258 research outputs found

    Quantifying domain uncertainty in linear elasticity

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    The present article considers the quantification of uncertainty for the equations of linear elasticity on random domains. To this end, we model the random domains as the images of some given fixed, nominal domain under random domain mappings, which are defined by a Karhunen-Loève expansion. We then prove the analytic regularity of the random solution with respect to the countable random input parameters which enter the problem through the Karhunen-Loève expansion of the random domain mappings. In particular, we provide appropriate bounds on arbitrary derivatives of the random solution with respect to those input parameters, which enable the use of state-of-the-art quadrature methods to compute deterministic statistics such as the mean and variance of quantities of interest such as the random solution itself or the random von Mises stress as integrals over the countable random input parameters in a dimensionally robust way. Numerical examples qualify and quantify the theoretical findings

    Journal of Computational and Applied Mechanics 18.

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    Static Analysis of Anisotropic Doubly-Curved Shell Subjected to Concentrated Loads Employing Higher Order Layer-Wise Theories

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    In the present manuscript, a Layer-Wise (LW) generalized model is proposed for the linear static analysis of doublycurved shells constrained with general boundary conditions under the influence of concentrated and surface loads. The unknown field variable is modelled employing polynomials of various orders, each of them defined within each layer of the structure. As a particular case of the LW model, an Equivalent Single Layer (ESL) formulation is derived too. Different approaches are outlined for the assessment of external forces, as well as for non-conventional constraints. The doubly-curved shell is composed by superimposed generally anisotropic laminae, each of them characterized by an arbitrary orientation. The fundamental governing equations are derived starting from an orthogonal set of principal coordinates. Furthermore, generalized blending functions account for the distortion of the physical domain. The implementation of the fundamental governing equations is performed by means of the Generalized Differential Quadrature (GDQ) method, whereas the numerical integrations are computed employing the Generalized Integral Quadrature (GIQ) method. In the post-processing phase, an effective procedure is adopted for the reconstruction of stress and strain through-the-thickness distributions based on the exact fulfillment of three-dimensional equilibrium equations. A series of systematic investigations are performed in which the static response of structures with various curvatures and lamination schemes, calculated by the present methodology, have been successfully compared to those ones obtained from refined finite element three-dimensional simulations. Even though the present LW approach accounts for a two-dimensional assessment of the structural problem, it is capable of well predicting the three-dimensional response of structures with different characteristics, taking into account a reduced computational cost and pretending to be a valid alternative to widespread numerical implementations

    PDE-Based Parameterisation Techniques for Planar Multipatch Domains

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    This paper presents a PDE-based parameterisation framework for addressing the planar surface-to-volume (StV) problem of finding a valid description of the domain's interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the StV problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem's governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain that, depending on the application, builds desired features into the computed harmonic map, such as homogeneous cell sizes or boundary layers

    Tailoring structures using stochastic variations of structural parameters.

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    Imperfections, meaning deviations from an idealized structure, can manifest through unintended variations in a structure’s geometry or material properties. Such imperfections affect the stiffness properties and can change the way structures behave under load. The magnitude of these effects determines how reliable and robust a structure is under loading. Minor changes in geometry and material properties can also be added intentionally, creating a more beneficial load response or making a more robust structure. Examples of this are variable stiffness composites, which have varying fiber paths, or structures with thickened patches. The work presented in this thesis aims to introduce a general approach to creating geodesic random fields in finite elements and exploiting these to improve designs. Random fields can be assigned to a material or geometric parameter. Stochastic analysis can then quantify the effects of variations on a structure for a given type of imperfection. Information extracted from the effects of imperfections can also identify areas critical to a structure’s performance. Post-processing stochastic results by computing the correlation between local changes and the structural performance result in a pattern, describing the effects of local changes. Perturbing the ideal deterministic geometry or material distribution of a structure using the pattern of local influences can increase performance. Examples demonstrate the approach by increasing the deterministic (without imperfections applied) linear buckling load, fatigue life, and post-buckling path of structures. Deterministic improvements can have a detrimental effect on the robustness of a structure. Increasing the amplitude of perturbation applied to the original design can improve the robustness of a structure’s response. Robustness analyses on a curved composite panel show that increasing the amplitude of design changes makes a structure less sensitive to variations. The example studied shows that an increase in robustness comes with a relatively small decrease in the deterministic improvement.Imperfektionen, d. h. die Abweichungen von einer idealisierten Struktur, können sich durch unbeabsichtigte Variationen in der Geometrie oder den Materialeigenschaften einer Struktur ergeben. Solche Imperfektionen wirken sich auf die Steifigkeitseigenschaften aus und können das Verhalten von Strukturen unter Last verändern. Das Ausmaß dieser Auswirkungen bestimmt, wie zuverlässig und robust eine Struktur unter Belastung ist. Kleine Änderungen der Geometrie und der Materialeigenschaften können auch absichtlich eingebaut werden, um ein verbessertes Lastverhalten zu erreichen oder eine stabilere Struktur zu schaffen. Beispiele hierfür sind Verbundwerkstoffe mit variabler Steifigkeit, die unterschiedliche Faserverläufe aufweisen, oder Strukturen mit lokalen Verstärkungen. Die in dieser Dissertation vorgestellte Arbeit zielt darauf ab, einen allgemeinen Ansatz zur Erstellung geodätischer Zufallsfelder in Finiten Elementen zu entwickeln und diese zur Verbesserung von Konstruktionen zu nutzen. Zufallsfelder können Material- oder Geometrieparametern zugeordnet werden. Die stochastische Analyse kann dann die Auswirkungen von Variationen auf eine Struktur für eine bestimmte Art von Imperfektion quantifizieren. Die aus den Auswirkungen von Imperfektionen gewonnenen Informationen können auch Bereiche identifizieren, die für das Tragvermögen einer Struktur kritisch sind. Die Auswertung der stochastischen Ergebnisse durch Berechnung der Korrelation zwischen lokalen Veränderungen und Strukturtragvermögen ergibt ein Muster, das die Auswirkungen lokaler Veränderungen beschreibt. Die Perturbation der idealen deterministischen Geometrie oder der Materialverteilung einer Struktur unter Verwendung des Musters der lokalen Einflüsse kann das Tragvermögen erhöhen. Anhand von Beispielen wird der Ansatz durch die Erhöhung der deterministischen (ohne Imperfektionen) linearen Knicklast, der Lebensdauer und des Nachknickverhaltens von Strukturen aufgezeigt. Deterministische Verbesserungen können sich zum Nachteil der Robustheit einer Struktur auswirken. Eine Vergrößerung der Amplitude der auf den ursprünglichen Designentwurf angewendeten Perturbation kann die Robustheit der Reaktion einer Struktur verbessern. Robustheitsanalysen an einer gekrümmten Verbundplatte zeigen, dass eine Struktur durch eine Vergrößerung der Amplitude der Entwurfsänderungen weniger empfindlich gegenüber Abweichungen wird. Das untersuchte Beispiel zeigt, dass eine Erhöhung der Robustheit mit einem relativ geringen Verlust der deterministischen Verbesserung eingeht

