproblem. Iterative Methods for the Eigenproblem. Notes. References. 20. Multigrid Methods for FEM and BEM Applications (Wolfgang Hackbusch). General Remarks on Multigrid Methods. Two-Grid Iteration. Multigrid Method. Application to Finite Element Equations. Additive Variant. Nested Iteration. Nonlinear Equations. Eigenvalue Problems. Applications to the Boundary Element Method (BEM). References. Further Reading. 21. Panel Clustering Techniques and Hierarchical Matrices for BEM and FEM (Wolfgang Hackbusch). Introduction. The panel clustering Method (First Version). The panel clustering Method (Second Version). Hierarchical Matrices. References. 22. Domain Decomposition Methods and Preconditioning (V. G. Korneev and U. Langer). Introduction. Domain Decomposition History. Fundamentals of Schwarz's Methods. Overlapping Domain Decomposition Methods. Nonoverlapping Domain Decomposition Methods. Acknowledgments. References. Further Reading. 23. Nonlinear Systems and Bifurcations (Werner C. Rheinboldt). Introduction. General Iterative Processes. Some Classes of Iterative Methods. Parameterized Systems. Bifurcation. References. Further Reading. 24. Adaptive Computational Methods for Parabolic Problems (K. Eriksson, C. Johnson and A. Logg). What is a Parabolic Problem?. Outline. References into the Literature. Introduction to Adaptive Methods for IVPs. Examples of Stiff IVPs. A Nonstiff IVP: The Lorenz System. Explicit Time-stepping for Stiff IVPs. Strong Stability Estimates for an Abstract Parabolic Model Problem. Adaptive Space--Time Galerkin Methods for the Heat Equation. A Priori and a Posteriori Error Estimates for the Heat Equation. Adaptive Methods/Algorithms. Reliability and Efficiency. Strong Stability Estimates for the Heat Equation. A Priori Error Estimates for the L2- and Elliptic Projections. Proof of the A Priori Error Estimates. Proof of the A Posteriori Error Estimates. Extension to Systems of Convection--Diffusion-reaction Problems. Examples of Reaction-- Diffusion Problems. Comparison with the Standard Approach to Time Step Control for odes. Software. References. Further Reading. 25. Time-dependent Problems with the Boundary Integral Equation Method (Martin Costabel). Introduction. Space-time Integral Equations. Laplace Transform Methods. Time-stepping Methods. References. 26. Finite Element Methods for Maxwell Equations (Leszek Demkowicz). Maxwell Equations. Variational Formulation. Exact Sequences. Projection-based Interpolation. De Rham Diagram. Additional Comments. Acknowledgment. Notes. References. Further Reading. Index. VOLUME 2: SOLIDS AND STRUCTURES. List of Contributors. Preface. 1. Solids, Introduction and Survey (Rene de Borst). Introduction. 2. Finite Element Method for Elasticity with Error-controlled Approximation and Model Adaptivity (Marcus Ruter and Erwin Stein). Introduction. Nonlinear and Linear Theory of Elasticity. Variational Problems and Their Finite Element Discretizations. Error Estimation and Adaptivity in Linearized Elasticity. Coupled a Posteriori Model and Discretization Error Estimation. A Posteriori Error Estimation in Finite Elasticity. Concluding Remarks. References. 3. Models and Finite Elements for Thin-walled Structures (W. A. Wall, M. Bischoff, K.-U. Bletzinger and E. Ramm). Introduction. Mathematical and Mechanical Foundations. Plates and Shells. Dimensional Reduction and Structural Models. Finite Element Formulation. Concluding Remarks. Acknowledgments. Related Chapters. References. Further Reading. Appendix. 4. Buckling of Structures (Eduard Riks). Introduction. Basic Concepts. Computation. References. Further Reading. 5. Linear and Nonlinear Structural Dynamics (Gregory M. Hulbert). Introduction. Formulation of Equations of Structural Dynamics. Time Integration Algorithms. Linear Structural Dynamics. Nonlinear Dynamics. Practical Considerations for Time Integration Algorithms. References. 6. Computational Contact Mechanics (P. Wriggers and G. Zavarise). Introduction. General Overview. Continuum Description of Contact. Contact Discretizations. Solution Algorithms. Conclusions. References. 7. Elastoplastic and Viscoplastic Deformations in Solids and Structures (F. Armero). Introduction. The Mechanical Problem. Infinitesimal Models of Elastoplasticity. Finite Deformation Elastoplasticity. Integration Algorithms. Primal Formulations of the Closest-Point Projection Equations. Dual Formulations of the Closest-Point Projection Equations. Augmented Lagrangian Formulations. Concluding Remarks. Acknowledgment. References. 