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[EN] The growing number of scientific publications on topology optimization (TO) shows the great interest that this technique has generated in recent years. Among the different methodologies for TO, this article focuses on the well-known solid isotropic material penalization (SIMP) method, broadly used because of its simple formulation and efficiency. Even so, the SIMP method has certain drawbacks, namely: lack of precision in definition of the edges of the optimized geometry and final results strongly influenced by the discretization used for the finite element (FE) analyses. In this article, we propose a combination of techniques to limit the effect of these drawbacks and, thus, to improve the behavior of TO. All these techniques are based on the use of the Cartesian grid finite element method (cgFEM), an immersed boundary method whose Cartesian grid structure and hierarchical data structure makes it specially appropriate for TO. All the proposed techniques are framed under the concept of mesh refinement. First, we propose the use of two meshes, the FE analysis mesh, and a finer mesh for integration and evaluation of sensitivities, to improve the resolution of the final solution at a marginal computational cost. Then we propose two h-adaptive mesh refinement strategies. The first one will tend to refine the elements having intermediate density values and will have the effect of sharpening the definition of the edges of the optimized geometry. We will clearly show that if the accuracy of the FE analyses is not taken into account, stress constrained TO will generate solutions that, once manufactured, will not satisfy the constraints. Hence, we also propose an h-refinement strategy based on the estimation of the discretization error in energy norm., The authors gratefully acknowledge the financial support of Conselleria d'Educacio, Investigacio, Cultura i Esport (Generalitat Valenciana, project Prometeo/2016/007), Ministerio de Economia, Industria y Competitividad (project DPI2017-89816-R) and Ministerio de Educacion (FPU16/07121).

[EN] This article presents a Topology Optimization (TO) method developed for maximizing the acoustic attenuation of a perforated dissipative muffler in the targeted frequency range by optimally distributing the absorbent material within the chamber. The Finite Element Method (FEM) is applied to the wave equation formulated in terms of acoustic pressure (chamber) and velocity potential (central duct, due to the existence of thermal gradients and mean flow) in order to evaluate the acoustic performance of the noise control device in terms of Transmission Loss (TL). Sound propagation through the chamber fibrous material is modelled considering complex equivalent acoustic properties, which vary spatially not only as a function of temperature, but also as a function of the lling density, since non-homogeneous density distributions are considered. The acoustic coupling at the perforated duct is performed by introducing a coordinate-dependent equivalent impedance. The objective function to maximize is expressed as the mean TL in the targeted frequency range. The sensitivities of this function with respect to the filling density of each element in the chamber are evaluated following the standard adjoint method. The Method of Moving Asymptotes (MMA) is used to update the design variables at each iteration of the TO process, keeping the weight of absorbent material equal or lower than a given value, while maximizing attenuation. Additionally, several particular designs inferred from the topology optimization results are analyzed. For example, the sizing optimization of a number of rings is carried out simultaneously with the aforementioned TO process (density layout). A reactive chamber is added in order to evaluate the TL of a hybrid muffler and its shape optimization is also carried out simultaneously with the aforementioned TO. Results show an increase in the muffler's mean TL at target frequencies, for all cases under study, while the amount of absorbent material used is maintained or even reduced.

[EN] The superconvergent patch recovery technique with constraints (SPR-C) consists in improving the accuracy of the recovered stresses obtained with the original SPR technique by considering known information about the exact solution, like the internal equilibrium equation, the compatibility equation or the Neumann boundary conditions, during the recovery process. In this paper the SPR-C is extended to consider the equilibrium around the contact area when solving contact problems with the Cartesian grid Finite Element Method. In the proposed method, the Finite Element stress fields of both bodies in contact are considered during the recovery process and the equilibrium is enforced by means of the continuity of tractions along the contact surface., The authors would like to thank Generalitat Valenciana (PROMETEO/2016/007), the Spanish Ministerio de Economía, Industria y Competitividad (DPI2017-89816-R), the Spanish Ministerio de Ciencia, Innovación y Universidades (FPU17/03993), and Universitat Politècnica de València (FPI2015) for the financial support to this work.

