In common engineering applications, the design and analysis of complex systems is bounded to undergo a phase of numerical studies, which are fundamental to fill the gap between preliminary results, usually based on simplified analytical models, and reality. Structural dynamics, in particular, is normally studied with the aid of the Finite Element Method (FEM), which consists in accurately reproducing the geometry of a real structure through the union of small elements, each one pertaining to a portion of the material of the whole structure. The more the model is complex and the more the model is rich in geometrical details, the more elements are required; this usually leads to models counting hundred of thousands or even millions of unknowns, whose numerical analysis can be extremely expensive in terms of computational resources and times. For these reasons, since the very origin of FEM (and even more then, when computational resources were much more limited than the ones available today), there has always been a struggle to develop strategies to reduce the complexity of the problem. First examples are to be found in the so-called modal analysis, where the full solution of the system is sought in terms of a weighted sum of a relatively low number of system eigenvectors (usually in the order of tents); this way, simulation times are greatly reduced. Modal analysis and its derivative methods, however, provide a solution only to problems featuring a dynamics that can be considered to be linear. When a structure undergoes a displacement that is relatively large with respect to its characteristic dimensions, usually the hypothesis of small deformations does not hold anymore: we talk in these cases of geometric nonlinearities and one has to resort to nonlinear dynamic analysis. The numerical complexity of the latter, however, is further increased by the fact some quantities (such as elastic internal forces and stiffness matrix) are now displacement-dependent and need to be evaluated at each analysis step/iteration of the adopted numerical scheme. For practical interest applications, this usually exponentially increases the computational times associated with the single analysis, often already demanding in the case of linear dynamics. Moreover, several phenomena that can be found in nonlinear dynamics and that have no counterpart in linear dynamics require specific numerical tools to be studied; the latter, however, usually can handle only a very limited number of unknowns. For these reasons, the development of NonLinear Reduced Order Models (NLROM, or ROM in short), is a research topic which gained ever-increasing momentum over the last decades. Several solutions are already available in the literature, which will be briefly addressed throughout the present dissertation. A theme that almost every method shares, however, is the fact that the construction of the ROM itself is usually an expensive process which, in fact, may hinder the efficiency of the method. A way to amortize these offline construction times is thus to have the ROM to be parametric (pROM), that is, valid over a set of one or more parameters rather than for a single instance of the structure. This way the pROM can be built once and used multiple times so that the attainable overall speedups can be very high. In the present dissertation, after introducing some rudiments of nonlinear FEM and of a class of ROMs, some nonlinear pROMs for the analysis of structures affected by the presence of defects are developed and discussed. Indeed, in many engineering applications, it is important to assess the performances of a system not only in the nominal-blueprint case, when the geometry is “perfect”, but also in the case in which small imperfections are present. Notable examples are the micro electro-mechanical system (MEMS) industry, where the characteristics of the sensors are directly related to the imperfections of the production process, and the aerospace field, where the presence of lightweight and slender structures undergoing high-amplitude vibrations usually triggers nonlinear responses strongly affected by the imperfect geometry of the real components.
Nella comune pratica ingegneristica, la progettazione e lo studio di sistemi complessi passa sempre e inevitabilmente attraverso l’analisi numerica, che sopperisce ai limiti di modelli analitici semplificati, generalmente utilizzati solo in fase preliminare. Lo studio strutturale di sistemi meccanici viene di prassi affrontato con l’ausilio del metodo agli elementi finiti (FEM), il quale permette di costruire modelli che riproducono accuratamente la geometria reale della struttura rappresentandola come unione di elementi, ognuno attenente ad una porzione di materiale dell’insieme. Più la complessità del problema cresce e il modello si arricchisce di dettagli geometrici, più elementi sono richiesti per descrivere il problema; nella pratica, ciò spesso porta a modelli che contano centinaia di migliaia o anche milioni di incognite, motivo per il quale l’analisi è spesso dispendiosa sia in termini di risorse computazionali che di tempo. Per questo motivo fin dalle origini del FEM, quando peraltro le risorse computazionali erano molto più limitate di quelle odierne, si sono sempre cercati dei metodi per ridurre la complessità del problema. I primi esempi si ritrovano nell’analisi modale, che va a ricostruire la soluzione dell’intero sistema come la somma di un numero molto ridotto di autovettori del sistema (in genere nell’ordine della decina); di conseguenza, i tempi di calcolo vengono estremamente velocizzati. L’analisi modale (e metodi derivati), tuttavia, è uno strumento valido nei casi in cui la dinamica può considerarsi lineare. Quando lo spostamento della struttura diventa relativamente grande rispetto alle sue dimensioni caratteristiche, l’ipotesi di piccoli spostamenti non è più valida: in questo caso si parla di nonlinearità geometrica, e si deve ricorrere ad analisi dinamiche nonlineari. La complessità computazionale di queste ultime è ulteriormente aggravata dal fatto che alcune quantità (forze elastiche interne e matrice di rigidezza) dipendono dallo spostamento corrente della struttura, e quindi vanno rivalutate ad ogni step o iterazione del metodo di risoluzione adottato. Per sistemi di