Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T15:42:19.742Z Has data issue: false hasContentIssue false
  • English
  • Français

Mechanics-Based Solution Verification for Porous Media Models

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  02 November 2016

M. Shabouei  and
K. B. Nakshatrala
M. Shabouei*
Affiliation:
Department of Civil & Environmental Engineering, University of Houston, USA
K. B. Nakshatrala*
Affiliation:
Department of Civil & Environmental Engineering, University of Houston, USA
*
*Corresponding author. Email addresses:[email protected] (M. Shabouei), [email protected] (K. B. Nakshatrala)
*Corresponding author. Email addresses:[email protected] (M. Shabouei), [email protected] (K. B. Nakshatrala)
Get access

Abstract

This paper presents a new approach to verify the accuracy of computational simulations. We develop mathematical theorems which can serve as robust a posteriori error estimation techniques to identify numerical pollution, check the performance of adaptive meshes, and verify numerical solutions. We demonstrate performance of this methodology on problems from flow thorough porous media. However, one can extend it to other models. We construct mathematical properties such that the solutions to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties include the total minimum mechanical power, minimum dissipation theorem, reciprocal relation, and maximum principle for the vorticity. All the developed theorems have firm mechanical bases and are independent of numerical methods. So, these can be utilized for solution verification of finite element, finite volume, finite difference, lattice Boltzmann methods and so forth. In particular, we show that, for a given set of boundary conditions, Darcy velocity has the minimum total mechanical power of all the kinematically admissible vector fields. We also show that a similar result holds for Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and Darcy-Brinkman velocities have the minimum total dissipation among their respective kinematically admissible vector fields. Using numerical examples, we show that the minimum dissipation and total mechanical power theorems can be utilized to identify pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman equations are shown to satisfy a reciprocal relation, which has the potential to identify errors in the numerical implementation of boundary conditions. It is also shown that the vorticity under both steady and transient Darcy-Brinkman equations satisfy maximum principles if the body force is conservative and the permeability is homogeneous and isotropic. A discussion on the nature of vorticity under steady and transient Darcy equations is also presented. Using several numerical examples, we will demonstrate the predictive capabilities of the proposed a posteriori techniques in assessing the accuracy of numerical solutions for a general class of problems, which could involve complex domains and general computational grids.

Keywords

Accuracy assessmentvalidation and verification (V&V)Darcy modelDarcy-Brinkman modelmechanical dissipationreciprocal relationsvorticitymaximum principles

