Composite Finite Elements for Trabecular Bone Microstructures

In many medical and technical applications, numerical simulations need to be performed for objects with interfaces of geometrically complex shape. We focus on the biomechanical problem of elasticity simulations for trabecular bone microstructures. The goal of this dissertation is to develop and implement an efficient simulation tool for finite element simulations on such structures, so-called composite finite elements. We will deal with both the case of material/void interfaces (complicated domains) and the case of interfaces between different materials (discontinuous coefficients). In classical finite element simulations, geometric complexity is encoded in tetrahedral and typically unstructured meshes. Composite finite elements, in contrast, encode geometric complexity in specialized basis functions on a uniform mesh of hexahedral structure. Other than alternative approaches (such as e.g. fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes, and extended finite element methods), the composite finite elements are tailored to geometry descriptions by 3D voxel image data and use the corresponding voxel grid as computational mesh, without introducing additional degrees of freedom, and thus making use of efficient data structures for uniformly structured meshes. The composite finite element method for complicated domains goes back to Wolfgang Hackbusch and Stefan Sauter and restricts standard affine finite element basis functions on the uniformly structured tetrahedral grid (obtained by subdivision of each cube in six tetrahedra) to an approximation of the interior. This can be implemented as a composition of standard finite element basis functions on a local auxiliary and purely virtual grid by which we approximate the interface. In case of discontinuous coefficients, the same local auxiliary composition approach is used. Composition weights are obtained by solving local interpolation problems for which coupling conditions across the interface need to be determined. These depend both on the local interface geometry and on the (scalar or tensor-valued) material coefficients on both sides of the interface. We consider heat diffusion as a scalar model problem and linear elasticity as a vector-valued model problem to develop and implement the composite finite elements. Uniform cubic meshes contain a natural hierarchy of coarsened grids, which allows us to implement a multigrid solver for the case of complicated domains. Besides simulations of single loading cases, we also apply the composite finite element method to the problem of determining effective material properties, e.g. for multiscale simulations. For periodic microstructures, this is achieved by solving corrector problems on the fundamental cells using affine-periodic boundary conditions corresponding to uniaxial compression and shearing. For statistically periodic trabecular structures, representative fundamental cells can be identified but do not permit the periodic approach. Instead, macroscopic displacements are imposed using the same set as before of affine-periodic Dirichlet boundary conditions on all faces. The stress response of the material is subsequently computed only on an interior subdomain to prevent artificial stiffening near the boundary. We finally check for orthotropy of the macroscopic elasticity tensor and identify its axes.Zusammengesetzte finite Elemente für trabekuläre Mikrostrukturen in Knochen In vielen medizinischen und technischen Anwendungen werden numerische Simulationen für Objekte mit geometrisch komplizierter Form durchgeführt. Gegenstand dieser Dissertation ist die Simulation der Elastizität trabekulärer Mikrostrukturen von Knochen, einem biomechanischen Problem. Ziel ist es, ein effizientes Simulationswerkzeug für solche Strukturen zu entwickeln, die sogenannten zusammengesetzten finiten Elemente. Wir betrachten dabei sowohl den Fall von Interfaces zwischen Material und Hohlraum (komplizierte Gebiete) als auch zwischen verschiedenen Materialien (unstetige Koeffizienten). In klassischen Finite-Element-Simulationen wird geometrische Komplexität typischerweise in unstrukturierten Tetraeder-Gittern kodiert. Zusammengesetzte finite Elemente dagegen kodieren geometrische Komplexität in speziellen Basisfunktionen auf einem gleichförmigen Würfelgitter. Anders als alternative Ansätze (wie zum Beispiel fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes und extended finite element methods) sind die zusammengesetzten finiten Elemente zugeschnitten auf die Geometriebeschreibung durch dreidimensionale Bilddaten und benutzen das zugehörige Voxelgitter als Rechengitter, ohne zusätzliche Freiheitsgrade einzuführen. Somit können sie effiziente Datenstrukturen für gleichförmig strukturierte Gitter ausnutzen. Die Methode der zusammengesetzten finiten Elemente geht zurück auf Wolfgang Hackbusch und Stefan Sauter. Man schränkt dabei übliche affine Finite-Element-Basisfunktionen auf gleichförmig strukturierten Tetraedergittern (die man durch Unterteilung jedes Würfels in sechs Tetraeder erhält) auf das approximierte Innere ein. Dies kann implementiert werden durch das Zusammensetzen von Standard-Basisfunktionen auf einem lokalen und rein virtuellen Hilfsgitter, durch das das Interface approximiert wird. Im Falle unstetiger Koeffizienten wird die gleiche lokale Hilfskonstruktion verwendet. Gewichte für das Zusammensetzen erhält man hier, indem lokale Interpolationsprobleme gelöst werden, wozu zunächst Kopplungsbedingungen über das Interface hinweg bestimmt werden. Diese hängen ab sowohl von der lokalen Geometrie des Interface als auch von den (skalaren oder tensorwertigen) Material-Koeffizienten auf beiden Seiten des Interface. Wir betrachten Wärmeleitung als skalares und lineare Elastizität als vektorwertiges Modellproblem, um die zusammengesetzten finiten Elemente zu entwickeln und zu implementieren. Gleichförmige Würfelgitter enthalten eine natürliche Hierarchie vergröberter Gitter, was es uns erlaubt, im Falle komplizierter Gebiete einen Mehrgitterlöser zu implementieren. Neben Simulationen einzelner Lastfälle wenden wir die zusammengesetzten finiten Elemente auch auf das Problem an, effektive Materialeigenschaften zu bestimmen, etwa für mehrskalige Simulationen. Für periodische Mikrostrukturen wird dies erreicht, indem man Korrekturprobleme auf der Fundamentalzelle löst. Dafür nutzt man affin-periodische Randwerte, die zu uniaxialem Druck oder zu Scherung korrespondieren. In statistisch periodischen trabekulären Mikrostrukturen lassen sich ebenfalls Fundamentalzellen identifizieren, sie erlauben jedoch keinen periodischen Ansatz. Stattdessen werden makroskopische Verschiebungen zu denselben affin-periodischen Randbedingungen vorgegeben, allerdings durch Dirichlet-Randwerte auf allen Seitenflächen. Die Spannungsantwort des Materials wird anschließend nur auf einem inneren Teilbereich berechnet, um künstliche Versteifung am Rand zu verhindern. Schließlich prüfen wir den makroskopischen Elastizitätstensor auf Orthotropie und identifizieren deren Achsen

