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Zoppello, Marta (2015) Controllability and optimization of deformable bodies in fluids: from biology to robotics. [Ph.D. thesis]

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Abstract (english)

This dissertation takes its cultural origins quite distant in time. My bachelor thesis in Mathematics consisted of a theoretical and numerical work to reassemble a conjecture Alberto Bressan, able to overcome the paradox of the "Scallop Theorem". Purcell in 1977 formalized this famous paradox according to which a 1-dimensional swimmer (scallop), which moves opening and closing periodically its valves, does not have a net shift in a viscous fluid (in the model Stokes), because of the reversibility in time of the equations of motion.
In the master thesis, done in collaboration with prof. Antonio De Simone of Sissa, we analyzed the system swimmer-fluid, involving the geometric theory of the mechanical control of finite dimensional swimmers model.
The thesis is divided into three parts, one was developed and carried out in collaboration with prof. Antonio De Simone of Sissa, Professor François Alouges of the Ecole Polytechnique in Paris and two other French researchers, Laetitia Giraldi and Pierre Martinon. With them we analyzed and defined a general kinematic structure to deal with the problem of self-propulsion in a fluid at low Reynolds numbers. We formulated the problem in terms of a gauge potential A, which provides the net rigid motion resulting from an arbitrary change of shape. To study in depth the implications which the controllability of the swimmer may have on the implementation of bio-inspired devices, we built a new model for slender micro swimmers that is more manageable.
We presented a discrete model of a slender swimmer which swims propagating bending waves along its body and in which the hydrodynamic interactions are treated using the local drag approximation of Resistive Force Theory (RFT). The mode ls easy to assemble and resolve and surprisingly accurate. Also we have proved that for the number of links N greater than 3 and for almost every length of the links, used to approximate the swimmer's body, this is controllable in the whole plane. The results obtained in this part have been published during my PhD in the following publications:

- "P.Martinon, L. Giraldi and M. Zoppello
"Optimal controllability and Strokes for N-Link Micro-swimmer"
Conference on Decision and Control, 2013.

- "F.Alouges, A. DeSimone, L. Giraldi, and M. Zoppello
"Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers".
International Journal of Non-Linear Mechanics, 2013.


Later we focused on finding a swimming strategy to steer the N-link swimmer between two fixed configurations in the minimum time and also to find the best design (ratio between the length of the link) to maximize displacement. Numerical simulations are consistent with our theoretical predictions for small deformations. More details can be found in the article:

- "P. Martinon, L. Giraldi and M. Zoppello.
"Optimal design of the three-link Purcell swimmer"
Physical Review E (2014)

In the second part of the thesis we study the geometric nature of the problem of swimming of a deformable continuous body immersed in a two-dimensional ideal incompressible and irrotational fluid. We faced a new and original problem: the study of the controllability properties of a dynamic system that starts with nonzero initial impulse. Reinterpreting the hydrodynamic forces exerted by the fluid on the body as a kinetic terms and describing shape changes with a finite number of parameters, we obtain the equations of motion. Then using classical techniques in control theory we are able to obtain interesting results on the controllability of this kind of systems.
In more detail:
(i) if the swimmer starts with zero initial impulse, we find results in the literature, more precisely we are always able to find a rate of deformation that leads the swimmer to move between two fixed configurations;
(ii) if instead the body starts with initial impulse different from zero, the swimmer can self-propel in almost every direction if he is able to deform with speed sufficiently high.
The fact that we take into account the presence of an initial impulse not null and the analysis of the controllability of the system seems innovative and makes the study of swimming in ideal fluids more accurate and complete. The results produced in this part, developed in the second year of PhD have been included in the work

- M. Zoppello, F. Cardin.
"Swim like motion of bodies immersed in an ideal fluid"
Submitted (2014).

Finally in the third part, done during the last year of PhD we performed a feasibility study for the engineering realization of artificial micro-swimmers, composed by a cargo head and a thin flexible tail formed by permanently magnetic material, set in motion by an external oscillating magnetic field. Our achievements are illustrated in the article:

- F. Alouges, A. DeSimone, L. Giraldi, M. Zoppello
"Can magnetic multilayers propel arts cial micro-swimmers mimicking sperm cells?"
SoftRobotics (2015).

indicate that, for a system characterized by geometrical parameters consistent with those obtainable by modern techniques of construction and from realistic values ​​of the magneto-elastic parameters (for example those of the Permalloy), may be obtained interesting performances swimming using magnetic fields easily producible in the laboratory. Our analysis shows that our magneto elastic swimmers move in a mechanism very different from those previously reported in the literature of the magneto elastic filaments. In fact the deformation of the swimmer consists of a global rotation and of a deformation flowing with constant spatial curvature, both oscillating in time with the same frequency as the external magnetic field, but with a phase shift.

Abstract (italian)

