zbMATH — the first resource for mathematics

PGD-based computational vademecum for efficient design, optimization and control. (English) Zbl 1354.65100
Summary: In this paper we are addressing a new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to high-dimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with high-dimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.

65J05 General theory of numerical analysis in abstract spaces
PDF BibTeX Cite
Full Text: DOI
[1] Ammar, A; Mokdad, B; Chinesta, F; Keunings, R, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J Non-Newton Fluid Mech, 139, 153-176, (2006) · Zbl 1195.76337
[2] Ammar, A; Ryckelynck, D; Chinesta, F; Keunings, R, On the reduction of kinetic theory models related to finitely extensible dumbbells, J Non-Newton Fluid Mech, 134, 136-147, (2006) · Zbl 1123.76309
[3] Ammar, A; Mokdad, B; Chinesta, F; Keunings, R, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. part II: transient simulation using space-time separated representation, J Non-Newton Fluid Mech, 144, 98-121, (2007) · Zbl 1196.76047
[4] Ammar, A; Chinesta, F; Joyot, P, The nanometric and micrometric scales of the structure and mechanics of materials revisited: an introduction to the challenges of fully deterministic numerical descriptions, Int J Multiscale Comput Eng, 6, 191-213, (2008)
[5] Ammar, A; Pruliere, E; Chinesta, F; Laso, M, Reduced numerical modeling of flows involving liquid-crystalline polymeres, J Non-Newton Fluid Mech, 160, 140-156, (2009) · Zbl 1274.76145
[6] Ammar, A; Normandin, M; Daim, F; Gonzalez, D; Cueto, E; Chinesta, F, Non-incremental strategies based on separated representations: applications in computational rheology, Commun Math Sci, 8, 671-695, (2010) · Zbl 1281.76022
[7] Ammar, A; Chinesta, F; Falco, A, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch Comput Methods Eng, 17, 473-486, (2010) · Zbl 1269.65120
[8] Ammar, A; Chinesta, F; Diez, P; Huerta, A, An error estimator for separated representations of highly multidimensional models, Comput Methods Appl Mech Eng, 199, 1872-1880, (2010) · Zbl 1231.74503
[9] Ammar, A; Normandin, M; Chinesta, F, Solving parametric complex fluids models in rheometric flows, J Non-Newton Fluid Mech, 165, 1588-1601, (2010) · Zbl 1274.76007
[10] Ammar, A; Cueto, E; Chinesta, F, Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions, Int J Numer Methods Biomed Eng, 28, 960-973, (2012)
[11] Ammar A, Cueto E, Chinesta F Non-incremental PGD solution of parametric uncoupled models defined in evolving domains. Int J Numer Methods Eng. doi:10.1002/nme.4413 · Zbl 1269.76079
[12] Barrault, M; Maday, Y; Nguyen, NC; Patera, AT, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations, C R Math, 339, 667-672, (2004) · Zbl 1061.65118
[13] Bellomo N (2008) Modeling complex living systems. Birkhäuser, Basel · Zbl 1140.91007
[14] Bernoulli Ch (1836) Vademecum des Mechanikers. Cotta, Stuttgart
[15] Bialecki, RA; Kassab, AJ; Fic, A, Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis, Int J Numer Methods Eng, 62, 774-797, (2005) · Zbl 1092.80010
[16] Bird, BB; Curtiss, CF; Armstrong, RC; Hassager, O, Dynamics of polymeric liquids, No. 2, (1987), New York
[17] Bognet, B; Leygue, A; Chinesta, F; Poitou, A; Bordeu, F, Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity, Comput Methods Appl Mech Eng, 201, 1-12, (2012) · Zbl 1239.74045
[18] Bordeu F, Leygue A, Modesto D, Gonzalez D, Cueto E, Chinesta F Real-time simulation techniques for augmented learning in science and engineering higher education. A PGD approach. Arch Comput Methods Eng, submitted · Zbl 1269.74209
[19] Bui-Thanh, T; Willcox, K; Ghattas, O; Bloemen Waanders, B, Goal-oriented, model-constrained optimization for reduction of large-scale systems, J Comput Phys, 224, 880-896, (2007) · Zbl 1123.65081
[20] Burkardt, J; Gunzburger, M; Lee, H-C, POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput Methods Appl Mech Eng, 196, 337-355, (2006) · Zbl 1120.76323
[21] Cancès E, Defranceschi M, Kutzelnigg W, Le Bris C, Maday Y (2003) Computational quantum chemistry: a primer. Handbook of numerical analysis, vol X. Elsevier, Amsterdam, pp 3-270 · Zbl 1070.81534
[22] Chaturantabut, S; Sorensen, DC, Nonlinear model reduction via discrete empirical interpolation, SIAM J Sci Comput, 32, 2737-2764, (2010) · Zbl 1217.65169
[23] Chinesta, F; Ammar, A; Cueto, E, Proper generalized decomposition of multiscale models, Int J Numer Methods Eng, 83, 1114-1132, (2010) · Zbl 1197.76093