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1127 - 1162
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] COMSOL Multiphysics User's Guide, Version 4.3b. COMSOL, Inc., Burlington, Massachusetts, http://www.comsol.com, 2013.Google Scholar
[2] Aage, N., Poulsen, T.H., Hansen, A. G., and Sigmund, O.. Topology optimization of large scale Stokes flow problems. Structural and Multidisciplinary Optimization, 35:175–180, 2008.CrossRefGoogle Scholar
[3] Ainsworth, M. and Oden, J. T.. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering, 142:1–88, 1997.CrossRefGoogle Scholar
[4] Babuška, I., Strouboulis, T., Mathur, A., and Upadhyay, C. S.. Pollution-error in the h-version of the finite-element method and the local quality of a-posteriori error estimators. Finite Elements in Analysis and Design, 17:273–321, 1994.Google Scholar
[5] Babuška, I., Ihlenburg, F., Strouboulis, T., and Gangaraj, S. K.. A posteriori error estimation for finite element solutions of Helmholtz equation-Part II: Estimation of the pollution error. International Journal for Numerical Methods in Engineering, 40:3883–3900, 1997.Google Scholar
[6] Babuška, I. and Oden, J. T.. Verification and validation in computational engineering and science: Basic concepts. Computer Methods in Applied Mechanics and Engineering, 193:4057–4066, 2004.Google Scholar
[7] Babuška, I. and Oh, H.-S.. Pollution problem of the p- and h-p versions of the finite element method. Communications in Applied Numerical Methods, 3:553–561, 1987.CrossRefGoogle Scholar
[8] Babuška, I. and Strouboulis, T.. The Finite Element Method and its Reliability. Oxford University Press, Oxford, 2001.Google Scholar
[9] Babuška, I., Strouboulis, T., Upadhyay, C. S., and Gangaraj, S. K.. A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method. International Journal for Numerical Methods in Engineering, 38:4207–4235, 1995.Google Scholar
[10] Batchelor, G. K.. An Introduction to Fluid Dynamics. Cambridge University Press, New York, 2000.Google Scholar
[11] Becker, R. and Rannacher, R.. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, 2001.Google Scholar
[12] Benamou, J.-D.. Computation of MULTI-VALUED TRAVELTIMES in the Marmousi model, 2014.Google Scholar
[13] Blottner, F. G.. Accurate Navier-Stokes results for the hypersonic flow over a spherical nosetip. Journal Spacecraft and Rockets, 27:113–122, 1990.Google Scholar
[14] Borrvall, T. and Petersson, J.. Topology optimization of fluids in Stokes flow. International Journal for Numerical Methods in Fluids, 41:77–107, 2003.CrossRefGoogle Scholar
[15] Botella, O. and Peyret, R.. Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids, 27:421–433, 1998.Google Scholar
[16] Botella, O. and Peyret, R.. Computing singular solutions of the Navier-Stokes equations with the Chebyshev-collocation method. International Journal for Numerical Methods in Fluids, 36:125–163, 2001.Google Scholar
[17] Brezzi, F. and Fortin, M.. Mixed and Hybrid Finite Element Methods, Volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.Google Scholar
[18] Brinkman, H. C.. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Scientific Research, A1:27–34, 1947.Google Scholar
[19] Brinkman, H. C.. On the permeability of the media consisting of closely packed porous particles. Applied Scientific Research, A1:81–86, 1947.Google Scholar
[20] Burggraf, O. R.. Analytical and numerical studies of the structure of steady separated flows. Journal of Fluid Mechanics, 24:113–151, 1966.Google Scholar
[21] Challis, V. J. and Guest, J. K.. Level set topology optimization of fluids in Stokes flow. International Journal for Numerical Methods in Engineering, 79:1284–1308, 2009.Google Scholar
[22] Darcy, H.. Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris, 1856.Google Scholar
[23] Erturk, E.. Discussions on driven cavity flow. International Journal for Numerical Methods in Fluids, 60:275–294, 2009.CrossRefGoogle Scholar
[24] Evans, L. C.. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 1998.Google Scholar
[25] Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order. Springer, New York, 2001.Google Scholar
[26] Guazzelli, E. and Morris, J. F.. A Physical Introduction to Suspension Dynamics. Cambridge University Press, Cambridge, 2012.Google Scholar
[27] Hansen, A. G., Sigmund, O., and Haber, R. B.. Topology optimization of channel flow problems. Structural and Multidisciplinary Optimization, 30:181–192, 2005.CrossRefGoogle Scholar
[28] Hassine, M.. Topology Optimization of Fluid Mechanics Problems. In Jones, S. A., editor, Advanced Methods for Practical Applications in Fluid Mechanics, pages 209–230. InTech, 2012.Google Scholar