A recovery-based error indicator developed to evaluate the quality of polygonal finite element approximations is presented in this paper. Generalisations of the finite element method to arbitrary polygonal meshes have been increasingly investigated in the last years, as they provide flexibility in meshing and improve solution accuracy. As any numerical approximation, they have an induced error which has to be accounted for in order to validate the approximate solution. Here, we propose a recovery type error measure based on a moving least squares fitting of the finite element stress field. The quality of the recovered field is improved by imposing equilibrium conditions and, for singular problems, splitting the stress field into smooth and singular parts. We assess the performance of the error indicator using three problems with exact solution, and we also compared the results with those obtained with standard finite element meshes based on simplexes. The results indicate good values for the local and global effectivities, similar to the values obtained for standard approximations, and are always within the recommended range.

[EN] Fictitious domain methods, like the Cartesian grid finite element method (cgFEM), are based on the use of unfitted meshes that must be intersected. This may yield to ill-conditioned systems of equations since the stiffness associated with a node could be small, thus poorly contributing to the energy of the problem. This issue complicates the use of iterative solvers for large problems. In this work, we present a new stabilization technique that, in the case of cgFEM, preserves the Cartesian structure of the mesh. The formulation consists in penalizing the free movement of those nodes by a smooth extension of the solution from the interior of the domain, through a postprocess of the solution via a displacement recovery technique. The numerical results show an improvement of the condition number and a decrease in the number of iterations of the iterative solver while preserving the problem accuracy., The authors wish to thank the Spanish "Ministerio de Economía y Competitividad," the "Generalitat Valenciana," and the "Universitat Politècnica de València" for their financial support received through the projects DPI2017-89816-R, Prometeo 2016/007 and the FPI2015 program, respectively.

[EN] This paper proposes a method of solving 3D large deformation frictional contact problems with the Cartesian Grid Finite Element Method. A stabilized augmented Lagrangian contact formulation is developed using a smooth stress field as stabilizing term, calculated by Zienckiewicz and Zhu Superconvergent Patch Recovery. The parametric definition of the CAD surfaces (usually NURBS) is considered in the definition of the contact kinematics in order to obtain an enhanced measure of the contact gap. The numerical examples show the performance of the method., The authors wish to thank the Spanish Ministerio de Economia y Competitividad the Generalitat Valenciana and the Universitat Politecnica de Valencia for their financial support received through the projects DPI2013-46317-R, Prometeo 2016/007 and the FPI2015 program.

[EN] We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of thematrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h- adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process., The authors wish to thank the Spanish Ministerio de Economia y Competitividad for the financial support received through the project DPI2013-46317-R and the FPI program (BES-2011-044080), and the Generalitat Valenciana through the project PROMETEO/2016/007.

[EN] Geometry plays a key role in contact and shape optimization problems in which the accurate representation of the exact geometry and the use of adaptive analysis techniques are crucial to obtaining accurate computationally-efficient Finite Element (FE) simulations. We propose a novel algorithm to generate 3D h-adaptive meshes for an Immersed Boundary Method (IBM) based on Cartesian grids and the so-called NEFEM (NURBS-Enhanced FE Method) integration techniques. To increase the accuracy of the results at the minimum computational cost we seek to keep the efficient Cartesian structure of the mesh during the whole analysis process while considering the exact boundary representation of domains given by NURBS or T-Splines. Within the framework of Cartesian grids, the two significant contributions of this paper are: (a) the methodology used for the mesh-geometry intersection, which represents a considerable challenge due to their independence; and (b) the robust procedure used to generate the integration subdomains that exactly represent the CAD model. The numerical examples given show the proper convergence of the method, its capacity to mesh complex 3D geometries and that Cartesian grid-based IBM can be considered a robust and reliable tool in terms of accuracy and computational cost., The authors wish to thank the Spanish Ministerio de Economia y Competitividad for the financial support received through Project DPI2013-46317-R and the FPI program (BES-2011-044080), also the Generalitat Valenciana for the assistance received through Project PROMETEO/2016/007.

