Proper Generalized Decomposition solutions within a Domain Decomposition strategy

"This is the peer reviewed version of the following article: Huerta, Antonio, Enrique Nadal, and Francisco Chinesta. 2018. Proper Generalized Decomposition Solutions within a Domain Decomposition Strategy. International Journal for Numerical Methods in Engineering 113 (13). Wiley: 1972 94. doi:10.1002/nme.5729, which has been published in final form at https://doi.org/10.1002/nme.5729. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane-out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.European Commission, Grant/Award Number: MSCA ITN-ETN 675919; ESI group, Grant/Award Number: ENSAM ESI Chair; Spanish Ministry of Economy and Competitiveness, Grant/Award Number: DPI2017-85139-C2-2-R; Generalitat de Catalunya, Grant/Award Number: 2014SGR1471Huerta, A.; Nadal, E.; Chinesta Soria, FJ. (2018). Proper Generalized Decomposition solutions within a Domain Decomposition strategy. International Journal for Numerical Methods in Engineering. 113(13):1972-1994. https://doi.org/10.1002/nme.5729S1972199411313Ammar, A., Mokdad, B., Chinesta, F., & Keunings, R. (2006). A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics, 139(3), 153-176. doi:10.1016/j.jnnfm.2006.07.007Chinesta, F., Leygue, A., Bordeu, F., Aguado, J. V., Cueto, E., Gonzalez, D., … Huerta, A. (2013). PGD-Based Computational Vademecum for Efficient Design, Optimization and Control. Archives of Computational Methods in Engineering, 20(1), 31-59. doi:10.1007/s11831-013-9080-xChinesta, F., Cueto, E., & Huerta, A. (2014). PGD for solving multidimensional and parametric models. CISM International Centre for Mechanical Sciences, 27-89. doi:10.1007/978-3-7091-1794-1_2Chinesta, F., Keunings, R., & Leygue, A. (2014). The Proper Generalized Decomposition for Advanced Numerical Simulations. SpringerBriefs in Applied Sciences and Technology. doi:10.1007/978-3-319-02865-1González, D., Ammar, A., Chinesta, F., & Cueto, E. (2009). Recent advances on the use of separated representations. International Journal for Numerical Methods in Engineering, n/a-n/a. doi:10.1002/nme.2710Ghnatios, C., Chinesta, F., & Binetruy, C. (2013). 3D Modeling of squeeze flows occurring in composite laminates. International Journal of Material Forming, 8(1), 73-83. doi:10.1007/s12289-013-1149-4Bognet, B., Bordeu, F., Chinesta, F., Leygue, A., & Poitou, A. (2012). Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity. Computer Methods in Applied Mechanics and Engineering, 201-204, 1-12. doi:10.1016/j.cma.2011.08.025Bognet, B., Leygue, A., & Chinesta, F. (2014). Separated representations of 3D elastic solutions in shell geometries. Advanced Modeling and Simulation in Engineering Sciences, 1(1), 4. doi:10.1186/2213-7467-1-4Ibáñez, R., Abisset-Chavanne, E., Chinesta, F., & Huerta, A. (2016). Simulating squeeze flows in multiaxial laminates: towards fully 3D mixed formulations. International Journal of Material Forming, 10(5), 653-669. doi:10.1007/s12289-016-1309-4Toselli, A., & Widlund, O. B. (2005). Domain Decomposition Methods — Algorithms and Theory. Springer Series in Computational Mathematics. doi:10.1007/b137868Dolean, V., Jolivet, P., & Nataf, F. (2015). An Introduction to Domain Decomposition Methods. doi:10.1137/1.9781611974065Nazeer, S. M., Bordeu, F., Leygue, A., & Chinesta, F. (2014). Arlequin based PGD domain decomposition. Computational Mechanics, 54(5), 1175-1190. doi:10.1007/s00466-014-1048-7Krause, R. H., & Wohlmuth, B. I. (2002). A Dirichlet-Neumann type algorithm for contact problems with friction. Computing and Visualization in Science, 5(3), 139-148. doi:10.1007/s00791-002-0096-2Farhat, C., & Roux, F.-X. (1991). A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32(6), 1205-1227. doi:10.1002/nme.1620320604Nitsche, J. (1971). Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36(1), 9-15. doi:10.1007/bf02995904Freud J Stenberg R On weakly imposed boundary conditions for second order problems 1995 Venice, ItalyStenberg, R. (1995). On some techniques for approximating boundary conditions in the finite element method. Journal of Computational and Applied Mathematics, 63(1-3), 139-148. doi:10.1016/0377-0427(95)00057-7Becker, R., Hansbo, P., & Stenberg, R. (2003). A finite element method for domain decomposition with non-matching grids. ESAIM: Mathematical Modelling and Numerical Analysis, 37(2), 209-225. doi:10.1051/m2an:2003023Iapichino, L., Quarteroni, A., & Rozza, G. (2012). A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Computer Methods in Applied Mechanics and Engineering, 221-222, 63-82. doi:10.1016/j.cma.2012.02.005Eftang, J. L., & Patera, A. T. (2013). Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. International Journal for Numerical Methods in Engineering, 96(5), 269-302. doi:10.1002/nme.4543Eftang, J. L., & Patera, A. T. (2014). A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation. Advanced Modeling and Simulation in Engineering Sciences, 1(1), 3. doi:10.1186/2213-7467-1-3Vallaghé, S., & Patera, A. T. (2014). The Static Condensation Reduced Basis Element Method for a Mixed-Mean Conjugate Heat Exchanger Model. SIAM Journal on Scientific Computing, 36(3), B294-B320. doi:10.1137/120887709Martini, I., Rozza, G., & Haasdonk, B. (2014). Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system. Advances in Computational Mathematics, 41(5), 1131-1157. doi:10.1007/s10444-014-9396-6Smetana, K. (2015). A new certification framework for the port reduced static condensation reduced basis element method. Computer Methods in Applied Mechanics and Engineering, 283, 352-383. doi:10.1016/j.cma.2014.09.020Smetana, K., & Patera, A. T. (2016). Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures. SIAM Journal on Scientific Computing, 38(5), A3318-A3356. doi:10.1137/15m1009603Iapichino, L., Quarteroni, A., & Rozza, G. (2016). Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries. Computers & Mathematics with Applications, 71(1), 408-430. doi:10.1016/j.camwa.2015.12.001Maday, Y., & Rønquist, E. M. (2002). Journal of Scientific Computing, 17(1/4), 447-459. doi:10.1023/a:1015197908587Phuong Huynh, D. B., Knezevic, D. J., & Patera, A. T. (2012). A Static condensation Reduced Basis Element method : approximation anda posteriorierror estimation. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1), 213-251. doi:10.1051/m2an/2012022Ammar, A., Huerta, A., Chinesta, F., Cueto, E., & Leygue, A. (2014). Parametric solutions involving geometry: A step towards efficient shape optimization. Computer Methods in Applied Mechanics and Engineering, 268, 178-193. doi:10.1016/j.cma.2013.09.003Zlotnik, S., Díez, P., Modesto, D., & Huerta, A. (2015). Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications. International Journal for Numerical Methods in Engineering, 103(10), 737-758. doi:10.1002/nme.4909Montlaur, A., Fernandez‐Mendez, S., & Huerta, A. (2008). Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations. International Journal for Numerical Methods in Fluids, 57(9), 1071-1092. doi:10.1002/fld.1716Ciarlet, P. G. (2002). The Finite Element Method for Elliptic Problems. doi:10.1137/1.9780898719208Szabó, B., & Babuška, I. (2011). Introduction to Finite Element Analysis. doi:10.1002/9781119993834Rozza G Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications Separated Representations and PGD-Based Model Reduction CISM International Centre for Mechanical Sciences: Courses and Lectures 554 Vienna Springer 2014 153 227Ammar, A., Chinesta, F., Diez, P., & Huerta, A. (2010). An error estimator for separated representations of highly multidimensional models. Computer Methods in Applied Mechanics and Engineering, 199(25-28), 1872-1880. doi:10.1016/j.cma.2010.02.012Maday, Y., & Ronquist, E. M. (2004). The Reduced Basis Element Method: Application to a Thermal Fin Problem. SIAM Journal on Scientific Computing, 26(1), 240-258. doi:10.1137/s106482750241993