[29] Hughes, T. J. R., Franca, L., and Balestra, M.. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering, 59:85–99, 1986.Google Scholar
[30] Kellogg, O. D.. Foundations of Potential Theory. Dover Publications, New York, 2010.Google Scholar
[31] Klimes, L.. Seismic Waves in Complex 3-D Structures (SW3D), Marmousi model and data set, Department of Geophysics, Charles University in Prague, 2014.Google Scholar
[32] Knupp, P. M. and Salari, K.. Verification of Computer Codes in Computational Science and Engineering. Chapman & Hall/CRC, Boca Raton, 2003.Google Scholar
[33] Masud, A. and Hughes, T. J. R.. A stabilized mixed finite element method for Darcy flow. Computer Methods in Applied Mechanics and Engineering, 191:4341–4370, 2002.Google Scholar
[34] Mudunuru, M. K. and Nakshatrala, K. B.. On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method. Journal of Computational Physics, 305:448–493, 2016.CrossRefGoogle Scholar
[35] Mudunuru, M. K., Shabouei, M., and Nakshatrala, K. B.. On local and global species conservation errors for nonlinear ecological models and chemical reacting flows. In ASME 2015 International Mechanical Engineering Congress and Exposition, pages V009T12A018–V009T12A018. American Society of Mechanical Engineers, 2015.Google Scholar
[36] Nakshatrala, K. B., Nagarajan, H., and Shabouei, M.. A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations. Communucation in Computational Physics, 19:53–93, 2016.Google Scholar
[37] Nakshatrala, K. B. and Rajagopal, K. R.. A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium. International Journal for Numerical Methods in Fluids, 67:342–368, 2011.Google Scholar
[38] Nakshatrala, K. B., Turner, D. Z., Hjelmstad, K. D., and Masud, A.. A stabilized mixed finite element formulation for Darcy flow based on a multiscale decomposition of the solution. Computer Methods in Applied Mechanics and Engineering, 195:4036–4049, 2006.Google Scholar
[39] Oberkampf, W. L. and Blottner, F. G.. Issues in computational fluid dynamics: Code verification and validation. AIAA Journal, 36:687–695, 1998.Google Scholar
[40] Oberkampf, W. L., Trucano, T. G., and Hirsch, C.. Verification, validation, and predictive capability in computational engineering and physics. Applied Mechanics Reviews, 57:345–384, 2004.Google Scholar
[41] Oden, J. T., Feng, Y., and Prudhomme, S.. Local and pollution error estimation for Stokesian flows. International Journal for Numerical Methods in Fluids, 27:33–39, 1998.3.0.CO;2-3>CrossRefGoogle Scholar
[42] Pao, C. V.. Nonlinear Parabolic and Elliptic Equations. Springer-Verlag, New York, USA, 1993.CrossRefGoogle Scholar
[43] Pingen, G., Evgrafov, A., and Maute, K.. Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Computers and Fluids, 38:910–923, 2009.CrossRefGoogle Scholar
[44] Rider, W., Witkowski, W., Kamm, J. R., and Wildey, T.. Robust verification analysis. Journal of Computational Physics, 307:146–163, 2016.Google Scholar
[45] Roache, P. J.. Quantification of uncertainty in computational fluid dynamics. Annual Review of Fluid Mechanics, 29:123–160, 1997.CrossRefGoogle Scholar
[46] Roache, P. J.. Verification and Validation in Computational Science and Engineering. Hermosa Publishers, New Mexico, 1998.Google Scholar
[47] Roy, C. J., Nelson, C. C., Smith, T. M., and Ober, C. C.. Verification of Euler/Navier-Stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44:599–620, 2004.Google Scholar
[48] Sadd, M. H.. Elasticity: Theory, Applications, and Numerics. Academic Press, Burlington, Massachusetts, 2009.Google Scholar
[49] Salari, K. and P. M. Knupp, . Code Verification by the Method of Manufactured Solutions. Technical Report SAND2000-1444, Sandia National Laboratories, Albuquerque, 2000.Google Scholar
[50] Truesdell, C. and Noll, W.. The Non-Linear Field Theories of Mechanics. Springer, Berlin, third edition, 2004.CrossRefGoogle Scholar
[51] Versteeg, R. J.. Sensitivity of prestack depth migration to the velocity model. Geophysics, 58:873–882, 1993.Google Scholar
[52] Versteeg, R. J. and Grau, G.. The Marmousi experience. In Practical aspects of seismic data inversion, European Association of Exploration Geophysicists, EAEG Workshop, 52nd EAEG Meeting, pages 1–194, Copenhagen, 1990.Google Scholar
[53] Versteeg, R. J. and Lailly, P.. In Practical aspects of seismic data inversion, European Association of Exploration Geophysicists, EAEG Workshop Report: First Break, pages 75–80, 1991.Google Scholar
[54] Wahlbin, L. B.. Local behavior in finite element methods. Handbook of Numerical Analysis, 2:353–522, 1991.Google Scholar