In this master thesis project we consider the wave equation, which is a model problem for the hyperbolic partial differential equations, and we deal with it especially in dimensions 1 and 2. The purpose of our work is to numerically solve the wave equation with a method with good approximation properties, which attains optimal error convergence rates, and that exactly preserves the energy, which is an important topic for long time simulations. For this purpose, we introduce a change of variables and rewrite the wave equation as a first order hyperbolic system formulation, composed by two coupled equations and completed with periodic boundary conditions. With such conditions, the continuous problem is energy conservative, hence we need to find a discretization scheme for the problem that maintains the above property. Making use of the mixed formulation of the wave equation, we propose a new method to solve the problem, by replacing the weak form of the second equation by the application of a projector. The mixed formulation above mentioned can be understood as writing the problem with the first equation in weak form while the second equation is solved in the strong form. In all this context we consider first the wave propagation velocity coefficient as a constant, then we let it depend on the spatial variable. For the spatial discretization of the fields we consider isogeometric methods. We recall that isogeometric analysis-based methods use a generalisation of Bézier curves, B-splines and NURBS, which are used to approximate partial differential equations. We will recall in our project the definitions of B-splines and NURBS, especially in the case of spline spaces with periodic conditions at the boundary and highest regularity. We want to use these spline spaces for the discretization of our system of partial differential equations, hence we need to project the equations in such spline spaces. In particular, for the discretization of the problem, we make use of the spline complex. These are the De Rham complexes for spline spaces, and they are fundamental for the discretization of our differential equations. The commutative properties of the diagrams are necessary in order to achieve the conservation of the energy, moreover, they will allow to more comfortably assemble the matrices associated with the discretized problem. As regards the projector, we make use of the local quasi-interpolants as defined in the article “Some examples of quasi-interpolants constructed from local spline projectors”, by B.-G. Lee, T. Lyche and K. Mørken, in “Mathematical methods for curves and surfaces: Oslo”, Citeseer, together with their commutative projectors. We will see in detail the construction of quasi-interpolants for univariate spline spaces of degree g = 2 and g = 3, referring to the necessary modifications for periodic spaces. These particular quasi-interpolants are shown to approximate functions with optimal convergence properties. We also show how to compute in the two-dimensional domain the quasi-interpolants above mentioned. Finally, along with the Crank-Nicolson method for time discretization, the scheme is proved to be energy conservative. In the numerical tests we solved the first order hyperbolic system for the wave equation, with the discretization scheme we presented, and compared the solutions with the ones obtained with a standard Galerkin discretization in space and again Crank-Nicolson in time. The convergence results we obtained are optimal. We also performed numerical tests for long time solutions of the system, and recorded the relative error between energy at initial and final time. In all the simulations we tried we have achieved the conservation of total energy, both in one dimensional and two dimensional case.
In questo progetto di tesi di laurea consideriamo l'equazione delle onde, come problema modello per le equazioni alle derivate parziali iperboliche, trattandola in particolare in dimensione 1 e 2. Lo scopo di questo lavoro è risolvere numericamente l'equazione delle onde con buone proprietà di approssimazione, come convergenze ottimali , e in particolare preservare l'energia, che è un argomento importante per le simulazioni a lungo termine. A tale scopo, introduciamo un cambio di variabili per riscrivere l'equazione delle onde come sistema iperbolico del primo ordine, composto da due equazioni accoppiate e completato con condizioni al contorno periodiche. Il problema continuo è conservativo, quindi è necessario trovare uno schema di discretizzazione che preservi l’energia. Facendo uso della formulazione mista dell'equazione delle onde, proponiamo un nuovo metodo per risolvere il problema, sostituendo la forma debole della seconda equazione con l'applicazione di un proiettore. La formulazione mista sopra menzionata consiste nello scrivere il problema con la prima equazione in forma debole mentre la seconda equazione è risolta in forma forte. In tutto questo contesto consideriamo prima il coefficiente di velocità di propagazione dell'onda come costante, poi lo lasciamo dipendere dalla variabile spaziale. Questo modo di affrontare il problema si basa su metodi isogeometrici. Ricordiamo che i metodi basati sull'analisi isogeometrica utilizzano B-spline e NURBS che vengono utilizzati per approssimare equazioni alle derivate parziali. Nel nostro progetto richiameremo le definizioni di B-spline e NURBS, soprattutto nel caso di spazi spline con condizioni periodiche al contorno e massima regolarità. Vogliamo utilizzare questi spazi per la discretizzazione del nostro sistema di equazioni alle derivate parziali, quindi dobbiamo proiettare le equazioni in tali spazi. In particolare, per la discretizzazione del problema, utilizziamo il complesso di spline. Questi sono i complessi di De Rham per gli spazi spline, e sono fondamentali per la discretizzazione delle nostre equazioni differenziali. Le proprietà commutative dei diagrammi sono molto importanti per ottenere la conservazione dell'energia, inoltre, permetteranno di assemblare più comodamente le matrici associate al problema discretizzato. Per quanto riguarda il proiettore, utilizziamo il quasi-interpolante locale come definito nell'articolo “Some examples of quasi-interpolants constructed from local spline projectors”, di B.