A stochastic computational multiscale approach; Application to MEMS resonators

NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering 294 (2015) 141–167, DOI: 10.1016/j.cma.2015.05.019

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Multi-scale; Stochastic; Finite-Elements; Polycrystalline; Resonance frequency; MEMS; LIMARC

Abstract :

[en] The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro-structure using a random Voronoi tessellation, each grain being assigned a random orientation. Then, we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso-scale. A random field of the meso-scale elasticity tensor can then be generated based on the information obtained from the SVE simulations. Finally, using a stochastic finite element method, these meso-scale uncertainties are propagated to the coarser scale. As an illustration we study the resonance frequencies of MEMS micro-beams made of poly-silicon materials, and we show that the stochastic multiscale approach predicts results in agreement with a Monte Carlo analysis applied directly on the fine finite-element model, i.e. with an explicit discretization of the grains.

Research center :

Computational & Multiscale Mechanics of Materials

Disciplines :

Materials science & engineering
Mechanical engineering

Author, co-author :

Lucas, Vincent ; Université de Liège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)

Golinval, Jean-Claude ; Université de Liège > Département d'aérospatiale et mécanique > LTAS - Vibrations et identification des structures

Paquay, Stéphane; Open-ENgineering S.A.

Nguyen, Van Dung ; Université de Liège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)

Noels, Ludovic ; Université de Liège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)

Wu, Ling ; Université de Liège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)

Language :

English

Title :

A stochastic computational multiscale approach; Application to MEMS resonators

Publication date :

01 September 2015

Journal title :

Computer Methods in Applied Mechanics and Engineering

ISSN :

0045-7825

eISSN :

1879-2138

Publisher :

Elsevier Science, Lausanne, Switzerland

Volume :

294

Pages :

141–167

Peer reviewed :

Peer Reviewed verified by ORBi

Tags :

CÉCI : Consortium des Équipements de Calcul Intensif

Additional URL :

http://dx.doi.org/10.1016/j.cma.2015.05.019

Name of the research project :

3SMVIB: The research has been funded by the Walloon Region under the agreement no 1117477 (CT-INT 2011-11-14) in the context of the ERA-NET MNT framework.

Funders :

Service public de Wallonie : Direction générale opérationnelle de l'économie, de l'emploi et de la recherche - DG06
CÉCI - Consortium des Équipements de Calcul Intensif [BE]

Available on ORBi :

since 01 June 2015

Statistics

Number of views

569 (162 by ULiège)