Regarding acoustic attenuation, reactive silencers show a good behaviour at low to mid frequencies, whereas dissipative silencers, i.e., those containing absorbent material, are used in exhaust system applications due to their good performance in wide frequency bands, essentially at high frequencies. The combination of both types of configurations results in the hybrid silencer, which shows considerable attenuation in almost all the frequency range of interest. As an initial approach, plane wave models with low computation cost can be employed for acoustic attenuation prediction mainly at low frequencies. However, for more general problems with complex geometries and heterogeneous absorbent material it is necessary the use of multidimensional numerical analysis methods, such as the Finite Element Method (FEM). In the present work, the axisymmetric FEM model using a pressure formulation [1] is used in order to predict sound attenuation in dissipative and hybrid silencers in terms of transmission loss (TL). Next, a Topology Optimization (TO) gradient-based algorithm is implemented in order to get the maximum attenuation in a certain target frequency range, assigning a different filling density to each element of the central chamber, while keeping constant the total mass of absorbent material. The adjoint method is used during the TO process in order to speed up the computation of the objective function sensitivities with respect to the design variables (bulk density of each element or group of elements). This TO method is combined with the geometry optimization of the dissipative and reactive chambers. Thus, the implemented algorithm is capable of obtaining not only the optimal absorbent material layout but also the optimal geometry of both chambers (radius and length) which maximizes the silencer transmission loss. In order to combine both techniques, the Method of the Moving Asymptotes (MMA) described in reference [2] is implemented. The FEM solver algorithm is validated with experimental results available in the bibliography. Finally, it is demonstrated that the optimization process results in substantial improvement of the silencer sound attenuation at the target frequency range.

Regarding shape optimization of structural components, topology optimization has become one of the most popular methods to achieve significant reductions in mass and volume, while maintaining stiffness. The basic topology optimization algorithm considered in this paper and a heuristic updating scheme are described in Bendsoe [1].

[EN] The definition of the surface plays an important role in the solution of contact problems, as the evaluation of the contact force is based on the measure of the gap between the solids. In this work three different methods to define the surface are proposed for the solution of contact problems within the framework of the 3D Cartesian grid finite elementmethod. A stabilized formulation is used to solve the contact problem and details of the kinematic description for each surface definition are provided. The three methods are compared solving some numerical tests involving frictionless contact with finite and small deformations., The authors wish to thank the Spanish Ministerio de Economia y Competitividad the Generalitat Valenciana and the Universitat Politècnica de València for their financial support received through the projects DPI2017-89816-R, Prometeo 2016/007 and the FPI2015 program.

Summary This paper proposes a novel Immersed Boundary Method where the embedded domain is exactly described by using its Computer-Aided Design (CAD) boundary representation with Non-Uniform Rational B-Splines (NURBS) or T-splines. The common feature with other immersed methods is that the current approach substantially reduces the burden of mesh generation. In contrast, the exact boundary representation of the embedded domain allows to overcome the major drawback of existing immersed methods that is the inaccurate representation of the physical domain. A novel approach to perform the numerical integration in the region of the cut elements that is internal to the physical domain is presented and its accuracy and performance evaluated using numerical tests. The applicability, performance, and optimal convergence of the proposed methodology is assessed by using numerical examples in three dimensions. It is also shown that the accuracy of the proposed methodology is independent on the CAD technology used to describe the geometry of the embedded domain. Copyright © 2015 John Wiley & Sons, Ltd.

Background: There exists a relation between the presence and location of the micro-gap and the loss of peri-implant bone. Several authors have shown that the treatments based on the use of platform switching result in less peri-implant bone loss and an increased tissue stability. The purpose of this study was to analyse the effect of the platform switching on the distribution of stresses on the peri-implant bone using the finite element method. Material and Methods: A realistic 3D full-mandible finite element model representing cortical bone and trabecular bone was used to study the distribution of the stress on the bone induced by an implant of diameter 4.1 mm. Two abutments were modelled. The first one, of diameter 4.1 mm, was used in the reference model to represent a conventional implant. The second one, of diameter 3.2 mm, was used to represent the implant with platform switching. Both models were subjected to axial and oblique masticatory loads. Results: The analyses showed that, although no relevant differences can be found for the trabecular bone, the use of platform switching reduces the maximum stress level in the cortical bone by almost 36% with axial loads and by 40% with oblique loads. Conclusions: The full 3D Finite Element model, that can be used to investigate the influence of other parameters (implant diameter, connection, ...) on the biomechanical behaviour of the implant, showed that this stress reduction can be a biomechanical reasons to explain why the platform switching seems to reduce or eliminate crestal bone resorption after the prosthetic restoration

