This book presents numerical algorithms for solving incompressible fluids, elastic structures and fluid-structure interactions. It collects some of the fundamental finite element methods as well as new approaches.For Stokes and Navier-Stokes equations, the mixed finite element method is employed. An arbitrary Lagrangian Eulerian framework is used for fluids in a moving domain. Schemes for linear and St Venant-Kirchhoff non-linear dynamic elasticity are presented. For fluid-structure interaction, two schemes are analyzed: the first is fully implicit and the second is semi-implicit, where the fluid domain is computed explicitly and consequently the computational time is considerably reduced.The stability of the schemes is proven in this self-contained book. Every chapter is supplied with numerical tests for the reader. These are aimed at Masters students in Mathematics or Mechanical Engineering.
1. Cover - p. ii 2. Title page - p. ii 3. Table of Contents - p. ii 4. Dedication - p. ii 5. Copyright - p. iv 6. Preface - p. ix 7. 1: Mixed Finite Elements for the Stokes Equations - p. 1 8. Abstract - p. 1 9. 1.1 Stokes equations - p. 1 10. 1.2 Existence and uniqueness for an abstract mixed problem - p. 4 11. 1.3 Existence and uniqueness for the Stokes equations - p. 10 12. 1.4 Finite element approximation of an abstract mixed problem - p. 14 13. 1.5 Mixed finite elements for the Stokes equations - p. 20 14. 1.6 Boundary conditions for the Stokes equations - p. 30 15. 1.7 Numerical tests - p. 36 16. 2: Numerical Schemes for the Navier-Stokes Equations - p. 45 17. Abstract - p. 45 18. 2.1 Navier-Stokes equations - p. 45 19. 2.2 Implicit Euler scheme, semi-implicit treatment of the nonlinear term - p. 48 20. 2.3 Implicit Euler scheme and implicit treatment of the nonlinear term - p. 54 21. 2.4 Implicit Euler scheme and explicit treatment of the nonlinear term - p. 56 22. 2.5 Some properties of the trilinear forms c and c˜ - p. 65 23. 2.6 Numerical tests: flow around a disc - p. 69 24. 3: The ALE Method for Navier-Stokes Equations in a Moving Domain - p. 79 25. Abstract - p. 79 26. 3.1 Lagrangian and Eulerian coordinates - p. 79 27. 3.2 Arbitrary Lagrangian Eulerian (ALE) coordinates - p. 84 28. 3.3 Weak non-conservative and conservative formulations - p. 89 29. 3.4 Discretization of order one in time for the weak conservative formulation - p. 94 30. 3.5 Stabilized discretization of order one in time for the non-conservative formulation - p. 101 31. 3.6 Finite element discretization in space - p. 104 32. 3.7 Numerical tests - p. 105 33. 4: Linear Elastodynamics - p. 111 34. 5: Nonlinear Elastodynamics - p. 133 35. 6: Numerical Schemes for the Fluid-Structure Interaction - p. 147 36. Abstract - p. 147 37. 6.1 Non-conservative and conservative weak formulations - p. 147 38. 6.2 Discretization in time of the conservative form: implicit domain calculation - p. 154 39. 6.3 Discretization in time of the non-conservative form: explicit domain calculation - p. 164 40. 6.4 Coupling strategies - p. 172 41. 6.5 The constants in the Poincaré, Korn and trace inequalities - p. 176 42. 6.6 Numerical tests - p. 181 43. Appendix: Functional Analysis Tools - p. 185 44. Bibliography - p. 193 45. Index - p. 197
This book presents numerical algorithms for solving incompressible fluids, elastic structures and fluid-structure interactions. It collects some of the fundamental finite element methods as well as new approaches.For Stokes and Navier-Stokes equations, the mixed finite element method is employed. An arbitrary Lagrangian Eulerian framework is used for fluids in a moving domain. Schemes for linear and St Venant-Kirchhoff non-linear dynamic elasticity are presented. For fluid-structure interaction, two schemes are analyzed: the first is fully implicit and the second is semi-implicit, where the fluid domain is computed explicitly and consequently the computational time is considerably reduced.The stability of the schemes is proven in this self-contained book. Every chapter is supplied with numerical tests for the reader. These are aimed at Masters students in Mathematics or Mechanical Engineering.
1. Cover - p. ii 2. Title page - p. ii 3. Table of Contents - p. ii 4. Dedication - p. ii 5. Copyright - p. iv 6. Preface - p. ix 7. 1: Mixed Finite Elements for the Stokes Equations - p. 1 8. Abstract - p. 1 9. 1.1 Stokes equations - p. 1 10. 1.2 Existence and uniqueness for an abstract mixed problem - p. 4 11. 1.3 Existence and uniqueness for the Stokes equations - p. 10 12. 1.4 Finite element approximation of an abstract mixed problem - p. 14 13. 1.5 Mixed finite elements for the Stokes equations - p. 20 14. 1.6 Boundary conditions for the Stokes equations - p. 30 15. 1.7 Numerical tests - p. 36 16. 2: Numerical Schemes for the Navier-Stokes Equations - p. 45 17. Abstract - p. 45 18. 2.1 Navier-Stokes equations - p. 45 19. 2.2 Implicit Euler scheme, semi-implicit treatment of the nonlinear term - p. 48 20. 2.3 Implicit Euler scheme and implicit treatment of the nonlinear term - p. 54 21. 2.4 Implicit Euler scheme and explicit treatment of the nonlinear term - p. 56 22. 2.5 Some properties of the trilinear forms c and c˜ - p. 65 23. 2.6 Numerical tests: flow around a disc - p. 69 24. 3: The ALE Method for Navier-Stokes Equations in a Moving Domain - p. 79 25. Abstract - p. 79 26. 3.1 Lagrangian and Eulerian coordinates - p. 79 27. 3.2 Arbitrary Lagrangian Eulerian (ALE) coordinates - p. 84 28. 3.3 Weak non-conservative and conservative formulations - p. 89 29. 3.4 Discretization of order one in time for the weak conservative formulation - p. 94 30. 3.5 Stabilized discretization of order one in time for the non-conservative formulation - p. 101 31. 3.6 Finite element discretization in space - p. 104 32. 3.7 Numerical tests - p. 105 33. 4: Linear Elastodynamics - p. 111 34. 5: Nonlinear Elastodynamics - p. 133 35. 6: Numerical Schemes for the Fluid-Structure Interaction - p. 147 36. Abstract - p. 147 37. 6.1 Non-conservative and conservative weak formulations - p. 147 38. 6.2 Discretization in time of the conservative form: implicit domain calculation - p. 154 39. 6.3 Discretization in time of the non-conservative form: explicit domain calculation - p. 164 40. 6.4 Coupling strategies - p. 172 41. 6.5 The constants in the Poincaré, Korn and trace inequalities - p. 176 42. 6.6 Numerical tests - p. 181 43. Appendix: Functional Analysis Tools - p. 185 44. Bibliography - p. 193 45. Index - p. 197
