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A Scaled Boundary Approach to Forward and Inverse Problems

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A Scaled Boundary Approach to Forward and Inverse Problems DISS. ETH NO. 26371 A SCALED BOUNDARY APPROACH TO FORWARD AND INVERSE PROBLEMS With Applications in Computational Fracture Mechanics, Damage Localization and Topology Optimization A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Adrian Walter Egger MSc ETH Civil Eng, ETH Zurich born on 30.03.1987 citizen of St. Gallen-Tablat, SG and Canada accepted on the recommendation of Prof. Dr. Eleni Chatzi (ETH Zurich, Zurich, Examiner) Prof. Dr. Savvas Triantafyllou (University of Nottingham, Nottingham, Co-examiner) Prof. Dr. Chongmin Song (University of New South Wales, Sydney, Co-examiner) 2019 Abstract The demand for sustainable design in, e.g., the aerospace, automotive and construction industries has lead to the development of lighter, stronger and more resilient structures, spawning the need to guard against failure processes by leveraging robust, economical and high-fidelity numerical simulations. Since its inception, the finite element method (FEM) has been advanced to handle a multitude of struc- tural analysis problems ranging from linear to nonlinear, static to dynamic, fracture and contact prob- lems among others. Within the context of fracture mechanics, it has been demonstrated that modeling of damage-related phenomena such as crack initiation, crack propagation and delamination can successfully be accomplished by means of the FEM. Nonetheless, various undesirable characteristics persist, which ren- der this method computationally prohibitive for more involved analyses. As a result, alternative methods have been pursued; the scaled boundary finite element method (SBFEM), is a little explored, yet highly capable alternative within the domain of linear elastic fracture mechanics (LEFM). Hence, the objective of this thesis is to accelerate computationally intensive numerical problems harnessing the merits and fur- ther extending the capabilities of the SBFEM to a wide class of forward and inverse problems, specifically with applications in computational fracture mechanics, damage localization and topology optimization. The increasing importance of sustainability implies the prudent use of existing resources, such that efficient computation schemes are sought. This thesis proposes several novel schemes to accomplish this goal. The Hamiltonian Schur decomposition is first adopted, to reverse the computational toll incurred during an SBFEM analysis, due to the linearization of the underlying quadratic eigen-problem. Further, an efficient recovery based error estimator is proposed, which additionally permits calculating the gener- alized stress intensity factors (gSIFs) at increased accuracy, using fewer degrees of freedom (DOF). The use of linear quadtree (QT) meshes, pioneered by previous authors, to overcome SBFEM’s unique mesh- ing requirements, can lead to reduced accuracy in calculated gSIFs for crack propagation problems. A method of internally elevating the approximation space of a crack tip element is proposed, which is shown to greatly improve the accuracy with which gSIFs are calculated on highly coarse QT meshes. These ap- proaches are exploited to develop the multiscale scaled boundary finite element method (MSBFEM), which harnesses the SBFEM to incorporate fracture on the fine scale and the enhanced multiscale finite element method (EMsFEM) to construct a coarse scale representation, where the governing equations are solved at a reduced computational cost. The newly developed MSBFEM is then extended to a highly efficient crack propagation scheme, which resolves only regions directly surrounding the crack tip, and incorporates the remaining domain via coarse scale macro-elements. In doing so, the amount of DOFs present during analysis are drastically reduced, while the crack path is still accurately captured. These novel insights in accelerating the forward problem are then applied to inverse analyses. Due to its domain specific advantage, SBFEM is subsequently applied to damage localization schemes. Taking advantage of the parallel nature with which heuristic algorithms approach damage localization, combined with precomputation of the undamaged domain by SBFEM and updating the effects of varying crack can- didates by reanalysis techniques, a highly efficient and effective scheme is devised to accelerate damage localization analyses to near real-time levels. Topology optimization (TO), which similarly to damage localization, is often marred by the repeated solution of an expensive forward problem, stands to benefit from efficient solvers. Automated adaptive analysis-ready meshes are achieved by harnessing image compression techniques. The proposed drop-in replacement for the forward solver, reduces