Fluid-Structure Interaction Problems in Hemodynamics:Parallel Solvers, Preconditioners, and Applications

In this work we aim at the description, study and numerical investigation of the fluid-structure interaction (FSI) problem applied to hemodynamics. The FSI model considered consists of the Navier-Stokes equations on moving domains modeling blood as a viscous incompressible fluid and the elasticity equation modeling the arterial wall. The fluid equations are derived in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Several existing formulations and discretizations are discussed, providing a state of the art on the subject. The main new contributions and advancements consist of: A description of the Newton method for FSI-ALE, with details on the implementation of the shape derivatives block assembling, considerations about parallel performance, the analytic derivation of the derivative terms for different formulations (conservative or not) and for different types of boundary conditions. The implementation and analysis of a new category of preconditioners for FSI (applicable also to more general coupled problems). The framework set up is general and extensible. The proposed preconditioners allow, in particular, a separate treatment of each field, using a different preconditioning strategy in each case. An estimate for the condition number of the preconditioned system is proposed, showing how preconditioners of this type depend on the coupling, and explaining the good performance they exhibit when increasing the number of processors. The improvement of the free (distributed under LGPL licence) parallel finite elements library LifeV. Most of the methods described have been implemented within this library during the period of this PhD and all the numerical tests reported were run using this framework. The simulation of clinical cases with patient-specific data and geometry, the comparison on simulations of physiological interest between different models (rigid, FSI, 1D), discretizations and methods to solve the nonlinear system. A methodology to obtain patient-specific FSI simulations starting from the raw medical data and using a set of free software tools is described. This pipeline from imaging to simulation can help medical doctors in diagnosis and decision making, and in understanding the implication of indicators such as the wall shear stress in the pathogenesis

