1,787 research outputs found
Near-optimal perfectly matched layers for indefinite Helmholtz problems
A new construction of an absorbing boundary condition for indefinite
Helmholtz problems on unbounded domains is presented. This construction is
based on a near-best uniform rational interpolant of the inverse square root
function on the union of a negative and positive real interval, designed with
the help of a classical result by Zolotarev. Using Krein's interpretation of a
Stieltjes continued fraction, this interpolant can be converted into a
three-term finite difference discretization of a perfectly matched layer (PML)
which converges exponentially fast in the number of grid points. The
convergence rate is asymptotically optimal for both propagative and evanescent
wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201
The motion of a deforming capsule through a corner
A three-dimensional deformable capsule convected through a square duct with a
corner is studied via numerical simulations. We develop an accelerated boundary
integral implementation adapted to general geometries and boundary conditions.
A global spectral method is adopted to resolve the dynamics of the capsule
membrane developing elastic tension according to the neo-Hookean constitutive
law and bending moments in an inertialess flow. The simulations show that the
trajectory of the capsule closely follows the underlying streamlines
independently of the capillary number. The membrane deformability, on the other
hand, significantly influences the relative area variations, the advection
velocity and the principal tensions observed during the capsule motion. The
evolution of the capsule velocity displays a loss of the time-reversal symmetry
of Stokes flow due to the elasticity of the membrane. The velocity decreases
while the capsule is approaching the corner as the background flow does,
reaches a minimum at the corner and displays an overshoot past the corner due
to the streamwise elongation induced by the flow acceleration in the downstream
branch. This velocity overshoot increases with confinement while the maxima of
the major principal tension increase linearly with the inverse of the duct
width. Finally, the deformation and tension of the capsule are shown to
decrease in a curved corner
Non degenerate anisotropic green's function for 3D magneto-electro-elasticity and bem shape sensitivity framework for 3D contact in anisotropic elasticity
The first part of the thesis presents a new expression for the magneto electro elastic (MEE) fundamental solution which is explicit in terms of the Stroh’s eigenvalues, remains welldefined for repeated Stroh’s eigenvalues and is exact. We then define a fast and robust numerical scheme to evaluate the function and its derivatives based on a double Fourier series representation. These newly developed expressions allow to compute the Fourier coefficients for any material symmetry or anisotropy, and is done only once for a given material. One evaluates the Green’s function and its derivatives through simple trigonometric formulas. Several results are presented for elastic, piezoelectric/piezomagnetic and magneto-eletro-elastic materials. The second part of the thesis provides a BEM-based formulation for shape sensitivity analysis of anisotropic elastic media, also including contact conditions, and based on the newly presented Green’s functions. The parameter sensitivity is evaluated using the complex step (CS) method: An approach similar to finite differentiation (FD), with the advantage of being step-size independent, therefore an extremely robust method. A convergence study on shape sensitivity is provided, proving the efficiency of the CS-BEM approach. We solve Hertz and non-Hertzian type contact problems as well as an application example of a dovetail joint found in gas turbines. We analyzed several parmeter sensitivities to shape variation, such as contact pressure, shear stress, as well as Von Mises stress, for both isotropic and anisotropic materials. The results showed good agreement with analytical solutions, as well as other works from the literature. In comparison with FD, which did not converged for an example case, the CS method showed excellent stability and precision for a broad range of step sizes.A primeira parte da tese apresenta uma nova expressĂŁo para a solução fundamental Magneto-Eletro-Elástica explĂcita em termos de autovalores de Stroh, bem definida para autovalores repetidos, e exata. Em seguida, uma sĂ©rie de Fourier dupla Ă© utilizada como uma forma rápida e robusta para avaliar a solução fundamental e as suas derivadas. As expressões recĂ©m-desenvolvidas permitem calcular os coeficientes de Fourier para qualquer simetria ou anisotropia de material, o que Ă© feito apenas uma vez para um dado material. Diversos resultados sĂŁo apresentados para materiais elásticos, piezoelĂ©tricos e magneto-eletro-elásticos. A segunda parte desta tese apresenta uma formulação completa para análise de sensibilidade em estruturas elasticas anisotrĂłpicas baseada nestas funções de Green recĂ©m apresentadas, incluindo condições de contato. A sensibilidade Ă parâmetros Ă© avaliada utilizando o mĂ©todo do incremento complexo, mĂ©todo extremamente robusto, similar a diferenciação finita (FD), mas independente do tamanho do incremento. Problemas de contato de Hertz e nĂŁo Hertzianos foram resolvidos, assim como um estudo de aplicação de uma palheta de turbinas a gás. Foi avaliada a sensibilidade Ă variação de forma das tensões de contato, tensões cisalhantes máximas e tambĂ©m nas tensões equivalentes de Von Mises, em diferentes materiais anisotrĂłpicos. Os resultados mostraram boa correlação com soluções analĂticas assim como em outros trabalhos da literatura. Quando comparado com FD, que nĂŁo obteve convergĂŞncia em um dos exemplos, o mĂ©todo CS demonstrou excelente estabilidade e precisĂŁo para uma larga faixa de tamanhos de incremento
Analysis and Visualization of Higher-Order Tensors: Using the Multipole Representation
Materialien wie Kristalle, biologisches Gewebe oder
elektroaktive Polymere kommen häufig in verschiedenen
Anwendung, wie dem Prothesenbau oder der Simulation von
kĂĽnstlicher Muskulatur vor.