interesse pratico, questo porta a far esplodere i tempi richiesti per una singola analisi. Inoltre, molti fenomeni legati alla dinamica nonlineare possono essere studiati soltanto utilizzando strumenti che, attualmente, sono in grado di processare soltanto un limitatissimo numero di incognite. Per queste ragioni, lo sviluppo di nuovi e diversi modelli ridotti nonlineari (NLROM, o solo ROM per brevità) è un tema che ha preso sempre più momento negli ultimi anni e che impegna diversi gruppi di ricerca in tutto il mondo. Già diverse soluzioni sono disponibili, brevemente ricapitolate nel corso di questo lavoro. Un tema che però accomuna quasi tutti i metodi è il fatto che la costruzione del ROM è tipicamente di per sè un processo oneroso in termini di tempo e risorse (anche se, sicuramente, più conveniente che affrontare la simulazione del sistema non ridotto). Un modo per ammortizzare ulteriormente i costi di costruzione del ROM è quello di fare sì che sia parametrico, ovvero valido per diversi valori di uno o più parametri. Utilizzando il ROM parametrico (pROM) molteplici volte al posto di analisi complete, si possono comparativamente ottenere fattori di velocizzazione notevoli. Nel presente lavoro, dopo aver introdotto i rudimenti necessari di FEM nonlineare e di una classe di ROM nonlineari, saranno presentati alcuni pROM nonlineari dedicati allo studio parametrico di strutture nelle quali sono presenti dei difetti che influiscono sulla risposta dinamica delle stesse. Questo tema è infatti di interesse in molti ambiti, come quello dell’industria dei sistemi micro elettro-meccanici (MEMS), dove sono intrensicamente presenti difetti di produzione che vanno ad impattare sulle performance dei prodotti, o nel settore aerospaziale, dove la diffusa presenza di strutture sottili soggette a spostamenti elevati spesso porta a risposte dinamiche nonlineari che sono fortemente influenzate dalla precisone geometrica dei vari componenti.
Parametric reduced order models for imperfect structures with geometric nonlinearities
Marconi, Jacopo
2021/2022
Abstract
In common engineering applications, the design and analysis of complex systems is bounded to undergo a phase of numerical studies, which are fundamental to fill the gap between preliminary results, usually based on simplified analytical models, and reality. Structural dynamics, in particular, is normally studied with the aid of the Finite Element Method (FEM), which consists in accurately reproducing the geometry of a real structure through the union of small elements, each one pertaining to a portion of the material of the whole structure. The more the model is complex and the more the model is rich in geometrical details, the more elements are required; this usually leads to models counting hundred of thousands or even millions of unknowns, whose numerical analysis can be extremely expensive in terms of computational resources and times. For these reasons, since the very origin of FEM (and even more then, when computational resources were much more limited than the ones available today), there has always been a struggle to develop strategies to reduce the complexity of the problem. First examples are to be found in the so-called modal analysis, where the full solution of the system is sought in terms of a weighted sum of a relatively low number of system eigenvectors (usually in the order of tents); this way, simulation times are greatly reduced. Modal analysis and its derivative methods, however, provide a solution only to problems featuring a dynamics that can be considered to be linear. When a structure undergoes a displacement that is relatively large with respect to its characteristic dimensions, usually the hypothesis of small deformations does not hold anymore: we talk in these cases of geometric nonlinearities and one has to resort to nonlinear dynamic analysis. The numerical complexity of the latter, however, is further increased by the fact some quantities (such as elastic internal forces and stiffness matrix) are now displacement-dependent and need to be evaluated at each analysis step/iteration of the adopted numerical scheme. For practical interest applications, this usually exponentially increases the computational times associated with the single analysis, often already demanding in the case of linear dynamics. Moreover, several phenomena that can be found in nonlinear dynamics and that have no counterpart in linear dynamics require specific numerical tools to be studied; the latter, however, usually can handle only a very limited number of unknowns. For these reasons, the development of NonLinear Reduced Order Models (NLROM, or ROM in short), is a research topic which gained ever-increasing momentum over the last decades. Several solutions are already available in the literature, which will be briefly addressed throughout the present dissertation. A theme that almost every method shares, however, is the fact that the construction of the ROM itself is usually an expensive process which, in fact, may hinder the efficiency of the method. A way to amortize these offline construction times is thus to have the ROM to be parametric (pROM), that is, valid over a set of one or more parameters rather than for a single instance of the structure. This way the pROM can be built once and used multiple times so that the attainable overall speedups can be very high. In the present dissertation, after introducing some rudiments of nonlinear FEM and of a class of ROMs, some nonlinear pROMs for the analysis of structures affected by the presence of defects are developed and discussed. Indeed, in many engineering applications, it is important to assess the performances of a system not only in the nominal-blueprint case, when the geometry is “perfect”, but also in the case in which small imperfections are present. Notable examples are the micro electro-mechanical system (MEMS) industry, where the characteristics of the sensors are directly related to the imperfections of the production process, and the aerospace field, where the presence of lightweight and slender structures undergoing high-amplitude vibrations usually triggers nonlinear responses strongly affected by the imperfect geometry of the real components.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/180025