The proper generalized decomposition for the simulation of delamination using cohesive zone model

The use of cohesive zone models is an efficient way to treat the damage, especially when the crack path is known a priori. This is the case in the modeling of delamination in composite laminates. However, the simulations using cohesive zone models are expensive in a computational point of view. When using implicit time integration scheme or when solving static problems, the non-linearity related to the cohesive model requires many iterations before reaching convergence. In explicit approaches, the time step stability condition also requires an important number of iterations. In this article, a new approach based on a separated representation of the solution is proposed. The Proper Generalized Decomposition is used to build the solution. This technique, coupled with a cohesive zone model, allows a significant reduction of the computational cost. The results approximated with the PGD are very close to the ones obtained using the classical finite element approach

The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics

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.Giner Maravilla, E.; Bognet, B.; Ródenas, J.; Leygue, A.; Fuenmayor Fernández, FJ.; Chinesta Soria, FJ. (2013). The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics. International Journal of Solids and Structures. 50(10):1710-1720. https://doi.org/10.1016/j.ijsolstr.2013.01.039S17101720501

First steps towards an advanced simulation of composites manufacturing by automated tape placement

International audienceComposite materials and their related manufacturing processes involve many modeling and simulation issues, mainly related to their multi-physics and multi-scale nature, to the strong couplings and the complex geometries. In our former works we developed a new paradigm for addressing the solution of such complex models, the so-called Proper Generalized Decomposition based model order reduction. In this work we are summarizing the most outstanding capabilities of such methodology and then all these capabilities will be put together for addressing efficiently the simulation of a challenging composites manufacturing process, the automated tape placement