-G. Lee, T. Lyche e K. Mørken, in “Mathematical methods for curves and surfaces: Oslo”, Citeseer, insieme ai loro proiettori commutativi. Vedremo in dettaglio la costruzione di quasi-interpolanti per spazi spline univariati di grado g = 2 e g = 3, facendo riferimento alle modifiche necessarie per spazi periodici. Questi particolari quasi-interpolanti sono indicati per approssimare funzioni con proprietà di convergenza ottimali. Mostriamo anche come calcolare nel dominio bidimensionale i quasi-interpolanti sopra menzionati. Infine, insieme al metodo di Crank-Nicolson per la discretizzazione in tempo, abbiamo dimostrato che lo schema preserva l’energia totale del sistema. In particolare abbiamo risolto il sistema iperbolico del primo ordine per l'equazione delle onde, con lo schema di discretizzazione che abbiamo proposto, e confrontato le soluzioni con quelle ottenute con una discretizzazione standard con il metodo di Galerkin in spazio e di nuovo Crank-Nicolson in tempo. I risultati di convergenza che abbiamo ottenuto sono ottimali. Abbiamo anche eseguito test numerici per soluzioni a lungo termine del sistema e registrato l'errore relativo tra l'energia al momento iniziale e quello finale. In tutte le simulazioni che abbiamo provato abbiamo ottenuto la conservazione dell'energia totale, sia nel caso unidimensionale che bidimensionale.
Energy conservative isogeometric techniques for the wave equation.
KUSHOVA, ALEN
2019/2020
Abstract
In this master thesis project we consider the wave equation, which is a model problem for the hyperbolic partial differential equations, and we deal with it especially in dimensions 1 and 2. The purpose of our work is to numerically solve the wave equation with a method with good approximation properties, which attains optimal error convergence rates, and that exactly preserves the energy, which is an important topic for long time simulations. For this purpose, we introduce a change of variables and rewrite the wave equation as a first order hyperbolic system formulation, composed by two coupled equations and completed with periodic boundary conditions. With such conditions, the continuous problem is energy conservative, hence we need to find a discretization scheme for the problem that maintains the above property. Making use of the mixed formulation of the wave equation, we propose a new method to solve the problem, by replacing the weak form of the second equation by the application of a projector. The mixed formulation above mentioned can be understood as writing the problem with the first equation in weak form while the second equation is solved in the strong form. In all this context we consider first the wave propagation velocity coefficient as a constant, then we let it depend on the spatial variable. For the spatial discretization of the fields we consider isogeometric methods. We recall that isogeometric analysis-based methods use a generalisation of Bézier curves, B-splines and NURBS, which are used to approximate partial differential equations. We will recall in our project the definitions of B-splines and NURBS, especially in the case of spline spaces with periodic conditions at the boundary and highest regularity. We want to use these spline spaces for the discretization of our system of partial differential equations, hence we need to project the equations in such spline spaces. In particular, for the discretization of the problem, we make use of the spline complex. These are the De Rham complexes for spline spaces, and they are fundamental for the discretization of our differential equations. The commutative properties of the diagrams are necessary in order to achieve the conservation of the energy, moreover, they will allow to more comfortably assemble the matrices associated with the discretized problem. As regards the projector, we make use of the local quasi-interpolants as defined in the article “Some examples of quasi-interpolants constructed from local spline projectors”, by B.-G. Lee, T. Lyche and K. Mørken, in “Mathematical methods for curves and surfaces: Oslo”, Citeseer, together with their commutative projectors. We will see in detail the construction of quasi-interpolants for univariate spline spaces of degree g = 2 and g = 3, referring to the necessary modifications for periodic spaces. These particular quasi-interpolants are shown to approximate functions with optimal convergence properties. We also show how to compute in the two-dimensional domain the quasi-interpolants above mentioned. Finally, along with the Crank-Nicolson method for time discretization, the scheme is proved to be energy conservative. In the numerical tests we solved the first order hyperbolic system for the wave equation, with the discretization scheme we presented, and compared the solutions with the ones obtained with a standard Galerkin discretization in space and again Crank-Nicolson in time. The convergence results we obtained are optimal. We also performed numerical tests for long time solutions of the system, and recorded the relative error between energy at initial and final time. In all the simulations we tried we have achieved the conservation of total energy, both in one dimensional and two dimensional case.È consentito all'utente scaricare e condividere i documenti disponibili a testo pieno in UNITESI UNIPV nel rispetto della licenza Creative Commons del tipo CC BY NC ND.
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https://hdl.handle.net/20.500.14239/12058