Micromechanics of Human Bone: Role of Architecture and Tissue Material Properties

Knowledge of the biomechanical behavior and failure mechanisms of human bone is fundamental to understanding the etiology of bone fractures as well as the mechanisms by which aging, disease, and treatment can alter the mechanical competence of bone. In this context, the focus of this dissertation was to enhance the current understanding of the biomechanical mechanisms of bone strength, and more specifically, to elucidate the role of architecture and tissue material properties in overall bone strength and whole-bone failure behavior.Using the latest advances in micro-computed tomography and high-resolution finite element modeling, we investigated the effect of typical population-variations in tissue-level ductility on human vertebral strength. We found that compared to the reference case, varying both cortical and trabecular tissue ultimate strains by ±1 SD from their mean values changed vertebral strength by at most ±8%, an effect that was relatively uniform across all the specimens. Overall strength changed similarly for similar (±1 SD) changes in trabecular versus cortical ductility. Further analysis revealed that only a tiny proportion of tissue failed (< 2%) when the whole bone reached its point of structure-level failure, and that the failure mode and location of this tiny amount were relatively insensitive to typical variations in tissue ductility. These findings suggest that it is the overall load transfer within the whole vertebral body —determined by bone volume fraction and microstructure— that dictates where failure occurs rather than typical variations in the ductility of the tissue. Together these findings suggest that typical variations in tissue ductility might have a relatively modest impact on vertebral strength compared to the multiple-fold variations in vertebral strength that are typically observed across any elderly population.Combining micro-computed tomography, high-resolution finite element modeling and biomechanical testing, we sought to provide further insight into the tissue modulus of trabecular bone and better elucidate its relation with bone volume fraction and trabecular microarchitecture. Our results indicated that effective tissue modulus of vertebral trabecular bone varied greatly among the specimens and was negatively correlated with bone volume fraction of each vertebra (R2 = 0.51, p < 0.05). These results suggest that there can be 3X variation in tissue modulus across the elderly human vertebrae, about 50% of which may be explained by variations in bone volume fraction. Together these findings suggest that as trabecular bone becomes older and thus more porous due to an imbalance between bone formation and resorption, the tissue may become stiffer to compensate for the bone loss.The work presented in this dissertation has also provided substantial insight into the structure–function relations for trabecular bone from different anatomic sites. We investigated the main structure–function relation —characterized by bone volume fraction versus on-axis yield stress— for human calcaneal trabecular bone and compared this relation to that for trabecular bone from other anatomic sites. We found that the relation between yield stress and bone volume fraction of the calcaneus was most similar to that of the proximal tibia. Furthermore, our results demonstrated that while there was no universal yield stress–bone volume fraction relation for trabecular bone across different anatomic sites for on-axis loading, the general (normalized) yield stress–bone volume fraction relation was similar for all sites. This similarity in the normalized relation suggests that a given percentage deviation from the mean bone mass has the same mechanical consequence at the calcaneus as it does at the other anatomic sites.In closure, this dissertation provides answers to some of the fundamental questions regarding the role of architecture and tissue material properties in explaining the variations in overall bone strength across individuals, and provides new insight into the etiology of age-related fractures. This work also outlines potential areas of future research to further advance our current understanding of overall bone strength and fracture etiology

Development of a finite volume method for elastic materials and fluid-solid coupled applications

