Full Record

Author | Letona Bolivar, Cristina Felicitas |

Title | On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types |

URL | http://hdl.handle.net/10919/73308 |

Publication Date | 2016 |

Date Accessioned | 2016-10-20 08:00:41 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | Virginia Tech |

Abstract | The methods for solving domain optimization problems depends on the case of study. There are methods that have been developed for the discretized problem, but not much is done in the infinite dimensional case. We analyze the theoretical aspects of the infinite dimensional case for a particular domain optimization problem where a portion of the boundary is parametrized, these results involve the existence of the solution to our problem and the calculation of the derivative of the shape functional. Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. We consider a special class of PDE-constrained shape optimization problems where different boundary condition types (Dirichlet and Neumann) are imposed on the same boundary segment. We also consider the case where the interface between these different boundary condition types may also be parameter dependent. This study also includes special cases where the shape of the region where the PDE is imposed does not change, but the domain of the partial differential operator is parameter dependent, due to the change in boundary condition type. Our treatment centers on the infinite dimensional formulation of the optimization problem. We consider existence of solutions as well as the calculation of derivatives of the associated shape functionals via adjoint solutions. These derivative formulations serve as a starting point for practical numerical approximations. |

Subjects/Keywords | Domain Optimization; Shape Derivatives; PDE Constraints; Mixed Boundary Conditions. |

Contributors | Borggaard, Jeffrey T. (committeechair); Zietsman, Lizette (committee member); Iliescu, Traian (committee member); Lin, Tao (committee member) |

Rights | In Copyright http://rightsstatements.org/vocab/InC/1.0/ |

Country of Publication | us |

Record ID | handle:10919/73308 |

Repository | vt |

Date Retrieved | 2020-10-09 |

Date Indexed | 2020-10-14 |

Grantor | Virginia Polytechnic Institute and State University |

Issued Date | 2016-10-19 00:00:00 |

Note | [degree] Ph. D.; |

Sample Search Hits | Sample Images | Cited Works

…include drag reduction, shielding electromagnetic fields, and estimating the shape of cavities
such as reservoirs or tumors. Variants of these applications can be posed as shape optimization problems with elliptic *PDE* *constraints*. As for any other…

…x5D;, [33], and
[35]. These studies vary in the choice of functionals or the form of the *PDE* *constraints*. The
typical study of the existence of solutions begins with an analysis of a shape optimization
problem subject to either…

…homogeneous Dirichlet boundary conditions or homogeneous Neumann boundary conditions. Few results on finding the minimizing domain subject to *PDE*
*constraints* with mixed boundary condition types are presented in the literature.
Since these problems deal with…

…domain is fixed.
3
1.3
Outline of the Dissertation
In this work we are going to consider shape optimization problems that contain specific *PDE*
*constraints*. Thus, the definition of the admissible sets needs to be chosen carefully since we
not only…

…continuity will be enough.
Chapter 2 presents a general approach of the abstract setting to the domain optimization
problem subject to *PDE* *constraints*. In that chapter, we begin with the consideration of
the unconstrained domain optimization problem. We…

…to *PDE* *constraints*. We define the set G
that contains the pair (Ωα , uα ) where α is a function whose graph describes Γ2 and so Ωα ,
and uα is the solution to the state problem on Ωα . We start describing the topology of G and
verify the…

…compactness property on G, then we can guarantee the existence of solutions using
the lower semicontinuity property. This framework allows us to formulate, to the best of our
knowledge, the first results that hold for *PDE* *constraints* with mixed boundary…

…design
problems with *PDE* *constraints*.
2.1
Unconstrained Domain Optimization Problem
The goal is to find the domain that minimizes a shape functional
Z
F (u, ∇u, x)dx,
J(Ω) =
(2.1)
Ω
where u is a given fixed function of…