191 research outputs found

    A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

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    The theories for thick plates and beams, namely Reissner–Mindlin’s and Timoshenko’s theories, are well known to suffer numerical locking when approximated using the standard Galerkin finite element method for small thicknesses. This occurs when the same interpolations are used for displacement and rotations, reason for which stabilization becomes necessary. To overcome this problem, a Variational Multiscale stabilization method is analyzed in this paper. In this framework, two different approaches are presented: the Algebraic Sub-Grid Scale formulation and the Orthogonal Sub-Grid Scale formulation. Stability and convergence is proved for both approaches, explaining why the latter performs much better. Although the numerical examples show that the Algebraic Sub-Grid Scale approach is in some cases able to overcome the numerical locking, it is highly sensitive to stabilization parameters and presents difficulties to converge optimally with respect to the element size in the L 2 norm. In this regard, the Orthogonal Sub-Grid Scale approach, which considers the space of the sub-grid scales to be orthogonal to the finite element space, is shown to be stable and optimally convergent independently of the thickness of the solid. The final formulation is similar to approaches developed previously, thus justifying them in the frame of the Variational Multiscale concept.This work was supported by Vicerrectoría de Investigación, Chile, Desarrollo e Innovación (VRIDEI) of the Univeridad de Santiago de Chile, and the National Agency for Research and Development (ANID) Doctorado Becas Chile/2019 - 72200128 of the Government of Chile. R. Codina acknowledges the support received from the ICREA Acadèmia Research Program of the Catalan Government, Spain .Peer ReviewedPostprint (published version

    Dg And Hdg Methods For Curved Structures

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    We introduce and analyze discontinuous Galerkin methods for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking. We also prove that, when polynomials of degree kk are used, {\em all} the numerical traces superconverge with a rate of order h 2k+1. Based on the superconvergent phenomenon and we show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree k are used, the post-processed approximation converges with order 2k+1 in the L2-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order k+1 only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small. Since the DG methods suffer from too many degree of freedoms we introduce and analyze a class of hybridizable discontinuous Galerkin (HDG) methods for Naghdi arches. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to approximations to the transverse and tangential displacement and bending moment at the element boundaries. The error analysis of the methods is based on the use of a projection especially designed to fit the structure of the numerical traces of the method. This property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Consequently, we prove that HDG methods have the same result as DG methods. At the end of the thesis, we talk a little bit of shell problems

    Meshless methods for shear-deformable beams and plates based on mixed weak forms

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    Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to overcome the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable first-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.Open Acces

    Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

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    There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering

    Stable numerical methodology for variational inequalities with application in quantitative finance and computational mechanics

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    Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed
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