8. Crystal Plasticity (Jan Schotte and Christian Miehe). Introduction. The Continuum Slip Theory of Crystal Plasticity. Stress Update Algorithms in Single Crystal Plasticity. Variational Formulation of Elastic--Plastic Crystals. Representative Numerical Examples. Conclusion. References. 9. Shakedown and Safety Assessment (Nestor Zouain). Introduction. Basic Notation. Classical Static and Kinematic Shakedown Formulations. Extremum Principles for Elastic Shakedown. Discrete Models for Elastic Shakedown. Preventing Alternating Plasticity. Preventing Simple Mechanisms of Incremental Collapse. Extensions Concerning Hardening and Temperature Effects. Other Extensions of Shakedown Theory. An Algorithm for Shakedown Analysis. Numerical Procedures for Shakedown Analysis. Applications. Conclusions. References. Further Reading. 10. Damage, Material Instabilities, and Failure (Rene de Borst). Introduction. Damage Mechanics. Material Instabilities and Mesh Sensitivity. Cohesive-Zone Models. Enhanced Continuum Descriptions. Discrete Failure Models. Concluding Remarks. References. Further Reading. 11. Computational Fracture Mechanics (Anthony R. Ingraffea). Introduction. A Taxonomy of Approaches for Representation of Cracking Processes. Geometrical Representation Approaches. Nongeometrical Representation Approaches. Summary. Acknowledgments. References. Further Reading. 12. Homogenization Methods and Multiscale Modeling (Tarek I. Zohdi). Introduction. Fundamental Micro--Macro Concepts. Testing Procedures. The Hill--Reuss--Voigt Bounds. More Refined Micro--Macro Approximations. Computational Homogenization. Microgeometrical Manufacturing Idealizations. Numerical Discretization. Overall Testing Process: Numerical Examples. Increasing Sample Size. Hierarchical/Multiscale Interpretations. Closing Comments and Future Directions. Notes. References. Further Reading. 13. Computational Modelling of Damage and Failures in Composite Laminates (J. N. Reddy and D. H. Robbins Jr.). Objectives and Scope. Introduction: the Multiscale Problem. Laminate Theories and Models. Review of Literature on Damage and Failures. Coupling Between the Microscale and the Mesoscale. Modeling of Progressive Damage in Composite Laminates. Acknowledgments. References. Further Reading. 14. Computational Modeling of Forming Processes (D. R. J. Owen and D. Peric). Introduction. Continuum Constitutive Modeling. Implicit Finite Element Solution Strategy. Contact--Friction Modeling. Element Technology. Refined Constitutive Models for Inelastic Solids at Finite Strains. Thermomechanical Coupling. Adaptive Strategies for Nonlinear Problems. Explicit Solution Strategies. Thin Sheet Forming Operations. Bulk Forming Operations. Metal Cutting Operations. Concluding Remarks. Acknowledgments. Note. References. Further Reading. 15. Computational Concrete Mechanics (Roman Lackner, Herbert A. Mang and Christian Pichler). Introduction. Basic Ingredients of Multiscale Modeling. Autogenous Shrinkage of Shotcrete in Hybrid Analysis of Tunnel Linings. Tension Stiffening in Cooling Tower Analysis. Concluding Remarks. References. 16. Computational Geomechanics Including Consolidation (John P. Carter and John C. Small). Introduction. Characteristics of Geotechnical Problems. Deterministic Geotechnical Analysis. Methods of Numerical Analysis. Limit Analysis Using Finite Elements. Constitutive Models for Geomaterials. Soil-Structure Interaction. Consolidation. Stochastic Techniques. Concluding Remarks. Acknowledgments. References. 17. Multifield Problems (B. A. Schrefler). Introduction. Partitioned Solution Procedures. Soil Dynamics. Monolithic Solution Procedure. Conclusions. Acknowledgments. References. Further Reading. 18. Computational Biomechanics of Soft Biological Tissue (Gerhard A. Holzapfel). Introduction. Mechanics of the Arterial Wall. Mechanics of the Heart Wall. Mechanics of the Ligament. Acknowledgments. References. 19. Identification of Material Parameters for Constitutive Equations (R. Mahnken). Introduction. General Framework for Development of Constitutive Models. Parameter Identification. Identification Methods. Optimization Methods. Instabilities in Least-Squares Problems. Stochastic Methods. Uniform Small Strain Problems. Parameter Identification For Nonuniform Large Strain Problems. Concluding Remarks. Acknowledgment. References. Further Reading. 