Osteoporosis, characterised by a decrease in bone mineral density, is responsible for millions of fractures. The reference test for this disease is bone densitometry. The diagnosis, based on statistical parameters obtained from an Xray image, evaluates the patient's risk of fracture. Their results are not accurate enough, since about 40% of patients with a low fracture risk diagnosis end up suffering from an osteoporotic fracture. Therefore, it seems reasonable to look for a technique that allows diagnoses of greater accuracy. The authors have developed an analysis technique, called the Cartesian Grid Finite Element Method (cgFEM), based on the use of Cartesian meshes, that allows the generation of Finite Element (FE) models directly from medical images. This method has been adapted to obtain an indication of the stress distribution on the femur that enriches the information available to evaluate the risk of osteoporotic failure. The multi-disciplinary collaboration of this work has allowed analysing densitometries from a large set of patients. The preliminary results obtained show that it will be able to improve the evaluation of the risk of fracture in femurs due to osteoporosis.

[EN] In patient specific biomedical simulation, the numerical model is usually created after cumbersome, time consuming procedures which often require highly specialized human work and a great amount of man-hours to be carried out. In order to make numerical simulation available for medical practice, it is of primary importance to reduce the cost associated to these procedures by making them automatic. In this paper a method for the automatic creation of Finite Element (FE) models from medical images is presented. This method is based on the use of a hierarchical structure of nested Cartesian grids in which the medical image is immersed. An efficient h-adaptive procedure conforms the FE model to the image characteristics by refining the mesh on the basis of the distribution of elastic properties associated to the pixel values. As a result, a problem with a reasonable number of degrees of freedom is obtained, skipping the geometry creation stage. All the image information is taken into account during the calculation of the element stiffness matrix, therefore it is straightforward to include the material heterogeneity in the simulation. The proposed method is an adapted version of the Cartesian grid Finite Element Method (cgFEM) for the FE analysis of objects defined by images. cgFEM is an immersed boundary method that uses h-adaptive Cartesian meshes non-conforming to the boundary of the object to be analysed. The proposed methodology, used together with the original geometry-based cgFEM, allows prosthesis geometries to be easily introduced in the model providing a useful tool for evaluating the effect of future implants in a preoperative framework. The potential of this kind of technology is presented by mean of an initial implementation in 2D and 3D for linear elasticity problems., With the support of the European Union Framework Programme (FP7) under grant agreement No. 289361 'Integrating Numerical Simulation and Geometric Design Technology (INSIST)', the Ministerio de Economia y Competitividad of Spain (DPI2010-20542) and the Generalitat Valenciana (PROMETEO/2016/007).

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model.

This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain's geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB., This work has been developed within the framework of research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia, and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. The authors also thank the support of the Framework Programme 7 Initial Training Network Funding under Grant no. 289361 "Integrating Numerical Simulation and Geometric Design Technology."

