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El Halabi F.,Aragon Institute of Engineering Research | El Halabi F.,Research Center Biomedica en Red en Bioingenieria Biomateriales y Nanomedicina | Gonzalez D.,Aragon Institute of Engineering Research | Chico-Roca A.,Ascamm Technology Center | And 3 more authors.
Computer Methods in Applied Mechanics and Engineering | Year: 2013

Multidimensional problems are found in almost every scientific field. In particular, this is standard in parametric design, inverse analysis, in optimization and in metamodeling analysis, whereby statistical or deterministic approximations of multiparametric solutions are built from the results of experimental campaigns or computer simulations. Multidimensional fitting or approximation of response functions exponentially increase their complexity and computational cost with the number of dimensions responding to the well-known " curse of dimensionality" To reduce the order of complexity and make the solution of many-parameter problems affordable, we propose to combine the model reduction technique known as Proper Generalized Decomposition (PGD) and the response surface (RSM) methodology. As a proof of concept we have used a simple fitting procedure as it is least squares, although other more complex fitting procedures may be easily included. The combined algorithm is presented and its capabilities discussed in a set of multidimensional examples. The data samples to be fit in each of these examples are obtained by means of appropriate discretizing the interval of interest for each design factor and then generating the output values (exact or stochastically modified) by means of virtual experiments. To have an idea of the number of discretization points needed along each direction, the Taguchi's design of experiments is used. The obtained results show evident improvements in computer time and accuracy when compared to other traditional multiparametric approximation techniques based on polynomial functions and the standard Levenberg-Marquardt algorithm, especially in problems with non-linear behavior and with high number of design parameters. Further comparison was done with the PARAFAC-ALS algorithm. The combination of PGD and RSM seems to be an appealing tool for accurately modelling multiparametric problems in almost real-time if a sufficient set of previous off-line results is available despite the intrinsic complexity of the problem and of the number of parameters involved. © 2012 Elsevier B.V. Source


El Halabi F.,Aragon Institute of Engineering Research | El Halabi F.,CIBER ISCIII | Gonzalez D.,Aragon Institute of Engineering Research | Chico A.,Ascamm Technology Center | And 3 more authors.
Computer Methods in Applied Mechanics and Engineering | Year: 2013

In recent years, a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials, where the complexity of solving numerically all microscale details is not feasible. In this paper we present a multiscale framework based on the decomposition of the displacement field into coarse (macro) and fine (micro) scales, earlier proposed in the Variational Multiscale approach or the hp-d method. The novelty of this work lies in solving the microscale step, where a multidimensional parametrized model of a generic RVE is solved, by means of the multidimensional model reduction technique, named as proper generalized decomposition (PGD). As result of this previous and off-line step, the displacement field over the RVE is obtained by simple algebraic operations for any combination of parameters (boundary condition, material properties, loads, etc.), allowing a significant reduction in computational cost when solving the macro scale problem. For problems where the detailed structure of the localized displacement, strain and stress fields are of interest, recovery of the fine scale components can be performed immediately. The basic concepts and several 1-D and 2-D linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving. © 2013 Elsevier B.V. Source


El Halabi F.,Aragon Institute of Engineering Research | El Halabi F.,Ascamm Technology Center | El Halabi F.,CIBER ISCIII | Rodriguez J.F.,Aragon Institute of Engineering Research | And 4 more authors.
Journal of the Mechanical Behavior of Biomedical Materials | Year: 2011

Cranial implants have experienced a significant evolution in the last decade in different aspects such as materials, method of fixation, and the structure. In addition, patient-specific cranial implants have recently been started to be developed. To achieve this objective, efficient mechanical characterization and numerical modeling of the implant are required to guarantee its functionality on each patient as well as to facilitate further developments. In this work, mechanical characterization and numerical models have been performed for patient-specific Polyaryletherketone (PEEK) scaffold cranial implants. Mechanical characterization has been performed at the scaffold and the whole implant levels under displacement control tests. Two different implant designs for the same patient but with different scaffold structure were experimentally characterized, and finite element models of the implants were developed within the framework of linear elasticity. Two types of finite element models were developed: a detailed finite element model with the actual scaffold geometry, and a solid shell-like model with effective material properties. These effective material properties were obtained by means of the Asymptotic Expansion Homogenization (AEH) theory which accounts for the periodicity of the underlying structure of the material. Experimental results showed a linear response of the material and the implant up to failure, therefore supporting the use of linear elastic models for simulation. Numerical models showed excellent agreement with experiments regarding load-displacement response. Models also showed a very consistent behavior with regard to the location and the value of the maximum principal stress in the implant when subjected to the maximum load of the experiments. The two numerical models were compared. The homogenized model gave results that were very close to those obtained with the detailed model, while reducing the number of degrees of freedom by 90%, and therefore the overall computational burden. The results showed that the models are able to reproduce experimental results conducted on actual implants, offering a valid alternative to be used in the design of customized cranial implants with a scaffold structure. © 2011 Elsevier Ltd. Source

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