Number of downloads

661 (92 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

Bibliography

Ghanem R., Spanos P. Stochastic Finite Elements: A Spectral Approach 1991, Springer Verlag.
Lemaitre O., Knio O. Spectral Methods for Uncertainty Quantification - With Applications to Computational Fluid Dynamics 2010, Springer.
Stefanou G. The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Engrg. 2009, 198(912):1031-1051. 0045-7825, 10.1016/j.cma.2008.11.007.
Shinozuka M., Deodatis G. Response variability of Stochastic finite element systems. J. Eng. Mech. 1988, 114(3):499-519. 10.1061/(ASCE)0733-9399(1988)114:3(499).
T. Harada, M. Shinozuka, The Scale of Correlation for Stochastic Fields-Technical Report, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY.
Baxter S., Graham L. Characterization of random composites using Moving-Window technique. Journal of Engineering Mechanics 2000, 126(4):389-397. 10.1061/(ASCE)0733-9399(2000)126:4(389).
Matthies H.G., Brenner C.E., Bucher C.G., Guedes Soares C. Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements. Struct. Saf. 1997, 19(3):283-336. devoted to the work of the Joint Committee on Structural Safety. 0167-4730, 10.1016/S0167-4730(97)00013-1.
Yamazaki F., Member A., Shinozuka M., Dasgupta G. Neumann expansion for Stochastic finite element analysis. J. Eng. Mech. 1988, 114(8):1335-1354. 10.1061/(ASCE)0733-9399(1988)114:8(1335).
Shinokuza M., Astill C.J. Random Eigenvalue problems in structural analysis. AIAA J. 1972, 10(4):456-462. 0001-1452, 10.2514/3.50119.
Castañeda P., Suquet P. Nonlinear composites. Advances in Applied Mechanics 1997, vol. 34:171-302. Elsevier. 10.1016/S0065-2156(08)70321-1. E. van der Giessen, T.Y. Wu (Eds.).
Doghri I., Ouaar A. Homogenization of two-phase elasto-plastic composite materials and structures: Study of tangent operators, cyclic plasticity and numerical algorithms. Int. J. Solids Struct. 2003, 40(7):1681-1712. 0020-7683, 10.1016/S0020-7683(03)00013-1.
Wu L., Noels L., Adam L., Doghri I. A combined incremental-secant mean-field homogenization scheme with per-phase residual strains for elasto-plastic composites. Int. J. Plast. 2013, 51:80-102. 0749-6419, 10.1016/j.ijplas.2013.06.006.
Moulinec H., Suquet P. A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructures. Solid Mechanics and Its Applications 1995, vol. 37:235-246. Springer Netherlands. 10.1007/978-94-011-0059-5_20. R. Pyrz (Ed.).
Ghosh S., Lee K., Moorthy S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. International Journal of Solids and Structures 1995, 32(1):27-62. 0020-7683, 10.1016/0020-7683(94)00097-G.
Michel J., Moulinec H., Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 1999, 172(1-4):109-143. 0045-7825, 10.1016/S0045-7825(98)00227-8.
Miehe C., Koch A. Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch. Appl. Mech. 2002, 72(4-5):300-317. 0939-1533, 10.1007/s00419-002-0212-2.
Kouznetsova V., Brekelmans W.A.M., Baaijens F.P.T. An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 2001, 27(1):37-48. 0178-7675, 10.1007/s004660000212.
Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 2003, 40(13-14):3647-3679. 0020-7683, 10.1016/S0020-7683(03)00143-4.
Geers M., Kouznetsova V., Brekelmans W. Multi-scale computational homogenization: Trends and challenges. J. Comput. Appl. Math. 2010, 234(7):2175-2182. 0377-0427, 10.1016/j.cam.2009.08.077.
Kalamkarov A.L., Andrianov I.V., Danishevskyy V.V. Asymptotic homogenization of composite materials and structures. Appl. Mech. Rev. 2009, 62(3):030802. 00036900, 10.1115/1.3090830.
Hill R. Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 1963, 11(5):357-372. 0022-5096, 10.1016/0022-5096(63)90036-X.
Ostoja-Starzewski M., Wang X. Stochastic finite elements as a bridge between random material microstructure and global response. Comput. Methods Appl. Mech. Engrg. 1999, 168(14):35-49. 0045-7825, 10.1016/S0045-7825(98)00105-4.
Fish J., Wu W. A nonintrusive stochastic multiscale solver. Internat. J. Numer. Methods Engrg. 2011, 88(9):862-879. 1097-0207, 10.1002/nme.3201.
Gusella V., Cluni F. Random field and homogenization for masonry with nonperiodic microstructure. J. Mech. Mater. Struct. 2006, 1(2):357-386. 1559-3959, 10.2140/jomms.2006.1.357.
Trovalusci P., Ostoja-Starzewski M., De Bellis M.L., Murrali A. Scale-dependent homogenization of random composites as micropolar continua. Eur. J. Mech. A 2015, 49(0):396-407. 0997-7538, 10.1016/j.euromechsol.2014.08.010.
Yin X., Chen W., To A., McVeigh C., Liu W. Statistical volume element method for predicting microstructure-constitutive property relations. Computer Methods in Applied Mechanics and Engineering 2008, 197(43-44):3516-3529. stochastic Modeling of Multiscale and Multiphysics Problems. 0045-7825, 10.1016/j.cma.2008.01.008.
