zbMATH — the first resource for mathematics

Hyper-reduction of generalized continua. (English) Zbl 1398.74337
Summary: This paper deals with the reduced order modeling of micromorphic continua. The reduced basis model relies on the proper orthogonal decomposition and the hyper-reduction. Two variants of creation of reduced bases using the proper orthogonal decomposition are explored from the perspective of additional micromorphic degrees of freedom. In the first approach, one snapshot matrix including displacement as well as micromorphic degrees of freedom is assembled. In the second approach, snapshots matrices are assembled separately for displacement and micromorphic fields and the singular value decomposition is performed on each system separately. Thereafter, the formulation is extended to the hyper-reduction method. It is shown that the formulation has the same structure as for the classical continua. The relation of higher order stresses introduced in micromorphic balance equations to creation of the reduced integration domain is examined. Finally, the method is applied to examples of microdilatation extension and clamped tension and to a size-dependent stress concentration in Cosserat elasticity. It is shown that the proposed approach leads to a good level of accuracy with significant reduction of computational time.
74S05 Finite element methods applied to problems in solid mechanics
74A60 Micromechanical theories
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
[1] Mindlin, RD, Micro-structure in linear elasticity, Arch Ration Mech Anal, 16, 3, (1964) · Zbl 0119.40302
[2] Eringen, AC; Suhubi, E, Nonlinear theory of simple micro-elastic solids—I, Int J Eng Sci, 2, 189-203, (1964) · Zbl 0138.21202
[3] Eringen, AC; Suhubi, E, Nonlinear theory of simple micro-elastic solids—II, Int J Eng Sci, 2, 389-404, (1964) · Zbl 0138.21202
[4] Forest S, Sievert R (2006) Nonlinear microstrain theories. Int J Solids Struct 43(24):7224-7245 · Zbl 1102.74003
[5] Almroth, B; Stern, P; Brogan, F, Automatic choice of global shape functions in structural analysis, AIAA Journal, 16, 525-528, (1978)
[6] Cuong NN, Veroy K, Patera AT (2005) Certified real-time solution of parametrized partial differential equations. In: Sidney Y (ed) Handbook of materials modeling: methods. Springer, Dordrecht, pp 1529-1564
[7] Ryckelynck, D, A priori hyperreduction method: an adaptive approach, J Comput Phys, 202, 346-366, (2005) · Zbl 1288.65178
[8] Ryckelynck, D, Hyper-reduction of mechanical models involving internal variables, Int J Numer Methods Eng, 77, 75-89, (2009) · Zbl 1195.74299
[9] Ryckelynck, D, Hyper reduction of finite strain elasto-plastic models, Int J Mater Form, 2, 567-571, (2009)
[10] Hernández, J; Oliver, J; Huespe, AE; Caicedo, M; Cante, J, High-performance model reduction techniques in computational multiscale homogenization, Comput Methods Appl Mech Eng, 276, 149-189, (2014) · Zbl 1423.74785
[11] Georgiou, I, Advanced proper orthogonal decomposition tools: using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods, Nonlinear Dyn, 41, 69-110, (2005) · Zbl 1142.74337
[12] Farhat, C; Avery, P; Chapman, T; Cortial, J, Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency, Int J Numer Methods Eng, 98, 625-662, (2014) · Zbl 1352.74348
[13] Forest, S, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J Eng Mech, 135, 117-131, (2009)
[14] Germain, P, La méthode des puissances virtuelles en mécanique des milieux continus, J Méc, 12, 236-274, (1973) · Zbl 0261.73003
[15] Germain, P, The method of virtual power in continuum mechanics. part 2: microstructure, SIAM J Appl Math, 25, 556-575, (1973) · Zbl 0273.73061
[16] Auffray, N; Quang, H; He, Q-C, Matrix representations for 3d strain-gradient elasticity, J Mech Phys Solids, 61, 1202-1223, (2013) · Zbl 1260.74012
[17] Eringen AC (1968) Mechanics of micromorphic continua. In Kröner E (ed) Mechanics of generalized continua: proceedings of the IUTAM-Symposium on The Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart (Germany) 1967. Springer, Berlin, pp 18-35 · Zbl 0987.76077
[18] Goodman, M; Cowin, S, A continuum theory for granular materials, Arch Ration Mech Anal, 44, 249-266, (1972) · Zbl 0243.76005
[19] Cosserat, E; Cosserat, F, Théorie des corps déformables, Paris, 3, 17-29, (1909) · JFM 40.0862.02
[20] Besson J, Cailletaud G, Chaboche J-L, Forest S (2009) Non-linear mechanics of materials, vol 167. Springer, Berlin · Zbl 0997.74002
[21] Sirovich, L, Turbulence and the dynamics of coherent structures. part I: coherent structures, Q Appl Math, 45, 561-571, (1987) · Zbl 0676.76047
[22] Iollo, A; Lanteri, S; Désidéri, J-A, Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations, Theor Comput Fluid Dyn, 13, 377-396, (2000) · Zbl 0987.76077
[23] Lutowska A (2012) Model order reduction for coupled systems using low-rank approximations. PhD thesis, Eindhoven University of Technology
[24] Fritzen, F; Hodapp, M; Leuschner, M, GPU accelerated computational homogenization based on a variational approach in a reduced basis framework, Comput Methods Appl Mech Eng, 278, 186-217, (2014) · Zbl 1423.74881
[25] Chaturantabut, S; Sorensen, DC, Nonlinear model reduction via discrete empirical interpolation, SIAM J Sci Comput, 32, 2737-2764, (2010) · Zbl 1217.65169
[26] Maday, Y; Rønquist, EM, A reduced-basis element method, J Sci Comput, 17, 447-459, (2002) · Zbl 1014.65119
[27] Ryckelynck, D; Gallimard, L; Jules, S, Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity, Adv Model Simul Eng Sci, 2, 1, (2015)
[28] Patzák, B; Bittnar, Z, Design of object oriented finite element code, Adv Eng Softw, 32, 759-767, (2001) · Zbl 0984.68526
[29] Patzák B (2012) OOFEM—an object-oriented simulation tool for advanced modeling of materials and structures. Acta Polytech 52(6):59-66 · Zbl 1423.74785
[30] Horák, M; Patzák, B; Jirásek, M, On design of element evaluators in OOFEM, Adv Eng Softw, 72, 193-202, (2014)
[31] Radermacher, A; Reese, S, Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring, Comput Mech, 54, 677-687, (2014) · Zbl 1311.74026
[32] Kaloni, PN; Ariman, T, Stress concentration effects in micropolar elasticity, Z Angew Math Phys, 18, 136-141, (1967)
[33] Dillard, T; Forest, S; Ienny, P, Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams, Eur J Mech A Solids, 25, 526-549, (2006) · Zbl 1094.74047
[34] Mindlin, R, Influence of couple-stresses on stress concentrations, Exp Mech, 3, 1-7, (1963)
[35] Cowin, SC, An incorrect inequality in micropolar elasticity theory, Z Angew Math Phys, 21, 494-497, (1970) · Zbl 0198.58503
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.