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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 Reviewed, Postprint (published version)

This paper reports on a laboratory experiment to study the effect of vehicle net charge on theinception of a positive leader from an aircraft exposed to high atmospheric electric fields. The experimentmodels the first stage of aircraft-triggered lightning in which a positive leader typically develops from thevehicle and is shortly afterwards followed by a negative leader. This mechanism of lightning initiationamounts to around 90% of strikes to aircraft. Aircraft can acquire net charge levels of the order of amillicoulomb from a number of sources including corona emission, charged particles in the engineexhaust, and charge transfer by collisions with particles in the atmosphere. In addition, aircraft couldpotentially be artificially charged through controlled charge emission from the surface. Experiments wereperformed on a model aircraft with a 1m wingspan, which was suspended between two parallel electrodesin a 1.45m gap with voltage difference of a few hundred kilovolts applied across it. In this configuration,it is found that the breakdown field can vary by as much as 30% for the range of charging levels tested. Theexperimental results show agreement with an electrostatic model of leader initiation from aircraft, and themodel indicates that the effect can be substantially stronger if additional negative charge is added to theaircraft. The results from this work suggest that flying uncharged is not optimal in terms of lightningavoidance and open up the possibility of developing risk-reduction strategies based on net charge control, Peer Reviewed, Postprint (author's final draft)

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/abstract, A 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 Reviewed, Postprint (author's final draft)

We present a robust method for generating high-order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high-order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high-order finite element analyses., Peer Reviewed, Postprint (author’s final draft)

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 Reviewed, Postprint (author’s final draft)

To evaluate the computational performance of high-order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high-order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two-dimensional and three-dimensional domains for simplices and parallelotopes. Moreover, both element-wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high-order methods are more efficient than linear ones., Peer Reviewed, Postprint (author's final draft)

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 technique, Peer Reviewed, Postprint (published version)

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 technique., Peer Reviewed, Postprint (published version)

A procedure to quantify the distortion (quality) of a high-order mesh composed by 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 by valid and high-quality high-order tetrahedral elements. To this end, we describe a method to simultaneously smooth and untangle high-order tetrahedral meshes by minimizing the proposed mesh distortion. Finally, we present several results to illustrate the two main applications of the proposed technique., Peer Reviewed, Postprint (published version)

We propose an a posteriori approach for generating curved meshes for viscous flow simulations composed by high-order tetrahedra. The proposed approach is performed in the following three steps: (1) generate a linear tetrahedral mesh for inviscid flow; (2) insert a boundary layer mesh, composed by linear tetrahedra, on the viscous part; and (3) convert the linear tetrahedral mesh to a curved and high-order mesh for viscous flow. This approach provides high-order tetrahedral meshes with boundary layer parts that are composed by elements that are: curved, valid, and of any interpolation degree., Peer Reviewed, Postprint (author's final draft)

The talk covered several issues motivated by a practical engineering wave propagation problem: real-time evaluation of wave agitation in harbors. The first part, presented the application of a reduced order model in the framework of a Helmholtz equation with non-constant coefficients in an unbounded domain. This problem requires large numbers of degrees of freedom (ndof) because relatively high frequencies with small (compared with the domain size) geometric features must be considered., Peer Reviewed, Postprint (published version)

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high-order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology., Peer Reviewed, Postprint (published version)

In this paper we study some recent formulations for the computation of upper bounds in limit analysis. We show that a previous formulation presented by the authors does not guarantee the strictness of the upper bound, nor does it provide a velocity field that satisfies the normality rule everywhere. We show that these deficiencies are related to the quadrature employed for the evaluation of the dissipation power. We derive a formulation that furnishes a strict upper bound of the load factor, which in fact coincides with a formulation reported in the literature. From the analysis of these formulations we propose a post-process which consists in computing exactly the dissipation power for the optimum upper bound velocity field. This post-process may further reduce the strict upper bound of the load factor in particular situations. Finally, we also determine the quadratures that must be used in the elemental and edge gap contributions so that they are always positive and their addition equals the global bound gap., Peer Reviewed, Postprint (published version)

This is the accepted version of the following article: [Huerta, A., Casoni, E. and Peraire, J. (2012), A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int. J. Numer. Meth. Fluids, 69: 1614–1632. doi:10.1002/fld.2654], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.2654/full, This article presents a novel shock-capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high-order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high-order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach., Peer Reviewed, Postprint (author's final draft)

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 Reviewed, Postprint (author's final draft)

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.

The iterative solution of the large systems of equations that result from discontinuous Galerkin (DG) discretizations require the ability to carry out fast matrix-vector products. DG matrices have a sparse block structure with a constant number of non-zero equal-sized non-overlapping blocks per row. General-purpose sparse matrix-vector product algorithms are not designed to exploit the speci c structure of the DG matrices and, as a consequence, result in sub-optimal performance. To address this issue, we propose a sparse matrix-vector product for DG discretizations based on a dense tensor contraction. A GPU implementation of the proposed algorithm for a hybridizable discontinuous Galerkin (HDG) method is tested on the NVIDIA GEFORCE GTX 285. The results show that the tensor contraction performs at about 20 to 25 GFLOP/s in double precision with a sustained efficiency of more than 40% (60 GBytes/s) of the peak memory bandwidth (160 GBytes/s). Moreover, for HDG matrices in double precision, the proposed method is 2 times faster than the general sparse matrix-vector products provided by the GPU library CUSPARSE and about 30 times faster than MATLAB running on a CPU., Peer Reviewed, Postprint (published version)

