Donald Brown, Ph.D.

Dr. Brown’s academic research focus is on multiscale modelling and simulation of porous and heterogenous materials and developing applicable mathematical tools and techniques.  The research has wide-ranging applications in the modeling of subsurface flows for oil and gas, lithium-ion batteries, composites, thermomechanical behavior, and corrosion.  Many complex materials processes have significant multiscale and multi-physical characteristics.  The multiscale nature of this media often makes solving the full problem intractable, and alternative techniques must be employed to bridge these scales.  Dr. Brown’s research is to develop computational tools and analysis techniques to deal with fundamental problems in multiscale processes of the complex physics. Dr. Brown has an interest in homogenization theory of flow and mechanics to obtain effective physical models, multilevel upscaling with complex nonlinearities and uncertainty, and development and analysis of multiscale methods for mechanics in heterogeneous materials.    

Dr. Brown’s recent interest includes application of these methods to real-world problems faced at Equity, particularly integration of large-scale finite element codes into the Equity Engineering Cloud, cloud-computing architectures, and prototyping compute environments.  The end goal is to provide Equity’s clients a coherent description and understanding of the challenges they face and fast delivery of solutions critical to day-to-day operations.

Awards:

1. Fraunhofer Institute for Industrial and Applied Mathematics & Interpore Society, Interpore-Fraunhofer Award for Young Researchers, Kaiserslautern, Germany, 2013.
2. Texas A&M University, L.F. Guseman Prize in Mathematics, College Station, TX, 2012.

Publications:

1. Brown, D.L., Gallistl, D., “Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds,” Computational Methods in Applied Mathematics 23.1 2023: 65-82, 2022.
2. Vasilyeva, M., Brown, D.L., et al., “Preconditioning Markov Chain Monte Carlo Method for Geomechanical Subsidence using multiscale method and machine learning technique,” Journal of Computational and Applied Mathematics 392: 113420, 2021.
3. Brown, D.L., Spring, D., Panzarella, C., “Level Three Assessment of Local Thin Areas in Pipelines Using Web-Enabled FEA,” Proceedings of the 2019 PVP Conference, San Antonio, TX, July 14-19, 2019.
4. Tyrylgin, A., Vasilyeva, M., Brown, D.L., “Generalized Multiscale Finite Element Method for Poroelasticity Problems in Heterogeneous Media,” 7th Conference on Finite Difference Methods: Theory and Applications, Lozenetz, Bulgaria, June 2018.
5. Pierce-Brown, J., Neves, L., Brown, D.L., “Uncertainty Quantification for Random Fields Estimated from Effective Modulus of Elasticity,” 8th International Workshop on Reliable Computing, Computing with Confidence, University of Liverpool, Liverpool, United Kingdom, July 2018.
6. Brown, D.L., Gedicke, J., Peterseim, D., “Numerical Homogenization of the Heterogeneous Fractional Laplacian,” Multiscale Modeling & Simulation 16.3 (2018): 1305-1332.
7. Brown, D.L., Hoang, V.H., “A Hierarchical Finite Element Monte Carlo Method for Stochastic Two-Scale Elliptic Equations,” Journal of Computational and Applied Mathematics Volume 323, pages 16-25, 2017.
8. Brown, D.L., Gallistl, D., Peterseim, D., “Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations,” LNCSE, Meshfree Methods for Partial Differential Equations VIII, pages 85-115, 2017.
9. Vignal, P., Collier, N., Dalcin, L., Brown, D.L., Calo, V., “An Energy-Stable Time-Integrator for Phase-Field Models,” Computer Methods in Applied Mechanics and Engineering 316, pages 1179-1214, 2017.
10. Collis, J., Brown, D.L., Reuben, O., Hubbard, M., “Effective Equations Governing an Active Poroelastic Medium,” Proceedings of the Royal Society A. Volume 473, Issue 2198, 2017.
11. Li, J., Brown, D.L., “Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media,” Geofluids Volume 2017, 2017.
12. Fellerman, J., Brown, D.L., Vasilyeva, M., “On Two-Scale Convergence of Fluid-Structure Interaction Problems with Applications to Poroelasticity,” Poromechanics VI, BIOT6 Poromechanics Conference Proceedings, Paris, France, 2017.
13. Brown, D.L., Peterseim, D., “A Multiscale Method for Porous Microstructures,” Multiscale Model. Simul., 14(3), pages 1123-1152, 2016.
14. Taralova, V., Brown, D.L., “On Multiscale Finite Elements for Neumann Problems in Porous Domains,” Disc. and Cont. Dyn. Sys Series S. 9(5), pages 1299-1326, 2016.
15. Brown, D.L., Vasilyeva, M., “Generalized Multiscale Finite Element Method for Poroelasticity Equations I:  Linear Problems,” Journal of Computational and Applied Mathematics 294, Issue C, pages 372-388, 2016.
16. Brown, D.L., Vasilyeva, M., “Generalized Multiscale Finite Element Method for Poroelasticity Equations II:  Nonlinear Coupling,” Journal of Computational and Applied Mathematics 297, 2016, pages 132-146, 2016.
17. Vignal, P., Collier, N., Dalcin, L., Brown, D.L., Calo, V., “An Energy-Stable Convex Splitting for the Phase-Field Crystal Equation,” Computers & Structures Volume 158, 1, pages 355-368, October 2015.
18. Brown, D.L., Li, G., Savatorova, V.L., Efendiev, Y., “Homogenization of Brinkmann Equations with High-Contrast Coefficients,” Multiscale Model. Simul., 13(2), pages 472-490, 2014.
19. Brown, D.L., Li, J., Calo, V., Ghommem, M., Efendiev, Y., “Lattice Boltzmann Flow Simulations with Applications of Reduced Order Modeling Techniques,” 7th International Petroleum Technology Conference, Doha, Qatar, January 2014.
20. Brown, D.L., Hoang, V.H., Efendiev, Y., “An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media,” SIAM MMS 11-1, pages 30-58, 2013.
21. Brown, D.L., Popov, P., Efendiev, Y., “Effective Equations for Fluid-Structure Interaction with Applications to Poroelasticity,” Applicable Analysis 93-4, pages 771-790, 2013.
22. Li, J., Brown, D.L., Calo, V., Efendiev, Y., Iliev, O., “On Multiscale Lattice Boltzmann Method for flow Simulations in Highly Heterogenous Porous Media,” SPE Reservoir Simulation Conference and Exhibition, Abu Dhabi, United Arab Emirates, September 2013.
23. Brown, D.L., Li, G., Savatorova, V.L., Efendeiv, Y., Popov, P., “Multiscale Modeling of High Contrast Brinkman Equations with Applications to Deformable Porous Media,” Poromechanics V, BIOT5 Poromechanics Conference, Vienna, Austria, July 2013.
24. Brown, D.L., Li, G., Savatorova, V.L., Efendeiv, Y., “Homogenization of High-Contrast Brinkman” extended abstract, “Mini Symposium Numerical Upscaling for Media with Deterministic and Stochastic Heterogeneity,” February 2013.

