The field of numerical simulation of wave propagation phenomena is of extreme interest for many scientific applications, and therefore, it continuously provides new advances in the development and analysis of numerical methods. From the modeling view points these phenomena are posed in the term of differential models that need to be suitably solved numerically. Concerning the space discretization, the Discontinuous Galerkin (DG) approach is broadly employed for its flexibility in dealing with complex geometries and heterogeneous media typically present in this field. Recently, the DG paradigm has been applied to the well-known Spectral Elements method, resulting in the Discontinuous Galerkin Spectral Element (DGSE) method. This technique inherits the accuracy properties of spectral elements while the drawback of employing a uniform polynomial order over the entire computational domain is overcome by the discontinuous paradigm. Regarding the time integration of the model, in engineering applications, (low order) explicit schemes are still preferred for their convenience in terms of computational requirements and ease of implementation, even if they require to respect suitably stability conditions that may seriously affect their computational efficiency. Space-time methods are still not very broadly employed, especially in engineering application, even if they provide very accurate approximations and high flexibility in terms of mesh generation, both in space and time. In this thesis, we present and analyze space-time Discontinuous Galerkin methods for the discretization of visco-elastic wave equation. We derive the space-time weak formulation, study well-posedness and prove a-priori error bounds. In order to increase the computational efficiency, we present an alternative Discontinuous Galerkin time integration method. We show that the resulting formulation is also well-posed, stable and provide optimal approximation properties. A wide set of two- and three-dimensional numerical experiments are also presented with the aim of validating the theoretical bounds and asses the practical performance of the proposed methods in real test cases.
La simulazione numerica di fenomeni di propagazione di onde è un argomento di estremo interesse in molte applicazioni scientifiche, e, pertanto, fornisce continuamente nuovi avanzamenti nello sviluppo e nell'analisi di metodi numerici. Dal punto di vista modellistico, questi fenomeni sono impostati in termini di modelli differenziali che richiedono di essere opportunamente risolti numericamente. Per quel che riguarda la discretizzazione spaziale, l'approccio Discontinuo di Galerkin (DG) è largamente utilizzato per la sua flessibilità nel trattare le geometrie complesse e i materiali eterogenei tipicamente presenti in questo ambito. Recentemente, il paradigma DG è stato anche applicato al noto metodo degli Elementi Spettrali, dando origine al metodo Discontinuo di Galerkin agli Elementi Spettrali (DGSE). Questa tecnica eredita le proprietà di accuratezza degli elementi spettrali, mentre lo svantaggio di utilizzare grado polinomiale uniforme su tutto il dominio computazionale è superato grazie all'utilizzo del paradigma discontinuo. Riguardo all'integrazione in tempo del modello, nelle applicazioni ingegneristiche si preferiscono ancora schemi espliciti (a basso ordine) per la loro convenienza in termini di requisiti computazionali e facilità di implementazione, anche se richiedono di rispettare opportune condizioni di stabilità che possono gravemente compromettere la loro efficienza computazionale. I metodi spazio-tempo sono tuttora poco utilizzati, specialmente nelle applicazioni ingegneristiche, anche se forniscono approssimazioni molto accurate e una grande flessibilità in termini di generazione della griglia, sia in spazio che in tempo. In questa tesi, presentiamo e analizziamo dei metodi spazio-tempo Discontinui di Galerkin per la discretizzazione dell'equazione dell'onda visco-elastica. Ricaviamo la formulazione debole spazio-temporale, studiamo la buona positura e proviamo stime a priori dell'errore. Per aumentare l'efficienza computazionale, presentiamo un metodo alternativo di integrazione in tempo Discontinuo di Galerkin. Mostriamo che la formulazione risultante è anch'essa ben posta, stabile e fornisce proprietà di convergenza ottimali. Presentiamo inoltre una vasta gamma di esperimenti numerici in due e tre dimensioni, con lo scopo di validare le stime teoriche e investigare il comportamento pratico dei metodi proposti in casi test reali.
Space-time discontinuous Galerkin methods for elastodynamics
MIGLIORINI, FRANCESCO
2020/2021
Abstract
The field of numerical simulation of wave propagation phenomena is of extreme interest for many scientific applications, and therefore, it continuously provides new advances in the development and analysis of numerical methods. From the modeling view points these phenomena are posed in the term of differential models that need to be suitably solved numerically. Concerning the space discretization, the Discontinuous Galerkin (DG) approach is broadly employed for its flexibility in dealing with complex geometries and heterogeneous media typically present in this field. Recently, the DG paradigm has been applied to the well-known Spectral Elements method, resulting in the Discontinuous Galerkin Spectral Element (DGSE) method. This technique inherits the accuracy properties of spectral elements while the drawback of employing a uniform polynomial order over the entire computational domain is overcome by the discontinuous paradigm. Regarding the time integration of the model, in engineering applications, (low order) explicit schemes are still preferred for their convenience in terms of computational requirements and ease of implementation, even if they require to respect suitably stability conditions that may seriously affect their computational efficiency. Space-time methods are still not very broadly employed, especially in engineering application, even if they provide very accurate approximations and high flexibility in terms of mesh generation, both in space and time. In this thesis, we present and analyze space-time Discontinuous Galerkin methods for the discretization of visco-elastic wave equation. We derive the space-time weak formulation, study well-posedness and prove a-priori error bounds. In order to increase the computational efficiency, we present an alternative Discontinuous Galerkin time integration method. We show that the resulting formulation is also well-posed, stable and provide optimal approximation properties. A wide set of two- and three-dimensional numerical experiments are also presented with the aim of validating the theoretical bounds and asses the practical performance of the proposed methods in real test cases.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/177658