Proper Generalized Decomposition method for incompressible flows in stream-vorticity formulation
DOI:
https://doi.org/10.13052/EJCM.19.591-617Keywords:
reduced order model, lid-driven cavity, separation of variablesAbstract
In this work, the Proper Generalized Decomposition (PGD) method will be considered in order to solve Navier-stokes equations with a stream-vorticity formulation by looking for the solution as a sum of tensor product functions. In the first stage, PGD will be applied to a model equation in order to test the capacity of the method to treat some timedependent problem. Then, we will solve the Navier-Stokes problem in the case of the liddriven cavity for different Reynolds numbers (Re = 100, 1000 and 10000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.
Downloads
References
Ammar A., Chinesta F., Diez P., Huerta A., An Error Estimator for Separated Representations
of Highly Multidimensional Models, Computer Methods in Applied Mechanics and
Engineering, vol. 199, p. 1872-1880, 2010.
Ammar A., Chinesta F., Falco A.Archives of Computational Methods in Engineering, In press,
n.d.
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 , Journal of Non-Newtonian Fluid Mechanics, vol. 139, n° 3, p. 153 - 176,
a.
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 modelling of
complex fluids: Part II: Transient simulation using space-time separated representations ,
Journal of Non-Newtonian Fluid Mechanics, vol. 144, n° 2-3, p. 98 - 121, 2007.
Ammar A., Ryckelynck D., Chinesta F., Keunings R., On the reduction of kinetic theory models
related to finitely extensible dumbbells, J. Non-Newtonian Fluid Mech., vol. 134, p. 136-
, 2006b.
Arnold D., Liu X., Local errors estimates for finite elemnt discretizations of the Stokes equations
, Math. Model. Numer. Anal., vol. 29, p. 367 - 389, 1995.
Bruneau C.-H., Saad M., The 2D lid-driven cavity problem revisited, Computers and Fluids,
vol. 35, n° 3, p. 326 - 348, 2006.
Chinesta F., Ammar A., Joyot P., The nanometric and micrometric scales of the structure and
mechanics of materials revisited: An introduction to he challenges of fully deterministic
numerical descriptions , International Journal for Multiscale Computational Engineering,
vol. 6/3, p. 191-213, 2008a.
Chinesta F., Ammar A., Lemarchand F., Beauchene P., Boust F., Alleviating mesh constraints:
Model reduction, parallel time integration and high resolution homogenization , Computer
Methods in Applied Mechanics and Engineering, vol. 197, n° 5, p. 400 - 413, 2008b. Enriched
Simulation Methods and Related Topics.
Chudanov V. V., Popkov A. G., Churbanov A. G., Vabishchevich P. N., Makarov M. M.,
Operator-splitting schemes for the stream function-vorticity formulation, Computers and
Fluids, vol. 24, n° 7, p. 771 - 786, 1995.
Dowell E., HAll K., «Modelling of fluid-structure interaction» Annal Review of Fluids
Mechanics, vol. 33, p. 445-490, 2001.
Ghia U., Ghia K., Shin C., High-re solutions for incompressible flow using the Navier-Stokes
equations and a multigrid method, J. of Computational physics, vol. 48, p. 387 - 411, 1982.
Kim Y., Kim D. W., Jun S., Lee J. H., Meshfree point collocation method for the streamvorticity
formulation of 2D incompressible Navier-Stokes equations , Computer Methods
in Applied Mechanics and Engineering, vol. 196, n° 33-34, p. 3095 - 3109, 2007.
KressW., Nilsson J., Boundary conditions and estimates for the linearized Navier-Stokes equations
on staggered grids , Computers and Fluids, vol. 32, n° 8, p. 1093 - 1112, 2003.
Ladeveze P., Non linear computational structural mechanics : new approaches and nonincremental
methods of calculation , Mechanical engineering series, Springer;, 1999.
Ladeveze P., Passieux J.-C., Neron D., The LATIN multiscale computational method and the
Proper Generalized Decomposition , Computer Methods in Applied Mechanics and Engineering,
vol. 199, n° 21-22, p. 1287 - 1296, 2010. Multiscale Models and Mathematical
Aspects in Solid and Fluid Mechanics.
Li J., Sun W. W., Spectral method of decoupling the vorticity and stream function for the
incompressible fluid flows, Computers and Mathematics with Applications, vol. 37, n° 1,
p. 35 - 40, 1999.
Liberge E., Hamdouni A., Reduced order modelling method via proper orthogonal decomposition
(POD) for flow around an oscillating cylinder, Journal of Fluids and Structures, vol.
, n° 2, p. 292 - 311, 2010.
Lieu T., Farhat C., Lesoinne M., Reduced-order fluid/structure modeling of a complete aircraft
configuration , Computer Methods in Applied Mechanics and Engineering, vol. 195, n° 41-
, p. 5730 - 5742, 2006. John H. Argyris Memorial Issue. Part II.
Mokdad B., Pruliere E., Ammar A., Chinesta F., On the simulation of kinetic theory models of
complex fluids using the FokkerâPlanck approach, Appl. Rheol., vol. 2, p. 1 - 14, 2007.
Nouy A., A generalized spectral decomposition technique to solve a class of linear stochastic
partial differential equations , Computer Methods in Applied Mechanics and Engineering,
vol. 196, n° 45-48, p. 4521 - 4537, 2007.
Nouy A., Maitre O. P. L., Generalized spectral decomposition for stochastic nonlinear problems
, Journal of Computational Physics, vol. 228, n° 1, p. 202 - 235, 2009.
Piller M., Stalio E., Finite-volume compact schemes on staggered grids , Journal of Computational
Physics, vol. 197, n° 1, p. 299 - 340, 2004.
Placzek A., Tran D.-M., Ohayon R., Hybrid proper orthogonal decomposition formulation for
linear structural dynamics , Journal of Sound and Vibration, vol. 318, n° 4-5, p. 943 - 964,
Ramsak M., Skerget L., Hribersek M., Zunic Z., A multidomain boundary element method
for unsteady laminar flow using stream function-vorticity equations , Engineering Analysis
with Boundary Elements, vol. 29, n° 1, p. 1 - 14, 2005.
Ryckelynck D., Reduction a priori de modeles thermomecaniques, Comptes Rendus
Mecanique, vol. 330, n° 7, p. 499-505, 2002.
Ryckelynck D., A priori hyperreduction method: an adaptive approach, Journal of Computational
Physics, vol. 202, p. 346-366, 2005.
Ryckelynck D., Hermanns L., Chinesta F., Alarcon E., An efficient a priori model reduction
for boundary element models , Engineering Analysis with Boundary Elements 29, vol. 29,
p. 796-801, 2005.
Taylor C., Hood P., A numerical solution of the Navier-Stokes equations using the finite element
technique , Computers and Fluids, vol. 1, p. 73 - 100, 1973.
Tezduyar T., Liou J., Ganjoo D., Incompressible flow computations based on the vorticitystream
function and velocity-pressure formulations , Computers and Structures, vol. 35,
n° 4, p. 445 - 472, 1990. Special Issue: Frontiers in Computational Mechanics.
Tokunaga H., Tanaka T., Ichinose K., Satofuka N., Numerical solutions of incompressible flows
in multiply connected domains by the vorticity stream function formulation , Computers and
Fluids, vol. 23, n° 2, p. 241 - 249, 1994.
Verdon N., Allery C., Beghein C., Hamdouni A., Ryckelynck D., Reduced-Order Modelling for
solving linear equations and non-linear equations , Communications in Numerical Methods
in Engineering, 2009.
