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.

Contents

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
Bibliography
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
References
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
References
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
References
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
References
A Results for Example 1
B Results for Example 2
Paper V….