The semiconductor sector is a competitive environment that forces constant improvements of its products to thrive and flourish from an industrial perspective. As a consequence, precise and accurate mathematical methods to predict the response of micro-electro-mechanical systems are essential tools since trial and error approaches are not sustainable. In this setting, full order solution methods are the most common techniques adopted in industry since they are reliable and accurate when it comes to complex domain geometries that cannot be modeled by analytic methods. Nevertheless, their applicability is rather poor when modeling devices that need to operate at resonance since steady state periodic solutions are computational demanding for large systems of differential equations. The result is that devices that operate at resonance are difficult to design and optimise using this class of methods. Consequently, one needs to adopt model order reduction strategies, i.e. techniques aimed at reducing the computational complexity of numerical models while retaining the maximum degree of information. Among the different methods available in literature, the parametrisation method for invariant manifolds represents one of the most appealing techniques for modeling nonlinear systems since it provides a mean to derive reduced models by parametrising the system motion along a low dimensional invariant set of the phase space. This approach has several benefits compared to other model order reduction strategies since it exactly addresses non-resonant coupling, the latter the cause of failure of energy based approaches as the proper orthogonal decomposition followed by hyper-reduction. In the present work, a direct formulation of the parametrisation method for invariant manifolds is proposed. The technique is based on the introduction of a coordinate change between nodal variables and normal coordinates, the latter corresponding to the coordinates defined over the parametrised set. Explicit relations and computational improvements of the method are detailed and the resulting algorithm shows performance that make it one of the most efficient dimensionality reduction method for nonlinear vibrating structures. The present work is structured in three parts. In Part I an introduction to the micro-electromechanical systems field is detailed, together with detailed description of the governing equations of this field. In the same part, an extensive treatment of the numerical methods adopted in this field is detailed. Discretisation methods for partial differential equations are briefly recalled together with a description of common numerical schemes for the computation of periodic orbits in dynamic system, the latter the target task of this work. In Part II the direct parametrisation method for invariant manifolds is derived for both autonomous and non-autonomous mechanical systems. All analytical derivations are reported and extensive analysis of the resulting equations is reported. Both autonomous and non-autonomous mechanical systems are then studied to show the remarkable results provided by this method. Finally, in Part III the method is extended to the case of structures actuated with piezoelectric materials and subjected to prestress. This represents common conditions met in the semiconductor industry and we extend the direct parametrisation method for invariant manifolds to address exactly the new terms introduced by the associated formulation. The method is finally applied to a set of real MEMS structures, hence highlighting how the application of the present method in an industrial setting should be considered a requirement for designing and optimize resonant MEMS structures.
Il settore dei semiconduttori è competitivo e migliorare costantemente i propri prodotti è una condizione necessaria per garantire la sopravvivenza di una industria in questo settore. Di conseguenza, metodi numerici precisi ed accurati per predirre la performance dei dispositivi micro-elettro-meccanici sono uno strumento essenziale in questo settore siccome un approccio "trial and error" non è ammissibile dati gli alti costi di fabbricazione dei dispositivi. Metodi full-order sono tipicamente utilizzati in quest’area data la loro affidabilitá e accuratezza nel caso di geometrie complesse che non possono essere modellate analiticamente. Tuttavia, l’applicabilitá di queste tecniche è scarsa se l’obiettivo è quello di modellare dispositivi che operano a risonanza dato che il calcolo di soluzioni periodiche stazionarie in sistemi di equazioni differenziali ad alta dimensionalitá rappresenta un compito computazionalmente proibitivo sia in termini di tempo che di memoria. Conseguentemente c’è la necessitá di usare tecniche di riduzione d’ordine, ovvero metodi che hanno come obiettivo quello di ridurre l’onere computazionale di modelli numerici mantenendo il massimo grado di informazione. Tra le varie tecniche disponibili in letteratura, il metodo di parametrizzazione di varietá invarianti si pone tra le tecniche piú interessanti per la modellazione di sistemi nonlineari dato che permette di derivare modelli ridotti tramite parametrizzazione del moto del sistema lungo set invarianti a bassa dimensionalitá. Questo approccio ha diversi vantaggi paragonato ad altre tecniche di riduzione d’ordine dato che riesce a trattare in modo esatto accoppiamento non risonante tra modi, quest’ultimo la causa di fallimento di tecniche di riduzione basate su stime energetiche come la decomposizione ai valori singolari accoppiata ad iper-riduzione. Nel presente lavoro è proposta una formulazione diretta del metodo di parametrizzazione per varietá invarianti. La tecnica si basa sulla introduzione di un cambio di coordinate tra variabili nodali e coordinate normali, ovvero le coordinate definite lungo il set parametrizzato. Vengono fornite relazioni esplicite e miglioramenti dello schema computazionale e