Journal Articles

The following journal papers can supplement the lecture notes. It is recommended that students read at least a couple of papers thoroughly. 

• [1965] A. E. Green, and P. M. Naghdi, A dynamical theory of interacting continua, International Journal of Engineering Science, 3: 231-241. 

• [1966] N. Mills, Incompressible mixtures of Newtonian fluids, International Journal of Engineering Science, 4: 97-112. 

• [1968] P. A. C. Raats, Forces acting upon the solid phase of a porous media, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 19: 606-613. 

• [1976] R. J. Atkin, and R. E. Craine, Continuum theories of mixtures: Basic theory and historical development, The Quarterly Journal of Mechanics and Applied Mathematics, 29: 209-244.

• [1978] D. S. Drumheller, The theoretical treatment of a porous solid using a mixture theory, International Journal of Solids and Structures, 14: 441--456.

• [1980] R. M. Bowen, Incompressible porous media model by use of the theory of mixtures, International Journal of Engineering Science, 18: 1129-1148. 

• [1982] R. M. Bowen, Compressible porous media model by use of the theory of mixtures, International Journal of Engineering Science, 20: 697-735.  

• [1983] A. Bedford, and D. S. Drumheller,  Theories of immiscible and structured mixtures, International Journal of Engineering Science, 21: 863-960.   

• [1989] A. C. Hansen,  Reexamining some basic definitions of modern mixture theory, International Journal of Engineering Science, 27: 1531-1544.   

• [1991] A. C. Hansen, R. L. Crane, M. H. Damson, R. P. Donovan, D. T. Horning, and J. L. Walker, Some notes on a volume fraction mixture theory and a comparison with the kinetic theory of gases, International Journal of Engineering Science, 29: 561-573. 

• [1994] A. C. Hansen, J. L. Walker, and R. P. Donovan, A finite element formulation for composite structures based on a volume fraction mixture theory, International Journal of Engineering Science, 32: 1-17.  

• [1990] S. Majid Hassanizadeh, and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Advances in Water Resources, 13: 169-186.   

• [2011] J. Niessner, S. Berg, and S. Majid Hassanizadeh, Comparison of two-phase Darcy's law with a thermodynamically consistent approach, Transport in Porous Media, 88: 133-148. 

• [2009] K. B. Nakshatrala, and A. J. Valocchi, Non-negative mixed finite element formulations for a tensorial diffusion equation, Journal of Computational Physics, 228: 6726-6752.   

• [2011] H. Nagarajan, and K. B. Nakshatrala, Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids, International Journal of Numerical Methods in Fluids, 67: 820-847. 

• [2013] K. B. Nakshatrala, M. K. Mudunuru, and A. J. Valocchi, A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and the non-negative constraint, Journal of Computational Physics, 253: 278-307.  

• [2016] M. K. Mudunuru, and K. B. Nakshatrala, 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] K. B. Nakshatrala, H. Nagarajan, and M. Shabouei, A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations, Communications in Computational Physics, 19(1): 53-93.

• [1992] L. P. Franca, S. L. Frey, and T. J. R. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model, Computer Methods in Applied Mechanics and Engineering, 95: 253-276.   

• [2009] P. W. Hsieh, and S. Y. Yang, On efficient least-squares finite element methods for convection-dominated problems, Computer Methods in Applied Mechanics and Engineering, 199: 183-196. 

• [2010] K. B. Nakshatrala, and A. J. Valocchi, Variational structure of the optimal artificial diffusion method for the advection–diffusion equation, International Journal of Computational Methods, 7: 559-572.

• [2011] D. Z. Turner, K. B. Nakshatrala, and K. D. Hjelmstad, A stabilized formulation for the advection–diffusion equation using the Generalized Finite Element Method, International Journal for Numerical Methods in Fluids, 66: 64-81.  

• [2016] M. K. Mudunuru, and K. B. Nakshatrala, 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. 

• [1997] T. Y. Hou, and X.-H. Wu, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Advances in Water Resources, 13: 169-186. 

• [2000] T. Arbogast, Comparison of two-phase Darcy's law with a thermodynamically consistent approach, Transport in Porous Media, 88: 133-148. 

• [1997] X. He, and L.-S. Luo, Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Physical Review E, 56(6): 6811.   

• [2000] L. Pierre, and L.-S.Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Physical Review E, 61(6): 6546. 

• [2002] S. Ansumali and I.V. Karlin, Entropy function approach to the lattice Boltzmann method, Journal of Statistical Physics, 107: 291-308.

• [2006] I. V. Karlin, S. Ansumali, C. E. Frouzakis, and S. S. Chikatamarla, Elements of the lattice Boltzmann method I: Linear advection equation, Communications in Computational Physics, 1(4): 616-655. 

• [2010] H. Yoshida, and M. Nagaoka, Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation, Journal of Computational Physics, 229: 7774-7795.         

• [2014] R. Huang, and H. Wu, A modified multiple-relaxation-time lattice Boltzmann model for convection–diffusion equation, Journal of Computational Physics, 274: 50-63.

Some references on hybrid methods will be added soon.