Conference & Training Presentations:

1. “Quieting a Chatterbox: Modeling and Simulation of PRD Chatter,” presented by D.L. Brown at the API Summit, January 2024, San Antonio, Texas.
2. “Numerical Homogenization of Non-Local Models,” presented by D.L. Brown at the UK-Interpore Meeting, University of Warwick, September 2017, Coventry, United Kingdom.
3. “Multiscale Methods in Elastodynamics,” presented by D.L. Brown at the MMR-Lab, August 2018, Yakutsk, Russia.
4. “On Two-Scale Convergence of Fluid-Structure Interaction Problems with Applications to Poroelasticity,” presented by D.L. Brown at Poromechanics VI, 2017, Paris, France.
5. “Multiscale Methods and Stability in Some High-Frequency Helmholtz Problems,” presented by D.L. Brown at the University of Augsburg, June 2017, Augsburg, Germany.
6. “A Hierarchical Finite Element MC Method for Stochastic Elliptic Equations,” presented by D.L. Brown at Interpore, May 2017, Rotterdam, the Netherlands.
7. “Generalized Multiscale FEM for Poroelasticity,” presented by D.L. Brown at MAFELAP, June 2016, London, United Kingdom.
8. “Multiscale LOD for Helmholtz Equations, Interpore,” presented by D.L. Brown, May 2016, Cincinnati, OH.
9. “Co-Organizer Mini-Symposium: Multiscale Methods with Applications to Fluids and Materials,” presented by D.L. Brown at ICIAM, August 2015, Beijing, China.
10. “Multiscale Methods in Porous Microstructures,” presented by D.L. Brown at Interpore, May 2015, Padua, Italy.
11. “Upscaling Lattice Boltzmann Simulation of Brinkmann Equations with Model Reductions Techniques,” presented by D.L. Brown at the International Conference on Numerical and Mathematical Modeling of Flow and Transport in Porous Media, September 2014, Dubrovnik, Croatia.
12. “Multilevel Upscaling with applications to Fluid-Structure Interaction,” presented by D.L. Brown at the Conference on Dynamical Systems, Differential Equations and Applications, July 2014, Madrid, Spain.
13. “Homogenization of High-Contrast Brinkman Flows with Applications to Deformable Porous Media,” presented by D.L. Brown at BIOT5, July 2013, Vienna, Austria.
14. “Multiscale Modeling of Deformable Porous Media,” presented by D.L. Brown at SIAM Geoscience, June 2013, Padua, Italy.
15. “Homogenization of High-Contrast Brinkman Flows,” presented by D.L. Brown at the Interpore Conference, May 2013, Prague, Czech Republic.
16. “Homogenization of High-Contrast Brinkman Flows,” presented by D.L. Brown at the Mini-Symposium on Numerical Upscaling, Oberwolfach Mathematics Institute, February 2012, Oberwolfach, Germany.
17. “Multilevel FEM for Stokes flow” and “Multiscale Fluid-Structure Interaction,” presented by D.L. Brown at the SIAM Annual Meeting, July 2012, Minneapolis, MN.