    Numerical quadrature for Gregory quads

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    We investigate quadrature rules in the context of quadrilateral Gregory patches, in short Gregory quads. We provide numerical and where possible symbolic quadrature rules for the space spanned by the twenty polynomial/rational functions associated with Gregory quads, as well as the derived spaces including derivatives, products, and products of derivatives of these functions. This opens up the possibility for a wider adoption of Gregory quads in numerical simulations

    Fully probabilistic deep models for forward and inverse problems in parametric PDEs

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    We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational inference to assemble a fully probabilistic coherent framework. In the posited probabilistic model, both the forward and inverse maps are approximated as Gaussian distributions with a mean and covariance parameterized by deep neural networks. The PDE residual is assumed to be an observed random vector of value zero, hence we model it as a random vector with a zero mean and a user-prescribed covariance. The model is trained by maximizing the probability, that is the evidence or marginal likelihood, of observing a residual of zero by maximizing the evidence lower bound (ELBO). Consequently, the proposed methodology does not require any independent PDE solves and is physics-informed at training time, allowing the real-time solution of PDE forward and inverse problems after training. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems such as linear and nonlinear Poisson problems, elastic shells with complex 3D geometries, and time-dependent nonlinear and inhomogeneous PDEs using a physics-informed neural network (PINN) discretization. We achieve up to three orders of magnitude speed-up after training compared to traditional finite element method (FEM), while outputting coherent uncertainty estimates

    A comparison of smooth basis constructions for isogeometric analysis

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    In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modeling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C1C^1, Analysis-Suitable G1G^1 and the Approximate C1C^1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C1C^1 and Analysis-Suitable G1G^1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C1C^1 and D-Patch provide relatively easy construction on complex geometries. The Almost-C1C^1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C1C^1 and Analysis-Suitable G1G^1 applicable to more complex geometries

    An EigenValue Stabilization Technique for Immersed Boundary Finite Element Methods in Explicit Dynamics

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    The application of immersed boundary methods in static analyses is often impeded by poorly cut elements (small cut elements problem), leading to ill-conditioned linear systems of equations and stability problems. While these concerns may not be paramount in explicit dynamics, a substantial reduction in the critical time step size based on the smallest volume fraction χ\chi of a cut element is observed. This reduction can be so drastic that it renders explicit time integration schemes impractical. To tackle this challenge, we propose the use of a dedicated eigenvalue stabilization (EVS) technique. The EVS-technique serves a dual purpose. Beyond merely improving the condition number of system matrices, it plays a pivotal role in extending the critical time increment, effectively broadening the stability region in explicit dynamics. As a result, our approach enables robust and efficient analyses of high-frequency transient problems using immersed boundary methods. A key advantage of the stabilization method lies in the fact that only element-level operations are required. This is accomplished by computing all eigenvalues of the element matrices and subsequently introducing a stabilization term that mitigates the adverse effects of cutting. Notably, the stabilization of the mass matrix Mc\mathbf{M}_\mathrm{c} of cut elements -- especially for high polynomial orders pp of the shape functions -- leads to a significant raise in the critical time step size Δtcr\Delta t_\mathrm{cr}. To demonstrate the efficacy of our technique, we present two specifically selected dynamic benchmark examples related to wave propagation analysis, where an explicit time integration scheme must be employed to leverage the increase in the critical time step size.Comment: 45 pages, 25 figure
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