[24] Chinesta, F; Ammar, A; Cueto, E, Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Arch Comput Methods Eng, 17, 327-350, (2010) · Zbl 1269.65106
[25] Chinesta, F; Ammar, A; Leygue, A; Keunings, R, An overview of the proper generalized decomposition with applications in computational rheology, J Non-Newton Fluid Mech, 166, 578-592, (2011) · Zbl 1359.76219
[26] Chinesta, F; Ladeveze, P; Cueto, E, A short review in model order reduction based on proper generalized decomposition, Arch Comput Methods Eng, 18, 395-404, (2011)
[27] Chinesta, F; Leygue, A; Bognet, B; Ghnatios, Ch; Poulhaon, F; Bordeu, F; Barasinski, A; Poitou, A; Chatel, S; Maison-Le-Poec, S, First steps towards an advanced simulation of composites manufacturing by automated tape placement, Int J Mater Forming, (2012)
[28] Cochelin, B; Damil, N; Potier-Ferry, M, The asymptotic numerical method: an efficient perturbation technique for nonlinear structural mechanics, Rev Eur Elem Finis, 3, 281-297, (1994) · Zbl 0810.73045
[29] Darema, F, Engineering/scientific and commercial applications: differences, similarities, and future evolution, No. 1, 367-374, (1994), Paris
[30] Dennis JE Jr., Schnabel RB (1996) Numerical methods for unconstrained optimization and nonlinear equations. Classics in applied mathematics, vol 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Corrected reprint of the 1983 original · Zbl 0847.65038
[31] Ghnatios, Ch; Chinesta, F; Cueto, E; Leygue, A; Breitkopf, P; Villon, P, Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion, Composites, Part A, 42, 1169-1178, (2011)
[32] Ghnatios, Ch; Masson, F; Huerta, A; Cueto, E; Leygue, A; Chinesta, F, Proper generalized decomposition based dynamic data-driven control of thermal processes, Comput Methods Appl Mech Eng, 213, 29-41, (2012)
[33] Girault, M; Videcoq, E; Petit, D, Estimation of time-varying heat sources through inversion of a low order model built with the modal identification method from in-situ temperature measurements, Int J Heat Mass Transf, 53, 206-219, (2010) · Zbl 1180.80055
[34] Gonzalez, D; Ammar, A; Chinesta, F; Cueto, E, Recent advances in the use of separated representations, Int J Numer Methods Eng, 81, 637-659, (2010) · Zbl 1183.65168
[35] Gonzalez, D; Masson, F; Poulhaon, F; Leygue, A; Cueto, E; Chinesta, F, Proper generalized decomposition based dynamic data-driven inverse identification, Math Comput Simul, 82, 1677-1695, (2012)
[36] Gunzburger, MD; Peterson, JS; Shadid, JN, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput Methods Appl Mech Eng, 196, 1030-1047, (2007) · Zbl 1121.65354
[37] http://www.epractice.eu/en/news/5304734
[38] http://www.ga-project.eu/ · Zbl 0810.73045
[39] http://www.humanbrainproject.eu/
[40] http://www.itfom.eu/ · Zbl 1359.76219
[41] http://robotcompanions.eu · Zbl 1269.65106
[42] http://www.futurict.eu · Zbl 1197.76093
[43] http://www.graphene-flagship.eu/ · Zbl 1191.65156
[44] Ladevèze, P, The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables, C R Acad Sci Paris, 309, 1095-1099, (1989) · Zbl 0677.73060
[45] Ladevèze, P; Nouy, A, A multiscale computational method with time and space homogenization, C R, Méc, 330, 683-689, (2002) · Zbl 1177.74316
[46] Ladevèze, P; Nouy, A; Loiseau, O, A multiscale computational approach for contact problems, Comput Methods Appl Mech Eng, 191, 4869-4891, (2002) · Zbl 1018.74036
[47] Ladevèze, P; Nouy, A, On a multiscale computational strategy with time and space homogenization for structural mechanics, Comput Methods Appl Mech Eng, 192, 3061-3087, (2003) · Zbl 1054.74701
[48] Ladevèze, P; Néron, D; Gosselet, P, On a mixed and multiscale domain decomposition method, Comput Methods Appl Mech Eng, 96, 1526-1540, (2007) · Zbl 1173.74379
[49] Ladevèze, P; Passieux, J-C; Néron, D, The Latin multiscale computational method and the proper generalized decomposition, Comput Methods Appl Mech Eng, 199, 1287-1296, (2010) · Zbl 1227.74111
[50] Ladevèze, P; Chamoin, L, On the verification of model reduction methods based on the proper generalized decomposition, Comput Methods Appl Mech Eng, 200, 2032-2047, (2011) · Zbl 1228.76089
[51] Lamari, H; Ammar, A; Cartraud, P; Legrain, G; Jacquemin, F; Chinesta, F, Routes for efficient computational homogenization of non-linear materials using the proper generalized decomposition, Arch Comput Methods Eng, 17, 373-391, (2010) · Zbl 1269.74187
[52] Lamari, H; Ammar, A; Leygue, A; Chinesta, F, On the solution of the multidimensional langer’s equation by using the proper generalized decomposition method for modeling phase transitions, Model Simul Mater Sci Eng, 20, (2012)
