Un método de captura de choques basado en las funciones de forma para Galerkin discontinuo de alto orden

En este artículo se presenta un método de alto orden de Galerkin discontinuo para problemas de flujo com- presible, en los cuales es muy frecuente la aparición de choques. La estabilización se introduce mediante una nueva base de funciones. Esta base tiene la flexibilidad de variar localmente (en cada elemento) entre un espacio de funciones polinómicas continuas o un espacio de funciones polinómicas a trozos. Así, el método propuesto proporciona un puente entre los métodos estándar de alto orden de Galerkin disconti- nuo y los clásicos métodos de volúmenes finitos, manteniendo la localidad y compacidad del esquema. La variación de las funciones de la base se define automáticamente en función de la regularidad de la solución y la estabilización se introduce mediante el operador salto, estándar en los métodos Galerkin disconti- nuo. A diferencia de los clásicos métodos de limitadores de pendiente, la estrategia que se presenta es muy local y robusta, y es aplicable a cualquier orden de aproximación. Además, el método propuesto no requiere refinamiento adaptativo de la malla ni restricción del esquema de integración temporal. Se consideran varias aplicaciones de las ecuaciones de Euler que demuestran la validez y efectividad del método, especialmente para altos órdenes de aproximación.Peer ReviewedPostprint (author's final draft

A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.Peer ReviewedPostprint (author's final draft

A critical comparison of two discontinuous galerkin methods for the navier-stokes equations using solenoidal aproximations

This paper compares two methods to solve incompressible problems, in particular the Navier-Stokes equations, using a discontinuous polynomial interpolation that is exactly divergence-free in each element. The first method is an Interior Penalty Method, whereas the second method follows the Compact Discontinuous Galerkin approach for the diffusive part of the problem. In both cases the Navier-Stokes equations are then solved using a fractional-step method, using an implicit method for the diffusion part and a semiimplicit method for the convection. Numerical examples compare the efficiency and the accuracy of the two proposed methods.Peer Reviewe

Strict upper and lower boundswith adaptive remeshing in limit state analysis

By writing the limit state analysis as an optimisation problem, and after resorting to suitable discretisations of the stress and velocity field, we compute strict bounds of the load factor. The optimisation problem is posed as a Second Order Conic Program (SOCP), which can be solved very efficiently using specific algorithms for conic programming. Eventually, the optimum stress and velocity fields of the lower and upper bound problem are used to construct an error measure (elemental gap) employed in an adaptive remeshing strategy. This technique is combined with an additional adaptive nodal remeshing that is able to reproduce fan-type mesh patterns around points with discontinuous surface loads. We paticularise the resulting formulation for twodimensional problems in plane strain, with VonMises andMohr-Coulomb plasticity. We demonstrate the effetiveness of the method with a set of numerical examples extracted from the literature

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: Application to the hybridizable discontinuous Galerkin method

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced allowing to derive alternative guaranteed bounds from nearly-arbitrary flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.Peer ReviewedPostprint (published version

Bounds and adaptivity for 3D limit analysis

In the present paper we compute upper and lower bounds for limit analysis in two and three dimensions. From the solution of the discretised upper and lower bound problems, and from the optimum displacement rate and stress fields, we compute an error estimate defined at the body elements and at their boundaries, which are applied in an adaptive remeshing strategy. In order to reduce the computational cost in 3D limit analysis, the tightness of the upper bound is relaxed and its computation avoided. Instead, the results of the lower bound are used to estimate elemental and edge errors. The theory has been implemented for Von Mises materials, and applied to two- and three-dimensions examples.Peer Reviewe

Validation and generation of high-order meshes on parameterized surfaces

We present a technique to extend Jacobian-based distortion (quality) measures for planar triangles to high-order isoparametric elements of any interpolation degree on CAD parameterized surfaces. The resulting distortion (quality) measures are expressed in terms of the parametric coordinates of the nodes. These extended distortion (quality) measures can be used to check the quality and validity of a high-order surface mesh. We also apply them to simultaneously smooth and untangle high-order surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes. Thus, the nodes always lie on the surface. Finally, we include several examples to illustrate the application of the proposed techniquePeer ReviewedPostprint (published version

Distortion and quality measures for validating and generating high-order tetrahedral meshes

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a -continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.Peer ReviewedPostprint (author’s final draft

A discontinuous galerkin formulation for solid dynamics

Postprint (published version

Optimal collapse simulator for three-dimensional frames

In this work a limit analysis for 3D structures software package is presented. The goal is to obtain for a certain structure the load factor λ that applied to the external loads induces collapse to the structure. The static theorem of limit analysis is the theoretical basis for the Structural Collapse Simulator (SCS), that is finding a stress distribution in equilibrium that does not violate yield criteria anywhere. The limit analysis is developed and written as a Linear Programming Problem, which consists of the maximization of the collapse load factor subject to equilibrium and yield criteria. The Structural Collapse Simulator has been applied to several types of structures to assess its capabilities on world applications