Main Research Projects
My research interests have links to three active areas of research : Phase field modeling, Bio-medical applications and electrically driven flows. My research is supported by New Mexico IDeA Networks of Biomedical Research (NM-INBRE). I am currently a NM-INBRE NIRF Faculty Fellow (NM-INBRE Research Faculty Program) 2024. Research for my students is supported by the NM-INBRE NISE (New Mexico INBRE Summer Experience) grant.
High Diffusion
Intermediate Diffusion
Low Diffusion
Bio-Medical Applications : this research is concerned with the better understanding of diseases in the body such as atherosclerosis. The above simulation describes the evolution of a drug concentration as it travels inside the artery and through the damaged region. Different diffusion coefficients are considered to account for effectiveness of drug near and past the damaged region.
Numerical Solution to the Cahn-Hilliard equation with variable mobility in 3 space dimensions: u(x,y,z,t) with uneven volume fractions of materials. Bottom 3 figures model even volume fraction.
Phase-field modeling: we study models for micro and nano-structure evolution of materials. Non-linear partial differential equations that have applications to materials science are studied (Cahn-Hilliard Equation, Phase-field crystal equation, Block-copolymer and Thin-film models). Our research is aimed at 1) developing efficient numerical methods with high order of accuracy and 2) modeling physical instabilities (forced - as a guiding tool or imbedded-in the nature of the model structure) to gain a better understanding of the material evolution with the purpose to improve the quality of materials.
Phase-field modeling continued: simulations for the solution to a higher order water-oil-surfactant model - collision of two drops - simulation done with a new proposed approach in 2023 - THM_{LX} - method using a 128^3 mesh grid.
Evolution of thin-film coatings: solution to the Thin-film Equation (TF) capturing the drop formation process from an initial configuration of five small droplets. An adaptive (variable) time-stepping algorithm is implemented. Picture on the right shows the different time steps used during the evolution of the drop formation.
Research Work with Students:
Current students:
Sampson Owusu - Industrial and Applied Mathematics Ph.D. student at NMT (Main Research Advisor/Academic Advisor) - research areas in computational math, modeling aspects of Wave Equations with variable damping coefficients. Upcoming projects: Phase Field Crystal PFC Equation and Diblock Copolymer BCP Equation. Recent work with the Functionalized Cahn-Hilliard FCH equation (under review).
Michael Millard - Accelerated M.S. at NMT (NM-INBRE-Advisor) - research areas in computational math, math biology, bio-medical applications (NM-INBRE 2024 funded project). Also GPU computations of phase field models. One paper in preparation.
M.Millard - Presentation at SRS New Mexico Tech 2024 - a simple approach for the CH equation using GPU architecture -Poster.
Ogochukwu Ifeacho - Ph.D. Student at NMT (Co-Advisor) - research areas computational math, math biology, mathematics education. One joint paper in the area of computational mathematics (under review).
Past students:
Logan Forman - B.S. Degree at NMT (Research Mentor) - research areas in computational math, GPU acceleration of computations and AI. One joint paper in the area of computational mathematics (under review).
Jesus Villegas - M.S. Degree at NMT (Co-Advisor)- research interest in computational math and biomedical applications. One joint paper in the area of mimetic methods (under review).
Stephen Jurgensen - B.S. Degree at NMT - (Directed Study) - On some numerical methods for ODEs. Project Report
New Students: Interested students in research work in the area of applied and computational mathematics are always welcomed. Send me an email or stop by the office to talk about interesting and challenging math problems.
Recent and Upcoming Talks/Conferences/Mini-Symposiums
7/19/2024: SIAM AN24 - VIRTUAL Mini-symposium organizer - MS7 Phase Field Models: Theoretical Results, Computational Advances and Applications. Session info
7/10/2024: SIAM AN24 - In person - Mini Symposium organizer - MS49 Workshop Celebrating Diversity: The Importance of Mathematical Modeling and Simulation for Time Dependent Problems - Session info
5/3/2024: New Mexico Tech Colloquium - "recent research results in the area of phase field modeling".
4/16/2024: NMT SRS 2024 - "A Simple Approach for CH Equation Using GPU Architecture" - M. Millard, S. Orizaga - Poster Presenter.
4/8/2024: Arizona State University - "introduction to computational mathematics + Methods for TF equations".