[53] Le Bris, C; Lelièvre, T; Maday, Y, Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constr Approx, 30, 621-651, (2009) · Zbl 1191.65156
[54] Leygue, A; Verron, E, A first step towards the use of proper general decomposition method for structural optimization, Arch Comput Methods Eng, 17, 465-472, (2010) · Zbl 1269.74182
[55] Leygue A, Chinesta F, Beringhier M, Nguyen TL, Grandidier JC, Pasavento F, Schrefler B Towards a framework for non-linear thermal models in shell domains. Int J Numer Methods Heat Fluid Flow. doi:10.1108/09615531311289105 · Zbl 1239.74045
[56] Maday, Y; Ronquist, EM, A reduced-basis element method, C R Acad Sci Paris, Ser I, 335, 195-200, (2002) · Zbl 1006.65128
[57] Maday, Y; Patera, AT; Turinici, G, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations, J Sci Comput, 17, 437-446, (2002) · Zbl 1014.65115
[58] Maday, Y; Ronquist, EM, The reduced basis element method: application to a thermal fin problem, SIAM J Sci Comput, 26, 240-258, (2004) · Zbl 1077.65120
[59] Néron, D; Ladevèze, P, Proper generalized decomposition for multiscale and multiphysics problems, Arch Comput Methods Eng, 17, 351-372, (2010) · Zbl 1269.74209
[60] Niroomandi, S; Alfaro, I; Cueto, E; Chinesta, F, Real-time deformable models of non-linear tissues by model reduction techniques, Comput Methods Programs Biomed, 91, 223-231, (2008)
[61] Niroomandi, S; Alfaro, I; Cueto, E; Chinesta, F, Model order reduction for hyperelastic materials, Int J Numer Methods Eng, 81, 1180-1206, (2010) · Zbl 1183.74365
[62] Niroomandi, S; Alfaro, I; Cueto, E; Chinesta, F, Accounting for large deformations in real-time simulations of soft tissues based on reduced order models, Comput Methods Programs Biomed, 105, 1-12, (2012)
[63] Niroomandi, S; Alfaro, I; Gonzalez, D; Cueto, E; Chinesta, F, Real time simulation of surgery by reduced order modelling and X-FEM techniques, Int J Numer Methods Biomed Eng, 28, 574-588, (2012) · Zbl 1243.92032
[64] Nouy, A, Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems, Arch Comput Methods Eng, 17, 403-434, (2010) · Zbl 1269.76079
[65] NSF Final Report (2006) DDDAS Workshop 2006, Arlington, VA, USA
[66] Oden JT, Belytschko T, Fish J, Hughes TJR, Johnson C, Keyes D, Laub A, Petzold L, Srolovitz D, Yip S (2006) Simulation-based engineering science: revolutionizing engineering science through simulation. NSF Blue Ribbon Panel on SBES
[67] Park, HM; Cho, DH, The use of the Karhunen-Loève decomposition for the modelling of distributed parameter systems, Chem Eng Sci, 51, 81-98, (1996)
[68] Passieux, J-C; Ladevèze, P; Néron, D, A scalable time-space multiscale domain decomposition method: adaptive time scale separation, Comput Mech, 46, 621-633, (2010) · Zbl 1358.74074
[69] Pruliere, E; Ferec, J; Chinesta, F; Ammar, A, An efficient reduced simulation of residual stresses in composites forming processes, Int J Mater Forming, 3, 1339-1350, (2010)
[70] Pruliere, E; Chinesta, F; Ammar, A, On the deterministic solution of multidimensional parametric models by using the proper generalized decomposition, Math Comput Simul, 81, 791-810, (2010) · Zbl 1207.65014
[71] Rozza, G; Huynh, DBP; Patera, AT, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations—application to transport and continuum mechanics, Arch Comput Methods Eng, 15, 229-275, (2008) · Zbl 1304.65251
[72] Ryckelynck, D; Hermanns, L; Chinesta, F; Alarcon, E, An efficient a priori model reduction for boundary element models, Eng Anal Bound Elem, 29, 796-801, (2005) · Zbl 1182.76913
[73] Ryckelynck, D; Chinesta, F; Cueto, E; Ammar, A, On the a priori model reduction: overview and recent developments, Arch Comput Methods Eng, 13, 91-128, (2006) · Zbl 1142.76462
[74] Schmidt, F; Pirc, N; Mongeau, M; Chinesta, F, Efficient mould cooling optimization by using model reduction, Int J Mater Forming, 4, 71-82, (2011)
[75] Various authors (2006) Final report. DDDAS workshop 2006 at Arlington, VA, USA Technical report, National Science Foundation · Zbl 1134.76326
[76] Veroy, K; Patera, A, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Int J Numer Methods Fluids, 47, 773-788, (2005) · Zbl 1134.76326
[77] Videcoq, E; Quemener, O; Lazard, M; Neveu, A, Heat source identification and on-line temperature control by a branch eigenmodes reduced model, Int J Heat Mass Transf, 51, 4743-4752, (2008) · Zbl 1154.80369
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.