20. Stochastic Finite Element Methods (Steen Krenk and Miguel A. Gutierrez). Introduction. Random Variables. Statically Determinate Problems. Representation of Random Fields. Basic Variable Sensitivity Computation. Perturbation Technique. Spectral Formulation. Reliability Methods. Status and Outlook. References. 21. Fluid-structure Interaction Problems (Roger Ohayon). Introduction. Structural-acoustic Problem. Structural-acoustic Equations. Incompressible Hydroelastic-sloshing Problem. Hydroelastic-sloshing Equations. Conclusion. References. Further Reading. 22. Acoustics (Peter M. Pinsky and Lonny L. Thompson). Introduction. Acoustic Field Equations. Time-harmonic Waves and the Helmholtz Equation. Discretization Methods for the Helmholtz Equation. The Exterior Problem in Unbounded Domains. The DtN Nonreflecting Boundary Condition. The Modified DtN Nonreflecting Boundary Condition. Infinite Elements. Perfectly Matched Layer (PML). Accelerated Multifrequency Solution Methods. Parallel Iterative Solution Methods. Domain Decomposition Methods. Direct Time-domain Methods for Acoustic Waves. Conclusions. Acknowledgments. References. 23. Boundary Integral Equation Methods for Elastic and Plastic Problems (Marc Bonnet). Introduction. Basic Integral Identities. The Boundary Element Method in Elasticity: Collocation. The Boundary Element Method in Elasticity: Symmetric Galerkin. Fast Solution Techniques. The Boundary Element Method for Fracture Mechanics. Boundary-Domain Integral Equations for Elastic--Plastic Problems. Shape Sensitivity Analysis. FEM--BEM Coupling. Related Chapters. References. 24. Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelastic Solids (M. Schanz, M. Kogl, L. Gaul and F. Moser). Viscoelastic Direct Boundary Element Formulation in Time Domain. Dual Reciprocity Method for Elastodynamics and Piezoelectricity. Nonsingular Hybrid Boundary Element Formulation for Elastodynamics. References. Index. VOLUME 3: FLUIDS. List of Contributors. Preface. 1. Fluids, Introduction and Survey (Thomas J. R. Hughes). 2. Multiscale and Stabilized Methods (Thomas J. R. Hughes, Leopoldo P. Franca and Guglielmo Scovazzi). Introduction. Dirichlet-to-Neumann Formulation. Variational Multiscale Method. Space--Time Formulations. Stabilized Methods for Advective--Diffusive Equations. Turbulence. Acknowledgments. References. 3. Spectral Element and hp Methods (Robert M. Kirby and George Em Karniadakis). Introduction. Polynomial Expansions on Unstructured Grids. Incompressible Flows. Compressible Flows. Plasma Flows. Discussion. Acknowledgments. References. Further Reading. 4. Discontinuous Galerkin Methods for Computational Fluid Dynamics (B. Cockburn). Introduction. Linear Hyperbolic Problems. Nonlinear Hyperbolic Problems. DG Methods for Second-order Elliptic Problems. DG Methods for Convection-dominated Problems. Concluding Remarks and Bibliographical Notes. Acknowledgments. References. 5. Vortex Methods (G. S. Winckelmans). Introduction. Vortex Methods for 2D Unbounded Inviscid Flows. Vortex Methods for 3D Unbounded Inviscid Flows. Viscous Flows. Improvement of the Truncation Error. Particle Redistribution. Efficient Velocity Evaluation: Fast Multipole Methods. Efficient Velocity Evaluation: Vortex-in-cell Methods. Flows with Solid Boundaries. Other Applications. Acknowledgments. References. Further Reading. 6. Incompressible Viscous Flows (Rolf Rannacher). Introduction. Mathematical Models. Discretization of Space. Discretization of Time. Error Control and Mesh Adaptation. Solution of the Algebraic Systems. Acknowledgment. References. Further Reading. 7. Computability and Adaptivity in CFD (J. Hoffman and C. Johnson). Introduction. Outline. References. The Incompressible Navier--Stokes Equations. Discretization: General Galerkin G$^{2}$. Adaptive Computation of the Drag of a Bluff Body. Drag of a Square Cylinder. The Drag of a Surface-mounted Cube. The Drag Versus the Total Dissipation. Reliability and Efficiency of the Adaptive Method. Averaged Navier--Stokes Equations and Reynolds Stresses. The Subgrid Model from Stabilization. Computability in Transition to Turbulence. Applications to Stationary Benchmark Problems in 3D. Acknowledgments. References. 8. Dynamic Multilevel Methods and Turbulence (T. Dubois, R. M. Temam and F. Jauberteau). Introduction. Turbulence: Multilevel Methods for the Navier--Stokes Equations. Multilevel Methods for the Shallow Water Equations. Renormalization of Small Eddies. Conclusion: Summary and Perspectives. References. Further Reading. 