624 research outputs found

    HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

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    This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

    Advances in contact algorithms and their application to tires

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    Currently used techniques for tire contact analysis are reviewed. Discussion focuses on the different techniques used in modeling frictional forces and the treatment of contact conditions. A status report is presented on a new computational strategy for the modeling and analysis of tires, including the solution of the contact problem. The key elements of the proposed strategy are: (1) use of semianalytic mixed finite elements in which the shell variables are represented by Fourier series in the circumferential direction and piecewise polynomials in the meridional direction; (2) use of perturbed Lagrangian formulation for the determination of the contact area and pressure; and (3) application of multilevel iterative procedures and reduction techniques to generate the response of the tire. Numerical results are presented to demonstrate the effectiveness of a proposed procedure for generating the tire response associated with different Fourier harmonics

    Multiscale Method for Elastic Wave Propagation in the Heterogeneous, Anisotropic Media

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    Seismic wave simulation in realistic Earth media with full wavefield methods is a fundamental task in geophysical studies. Conventional approaches such as the finite-difference method and the finite-element method solve the wave equation in geological models represented with discrete grids and elements. When the Earth model includes complex heterogeneities at multiple spatial scales, the simulation requires fine discretization and therefore a system with many degrees of freedom, which often exceeds current computational abilities. In this dissertation, I address this problem by proposing new multiscale methods for simulating elastic wave propagation based on previously developed algorithms for solving the elliptic partial differential equations and the acoustic wave equation. The fundamental motivation for developing the multiscale method is that it can solve the wave equation on a coarsely discretized mesh by incorporating the effects of fine-scale medium properties using so-called multiscale basis functions. This can greatly reduce computation time and degrees of freedom compared with conventional methods. I first derive a numerical homogenization method for arbitrarily heterogeneous, anisotropic media that utilizes the multiscale basis functions determined from a local linear elasticity equation to compute effective, anisotropic properties, and these equivalent elastic medium parameters can be used directly in existing elastic modeling algorithms. Then I extend the approach by constructing multiple basis functions using two types of appropriately defined local spectral linear elasticity problems. Given the eigenfunctions determined from local spectral problems, I develop a generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media in both continuous Galerkin (CG) and discontinuous Galerkin (DG) formulations. The advantage of the multiscale basis functions is they are model-dependent, unlike the predefined polynomial basis functions applied in conventional finite-element methods. For this reason, the GMsFEM can effectively capture the influence of fine-scale variation of the media. I present results for several numerical experiments to verify the effectiveness of both the numerical homogenization method and GMsFEM. These tests show that the effectiveness of the multiscale method relies on the appropriate choice of boundary conditions that are applied for the local problem in numerical homogenization method and on the selection of basis functions from a large set of eigenfunctions contained in local spectral problems in GMsFEM. I develop methods for solving both these problems, and the results confirm that the multiscale method can be powerful tool for providing accurate full wavefield solutions in heterogeneous, anisotropic media, yet with reduced computation time and degrees of freedom compared with conventional full wavefield modeling methods. Specially, I applied the DG-GMsFEM to the Marmousi-2 elastic model, and find that DG-GMsFEM can greatly reduce the computation time compared with continuous Galerkin (CG) FEM

    A Virtual Element Method for elastic and inelastic problems on polytope meshes

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    We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included

    Nonstandard Finite Element Methods

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    Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

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    Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

    SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

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    Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells
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