|
| Titre : |
Theoretical and numerical study of some interaction problems in continuum mechanics |
| Type de document : |
document électronique |
| Auteurs : |
Ahmed ayoub Boukelia, Auteur ; Mohamed Kara, Directeur de thèse |
| Editeur : |
Setif:UFA |
| Année de publication : |
2026 |
| Importance : |
1 vol (119 f.) |
| Format : |
29 cm |
| Langues : |
Anglais (eng) |
| Catégories : |
Thèses & Mémoires:Mathématique
|
| Mots-clés : |
Fluid Mechanics
Fluid–Structure Interaction
FVM (Finite Volume Method)
Continuum Mechanics
Interaction Problems |
| Index. décimale : |
510 - Mathématique |
| Résumé : |
This thesis investigates elastic–elastic and fluid–structure interaction (FSI) problems under
generalized interface conditions that extend beyond classical continuity and balance
laws. The work develops a rigorous functional framework, establishing weak formulations,
existence, and stability results for systems where coupling laws incorporate impedance,
stiffness, or damping effects through positive definite operators. These additional terms
modify the natural energy balance and lead to new conditions for well-posedness. On
the computational side, finite volume schemes are designed on admissible meshes, with
particular attention to conservative and stable treatment of interface terms. Penalization
techniques are introduced to enforce generalized interface laws, and convergence to
the original model is demonstrated. Theoretical findings are validated through a set of
numerical experiments, which confirm convergence, illustrate the influence of interface
parameters, and highlight the role of generalized coupling in dynamic responses. Overall,
the thesis bridges rigorous mathematical analysis with efficient numerical strategies,
offering a unified framework for analyzing and simulating complex coupled systems. |
| Note de contenu : |
Sommaire
List of Figures i
List of Tables ii
Notations 1
Introduction 3
1 Functional Analysis and Elementary Concepts 11
1.1 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 Spaces of Banach-valued functions . . . . . . . . . . . . . . . . . . . 14
1.1.3 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.4 Trace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Finite volume method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.1 Admissible meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.2 Discrete spaces and operators . . . . . . . . . . . . . . . . . . . . . 23
2 Fundamentals of Elasticity and Fluid Mechanics 28
2.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1 Kinematics of Deformation . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Strain Tensors and the Small Deformation Assumption . . . . . . . 30
2.1.3 Stress Tensor and Hooke’s Law . . . . . . . . . . . . . . . . . . . . 31
2.1.4 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . 32
2.1.5 Time-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.6 Space-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.7 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Incompressible Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.1 Kinematics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Newtonian Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.4 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 44
2.2.5 Special Case: Stokes Equations . . . . . . . . . . . . . . . . . . . . 45
2.2.6 Remarks on Coupling and Well-Posedness . . . . . . . . . . . . . . 47
2.2.7 Discretization of Stokes Problem . . . . . . . . . . . . . . . . . . . 49
2.2.8 Discretization of Navier-Stokes Problem . . . . . . . . . . . . . . . 51
2.2.9 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Elastic–Elastic Interactions 62
3.1 A well-posed elastic–elastic problem with jump embedded boundary conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The elastic/elastic problem with interface jump conditions of Spring-Law
type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Penalized problem of elastic–elastic model with J.E.B.C. . . . . . . . . . . 69
3.4 Finite Volume discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1 Admissible meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Treatement of Internal cells . . . . . . . . . . . . . . . . . . . . . . 74
3.4.3 Treatment of the Interface Conditions on Σ . . . . . . . . . . . . . 75
3.4.4 Linear algebraic system . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.1 From Penalized Interface to Continuity Condition . . . . . . . . . . 77
3.5.2 From Penalized Interface to Discontinuity Condition . . . . . . . . . 81
4 Fluid–Structure Interactions 83
4.1 Fluid-Structure Interaction Model . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 Governing equations, initial and boundary conditions . . . . . . . . 84
4.1.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.3 Functional Spaces, Inner Products and Norms . . . . . . . . . . . . 89
4.2 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1 Auxiliary Problem and Abstract Nonlinear Evolution . . . . . . . . 93
Conclusion 109 |