l’algoritmo che ne risulta esibisce prestazioni che lo rendono una delle tecniche di riduzione piú efficienti per strutture nonlineari eccitate a risonanza. Il presente lavoro è strutturato in tre parti. In Parte I viene fornita una introduzione al campo dei sistemi micro-elettro-meccanici, complementata con una descrizione dettagliata delle equazioni che governano questo campo. Contestualmente viene riportata una trattazione degli schemi numerici tipici di questo ambiti viene riportata. In particolare, vengono spiegati i metodi di discretizzazione per equazioni alle derivate parziali insieme alle tecniche di calcolo di orbite periodiche in sistemi dinamici, quest’ultimo il fine ultimo dei metodi sviluppati in questo lavoro. Nella Parte II viene derivato il metodo di parametrizzazione diretta di varietá invarianti sia per sistemi meccanici autonomi che non autonomi. Tutte le derivazioni analitiche sono riportate e le equazioni derivate sono analizzate in modo estensivo. Successivamente sono analizzati una serie di esempi di sistemi autonomi e non autonomi con lo scopo di mostrare i risultati impressionanti di questo metodo. Finalmente, nella Parte III il metodo viene esteso al caso di strutture attuate con materiali piezoelettrici e soggette a prestress, ovvero condizioni tipiche nell’industria dei semiconduttori. Il metodo di parametrizzazione di varietá invarianti diretto viene esteso per trattare in modo esatto i nuovi termini associati a questa formulazione. Il metodo viene finalmente applicato ad una serie di strutture MEMS reali, evidenziando come l’applicazione di questo metodo in ambito industriale dovrebbe essere considerato un requisito per il design e l’ottimizzazione di strutture micro-elettro-meccaniche.
The direct parametrisation method for invariant manifolds : developments and application to large dimensional finite element models of MEMS structures
Opreni, Andrea
2021/2022
Abstract
The semiconductor sector is a competitive environment that forces constant improvements of its products to thrive and flourish from an industrial perspective. As a consequence, precise and accurate mathematical methods to predict the response of micro-electro-mechanical systems are essential tools since trial and error approaches are not sustainable. In this setting, full order solution methods are the most common techniques adopted in industry since they are reliable and accurate when it comes to complex domain geometries that cannot be modeled by analytic methods. Nevertheless, their applicability is rather poor when modeling devices that need to operate at resonance since steady state periodic solutions are computational demanding for large systems of differential equations. The result is that devices that operate at resonance are difficult to design and optimise using this class of methods. Consequently, one needs to adopt model order reduction strategies, i.e. techniques aimed at reducing the computational complexity of numerical models while retaining the maximum degree of information. Among the different methods available in literature, the parametrisation method for invariant manifolds represents one of the most appealing techniques for modeling nonlinear systems since it provides a mean to derive reduced models by parametrising the system motion along a low dimensional invariant set of the phase space. This approach has several benefits compared to other model order reduction strategies since it exactly addresses non-resonant coupling, the latter the cause of failure of energy based approaches as the proper orthogonal decomposition followed by hyper-reduction. In the present work, a direct formulation of the parametrisation method for invariant manifolds is proposed. The technique is based on the introduction of a coordinate change between nodal variables and normal coordinates, the latter corresponding to the coordinates defined over the parametrised set. Explicit relations and computational improvements of the method are detailed and the resulting algorithm shows performance that make it one of the most efficient dimensionality reduction method for nonlinear vibrating structures. The present work is structured in three parts. In Part I an introduction to the micro-electromechanical systems field is detailed, together with detailed description of the governing equations of this field. In the same part, an extensive treatment of the numerical methods adopted in this field is detailed. Discretisation methods for partial differential equations are briefly recalled together with a description of common numerical schemes for the computation of periodic orbits in dynamic system, the latter the target task of this work. In Part II the direct parametrisation method for invariant manifolds is derived for both autonomous and non-autonomous mechanical systems. All analytical derivations are reported and extensive analysis of the resulting equations is reported. Both autonomous and non-autonomous mechanical systems are then studied to show the remarkable results provided by this method. Finally, in Part III the method is extended to the case of structures actuated with piezoelectric materials and subjected to prestress. This represents common conditions met in the semiconductor industry and we extend the direct parametrisation method for invariant manifolds to address exactly the new terms introduced by the associated formulation. The method is finally applied to a set of real MEMS structures, hence highlighting how the application of the present method in an industrial setting should be considered a requirement for designing and optimize resonant MEMS structures.| File | Dimensione | Formato | |
|---|---|---|---|
|
Andrea_Opreni_PhD_Thesis.pdf
non accessibile
Descrizione: Tesi
Dimensione
21.89 MB
Formato
Adobe PDF
|
21.89 MB | Adobe PDF | Visualizza/Apri |
I documenti in POLITesi sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/10589/188966