The work presented in this dissertation was carried out at the Division of Mechanics, Department of Management and Engineering at Link¨oping University,between 2003 and 2008. It was supervised by Prof. Anders Klarbring,head of the division, and financially supported by the National GraduateSchool of Scientific Computing (NGSSC) and the Swedish Research Council(VR). There are many people to whom I would like to express my gratitude…

*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 conclusion

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

1 Introduction

2 General multi–ﬁeld ﬂow model

2.1 Planar ﬂuid ﬂow

2.2 Planar heat ﬂow/diffusion

3 Model for mixed ﬂuid and heat ﬂow

References

A Derivation of the heat equation

B Derivation of the diﬀusion equation

Author: Wiker, Niclas

Source: Linkoping University

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