3/1/2024 - University Wide Research Colloquium: NMT, Sandia National Laboratories (SNL) and Los Alamos National Laboratory (LANL). Research talk "on some recent advances for phase field models and computation".
1/31/2024 - San Diego State University : Virtual Latin America Colloquium. Research talk "on TF equations using new efficient methods".
Recent Research Grant Proposal Activities
2/02/2024 - NM NIRF Fellows Proposal Application Submitted (Single PI - 1/9 summer support - Awarded).
1/25/2024 - NSF-DMS Research Grant Proposal Submitted (Single PI - 250k - pending).
1/08/2024 - NM INBRE Student Experience NISE (Summer Student Funding: PI/Mentor - 6k - Awarded).
1/04/2024 - NSF Office hours - meeting with NSF officers for the DMS Webinar part 2.
12-23-2023 -Travel Research Grant Submitted (Single PI - 50 k - pending).
12/14/2023 - NSF Office hours - meeting with NSF officers for the DMS Webinar part 1.
06/10/2022 - Academic Careers Workshop - Chicago, Illinois.
---Recent Research Work---
Papers (In preparation)
A simple approach for CH equation using GPU architecture [w/M. Fabien - University of Wisconsin at Madision]. To be Submitted - soon. 🖥️💾
On drug delivery in a catheterized artery II [w/ Daniel N. Riahi, M. Millard (NM-INBRE) - UIUC/UTRGV].
A review of phase field models [w/K. Glasner - University of Arizona].
Mathematical modelling and simulation for lighting return stroke [w/C. da Silva - NMT]
Some physical characterizations in materials science problems [w/T. Witelski - Duke University]
Papers Submitted (under review)
Mimetic methods for Allen-Cahn equation [ w/ Gilberto Gonzalez Parra, Logan Forman and Jesus V.- NMT ]. Under Revision
Publications in 2024
On an efficient numerical procedure for the Functionalized Cahn-Hilliard (FCH) equation [w/ Ogochukwu Ifeacho and Sampson Owusu- NMT ], AIMS -Mathematics (advances in mathematical modeling and simulation of complex systems), 2024.
IMEX Methods for Thin-Film Equations and Cahn-Hilliard equations with variable mobility [w/Thomas Witelski - Duke University] - Computational Materials Science (Elsevier), 2024.
On modeling arterial blood flow with or without solute transport and in presence of atherosclerosis, Special Issue: The importance of simulations in biophysical applications. AIMS Biophysics - 2024.
Publications in 2023
On viscoelastic fiber jet formation by force-spinning at high rotation rate, Journal of Theoretical and Applied Mechanics (JTAM) - 2023.
Modeling and computation for unsteady blood flow and solute concentration in a constricted porous artery, AIMS Bioengineering - 2023.
Publications in 2022
D. N. Riahi, S. Orizaga, On three-dimensional rotating viscoelastic jets in the Giesekus model, Partial Differential Equations in Applied Mathematics, Elsevier, 2022.
D. N. Riahi, S. Orizaga, On rotationally driven nonlinear inclined polymeric jet with gravity effect[J], Mathematics in Engineering, 2022, 4(2): 1-18.
Publications in 2021
R. Fisher, M. Garcia, N. Rudrapatna, S. Orizaga and L. Wang, Comparing Numerical Solution Methods for the Cahn-Hilliard Equation , SIAM Undergraduate Research Online SIURO Volume 14, (https://www.siam.org/publications/siuro/volume-14) (2021).
Publications in 2020
S. Orizaga., Riahi, D. N. , Soto, J. R., Drug delivery in catheterized arterial blood flow with atherosclerosis, Results in Applied Mathematics (Elsevier), 7, 100117 (2020).
Recent Selected Publications
K. Glasner, S. Orizaga, Multidimensional equilibria and their stability in copolymer solvent mixtures, Physica D: Nonlinear Phenomena 373 (2018).
S. Orizaga, D.N. Riahi, L. Steven Hou, Triad resonant wave interactions in electrically charged jets, Applied Mathematics and Mechanics (2017).
S. Orizaga, K. Glasner, Instability and reorientation of block copolymer microstructure by imposed electric fields, Physical Review E, 93 052504 (2016).
Karl Glasner, S. Orizaga, Improving the accuracy of convexity splitting methods for gradient flow equations, Journal of Computational Physics 315 (2016) 5264.