Questa tesi di dottorato prende culturalmente le origini abbastanza lontane nel tempo. La mia tesi 3-nnale in Matematica consistette in un lavoro di ricomposizione teorica e numerica di una congettura di Alberto Bressan, atta al superamento del paradosso dello "Scallop Theorem". Purcell nel 1977 formalizzo questo famoso paradosso secondo cui un nuotatore 1-dim (scallop), che si muove aprendo e chiudendo alternativamente le sue valve, non ha uno spostamento netto in un fluido viscoso (nel modello Stokes), data la reversibilita nel tempo delle equazioni del moto.
Nella tesi di laurea magistrale, condotta in collaborazione anche con il prof. Antonio De Simone della Sissa, si entrò in dettaglio nello studio del sistema nuotatore-fluido, coinvolgendo la teoria geometrica del controllo meccanico di nuotatori modellizzati finito dimensionalmente.
La tesi e divisa in tre parti, una e stata sviluppata e portata avanti in collaborazione con il prof. Antonio De Simone della Sissa, il prof Francosì Alouges dell'Ecole Polytechnique di Parigi e altri due ricercatori francesi, Laetitia Giraldi e Pierre Martinon. Con loro abbiamo analizzato e definito una struttura cinematica generale per trattare il problema dell'auto-propulsione in un fluido a bassi numeri di Reynolds. Abbiamo formulato il problema in termini di un potenziale di gauge A, che ci fornisce il moto rigido netto risultante da un arbitrario cambiamento di forma. Per studiare a fondo le implicazioni che la controllabilità del nuotatore può avere sulla realizzazione di dispositivi bio-ispirati, abbiamo costruito un modello nuovo per micro nuotatori filiformi, che e piu maneggevole.
Abbiamo presentato un modello discreto di un nuotatore filiforme che nuota propagando bending waves lungo il suo corpo e in cui le interazioni idrodinamiche sono trattate usando l'approssimazione locale della Resistive Force Theory (RFT). Il modello e facile da assemblare e risolvere e sorprendentemente accurato. Inoltre abbiamo provato che per il numero di link N maggiore di 3 e per quasi ogni lunghezza dei link, usati per approssimare il corpo del nuotatore, questo e controllabile in tutto il piano. I risultati ottenuti in questa parte sono stati pubblicati durante il dottorato nelle seguenti pubblicazioni:

- P.Martinon, L. Giraldi and M. Zoppello
"Controllability and Optimal Strokes for N-link Micro-swimmer"
Conference on Decision and Control 2013.

- F.Alouges, A. DeSimone, L. Giraldi, and M. Zoppello
"Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers".
International Journal of Non-Linear Mechanics, 2013.


Successivamente ci siamo focalizzati sul trovare una strategia di nuoto che facesse muovere l'N-link swimmer tra due configurazioni fissate nel minimo tempo e sul trovare anche il design migliore (rapporto tra le lunghezze dei link) per massimizzare lo spostamento. Le simulazioni numeriche sono consistenti con le nostre predizioni teoriche per piccole deformazioni. Maggiori dettagli si trovano nell'articolo:

- P.Martinon, L. Giraldi and M. Zoppello.
"Optimal design of the three-link Purcell swimmer"
Physical Review E (2014)

Nella seconda parte della tesi si studia la natura geometrica del problema del nuoto di un corpo continuo deformabile immerso in un fluido bidimensionale ideale incomprimibile e irrotazionale. Si affronta un problema nuovo ed originale: lo studio delle proprieta di controllabilità di un sistema dinamico che parte con impulso iniziale non nullo. Reinterpretando le forze idrodinamiche esercitate dal fluido sul corpo come termini cinetici e descrivendo i cambiamenti di forma con un numero nito di parametri, si ottengono le equazioni del moto. Usando poi tecniche classiche in teoria del controllo si è in grado di ottenere risultati interessanti sulla controllabilità di questo tipo di sistemi.
In piu dettaglio:
(i) se il nuotatore parte con impulso iniziale nullo, ritroviamo risultati presenti in letteratura, piu precisamente siamo sempre in grado di trovare una velocita di deformazione appropriata che fa spostare il nuotatore tra due configurazioni fissate;
(ii) se invece il corpo parte con impulso iniziale diverso da zero, il nuotatore può auto-propellersi in quasi ogni direzione se e in grado di deformarsi con velocita sufficientemente elevata.
Il fatto che teniamo conto della presenza di un impulso iniziale non nullo e l'analisi della controllabilità del sistema sembra innovativo e rende lo studio del nuoto in fluidi ideali piu accurato e completo. I risultati prodotti in questa parte, sviluppati nel secondo anno di dottorato, sono stati inseriti nel lavoro

- M. Zoppello, F. Cardin.
"Swim like motion of bodies immersed in an ideal fluid"
Submitted (2014).

Infine nella terza parte, affrontata durante l'ultimo anno di dottorato, abbiamo svolto uno studio di fattibilita per la realizzazione ingegneristica di nuotatori microscopici articiali, formati da una testa, usata come contenitore, e da una sottile coda flessibile costituita da materiale permanentemente magnetico, messi in moto da un campo magnetico esterno oscillante. I nostri risultati, illustrati nell'articolo:

- F. Alouges, A. DeSimone, L. Giraldi, M. Zoppello
"Can magnetic multilayers propel articial micro-swimmers mimicking sperm cells?"
SoftRobotics (2015).

indicano che, per un sistema caratterizzato da parametri geometrici consistenti con quelli ottenibili dalle moderne tecniche di costruzione e da valori realistici dei parametri magneto-elastici (ad esempio quelli del Permalloy), possono essere ottenute interessanti performance di nuoto usando campi magnetici facilmente producibili in laboratorio. La nostra analisi mostra che i nuotatori magneto elastici da noi descritti si muovono con un meccanismo molto diverso da quelli precedentemente riportati nella letteratura sui filamenti magneto elastici. Infatti la deformazione del nuotatore consiste di una rotazione globale e di una deformazione fluente con curvatura spaziale costante, entrambe oscillanti nel tempo alla stessa frequenza del campo magnetico esterno, ma con una traslazione di fase.

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EPrint type:Ph.D. thesis
Tutor:Cardin, Franco
Ph.D. course:Ciclo 28 > Scuole 28 > SCIENZE MATEMATICHE > MATEMATICA
Data di deposito della tesi:14 January 2016
Anno di Pubblicazione:31 December 2015
Key Words:Fluid dynamics control micro-swimmers Resistive Force Theory
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/07 Fisica matematica
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica
Codice ID:9029
Depositato il:06 Oct 2016 15:18
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