Diese und viele weitere Materialien haben gemeinsam, dass sie
unter gewissen Umständen ihre Form und andere
Materialeigenschaften ändern.
Um diese Veränderung beschreiben zu können, werden, abhängig
von der Anwendung, verschiedene Tensoren unterschiedlicher
Ordnung benutzt.
Durch die Komplexität und die starke Abhängigkeit der
Tensorbedeutung von der Anwendung, gibt es bisher kein
Verfahren Tensoren höherer Ordnung darzustellen, welches
standardmäßig benutzt wird.
Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig
Arbeiten, die sich mit der visuellen Darstellung dieser
Tensoren auseinandersetzt.
Diese Arbeit beschäftigt sich mit diesem Problem.
Es werden drei verschiedene Methoden präsentiert, Tensoren
höherer Ordnung zu analysieren und zu visualisieren.
Alle drei Methoden basieren auf der sogenannte deviatorischen
Zerlegung und der Multipoldarstellung.
Mit Hilfe der Multipole können die Symmetrien des Tensors
und damit des beschriebenen Materials bestimmt werden.
Diese Eigenschaft wird in fĂĽr die Visualisierung
des Steifigkeitstensors benutzt.
Die zweite Methode basiert direkt auf den Multipolen und kann
damit beliebige Tensoren in drei Dimensionen darstellen.
Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor
dritter Ordnung, vorgestellt.
Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren.
Das dritte Verfahren ist ein erstes globales
Tensorvisualisierungsverfahren, welches Tensoren beliebiger
Ordnung und Symmetry in drei Dimensionen mit Hilfe eines
linienbasierten Verfahrens darstellt.Materials like crystals, biological tissue or electroactive
polymers are frequently used in applications like prosthesis
construction or the simulation of artificial musculature.
These and many other materials have in common that they
change their shape and other material properties under
certain circumstances.
To describe these changes, different tensors of different
order, dependent of the application, are used.
Due to the complexity and the strong dependency of the
tensor meaning of the application, there is, by now, no
visualization method that is used by default.
Also for specific applications there are only a few methods
that address the visual analysis of higher-order tensors.
This work adresses this problem.
Three different methods to analyse and visualize tensors of
higher order will be provided.
All three methods are based on the so called deviatoric
decomposition and the multipole representation.
Using the multipoles the symmetries of a tensor and, therefore,
of the described material, can be calculated.
This property is used to visualize the stiffness tensor.
The second method uses the multipoles directly and can be
used for each tensor of any order in three dimensions.
This method is presented by analysing the third-order
coupling tensor.
These two techniques are glyph-based visualization methods.
The third one, a line-based method, is, according to our
knowledge, a first global visualization method that can be
used for an arbitrary tensor in three dimensions
Nanoindentation testing of soft polymers : computation, experiments and parameters identification
Since nanoindentation technique is able to measure the mechanical properties of extremely thin layers and small volumes with high resolution, it also became one of the important testing techniques for thin polymer layers and coatings. This dissertation is focusing on the characterization of polymers using nanoindentation, which is dealt with numerical computation, experiments and parameters identification. An analysis procedure is developed with the FEM based inverse method to evaluate the hyperelasticity and time-dependent properties. This procedure is firstly verified with a parameters re-identification concept. An important issue in this dissertation is to take the error contributions in real nanoindentation experiments into account. Therefore, the effects of surface roughness, adhesion force, friction and the real shape of the tip are involved in the numerical model to minimize the systematic error between the experimental responses and the numerical predictions. The effects are quantified as functions or models with corresponding parameters to be identified. Finally, data from uniaxial or biaxial tensile tests and macroindentation tests are taken into account. The comparison of these different loading situations provides a validation of the proposed material model and a deep insight into nanoindentation of polymers.Da Nanoindentation die Messung der mechanischen Eigenschaften von dünnen Schichten und kleinen Volumen mit hoher Auflösung ermöglicht, hat sich diese Messmethode zu einer der wichtigsten Testmethoden für dünne Polymerschichten und -beschichtungen entwickelt. Diese Dissertation konzentriert sich auf die Charakterisierung von Polymeren mittels Nanoindentation, die in Form von numerischen Berechnungen, Experimenten und Parameteridentifikationen behandelt wird. Es wurde ein Auswertungsverfahren mit einer FEM basierten inversen Methode zur Berechnung der Hyperelastizität und der zeitabhängigen Eigenschaften entwickelt. Dieses Verfahren wird zunächst mit einem Konzept der Parameter Re-Identifikation verifiziert. Fehlerquellen wie Oberflächenrauheit, Adhäsionskräfte, Reibung und die tatsächlichen Form der Indenterspitze werden in das numerische Modell eingebunden, um die Abweichungen der numerischen Vorhersagen von den experimentellen Ergebnissen zu minimieren. Diese Einflüsse werden als Funktionen oder Modelle mit dazugehörigen, zu identifizierenden Parametern, quantifiziert. Abschließend werden Messwerte aus uni- oder biaxialen Zugversuchen und Makroindentationsversuchen betrachtet. Der Vergleich dieser verschiedenen Belastungszustände liefert eine Bestätigung des vorgeschlagenen Materialmodells und verschafft einen tieferen Einblick in die bei der Nanoindentation von Polymeren ablaufenden Mechanismen
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