
Applied Shape Optimization for Fluids
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Content
- Intro
- Contents
- 1 Introduction
- 2 Optimal shape design
- 2.1 Introduction
- 2.2 Examples
- 2.2.1 Minimum weight of structures
- 2.2.2 Wing drag optimization
- 2.2.3 Synthetic jets and riblets
- 2.2.4 Stealth wings
- 2.2.5 Optimal breakwater
- 2.2.6 Two academic test cases: nozzle optimization
- 2.3 Existence of solutions
- 2.3.1 Topological optimization
- 2.3.2 Suficient conditions for existence
- 2.4 Solution by optimization methods
- 2.4.1 Gradient methods
- 2.4.2 Newton methods
- 2.4.3 Constraints
- 2.4.4 A constrained optimization algorithm
- 2.5 Sensitivity analysis
- 2.5.1 Sensitivity analysis for the nozzle problem
- 2.5.2 Numerical tests with freefem++
- 2.6 Discretization with triangular elements
- 2.6.1 Sensitivity of the discrete problem
- 2.7 Implementation and numerical issues
- 2.7.1 Independence from the cost function
- 2.7.2 Addition of geometrical constraints
- 2.7.3 Automatic differentiation
- 2.8 Optimal design for Navier-Stokes flows
- 2.8.1 Optimal shape design for Stokes flows
- 2.8.2 Optimal shape design for Navier-Stokes flows
- References
- 3 Partial differential equations for fluids
- 3.1 Introduction
- 3.2 The Navier-Stokes equations
- 3.2.1 Conservation of mass
- 3.2.2 Conservation of momentum
- 3.2.3 Conservation of energy and and the law of state
- 3.3 Inviscid flows
- 3.4 Incompressible flows
- 3.5 Potential flows
- 3.6 Turbulence modeling
- 3.6.1 The Reynolds number
- 3.6.2 Reynolds equations
- 3.6.3 The k - e model
- 3.7 Equations for compressible flows in conservation form
- 3.7.1 Boundary and initial conditions
- 3.8 Wall laws
- 3.8.1 Generalized wall functions for u
- 3.8.2 Wall function for the temperature
- 3.8.3 k and e
- 3.9 Generalization of wall functions
- 3.9.1 Pressure correction
- 3.9.2 Corrections on adiabatic walls for compressible flows
- 3.9.3 Prescribing ?[sub(w)]
- 3.9.4 Correction for the Reichardt law
- 3.10 Wall functions for isothermal walls
- References
- 4 Some numerical methods for fluids
- 4.1 Introduction
- 4.2 Numerical methods for compressible flows
- 4.2.1 Flux schemes and upwinded schemes
- 4.2.2 A FEM-FVM discretization
- 4.2.3 Approximation of the convection fluxes
- 4.2.4 Accuracy improvement
- 4.2.5 Positivity
- 4.2.6 Time integration
- 4.2.7 Local time stepping procedure
- 4.2.8 Implementation of the boundary conditions
- 4.2.9 Solid walls: transpiration boundary condition
- 4.2.10 Solid walls: implementation of wall laws
- 4.3 Incompressible flows
- 4.3.1 Solution by a projection scheme
- 4.3.2 Spatial discretization
- 4.3.3 Local time stepping
- 4.3.4 Numerical approximations for the k - e equations
- 4.4 Mesh adaptation
- 4.4.1 Delaunay mesh generator
- 4.4.2 Metric definition
- 4.4.3 Mesh adaptation for unsteady flows
- 4.5 An example of adaptive unsteady flow calculation
- References
- 5 Sensitivity evaluation and automatic differentiation
- 5.1 Introduction
- 5.2 Computations of derivatives
- 5.2.1 Finite differences
- 5.2.2 Complex variables method
- 5.2.3 State equation linearization
- 5.2.4 Adjoint method
- 5.2.5 Adjoint method and Lagrange multipliers
- 5.2.6 Automatic differentiation
- 5.2.7 A class library for the direct mode
- 5.3 Nonlinear PDE and AD
- 5.4 A simple inverse problem
- 5.5 Sensitivity in the presence of shocks
- 5.6 A shock problem solved by AD
- 5.7 Adjoint variable and mesh adaptation
- 5.8 Tapenade
- 5.9 Direct and reverse modes of AD
- 5.10 More on FAD classes
- References
- 6 Parameterization and implementation issues
- 6.1 Introduction
- 6.2 Shape parameterization and deformation
- 6.2.1 Deformation parameterization
- 6.2.2 CAD-based
- 6.2.3 Based on a set of reference shapes
- 6.2.4 CAD-free
- 6.2.5 Level set
- 6.3 Handling domain deformations
- 6.3.1 Explicit deformation
- 6.3.2 Adding an elliptic system
- 6.3.3 Transpiration boundary condition
- 6.3.4 Geometrical constraints
