1,543 research outputs found

    A spectral collocation technique based on integrated Chebyshev polynomials for biharmonic problems in irregular domains

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    In this paper, an integral collocation approach based on Chebyshev polynomials for numerically solving biharmonic equations [N. Mai-Duy, R.I. Tanner, A spectral collocation method based on integrated Chebyshev polynomials for biharmonic boundary-value problems, J. Comput. Appl. Math. 201 (1) (2007) 30–47] is further developed for the case of irregularly shaped domains. The problem domain is embedded in a domain of regular shape, which facilitates the use of tensor product grids. Two relevant important issues, namely the description of the boundary of the domain on a tensor product grid and the imposition of double boundary conditions, are handled effectively by means of integration constants. Several schemes of the integral collocation formulation are proposed, and their performances are numerically investigated through the interpolation of a function and the solution of 1D and 2D biharmonic problems. Results obtained show that they yield spectral accuracy

    Spectral methods for exterior elliptic problems

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    Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases

    Black hole evolution by spectral methods

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    Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation (PSC) method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein's equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast to finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the extension of this method to three spatial dimensions.Comment: 20 pages, 17 figures, submitted to PR

    The strain-based beam finite elements in multibody dynamics

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    We present a strain-based finite-element formulation for the dynamic analysis of flexible elastic planar multibody systems, composed of planar beams. We consider finite displacements, rotations and strains. The discrete dynamic equations of motion are obtained by the collocation method. The strains are the basic interpolated variables, which makes the formulation different from other formulations. The further speciality of the formulation is the strong satisfaction of the cross-sectional constitutive conditions at collocation points. In order to avoid the nested integrations, a special algorithm for the numerical integration over the length of the finite element is proposed. The midpoint scheme is used for the time integration. The performance of the formulation is illustrated via numerical examples, including a stiff multibody system. (c) 2007 Elsevier Ltd. All rights reserved

    An introduction to programming Physics-Informed Neural Network-based computational solid mechanics

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    Physics-informed neural network (PINN) has recently gained increasing interest in computational mechanics. In this work, we present a detailed introduction to programming PINN-based computational solid mechanics. Besides, two prevailingly used physics-informed loss functions for PINN-based computational solid mechanics are summarised. Moreover, numerical examples ranging from 1D to 3D solid problems are presented to show the performance of PINN-based computational solid mechanics. The programs are built via Python coding language and TensorFlow library with step-by-step explanations. It is worth highlighting that PINN-based computational mechanics is easy to implement and can be extended for more challenging applications. This work aims to help the researchers who are interested in the PINN-based solid mechanics solver to have a clear insight into this emerging area. The programs for all the numerical examples presented in this work are available on https://github.com/JinshuaiBai/PINN_Comp_Mech.Comment: 32 pages, 20 figures are include in this manuscrip

    NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

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    Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p 648), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace\u2013Beltrami and Allen\u2013Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA)

    Bezmrežni postupak kao alternativa metodi konačnih elemenata pri numeričkom modeliranju u mehanici čvrstih tijela

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    Meshless approaches enable discretizations of a computational model only by a set of nodes, which do not need to be connected to elements. This paper presents the meshless local Petrov-Galerkin method, which belongs to truly meshless approaches, as it does not require any kind of mesh or background cells for either interpolation or integration. Full displacement and mixed formulations are presented. The full displacement approach is used for the solution of a three-dimensional elasto-static problem, while the mixed approach is applied for the modeling of deformation responses of shell-like structures. The modeling of material discontinuities is performed by the mixed meshless local Petrov-Galerkin approach by employing the collocation method. The efficiency and accuracy of all the presented methods are tested and compared with finite element formulations in numerical examples. It is demonstrated that the meshless approaches may be considered an alternative to the well-known finite element method regarding certain problems.Bezmrežni postupak omogućuje diskretizaciju računalnog modela samo s čvorovima koji ne trebaju biti povezani s konačnim elementima. U ovom članku prikazuje se lokalna Petrov-Galerkinova formulacija koja u potpunosti spada u bezmrežne postupke jer ne zahtijeva niti jednu vrstu mreže ili tzv. popratnih ćelija, kako za interpolaciju tako i za integraciju. Prikazane su puna formulacija pomaka i mješovita formulacija. Postupak punog pomaka se primjenjuje za rješavanje trodimenzijskog elasto-statičkog problema, dok se mješovita formulacija koristi za modeliranje deformiranja ljuskastih konstrukcija. Modeliranje materijalnog diskontinuiteta se provodi mješovitim bezmrežnim lokalnim Petrov-Galerkinovim postupkom koji uključuje kolokacijsku metodu. U numeričkim primjerima, učinkovitost i točnost svih prikazanih metoda je testirana i uspoređena s formulacijama metode konačnih elemenata. Pokazano je da se bezmrežni postupci mogu smatrati alternativom za dobro poznatu metodu konačnih elemenata
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