This is the peer reviewed version of the following article: Rodenas, J. J., Bugeda, G., Albelda, J. and Onate, E. (2011), On the need for the use of error-controlled finite element analyses in structural shape optimization processes. Int. J. Numer. Meth. Engng., 87: 1105–1126, which has been published in final form at http://dx.doi.org/10.1002/nme.3155. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Various numerical methods are used in engineering analysis, among which the finite element method (FEM) is the most prominent. This method relies heavily on a mesh structure, and the generation of this mesh generation accounts for about 80% of a typical analysis time for practical engineering problems. Various ideas have been proposed in the literature to avoid these problems by either simplifying this mesh generation process, or relaxing some of the constraints associated with the very presence of a mesh. This paper reviews recent advances in this direction by focusing on what the authors consider the most versatile and prominent approaches to overcome the mesh burden in computational science, namely: * Meshfree methods (MMs) somewhat reduce the constraints posed by the mesh by generalizing the concept of elements, simplifying. * Isogeometric analysis whose focus is to closely tie the geometry, i.e. computer aided design data to the analysis, e.g. FEM by using the same functions for both. If alterations to the geometry are required, the FE model is automatically modified, without changing the mesh, which greatly simplifies the design iterations. * The extended/generalized FEM (X/GFEM) where one of the aims is to increase the independence between the problem solved and the mesh. These methods are particularly attractive as they afford the modelling of crack propagation without remeshing. Inclusions and holes as well as the domain boundary can also be treated independently of the mesh. * Geometry independent methods, implicit meshing, immersed FEM, immersed boundary, fictitious domain/fixed grid FEM are alternatives to the FEM where the geometrically complex domain is implicitly embedded in a much simpler domain which is easily meshed using a regular, Cartesian grid. Geometry independent techniques have been used both in the context of finite volume method (FVM) and FEM. The literature on this type of technique goes back, according to, to the 1960s in the Russian language publication by Saulev and have subsequently been applied in different fields. Among other references in each discipline, we can mention applications in acoustics, fluid dynamics and fluid structure interaction, biomechanics, convection diffusion, optimization, etc. * Strain smoothing in finite elements allows decreasing the negative effects of mesh distortion. These methods have been used to solve a variety of mechanics problems including plates, shells, cracking, and three-dimensional viscoelastic deformation. The main goal of these methods is to rely on simplex meshes (tetrahedral, triangles) which are easier to generate than the more robust and accurate hexahedral meshes, and to do so without sacrificing accuracy. The relative merits and shortcomings of these methods are critically reviewed and a few examples are given for illustrative purposes. Other methods which are not covered here include the boundary element method and the scaled boundary finite element method.

[EN] This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by structural shape optimization algorithms over the behaviour of the algorithm. If the FE analyses are not accurate enough, the final solution will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper also proposes the use of the algorithm described in [1] in order to reduce the computational cost associated to the adaptive FE analysis of each geometrical configuration when evolutive optimization algorithms are used.¿, The authors have been sponsored by the Spanish Ministerio de Ciencia e Innovación through the projects DPI2008-05250 (first author) and DPI2007-66773-C02-01 (second and third authors). The second and third authors have also been sponsored by the Generalitat Valenciana and the Universidad Politécnica de Valencia.

[EN] The superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field sigma* from the finite element Solution sigma(fe). Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR-C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch sigma(p)* that locally satisfy the equations that should be satisfied by the exact Solution. As a result the evaluated expressions for sigma(p)* will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field sigma* and therefore the local effectivity of the ZZ error estimator., This work has been carried out within the framework of the research project DPI2004-07782-C02-02 of the Ministerio de Educación y Ciencia (Spain).The financial support of the Generalitat Valenciana and the Universidad Politécnica de Valencia is also acknowledged.

Several error estimators have been developed in order to control the accuracy of the Finite Element simulations. Most of these techniques can be divided in three main groups: (i) residual-based error estimators, (ii) dual techniques and (iii) recovery-based error estimators. In this work we will introduce the main ideas of the recovery-based error estimators.

The aim of this work is to propose a formulation to solve both small and large deformation contact problems using the finite element method. We consider both standard finite elements and the so-called immersed boundary elements. The method is derived from a stabilized Nitsche formulation. After introduction of a suitable Lagrange multiplier discretization the method can be simplified to obtain a modified perturbed Lagrangian formulation. The stabilizing term is iteratively computed using a smooth stress field. The method is simple to implement and the numerical results show that it is robust. The optimal convergence rate of the finite element solution can be achieved for linear elements., The authors wish to thank the Spanish Ministerio de Economia y Competitividad for the financial support received through the DPI Project DPI2013-46317-R.

In this paper we presenta methodology to performelasticity analyses of trabecular bone fromhuman wrist micro CT scans, with the final purpose of providing datawhich can be used, together with other measures, to develop new diagnostic tools to improve the characterisation of bone quality and its structural behaviour.This analysis is carried out by using a version of the Cartesian grid Finite Element Method (cgFEM) for medical images. As inimage based methods, which assign one element to each voxel, cgFEM takes advantage of the fact that both the mesh and the image have compatible Cartesian structures in order to create the FE modelfrom the image, in an automatic way, without any intermediate step. In contrast to the image based methods, cgFEM can keep the number of degrees of freedom relatively low, by using an efficient h-adaptive process to adapt the mesh to the heterogeneity of the image.H-adaptivity allows cgFEM to capture the borders of the bodies represented in the medical images without the necessity of creatingthem explicitly as in the usual geometry based methods. As a consequence,it does not require the cumbersome, often manual, process of creatingintermediate geometrical modelsfrom the bitmap which areusually one of the most expensive stages in patient specific simulations.Here, in particular, we show results of trabecular bone overall stiffness tensor evaluation obtained by using cgFEM in a homogenisation procedure.