X. Yin, S. Lee, W. Chen, W.K. Liu, M.F. Horstemeyer, Efficient random field uncertainty propagation in design using multiscale analysis, J. Mech. Des. (ISSN 1050-0472) 131 (2). http://dx.doi.org/10.1115/1.3042159.
Clément A., Soize C., Yvonnet J. Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis. International Journal for Numerical Methods in Engineering 2012, 91(8):799-824. 1097-0207, 10.1002/nme.4293.
Soize C. Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J. Acoust. Soc. Am. 2001, 109(5):1979-1996. 10.1121/1.1360716.
Soize C. Random matrix theory for modeling uncertainties in computational mechanics. Computer Methods in Applied Mechanics and Engineering 2005, 194(12-16):1333-1366. special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis. 0045-7825, 10.1016/j.cma.2004.06.038.
Das S., Ghanem R. A bounded random matrix approach for Stochastic upscaling. Multiscale Model. Simul. 2009, 8(1):296-325. 10.1137/090747713.
Guilleminot J., Noshadravan A., Soize C., Ghanem R. A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Engrg. 2011, 200(17-20):1637-1648. 0045-7825, 10.1016/j.cma.2011.01.016.
Noshadravan A., Ghanem R., Guilleminot J., Atodaria I., Peralta P. Validation of a probabilistic model for mesoscale elasticity tensor or random polycrystals. Int. J. Uncertain. Quantif. 2013, 3(1):73-100. 2152-5080.
Cho H., Venturi D., Karniadakis G. Karhunen-Loève expansion for multi-correlated stochastic processes. Probab. Eng. Mech. 2013, 34(0):157-167. 0266-8920, 10.1016/j.probengmech.2013.09.004.
Shinozuka M. Simulation of multivariate and multidimensional random processes. The Journal of the Acoustical Society of America 1971, 49(1B):357-368. 10.1121/1.1912338.
Shinozuka M., Jan C.-M. Digital simulation of random processes and its applications. J. Sound Vib. 1972, 25(1):111-128. 0022-460X, 10.1016/0022-460X(72)90600-1.
Popescu R., Deodatis G., Prevost J. Simulation of homogeneous nonGaussian stochastic vector fields. Probab. Eng. Mech. 1998, 13(1):1-13. 0266-8920, 10.1016/S0266-8920(97)00001-5.
Soize C. Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. Computer Methods in Applied Mechanics and Engineering 2010, 199(33-36):2150-2164. 0045-7825, 10.1016/j.cma.2010.03.013.
Der Kiureghian A., Ke J. The stochastic finite element method in structural reliability. Probabilistic Engineering Mechanics 1988, 3(2):83-91. 0266-8920, 10.1016/0266-8920(88)90019-7.
Lepage S. Stochastic Finite Element Method for the Modeling of Thermoelastic Damping in Micro-resonators 2007, Leloup.
Huet C. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 1990, 38(6):813-841. 0022-5096, 10.1016/0022-5096(90)90041-2.
Nguyen V.-D., Béchet E., Geuzaine C., Noels L. Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput. Mater. Sci. 2012, 55:390-406. 10.1016/j.commatsci.2011.10.017.
Hazanov S., Huet C. Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. Journal of the Mechanics and Physics of Solids 1994, 42(12):1995-2011. 0022-5096, 10.1016/0022-5096(94)90022-1.
Nguyen V.D., Becker G., Noels L. Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation. Comput. Methods Appl. Mech. Engrg. 2013, 260:63-77. 0045-7825, 10.1016/j.cma.2013.03.024.
Hopcroft M., Nix W., Kenny T. What is the Young's modulus of Silicon?. J. Microelectromech. Syst. 2010, 19(2):229-238. 1057-7157, 10.1109/JMEMS.2009.2039697.
G.D. Geers M., W.C. Coenen E., Kouznetsova V.G. Multi-scale computational homogenization of structured thin sheets. Modelling Simul. Mater. Sci. Eng. 2007, 15(4):S393. 10.1088/0965-0393/15/4/S06.
Sena M.P., Ostoja-Starzewski M., Costa L. Stiffness tensor random fields through upscaling of planar random materials. Probab. Eng. Mech. 2013, 34:131-156. 10.1016/j.probengmech.2013.08.008.
Reddy J. On locking-free shear deformable beam finite elements. Comput. Methods Appl. Mech. Engrg. 1997, 149(1-4):113-132. containing papers presented at the Symposium on Advances in Computational Mechanics. 0045-7825, 10.1016/S0045-7825(97)00075-3.
Mulay S., Becker G., Vayrette R., Raskin J.-P., Pardoen T., Galceran M., Godet S., Noels L. Multiscale modelling framework for the fracture of thin brittle polycrystalline films: application to polysilicon. Comput. Mech. 2015, 55:73-91. 0178-7675, 10.1007/s00466-014-1083-4.
Geuzaine C., Remacle J.-F. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Internat. J. Numer. Methods Engrg. 2009, 79(11):1309-1331. 10.1002/nme.2579.
Soize C. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput. Methods Appl. Mech. Engrg. 2006, 195(13):26-64. 0045-7825, 10.1016/j.cma.2004.12.014.