93 research outputs found

    An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

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    We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

    An isogeometric finite element formulation for phase transitions on deforming surfaces

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    This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using C1C^1-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise C1C^1-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-α\alpha scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to the text, added supplementary movie file

    Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

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    Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness (2ε2\varepsilon) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency kk and the half-thickness ε\varepsilon. Optimizing kk for each chosen ε\varepsilon, we find power laws for kk in function of ε\varepsilon that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as ε\varepsilon tends to 00. Its angular frequency exhibits a power law relationof the form k=γεβk=\gamma \varepsilon^{-\beta} with β=14\beta=\frac14 in the parabolic case (cylinders and trimmed cones), and the various β\betas 25\frac25, 37\frac37, and 13\frac13 in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented

    Localization in musical steelpans

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    The steelpan is a pitched percussion instrument that takes the form of a concave bowl with several localized dimpled regions of varying curvature. Each of these localized zones, called notes, can vibrate independently when struck, and produces a sustained tone of a well-defined pitch. While the association of the localized zones with individual notes has long been known and exploited, the relationship between the shell geometry and the strength of the mode confinement remains unclear. Here, we explore the spectral properties of the steelpan modeled as a vibrating elastic shell. To characterize the resulting eigenvalue problem, we generalize a recently developed theory of localization landscapes for scalar elliptic operators to the vector-valued case, and predict the location of confined eigenmodes by solving a Poisson problem. A finite element discretization of the shell shows that the localization strength is determined by the difference in curvature between the note and the surrounding bowl. In addition to providing an explanation for how a steelpan operates as a two-dimensional xylophone, our study provides a geometric principle for designing localized modes in elastic shells.Comment: 17 pages, 5 figure

    An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

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    We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1C^1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α\alpha method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.Comment: In this version, typos were fixed, Fig. 16 is added, the literature review is extended and clarifying explanations and remarks are added at several places. Supplementary movies are available at https://av.tib.eu/series/641/supplemental+videos+of+the+paper+an+adaptive+space+time+phase+field+formulation+for+dynamic+fracture+of+brittle+shells+based+on+lr+nurb

    Finite Thermal Wave Propagation in a Half-Space Due to Variable Thermal Loading

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    The thermoelastic interaction for the dual-phase-lag (DP) heat conduction in a thermoelastic half space is studied in the light of two-temperature generalized thermoelasticity theory (2TT). The medium is assumed to be initially quiescent. Using Laplace transform, the fundamental equations are expressed in the form of a vector-matrix differential equation which is then solved by statespace approach. The obtained general solution is then applied to the mechanical loading and various types of thermal loading (the thermal shock and the ramp-type heating). The numerical inversion of the Laplace transforms are carried out by the method of Fourier series expansion technique. The numerical results are computed for copper like material. Significant dissimilarities between two models (the two-temperature Lord-Shulman (2TLS) and the two temperature Dual-phase-lag model (2TDP)) are shown graphically. Because of the short duration of the second sound effect, the small-time solutions are analyzed and the discontinuities that occur at the wave fronts are also discussed. The effects of two-temperature and ramping parameters are studied

    Least-squares variational principles and the finite element method: theory, formulations, and models for solid and fluid mechanics

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    We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures
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