the amount of DOF during present during analysis by over an order of magnitude. This approach is successfully extended to 3D problems. i Kurzfassung Die Nachfrage nach nachhaltigem Design in z.B. der Luft- Raumfahrt-, Automobil sowie der Bauindustrie haben zur Entwicklung leichterer, beständiger und belastbarerer Strukturen geführt, deren Simulation mittels robuster, wirtschaftlicher und präziser numerischer Simulationen gewährleist werden müssen. Seit ihrer Einführung wurde die Finite-Elemente-Methode (FEM) weiterentwickelt, um unter anderem eine Vielzahl von Strukturanalyseproblemen zu lösen, die von linearen über nichtlineare, statische bis hin zu dynamischen, Bruch- und Kontaktproblemen reichen. Auf dem Gebiet der Bruchmechanik wurde gezeigt, dass die Modellierung von schadensbedingten Phänomenen wie Rissentstehung, Rissausbreitung und Delaminierung mit der FEM erfolgreich simuliert werden kann. Dennoch bleiben verschiedene limi- tierende Eigenschaften bestehen, die diese Methode für aufwendigere Analysen unatraktiv machen. Da- her wurden Alternativemethoden erforscht: Eine solche Methode ist die skalierte Rand-Finite Elemente Methode (SBFEM), eine wenig erforschte und dennoch hochleistungsfähige Alternative im Bereich der linear-elastischen Bruchmechanik (LEFM). Ziel dieser Arbeit ist es daher, die Vorzüge der SBFEM zur Beschleunigung von rechenintensiven numerischen Problemen zu nutzen sowie ihre Anwendbarkeit auf eine breite Klasse an Vorwärts- und Rückwärtsproblemen auszuweiten, insbesondere bei Anwendungen in der rechnergestützten Bruchmechanik, Schadenslokalisierung (SL) und Topologieoptimierung (TO). Die zunehmende Bedeutung der Nachhaltigkeit setzt den umsichtigen Einsatz vorhandener Ressourcen voraus, sodass effiziente Berechnungsschemata angestrebt werden. Diese Dissertation schlägt mehrere neuartige Schemata vor, um dieses Ziel zu erreichen. Die Hamiltonsche Schur-Zerlegung wird zuerst angewendet, um den während einer SBFEM-Analyse aufgrund der Linearisierung des quadratischen Eigenproblems anfallenden Rechenaufwandes zu beseitigen. Ferner wird ein effizienter Fehlerabschätzer vorgeschlagen, der es zusätzlich ermöglicht, die verallgemeinerten Spannungsintensitätsfaktoren (gSIF) mit erhöhter Genauigkeit selbst unter Verwendung von wenigen Freiheitsgraden (DOF) zu berechnen. Die Verwendung von linearen Quadtree-Netzen (QT), die von früheren Autoren entwickelt wurden, um die einzigartigen Netzanforderungen von SBFEM zu überwinden, kann zu einer verringerten Genauigkeit der berechneten gSIFs bei Rissausbreitungssimulationen führen. Es wird ein Verfahren zum internen Erhöhen des Approximationsraums eines Rissspitzenelements vorgeschlagen, das die Genauigkeit, mit der gSIFs auf äusserst groben QT-Netzen berechnet werden, erheblich verbessert. Diese Ansätze werden genutzt, um die Multiskalen-SBFEM (MSBFEM) zu entwickeln, welche mittels SBFEM Diskontinuitäten in der Mikroskala berücksichtigt, und die erweiterte Multiskalen-Finite-Elemente-Methode (EMsFEM) verwendet, um eine Makroskala-Representation des Systems zu erstellen, welches anschliessend mit re- duziertem Rechenaufwand gelöst wird. Das neu entwickelte MSBFEM wird dann auf ein hocheffizientes Rissausbreitungsschema erweitert, das nur Regionen auflöst, welche die Rissspitze direkt umgeben, und die restliche Domäne über Makroelemente berücksichtigt. Auf diese Weise wird die Anzahl der DOFs drastisch reduziert, während der Risspfad ausreichend genau erfasst wird. Diese neuartigen Erkenntnisse zur Beschleunigung des Vorwärtsproblems werden dann auf Rück- wärtsprobleme angewendet. Aufgrund seines domänenspezifischen Vorteils wird SBFEM folglich auf SL- Schemata angewendet. Ausgehend von der Parallelität, mit der heuristische Algorithmen SL begegnen, kombiniert mit der Vorberechnung der unbeschädigten Domäne durch SBFEM und der Aktualisierung der Auswirkungen verschiedener Risskandidaten durch Reanalysetechniken, wird ein hocheffizientes und effektives Schema entwickelt, um die SL-Analysen fast auf Echtzeitberechungen zu beschleunigen. Die TO, die ähnlich wie die SL häufig durch die wiederholte Lösung eines teuren Vorwärtsproblems beeinträchtigt wird, profitiert von effizienten Solvern. Automatisierte, adaptive, analysebereite Netze wer- den durch die Nutzung von Bildkomprimierungstechniken erzielt. Der vorgeschlagene Drop-In-Ersatz für den Vorwärtssolver verringert die Anzahl an DOF, welche während der Analyse behandelt werden müssen um eine Grössenordnung. Dieser Ansatz wird erfolgreich auf 3D-Probleme ausgeweitet. iii Acknowledgements First and foremost, I would like to thank my supervisors and co-examiners Prof. Dr. Eleni Chatzi, Prof. Dr. Savvas Triantafyllou and Prof. Dr. Chongmin Song. To this day, I am supremely thankful to both Eleni and Savvas for repeatedly taking action in my best interests, starting with
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