In this thesis we study the nature of the different vector Laplacian forms by showing a case of its derivation from an energy associated to a nematic liquid crystal model. In particular we will study how the Hodge or the Bochner Laplacians develop and are related through Weitzenböck identity. Then we study the discretization of the latter by means of the ISFEM (intrinsic finite element method), by extending the latter to the vector case and using the building blocks developed for the scalar case. Finally, a few examples on simple surfaces will be numerically solved to test the accuracy and efficiency of the proposed extension of scalar ISFEM to the vector Laplacian.
In questa tesi studiamo la natura delle diverse forme di Laplaciani vettoriali mostrando un caso della sua derivazione da un'energia associata a un modello di cristallo liquido nematico. In particolare studieremo come si sviluppano i Laplaciani di Hodge o di Bochner e come sono correlati attraverso l'identità di Weitzenböck. Poi si studia la discretizzazione di questi ultimi per mezzo dell'ISFEM (metodo intrinseco degli elementi finiti), estendendo quest'ultimo al caso vettoriale e utilizzando i blocchi di costruzione sviluppati per il caso scalare. Infine, alcuni esempi su superfici semplici saranno risolti numericamente per testare la precisione e l'efficienza dell'estensione proposta dell'ISFEM scalare al Laplaciano vettoriale.
Intrinsic FEM for Vector Laplacian equations
FAVERO, LUCA
2021/2022
Abstract
In this thesis we study the nature of the different vector Laplacian forms by showing a case of its derivation from an energy associated to a nematic liquid crystal model. In particular we will study how the Hodge or the Bochner Laplacians develop and are related through Weitzenböck identity. Then we study the discretization of the latter by means of the ISFEM (intrinsic finite element method), by extending the latter to the vector case and using the building blocks developed for the scalar case. Finally, a few examples on simple surfaces will be numerically solved to test the accuracy and efficiency of the proposed extension of scalar ISFEM to the vector Laplacian.File | Dimensione | Formato | |
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https://hdl.handle.net/20.500.12608/29709