| Côte titre : |
DM/0218 |
Theoretical and numerical study of some interaction problems in continuum mechanics [document électronique] / Ahmed ayoub Boukelia, Auteur ; Mohamed Kara, Directeur de thèse . - [S.l.] : Setif:UFA, 2026 . - 1 vol (119 f.) ; 29 cm. Langues : Anglais ( eng)
| Catégories : |
Thèses & Mémoires:Mathématique
|
| Mots-clés : |
Fluid Mechanics
Fluid–Structure Interaction
FVM (Finite Volume Method)
Continuum Mechanics
Interaction Problems |
| Index. décimale : |
510 - Mathématique |
| Résumé : |
This thesis investigates elastic–elastic and fluid–structure interaction (FSI) problems under
generalized interface conditions that extend beyond classical continuity and balance
laws. The work develops a rigorous functional framework, establishing weak formulations,
existence, and stability results for systems where coupling laws incorporate impedance,
stiffness, or damping effects through positive definite operators. These additional terms
modify the natural energy balance and lead to new conditions for well-posedness. On
the computational side, finite volume schemes are designed on admissible meshes, with
particular attention to conservative and stable treatment of interface terms. Penalization
techniques are introduced to enforce generalized interface laws, and convergence to
the original model is demonstrated. Theoretical findings are validated through a set of
numerical experiments, which confirm convergence, illustrate the influence of interface
parameters, and highlight the role of generalized coupling in dynamic responses. Overall,
the thesis bridges rigorous mathematical analysis with efficient numerical strategies,
offering a unified framework for analyzing and simulating complex coupled systems. |
| Note de contenu : |
Sommaire
List of Figures i
List of Tables ii
Notations 1
Introduction 3
1 Functional Analysis and Elementary Concepts 11
1.1 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 Spaces of Banach-valued functions . . . . . . . . . . . . . . . . . . . 14
1.1.3 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.4 Trace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Finite volume method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.1 Admissible meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.2 Discrete spaces and operators . . . . . . . . . . . . . . . . . . . . . 23
2 Fundamentals of Elasticity and Fluid Mechanics 28
2.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1 Kinematics of Deformation . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Strain Tensors and the Small Deformation Assumption . . . . . . . 30
2.1.3 Stress Tensor and Hooke’s Law . . . . . . . . . . . . . . . . . . . . 31
2.1.4 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . 32
2.1.5 Time-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.6 Space-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.7 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Incompressible Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.1 Kinematics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Newtonian Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.4 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 44
2.2.5 Special Case: Stokes Equations . . . . . . . . . . . . . . . . . . . . 45
2.2.6 Remarks on Coupling and Well-Posedness . . . . . . . . . . . . . . 47
2.2.7 Discretization of Stokes Problem . . . . . . . . . . . . . . . . . . . 49
2.2.8 Discretization of Navier-Stokes Problem . . . . . . . . . . . . . . . 51
2.2.9 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Elastic–Elastic Interactions 62
3.1 A well-posed elastic–elastic problem with jump embedded boundary conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The elastic/elastic problem with interface jump conditions of Spring-Law
type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Penalized problem of elastic–elastic model with J.E.B.C. . . . . . . . . . . 69
3.4 Finite Volume discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1 Admissible meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Treatement of Internal cells . . . . . . . . . . . . . . . . . . . . . . 74
3.4.3 Treatment of the Interface Conditions on Σ . . . . . . . . . . . . . 75
3.4.4 Linear algebraic system . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.1 From Penalized Interface to Continuity Condition . . . . . . . . . . 77
3.5.2 From Penalized Interface to Discontinuity Condition . . . . . . . . . 81
4 Fluid–Structure Interactions 83
4.1 Fluid-Structure Interaction Model . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 Governing equations, initial and boundary conditions . . . . . . . . 84
4.1.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.3 Functional Spaces, Inner Products and Norms . . . . . . . . . . . . 89
4.2 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1 Auxiliary Problem and Abstract Nonlinear Evolution . . . . . . . . 93
Conclusion 109 |
| Côte titre : |
DM/0218 |
|
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