- 6.4 Mesh adaption
- 6.5 Fluide-structure coupling
- References
- 7 Local and global optimization
- 7.1 Introduction
- 7.2 Dynamical systems
- 7.2.1 Examples of local search algorithms
- 7.3 Global optimization
- 7.3.1 Recursive minimization algorithm
- 7.3.2 Coupling dynamical systems and distributed computing
- 7.4 Multi-objective optimization
- 7.4.1 Data mining for multi-objective optimization
- 7.5 Link with genetic algorithms
- 7.6 Reduced-order modeling and learning
- 7.6.1 Data interpolation
- 7.7 Optimal transport and shape optimization
- References
- 8 Incomplete sensitivities
- 8.1 Introduction
- 8.2 Efficiency with AD
- 8.2.1 Limitations when using AD
- 8.2.2 Storage strategies
- 8.2.3 Key points when using AD
- 8.3 Incomplete sensitivity
- 8.3.1 Equivalent boundary condition
- 8.3.2 Examples with linear state equations
- 8.3.3 Geometric pressure estimation
- 8.3.4 Wall functions
- 8.3.5 Multi-level construction
- 8.3.6 Reduced order models and incomplete sensitivities
- 8.3.7 Redefinition of cost functions
- 8.3.8 Multi-criteria problems
- 8.3.9 Incomplete sensitivities and the Hessian
- 8.4 Time-dependent flows
- 8.4.1 Model problem
- 8.4.2 Data mining and adjoint calculation
- References
- 9 Consistent approximations and approximate gradients
- 9.1 Introduction
- 9.2 Generalities
- 9.3 Consistent approximations
- 9.3.1 Consistent approximation
- 9.3.2 Algorithm: conceptual
- 9.4 Application to a control problem
- 9.4.1 Algorithm: control with mesh refinement
- 9.4.2 Verification of the hypothesis
- 9.4.3 Numerical example
- 9.5 Application to optimal shape design
- 9.5.1 Problem statement
- 9.5.2 Discretization
- 9.5.3 Optimality conditions: the continuous case
- 9.5.4 Optimality conditions: the discrete case
- 9.5.5 Definition of ?[sub(h)]
- 9.5.6 Implementation trick
- 9.5.7 Algorithm: OSD with mesh refinement
- 9.5.8 Orientation
- 9.5.9 Numerical example
- 9.5.10 A nozzle optimization
- 9.5.11 Theorem
- 9.5.12 Numerical results
- 9.5.13 Drag reduction for an airfoil with mesh adaptation
- 9.6 Approximate gradients
- 9.6.1 A control problem with domain decomposition
- 9.6.2 Algorithm
- 9.6.3 Numerical results
- 9.7 Conclusion
- 9.8 Hypotheses in Theorem 9.3.2.1
- 9.8.1 Inclusion
- 9.8.2 Continuity
- 9.8.3 Consistency
- 9.8.4 Continuity of ?
- 9.8.5 Continuity of ?[sub(h)](a[sub(h)])
- 9.8.6 Convergence
- References
- 10 Numerical results on shape optimization
- 10.1 Introduction
- 10.2 External flows around airfoils
- 10.3 Four-element airfoil optimization
- 10.4 Sonic boom reduction
- 10.5 Turbomachines
- 10.5.1 Axial blades
- 10.5.2 Radial blades
- 10.6 Business jet: impact of state evaluations
- References
- 11 Control of unsteady flows
- 11.1 Introduction
- 11.2 A model problem for passive noise reduction
- 11.3 Control of aerodynamic instabilities around rigid bodies
- 11.4 Control in multi-disciplinary context
- 11.4.1 A model problem
- 11.4.2 Coupling strategies
- 11.4.3 Low-complexity structure models
- 11.5 Stability, robustness, and unsteadiness
- 11.6 Control of aeroelastic instabilities
- References
- 12 From airplane design to microfluidics
- 12.1 Introduction
- 12.2 Governing equations for microfluids
- 12.3 Stacking
- 12.4 Control of the extraction of infinitesimal quantities
- 12.5 Design of microfluidic channels
- 12.5.1 Reduced models for the flow
- 12.6 Microfluidic mixing device for protein folding
- 12.7 Flow equations for microfluids
- 12.7.1 Coupling algorithm
- References
- 13 Topological optimization for fluids
- 13.1 Introduction
- 13.2 Dirichlet conditions on a shrinking hole
- 13.2.1 An example in dimension 2
- 13.3 Solution by penalty
- 13.3.1 A semi-analytical example
- 13.4 Topological derivatives for fluids
- 13.4.1 Application
- 13.5 Perspective
- References
- 14 Conclusions and prospectives
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- R
- S
- T
- U
- V
- W
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