A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

The present paper extends in-plane-out-of-plane separated representations successfully used for addressing fully 3D model solutions defined in plate-like domain, to dynamics. Common time integration are performed using explicit or implicit strategies. Even if the implementation of implicit integration schemes into a 3D in-plane-out-of-plane separated representation does not imply major difficulties, the use of explicit integration preferable in many applications becomes a tricky issue. In fact the mesh employed for discretizing the out-of-plane dimension (thickness) determines the maximum time-step ensuring stability. In this paper we introduce a new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-plane discretization

Tetrodotoxin-Bupivacaine-Epinephrine Combinations for Prolonged Local Anesthesia

Currently available local anesthetics have analgesic durations in humans generally less than 12 hours. Prolonged-duration local anesthetics will be useful for postoperative analgesia. Previous studies showed that in rats, combinations of tetrodotoxin (TTX) with bupivacaine had supra-additive effects on sciatic block durations. In those studies, epinephrine combined with TTX prolonged blocks more than 10-fold, while reducing systemic toxicity. TTX, formulated as Tectin, is in phase III clinical trials as an injectable systemic analgesic for chronic cancer pain. Here, we examine dose-duration relationships and sciatic nerve histology following local nerve blocks with combinations of Tectin with bupivacaine 0.25% (2.5 mg/mL) solutions, with or without epinephrine 5 µg/mL (1:200,000) in rats. Percutaneous sciatic blockade was performed in Sprague-Dawley rats, and intensity and duration of sensory blockade was tested blindly with different Tectin-bupivacaine-epinephrine combinations. Between-group comparisons were analyzed using ANOVA and post-hoc Sidak tests. Nerves were examined blindly for signs of injury. Blocks containing bupivacaine 0.25% with Tectin 10 µM and epinephrine 5 µg/mL were prolonged by roughly 3-fold compared to blocks with bupivacaine 0.25% plain (P < 0.001) or bupivacaine 0.25% with epinephrine 5 µg/mL (P < 0.001). Nerve histology was benign for all groups. Combinations of Tectin in bupivacaine 0.25% with epinephrine 5 µg/mL appear promising for prolonged duration of local anesthesia

On the coupling of local 3D solutions and global 2D shell theory in structural mechanics

Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane-out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity of 2D simulations. However, that procedure it was revealed to be too intrusive for being introduced into existing commercial softwares. Moreover, experience indicated that such enriched descriptions are only compulsory locally, in some regions or structure components. In the present paper we propose an enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations. The proposed strategy will be extended to inelastic behaviors and structural dynamics

Stratégies numériques avancées pour la simulation de modèles définis sur des géométries de plaques et coques : solutions 3D avec une complexité 2D

Nowadays, most of the engineering products for transports (naval, aeronautical, automotive, ...), energy (wind power, ...), and civil engineering widely uses parts of small thickness: plates end shells. Metallic materials are still very used although composite materials are more and more used. The design and dimensioning of metal and composite parts therefore requires adapted and efficient simulation tools. The here chosen approach is to perform 3D mechanical simulations combined with a PGD (Proper Generalized Decomposition) based model reduction method to solve the problem in separated space variables. This method consists in looking for the solution under the form of a finite sum of products of functions involving the mean surface's coordinates and functions involving the coordinate of the thickness. The finite element method is used to solve the 2D (based on the coordinates of the mean surface) and 1D (depending on the thickness coordinate) problems from the variables separation. Thanks to this method, the 3D solution of the problem is iteratively built, with a complexity that scales like the complexity of a 2D problem. Additional variables are added as coordinates of the problem in order to include possible uncertainties, variabilities, design parameters or process parameters in the simulations. These multidimensional simulations therefore provide numeric charts, which can then be used for design and optimization.La plupart des produits d'ingénierie actuels, que ce soit dans le domaine des transports (naval, aéronautique, automobile, ...), de l'énergie (éolien, ...) ou du génie civil, font massivement appel à des pièces de faible épaisseur de formes variées : les plaques et les coques. Les matériaux métalliques sont toujours très utilisés, bien que l'utilisation des matériaux composites augmente fortement. La conception et le dimensionnement des pièces métalliques et composites nécessite par conséquent des outils de calculs adaptés et performants. L'approche retenue est d'effectuer des simulations mécaniques 3D et d'utiliser la méthode de réduction de modèle PGD (Proper Generalized Decomposition) pour résoudre le problème en variables d'espace séparées. Cette méthode consiste à chercher la solution sous la forme d'une somme finie de produits de fonctions des coordonnées de la surface moyenne et de fonctions de la coordonnée de l'épaisseur. La résolution par la méthode des éléments finis des problèmes 2D (fonction des coordonnées de la surface moyenne) et 1D (fonction de la coordonnée de l'épaisseur) issus de la séparation des variables permet de construire de façon itérative la solution 3D du problème avec une complexité qui reste celle d'un problème 2D. Des variables supplémentaires sont également ajoutées en tant que coordonnées du problème afin d'inclure dans les simulations d'éventuelles incertitudes, variabilités, des paramètres de conception ou des paramètres du procédé d'élaboration. Ces simulations multidimensionnelles fournissent donc des abaques numériques qui peuvent ensuite être utilisées pour la conception et l'optimisation