This thesis presents the development of a parallel finite volume numerical method to analyse thermoelastic and hyperelastic materials and applied problems with mutual interaction between a fluid and a structure. The solid problem follows a cell-centred finite volume formulation for three-dimensional unstructured grids under the same framework that is frequently devoted to computational fluid dynamics. Second-order accurate schemes are used to discretise both in time and space. A direct implicit time integration promotes numerical stability when facing vibration and quasi-static scenarios. The geometrical non-linearities, encountered with the large displacements of both Saint Venant-Kirchhoff and neo-Hookean models, are tackled by means of an updated Lagrangian approach. Verification of the method is conducted with canonical cases which involve: static equilibrium, thermal stress, vibration, structural damping, large deformations, nearly incompressible materials and high memory usage. Significant savings in computation time are achieved owing to the acceleration strategies implemented within the system resolution, namely a segregated algorithm with Aitken relaxation and a block-coupled system arrangement. The similarities between the block-coupled method and the displacement-based finite element method, with regards to the matrix form of the resulting equations, allow for including Rayleigh viscous damping within a finite volume solver. The program for structures is to be coupled with the in-house fluid numerical models in order to produce a unified fluid-structure interaction platform, where an arbitrary Lagrangian-Eulerian approach is used to solve the flow in a conforming grid. As a first step, the method for incompressible Newtonian fluids is adapted to deal with structure-coupled problems. To do so, the Lagrangian-Eulerian version of the Navier-Stokes equations is presented, and automatic moving mesh techniques are developed. These techniques are designed to mitigate the mesh quality deterioration and to satisfy the space conservation law. Besides, a semi-implicit coupling algorithm, which only implicitly couples the fluid pressure term to the structure, is implemented. As a result, numerical stability for strongly coupled phenomena at a reduced computational cost is obtained. These new tools are tested on an applied case, consisting of the turbulent flow through self-actuated flexible valves. Finally, a pioneering coupled numerical model for the thermal and structural analysis of packed-bed thermocline storage tanks is developed. This thermal accumulation system for concentrated solar power plants has attracted the attention of the industry due to the economic advantage compared to the usual two-tank system. Dynamic coupling among the thermoelastic equations for the tank shell and the numerical models for all other relevant elements of the system is considered. After validating the model with experimental results, the commercial viability of the thermocline concept, regarding energetic effectiveness and structural reliability, is evaluated under real operating conditions of the power plants.Esta tesis presenta el desarrollo de un método numérico paralelo basado en volúmenes finitos para analizar materiales termoelásticos e hiperelásticos y problemas con una interacción mutua entre un fluido y una estructura. El problema del sólido sigue una formulación de volúmenes finitos centrada en las celdas para mallas no-estructuradas tridimensionales, bajo el mismo marco que se suele emplear en la dinámica de fluidos computacional. Se utilizan esquemas de segundo orden de precisión para discretizar el tiempo y el espacio. Una integración temporal directa implícita asegura estabilidad numérica al afrontar escenarios casi-estáticos o de vibración. Las no linealidades, que aparecen con los amplios desplazamientos de los modelos de Saint Venant-Kirchhoff y de neo-Hookean, son abordadas con un enfoque Lagrangiano actualizado. La verificación del método se realiza a través de casos canónicos que involucran: equilibrio estático, tensiones térmicas, vibración, amortiguación estructural, grandes deformaciones, materiales casi incompresibles y altos requerimientos de memoria. Se registra un ahorro significativo en el tiempo de cálculo gracias a las estrategias de aceleración implementadas dentro de la resolución del sistema, principalmente un algoritmo segregado con relajación Aitken y una disposición acoplada en bloques del sistema. Las similitudes entre este método acoplado en bloques y el método de los elementos finitos basados en el desplazamiento, con respecto a la forma matricial de las ecuaciones resultantes, permiten incluir la amortiguación viscosa tipo Rayleigh dentro de un solucionador de volúmenes finitos. El programa para estructuras se acoplará con los modelos numéricos internos para fluidos con el objetivo de generar una plataforma unificada de interacción fluido-estructura, donde se usa un enfoque arbitrario Lagrangiano-Euleriano sobre una malla conforme para resolver el fluido. Como primer paso, el método para flujos incompresibles Newtonianos se adapta para lidiar con problemas acoplados a una estructura. Para ello, se presenta la versión Lagrangiana-Euleriana de las ecuaciones de Navier-Stokes y se desarrollan técnicas automáticas de movimiento de malla. El diseño de estas técnicas se centra en mitigar el deterioro de la calidad de la malla y satisfacer la ley de conservación del espacio. Además, se implementa un algoritmo de acoplamiento semi-implícito, que sólo acopla implícitamente el término fluido de presión a la estructura. Como resultado, se obtiene estabilidad numérica para fenómenos fuertemente acoplados a un coste computacional reducido. Estas nuevas herramientas se prueban en un caso aplicado, que consiste el flujo turbulento a través de válvulas flexibles autoactivadas. Finalmente, se desarrolla un modelo numérico acoplado pionero para analizar estructuralmente y térmicamente los tanques termoclina de almacenamiento térmico. Este sistema de acumulación para centrales termosolares ha atraído la atención de la industria debido al ahorro económico comparado con el sistema de doble tanque habitual. Se tiene en cuenta el acoplamiento dinámico entre las ecuaciones gobernantes de la pared del tanque y las de todos los elementos relevantes del sistema. Tras validar el modelo con datos experimentales, se evalúa la viabilidad comercial de estos tanques, en cuanto a rendimiento energético y fiabilidad estructural, bajo condiciones reales de operación de las centrales.Postprint (published version