This contribution proposes a 3Dimmersed boundary method,based on Cartesian grids,modified to improve the accuracy along the boundary and the efficiency during the resolution.Onthe one hand, the embedded domain will be definedby its CAD boundary representation with NURBS or T-Splines, instead of using an approximated faceted representation. The exact boundary representation of the embedded domain allows overcomingthe major drawbackof existing immersed methods that is the inaccurate representation of the physical domain.A novel approachto perform the numerical integration taking into consideration the exact representation of thephysical domain ispresented and its accuracy and performance evaluated using numerical tests.Onthe other hand, and now from the analysis point of view, we propose to use a Nested Domain Decomposition(NDD) reordering technique.When a direct solver is used to solve a system of equations a previous reordering is usually used in order to improve theperformance of the solver. Usually this reordering is obtained via an optimizationprocess which not always obtains the best reordering for the system ofequations. In our case, the NDDis based on the Cartesian gridstructure,intimately related to the mesh topology, thus providing an optimal reordering.This reordering will yield in a significantreduction of the computationalcost associated to the resolution of the system of equations.

[EN] Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recovery-based estimates is proposed. The idea is (1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, (2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and (3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications., The authors also thank the support of the Framework Programme 7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology".

The velocity field in shape sensitivity analysis is not uniquely defined although it must meet numerous theoretical and practical criteria. These practical criteria can be used to compare the existing velocity field computation methods which meet the theoretical criteria, but only in qualitative terms. When the FEM is used in design sensitivity analysis (DSA), due to the FE discretization error, the DSA errors will depend on the design velocity field considered. This paper presents a numerical methodology for quality evaluation of design velocity field computation methods in quantitative terms based on the analysis of the DSA discretization error. The sensitivity of the squared energy norm (χm = ∂∥u∥2/∂am, am being a design variable) has been taken as the magnitude to measure the error of the DSA. For h-refinements, the squared error in energy norm (∥e(u) ∥2) and the error in χm(e(χm)) are theoretically related by a constant which is independent of the refinement degree of the FE model. The quality of the design velocity field computation methods can therefore be assessed in terms of the stability of e(χm) /∥e(u) ∥2 in sequences of meshes. An example of the use of this methodology, where six design velocity field computation methods are compared, is presented. Copyright © 2004 John Wiley & Sons, Ltd.

[EN] Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool., This work was supported by the EPSRC Grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has received partial support from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. All authors also thank the partial support of the Framework Programme 7 Initial Training Network Funding under Grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology."

Patient specific Finite Element (FE) simulations are usually expensive. Time consuming geometry creation procedures are normally necessary to use standard FE meshing software, while direct pixel-based meshing techniques typically lead to a large number of degrees of freedom hence introducing a high computational cost. Image-based Cartesian grid Finite Element Method (image-based cgFEM) allows accurate models to be automatically obtained with a low computational cost without the necessity of defining geometries. In cgFEM the image is directly immersed into a Cartesian mesh which is h-adapted on the basis of the pixel value distribution. A hierarchical structure of nested Cartesian grids guarantees the efficiency of the process. In each element, the material elastic properties are heterogeneous, therefore a critical aspect of image-based cgFEM is the integration of the element stiffness matrices which homogenize the material elastic behavior at the element level. This paper compares accuracy and computational cost of different integration strategies: pixel direct integration schemes (Riemann sum and subdomain Gauss quadrature) and recovery based schemes (Least Squares fitting and Superconvergent Patch Recovery).