9. Turbulence Direct Numerical Simulation and Large-eddy Simulation (Pierre Sagaut). Introduction. Mathematical Models and Governing Equations. Basic Numerical Issues for DNS and LES. Subgrid-scale Modeling for the Incompressible Case. Extension of Subgrid Models for the Compressible Case. Boundary Conditions for LES. Applications of DNS and LES. Related Chapters. Acknowledgment. References. Further Reading. 10. Turbulence Closure Models for Computational Fluid Dynamics (Paul A. Durbin). Introduction: The Role of Statistical Closure. Reynolds-averaged Navier--Stokes Equations. Models With Scalar Variables. Second Moment Transport. Reynolds-averaged Computation. Notes. References. Further Reading. 11. Aerodynamics (Antony Jameson). Focus and Historical Background. Mathematical Models of Fluid Flow. Potential Flow Methods. Shock-capturing Algorithms for the Euler and Navier--Stokes Equations. Discretization Scheme for Flows in Complex Multidimensional Domains. Time-stepping Schemes. Aerodynamic Shape Optimization. Acknowledgment. References. 12. Industrial Aerodynamics (Frederic L. Chalot). Introduction. The Historical Development of Computational Flow Mechanics. Numerical Codes. Euler Code. Navier--Stokes Code. Reynolds-averaged Turbulence Modeling. Large Eddy Simulation. Mesh Generation. Validation. Code Interface. Military Applications. Civil Applications. Space Applications. Fundamental Studies and Research Work. Shape Optimization. Aeroacoustics. Multidisciplinary Applications. Conclusion. Acknowledgment. References. Further Reading. 13. CFD-based Nonlinear Computational Aeroelasticity (Charbel Farhat). Introduction. The Three-field Formulation of Nonlinear Aeroelastic Problems. Recent Computational Advances. The Aero Simulation Platform. Sample Recent Validation Results. Conclusions. Acknowledgments. References. 14. Mixed Finite Element Methods for Non-newtonian Fluid (Patrick D. Anderson, Martien A. Hulsen and Frank P. T. Baaijens). Introduction. Mathematical Formulation. Steady Flow: Variational Formulations. Time Dependent Flows. Integral and Stochastic Constitutive Models. Governing Equations. The Deformation Fields Method. Brownian Configuration Fields. Numerical Methods. Results. Conclusions and Discussion. References. 15. Combustion (T. J. Poinsot and D. P. Veynante). Introduction. Combustion Regimes. Governing Equations. Combustion Terminology and Basics. Homogeneous Reactors and Laminar Flames. Turbulent Flames. Conclusions. References. 16. Blood Flow (Charles A. Taylor). Introduction. Overview of the Cardiovascular System. Lumped Parameter Models. One-Dimensional Wave Propagation Models. Three-Dimensional Equations of Blood Flow. Future Work. Acknowledgments. Literature Cited. 17. Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces (Tayfun E. Tezduyar). Introduction. Governing Equations. Stabilized Formulations. DSD/SST Finite Element Formulation. Calculation of the Stabilization Parameters for Incompressible Flows. Discontinuity-Capturing Directional Dissipation (DCDD). Calculation of the Stabilization Parameters for Compressible Flows and Shock-Capturing. Mesh Update Methods. Shear--Slip Mesh Update Method (SSMUM). DSD/SST Formulation for Fluid--Object Interactions in Spatially Periodic Flows. Space--Time Contact Technique (STCT). Fluid--Object Interactions Subcomputation Technique (FOIST). Enhanced-Discretization Interface-Capturing Technique (EDICT). Extensions and Offshoots of Edict. Mixed Interface-Tracking/Interface-Capturing Technique (MITICT). Edge-Tracked Interface Locator Technique (ETILT). Line-Tracked Interface Update Technique (LTIUT). Iterative Solution Methods. Enhanced Solution Techniques. Mixed Element-Matrix-Based/Element-Vector-Based Computation Technique (MMVCT). Enhanced-Discretization Successive Update Method (EDSUM). Examples of Flow Simulations. Concluding Remarks. Acknowledgment. References. 18. Ship Hydrodynamics (Sergio R. Idelsohn, Julio Garcia and Eugenio Onate). Introduction. The Navier--Stokes Equations for Incompressible Flows. ALE Formulation. About the Finite Element Solution of the Navier--Stokes Equations. Basic Concepts of the Finite Calculus (FIC) Method. FIC Equations for Viscous Incompressible Flow. ALE Formulation. Finite Element Discretization. Fluid-ship Interaction. A Simple Algorithm for Updating the Mesh Nodes. Modeling of the Transom Stern Flow. Lagrangian Flow Formulation. Modeling the Structure as a Viscous Fluid. Computation of the Characteristic Lengths. Turbulence Modeling. Examples. Concluding Remarks. Acknowledgments. References. Further Reading. Index.