Optimization in Continuum Flow Problems

Different kinds of flow systems are found all over the place: in the water streaming in the ground, in lakes and rivers, and in the air in the atmosphere, in the dissolved sugar spreading in a cup of tea , or in the transport of heat from stove plate to frying pan, to note just some. On a macroscopic level, systems such as these can, with a high level of accuracy, be treated as continuous (instead of discrete) throughout their occupied region in space. Within this dissertation, mathematical models of such continuum flow systems are made. Depending on these models, optimisation issues are then developed using well developed approaches from the field of topology optimization.


1 Introduction
2 The Darcy flow sample problem
2.1 State problem
2.1.1 Derivation of the state equations
2.1.2 Defining the state problem and variational form
2.2 Optimization problem formulation
2.2.1 Introducing a design variable
2.2.2 Choosing an objective function
2.2.3 Considering constraints and interpolation function
2.2.4 Formulating the optimization problem
3 Developments
3.1 Darcy flow
3.2 Stokes flow
3.2.1 Applications of Stokes flow optimization
3.2.2 Design dependent viscosity parameter
3.2.3 Constructing bottom profiles in Darcy–Stokes related flow
3.2.4 Designing porous materials with optimized properties
3.3 Navier–Stokes flow
3.3.1 Alternative methods for flow simulation
3.3.2 Changing fluid distribution description
3.3.3 Tackling real engineering problems
3.4 Multiphysic flow
3.4.1 From temperature to concentration
3.4.2 Considering fluid flow and elasticity
Paper I
1 Introduction
2 State problem
2.1 Continuum mechanical background
2.2 State problem definition and variational formulation
3 Optimization problem
4 Existence proof
5 Discrete designs
6 Finite element discretization
6.1 Matrix-vector formulations
6.2 Solution algorithm and sensitivity analysis
7 Numerical examples
7.1 Preliminaries
7.2 A comparison with a pure Darcy problem
7.3 The influence of Γt and R
7.4 The influence of γ and solution procedures
7.5 The influence of α/ and
8 Summary and conclusions
Paper II
1 Introduction
2 State problem
2.1 Geometry and Reynolds’ transport theorem
2.2 Derivation of non-linear state equations
2.3 Assumptions and simplifications
2.4 State problem definition and variational formulation
3 Optimization problem
4 Discrete formulation
5 Numerical results
5.1 Drainage problem
5.2 Pole–in–a–river problem
6 Summary and conclusions
A Motivation of assumption (26)
Paper III
1 Introduction
2 Design method formulation
2.1 State problem
2.2 Optimization problem
2.3 Numerical solution and implementation
3 Example 1: River delta
3.1 Varying the parameters A and γpres
4 Example 2: Spillway
4.1 Varying the parameters A and γopt
5 Summary and conclusions
Paper IV
1 Introduction
2 Moisture flow model for porous material with cracks
2.1 Flow equation
2.2 Diffusion equation
2.3 Including a material distribution variable
2.4 Dimensionless form with length scale parameter
2.5 Boundary conditions and variational formulation
3 Optimization problem formulation
3.1 Objective function
3.2 Constraints
3.3 Regularization and problem formulation
4 Discrete formulation
4.1 Approximate formulation
4.2 Matrix formulation
5 Solution strategy
5.1 Sensitivity analysis
5.2 Solution scheme
6 Numerical examples
6.1 Practicalities
6.2 Example 1: the optimal domain dimensions
6.3 Example 2: dependence on flow velocity
7 Discussion and future work
A Results for Example 1
B Results for Example 2
Paper V….

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