Nowadays, when it comes to generation of patient-specific Finite Element model, there are two main alternatives. On the one hand, it is possible to generate geometrical models through segmentation, whereupon FE models would be obtained using standard mesh generators. On the other hand, we can create a Cartesian grid of uniform hexahedra in which the elements fit each pixel/voxel perfectly. In both cases, geometries will take part during the analysis either as complete models, in the first case, or as auxiliary entities, to apply boundary conditions properly for instance, in the second case. In any case, once the geometrical entities have been obtained from the medical image, the efficient generation of an accurate Finite Element model for numerical simulation in not trivial. The aim of this paper is to propose an efficient integration strategy, using Cartesian meshes, of 3D geometries defined by parametric surfaces, i.e. NURBS, obtained from medical images.

[EN] An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a ¿smooth + singular¿ decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features., Stephane Bordas would like to thank the partial financial support of the Royal Academy of Engineering and of the Leverhulme Trust for his Senior Research Fellowship Towards the next generation surgical simulators as well as the financial support for Octavio A. Gonzalez-Estrada and Stephane Bordas from the UK Engineering Physical Science Research Council (EPSRC) under grant EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method. Stephane Bordas also thanks partial financial support of the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) and the FP7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology, INSIST". This work has been carried out within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia e Innovacion (Spain). The financial support from Universitat Politecnica de Valencia, PROMETEO/2012/023 and Generalitat Valenciana are also acknowledged.

[EN] Goal oriented error estimation and adaptive procedures are essential for the accurate and efficient evaluation of finite element numerical simulations that involve complex domains. By locally improving the approximation quality, for example, by using the extended finite element method (XFEM), we can solve expensive problems which could result intractable otherwise. Here, we present an error estimation technique for enriched finite element approximations that is based on an equilibrated recovery technique, which considers the stress intensity factor as the quantity of interest. The locally equilibrated superconvergent patch recovery is used to obtain enhanced stress fields for the primal and dual problems defined to evaluate the error estimate., This work was supported by the EPSRC grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has been carried out within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia e Innovacion (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged.

In this work, we present a new approach to solve linear elastic crack problems in plates using the so-called Proper Generalized Decomposition (PGD). In contrast to the standard FE method, the method enables to solve the crack problem in an efficient way by obtaining a single solution in which the Poisson's ratio v and the plate thickness B are non-fixed parameters. This permits to analyze the influence of v and B in the 3D solutions at roughly the cost of a series expansion of 2D analyses. Computationally, the PGD solution is less expensive than a full 3D standard FE analysis for typical discretizations used in practice to capture singularities in 3D crack problems. In order to verify the effectiveness of the proposed approach, the method is applied to cracked plates in mode I with a straight-through crack and a quarter-elliptical corner crack, validating J-integral results with different reference solutions., The authors thank the Ministerio de Ciencia y Tecnologia for the support received in the framework of the projects DPI2010-20990, DPI2010-20542 and to the Generalitat Valenciana, Programme PROMETEO 2012/023.

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. In the techniques based on the superconvergent patch recovery (SPR) the continuity of the recovered field is provided by the shape functions of the underlying mesh. We explore the capabilities of a recovery technique based on an MLS fitting, more flexible than SPR techniques as it directly provides continuous interpolated fields without relying on any FE mesh, to obtain estimates of the error in energy norm as an alternative to SPR. In the enhanced MLS proposed in the paper, boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results indicate the high accuracy of the proposed error., This work has been carried within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia y e Innovacion (Spain). The financial support of the Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) is also acknowledged. Support from the EPSRC grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method", the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) "RealTCut Towards real time multi-scale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery" and partial support of the Framework Programme 7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology (INSIST)" are acknowledged.