Aeronautical engineering: A continuing bibliography with indexes (supplement 245)

This bibliography lists 537 reports, articles, and other documents introduced into the NASA scientific and technical information system in October, 1989. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Parallel simulation of volume-coupled multi-field problems with special application to soil dynamics

Zur Lösung vieler ingenieur- und naturwissenschaftlichen Problemstellungen sind numerische Simulationen ein wichtiges Hilfsmittel. Sie dienen beispielsweise der Wettervorhersage in der Meteorologie oder der Strukturanalyse und Strukturoptimierung im Maschinenbau. In vielen Aufgabenstellungen kann das untersuchte Problem, aufgrund seiner starken Wechselwirkung mit den angrenzenden Systemen, nicht losgelöst betrachtet werden, so dass eine gesamtheitliche Betrachtungsweise notwendig wird. Diese Systeme werden in der Literatur als gekoppelte Probleme bezeichnet. Aufgrund der Komplexität der betrachteten Probleme sind zur effizienten Lösung der zugrunde liegenden Gleichungen parallele Lösungsstrategien von Vorteil. Hierbei wird das Gesamtproblem in kleinere Teilprobleme zerlegt, die gleichzeitig auf verschiedenen Rechnern oder Prozessoren gelöst werden. Um die Vorteile dieses Lösungsverfahrens bestmöglich nutzen zu können, sind erhebliche Anstrengungen zunächst für die initiale Entwicklung und Umsetzung eines effizienten Lösungsverfahrens sowie anschließend für dessen kontinuierliche Weiterentwicklung notwendig. Die vorliegende Monographie beschreibt einen Ansatz zur Kosimulation numerischer Probleme zwischen dem kommerziellen auf der Finite-Elemente-Methode (FEM) basierenden Programmpaket Abaqus und dem für die Forschung entwickelten Löser PANDAS. Durch die Entwicklung einer allgemeinen Schnittstelle können die Materialmodelle von PANDAS direkt, ohne eine langwierige und fehleranfällige Reimplementierung, in eine für die industrielle Anwendung wichtige Simulationsumgebung überführt werden. Hierbei kann direkt auf die umfangreiche Materialmodellbibliothek von PANDAS zurückgegriffen werden. Zur Illustration der Anwendungsmöglichkeiten der Abaqus-PANDAS-Kopplung wird diese exemplarisch zur Simulation verschiedener volumengekoppelter Mehrfeldprobleme herangezogen. Als bodenmechanisches Anwendungsbeispiel wird die Tragfähigkeit eines flüssigkeitgesättigten granularen Materials unter quasi-statischen und dynamischen zyklischen Belastungen untersucht. Weiterhin werden mehrphasige Strömungsprozesse, wie sie z. B. im Produktionsprozess von faserverstärkten Kunststoffen auftreten, numerisch simuliert. Im sogenannten Vaccum-Assisted-Resin-Transfer-Moulding (VARTM), wird ein zunächst trockenes (gasgesättigtes) Fasergewebe kontinuierlich mit Harz getränkt, wobei für die praktische Anwendung insbesondere die Zeit bis zur vollständigen Sättigung und der sich einstellende Faservolumenanteil im fertigen Bauteil von großem Interesse sind. Weiterhin werden die Effizienz und die parallele Skalierbarkeit des vorgeschlagenen Kosimulationsansatzes untersucht