An interior penalty method and a compact discontinuous Galerkin method are proposed and compared for the solution of the steady incompressible Navier–Stokes equations. Both compact formulations can be easily applied using high-order piecewise divergence-free approximations, leading to two uncoupled problems: one associated with velocity and hybrid pressure, and the other one only concerned with the computation of pressures in the elements interior. Numerical examples compare the efficiency and the accuracy of both proposed methods., Peer Reviewed, Postprint (published version)

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 Reviewed

This is the pre-peer reviewed version of the following article: Muñoz, José J. [et al.]. Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading. "International journal for numerical methods in engineering", Agost 2008, vol. 77, núm. 4, p. 471-501., which has been published in final form at http://www3.interscience.wiley.com/journal/121370765/abstract, Peer Reviewed, Postprint (author’s final draft)

Postprint (published version)

We present an a posteriori method for computing rigorous upper and lower bounds for the J-integral in two-dimensional linear elasticity. The J-integral, which is typically expressed as a contour integral, is recast as a quadratic continuous functional of the displacement involving only area integration. By expanding the quadratic output about an approximate finite element solution, the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error. The quadratic component is bounded by the energy norm of the error scaled by a continuity constant, which is determined explicitly. The linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution, and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint. Upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields. The method is illustrated with two fracture problems in plane strain elasticity. An important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations., Peer Reviewed, Postprint (author’s final draft)

We present a method for the computation of upper and lower bounds for linear-functional outputs of the exact solutions to the two dimensional elasticity equations. The method can be regarded as a generalization of the well known complementary energy principle. The desired output is cast as the supremum of a quadratic-linear convex functional over an infinite dimensional domain. Using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary, or dual, problem. In order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced. First, the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle. The Lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation. Second, the solution of the adjoint problem is approximated over the triangulation using a standard Galerkin finite element approach. A lower bound for the output of interest is computed by repeating the process for the negative of the output. Reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output. The method can be easily generalized to three dimensions. However, a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions. The computed bound gaps are found to converge optimally, that is, at the same rate as the finite element approximation. An attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds. Using this data set and a simple algorithm, the correctness of the computed bounds can be established without recourse to the original code used to compute them. In the present paper, only c, Peer Reviewed, Postprint (author’s final draft)

El análisis límite nace de la necesidad de complementar la teoría de elasticidad lineal para conseguir resultados más realistas con respecto a la evolución hacia el colapso de un cuerpo. El análisis límite es una teoría ampliamente estudiada y empleada en diversos campos ingenieriles como el de la mecánica de suelos o de sólidos rígidos. En nuestro caso nos centraremos en el estudio de suelos a partir del modelo de rotura de Mohr-Coulomb. La teoría de estado límite se basa en el cálculo y modelado del colapso de un determinado sistema a partir de una distribución de carga uniforme y estática.

En la actualidad, el cálculo en estado límite etá cogiendo cada vez más relevancia por la necesidad de obtener resultados más realistas. Surge la necesidad de complementar los resultados en elasticidad lineal con modelos de cálculo más complejos, que permitan, por ejemplo, una plastificación de parte del sólido antes de la rotura, acercándose mucho más al comportamiento real del sólido. Trabajamos con un sólido rígido, perfectamente plástico y sometido a una distribución fija de carga. El problema básico que aborda el análisis en estado límite es la consecución del mínimo múltiplo de esta distribución de carga que provoque la rotura del sólido. Estas necesidades ingenieriles combinadas con los grandes avances de los ordenadores y el surgimiento de nuevos, eficientes y robustos algoritmos de optimización no lineal hacen del problema de estado límite un campo asequible, por lo que se encuentra en proceso de pleno estudio

We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact weak solution of the infinite-dimensional continuum problem with piecewise-polynomial forcing. The method results from exploiting the Lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem. Localization is achieved by Lagrangian relaxation and the bounds are computed by appeal to a local dual problem. The proposed method computes approximate Lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers. The computed bounds hold uniformly for any level of refinement, and in the asymptotic convergence regime of the finite element method, the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization. The elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators, as we demonstrate with a numerical example., Peer Reviewed, Postprint (published version)

A finite element a posteriori method to compute rigorous inexpensive upper and lower bounds for engineering outputs of interest, in the field of large strain elasticity, is presented. Based on a general Lagrangian formulation, the bound computation requires global calculations on a finite element coarse mesh and the solution of local decoupled Neumann problems over each element of previous mesh. A new procedure to solve local problems is introduced, whereby bounds are computed locally over the continua for the exact solution in a completely analytical manner. The resulting method shows a considerable gain in computational performance and, besides being robust, can easily be incorporated into existing finite element codes without major modifications.

We present an implicit a posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed relies on the existence of a potential energy functional whose local minima, over a space of suitably chosen continuous functions, corresponds to the problem solution. The output of interest is cast as a constrained minimization problem over an enlarged discontinuous finite element space. A Lagrangian is formed were the multipliers are an adjoint solution, which enforces equilibrium, and hybrid fluxes, which constrain the solution to be continuous. By computing approximate values for the multipliers on a coarse mesh, strict upper and lower bounds for the output of interest on a suitably refined mesh, are obtained. This requires a minimization over a discontinuous space, which can be carried out locally at low cost. The computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. The method is demonstrated with two applications involving large strain plane stress incompressible neo-Hookean hyperelasticity., Peer Reviewed, Postprint (author’s final draft)

A Neumann subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, constant-free upper and lower bounds for non-linear outputs of the Helmholtz equation in two-dimensional exterior domains is presented. The bound procedure is firstly formulated, with particular emphasis on appropriate extension to complex-valued equations; then, illustrative numerical examples for outputs, such as the intensity of the scattered wave over a small segment of the domain boundary, are provided., Peer Reviewed, Postprint (author’s final draft)

Postprint (published version)