[ES] Este trabajo analiza la influencia del nivel de error de discretización de los análisis por elementos finitos (MEF) de las distintas configuraciones geométricas propuestas por los algoritmos de optimización de forma estructural sobre el comportamiento de dichos algoritmos. El artículo demuestra claramente que si los análisis mediante el MEF no presentan la suficiente precisión, la solución final proporcionada por el algoritmo de optimización no será ni el óptimo buscado ni cumplirá con las restricciones impuestas. Se demostrará la necesidad de la utilización de técnicas del MEF basadas en remallados adaptables. El artículo propone la combinación de dos estrategias para reducir el coste computacional relacionado con la utilización de dichos remallados adaptables junto con algoritmos evolutivos de optimización: (a) la utilización del algoritmo descrito por Bugeda et al. [1] que reduce el coste computacional asociado al análisis de cada configuración geométrica mediante el MEF con remallados adaptables y, (b) el incremento paulatino de la precisión requerida a cada análisis a fin de obtener una reducción sustancial del coste computacional en las etapas iniciales del proceso de optimización., Los autores primero, tercero y quinto agradecen el soporte económico recibido para el desarrollo de este artículo a través de los proyectos de investigación DPI2007-66773-C02-01 y DPI2010-20542 del Ministerio de Educación y Ciencia (Espana) y los fondos ˜ recibidos de la Generalitat Valenciana y de la Universitat Politècnica de València. El segundo autor agradece el soporte económico recibido para el desarrollo de este artículo a través del proyecto de investigación DPI2008-05250 del Ministerio de Educación y Ciencia (Espana).

Purpose - The purpose of this paper is to assess the effect of the statistical admissibility of the recovered solution and the ability of the recovered solution to represent the singular solution; also the accuracy, local and global effectivity of recovery-based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM). Design/methodology/approach - The authors study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution. Findings - Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statistically admissible recovered solutions. Originality/value - The paper shows that both extended recovery procedures and statistical admissibility are key to an accurate assessment of the quality of enriched finite element approximations. © Emerald Group Publishing Limited.

Purpose: This paper aims at assessing the effect of (1) the statical admissibility of the recovered solution; (2) the ability of the recovered solution to represent the singular solution; on the accuracy, local and global effectivity of recovery-based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM). Design/methodology/approach: We study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution. Findings: Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions. Originality/value: This work shows that both extended recovery procedures and statical admissibility are key to an accurate assessment of the quality of enriched finite element approximations., Comment: Submitted to Engineering Computations

[EN] This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM) The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework The upper bound property requires the recovered solution to be equilibrated and continuous The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch) Then a postprocess based on the partition of unity concept is used to obtain continuity The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests, This work has been carried out within the framework of the research projects DPI2007-66773-C02-01, DPI2007-66995-C03-02 and DPI2007-62395 of the Ministerio de Educacion y Ciencia (Spain). The financial support of the Generalitat Valenciana and the Universidad Politecnica de Valencia is also acknowledged.

Evolutive methods are a powerful and robust tool for the resolution of structural shape optimization problems. Nevertheless, the use of these methods requires the analysis of an important number of different designs. The computational cost and the quality of the solutions are very much dependent on the quality of the finite element meshes used for the analysis. One important ingredient of the numerical analysis is the strategy for the generation of a proper mesh for each design. Here we can see two types of strategies: 1. To adapt a single existing mesh to the geometries of all different designs. Some existing strategies allow adapting an existing mesh for very big modifications of the boundary shape preventing the elements from being too much distorted. Nevertheless, despite the fact that this type of strategies provides a valid mesh for each design, there is no control of the discretization error contained in the results of each analysis. 2. To perform a classical adaptive remeshing procedure for the analysis of each different design. Of course, this procedure ensures good quality results in the numerical analysis of each design, but the total computational cost grows significantly because each design is computed more than once. This work presents a new strategy that allows generating an adapted mesh for each design without the necessity of performing a full adaptive remeshing procedure. It is based on the use of sensitivity analysis of all magnitudes related with adaptive remeshing (location of nodes, error estimation,...) with respect to the design variables. This sensitivity analysis is performed only once using a geometry of reference and it is used to project the results of the corresponding analysis to all other designs to be analyzed. The projected information allows generating an appropriate adapted mesh for each new design usually in one shot, greatly reducing the computational cost compared with the described strategy 2. This method was developed and used in the context of the solution of shape optimization problems using deterministic methods.

This work analyzes the influence of the discretization error contained in the finite element (FE) analyses of each design configuration proposed by structural shape optimization algorithms over the behavior of the algorithm. If the FE analyses are not accurate enough, the final solution will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper also proposes the use of the algorithm described in the previous study conducted in order to reduce the computational cost associated to the adaptive FE analysis of each geometrical configuration when evolutive optimization algorithms are used.