Research Areas

Mission Statement

The overarching goal of my research group (CAML) is multi-physics modeling: coupling solid deformation, fluid flow, chemical reactive-transport, and thermal processes. We utilize continuum theories, computational mechanics, and applied mathematics. For validation studies, we often collaborate with experimental groups.

CAML approaches a research problem on five fronts:

1. 1. Develop mathematical models or generalize existing ones using continuum theories, emphasizing coupled problems.
  2. Apply mathematical analysis to gain a deeper understanding of the predictive nature of these models: derive analytical solutions (if possible) and establish qualitative properties that solutions under these models satisfy.
  3. Devise robust numerical algorithms that respect the models' underlying physical constraints and mathematical properties even in the discrete setting.
  4. Leverage high-performance computing and machine learning tools to tackle large-scale practical problems.
  5. Validate the models using experiments by collaborating with experimental groups.

One can find descriptions of the four research areas currently pursued by CAML.

Collaborators

We are fortunate to collaborate actively with three excellent researchers who made our scientific journey enjoyable.

Prof. Jason Patrick, Assistant Professor, North Carolina State University (NCSU)
- - An experimentalist and a leading expert in fiber-reinforced polymer multi-functional composites and self-healing in synthetic materials.
Dr. Maruti Mudunuru, Staff Scientist, Pacific Northwest National Laboratory (PNNL)
- - An alumnus of CAML and an expert in subsurface modeling and machine learning modeling.
Professor Yi-Lung Mo, Moores Professor, University of Houston
- - An experimentalist and structural engineer with renowned expertise in seismic isolation and earthquake engineering.

Vascular-based Thermal Regulation

Background and Motivation. Vascular-based thermal modulation plays a crucial role in the functionality of biological systems, such as maintaining homeostasis, which ensures controlled physical and chemical conditions within the body and organs. Even in the synthetic world, leveraging bio-inspired temperature modulation is indispensable for the advancement of various modern technologies, including space probes, hypersonic aviation, electronic packaging, and implantable medical devices, among others. Given its diverse applications and its fundamental importance to physiological understanding, identifying the principal properties of thermal regulation holds significant potential to benefit researchers and propel the field forward. At CAML, we specialize in unraveling the mysteries of fluid-induced vascular-based thermal regulation.

We address the following scientific questions:

How to develop predictive “models” for thermal regulation in vascular systems?
What qualitative properties do solution fields satisfy under thermal regulation?
How do (material) properties and input parameters affect performance (e.g., thermal efficiency)?
How to steer heat in synthetic material systems?
What are the interactions between mechanical and thermal responses under vascular-based thermal regulation?
How to design efficient thermal regulation systems?

Selected publications

K. B. Nakshatrala, "Modeling thermal regulation in thin vascular systems: A mathematical analysis," Communications in Computational Physics, 33(4): 1035-1068, 2023. [Journal link] [arXiv link]
K. B. Nakshatrala, K. Adhikari*, S. R. Kumar, and J. F. Patrick, "Configuration-independent thermal invariants under flow reversal in thin vascular systems," PNAS Nexus, 2(8): pgad266, 2023. [Journal link] [UH eNews Coverage]
N. V. Jagtap*, M. K. Mudunuru, and K. B. Nakshatrala, "CoolPINNs: A physics-informed neural network modeling of active cooling in vascular systems," Applied Mathematical Modelling, 122: 265-287, 2023. [Journal link] [arXiv link]

An experimental validation of a reduced-order model developed at CAML using thermal regulation experiments on microvascular composites.

Degradation, Fracture, and Self-Healing in Composite Materials

Background and Motivation. Fiber-reinforced composites combine high strength and stiffness with low weight, but moisture, temperature variations, and mechanical loading can cause degradation, cracking, and delamination. Such damage is often difficult to detect and costly to repair. Self-healing composites can restore fracture resistance, yet repeated repair is limited by irreversible damage and changes in healing capacity. The central question is: How much mechanical performance can be recovered, how often, and under what conditions? Our research connects degradation and healing mechanisms with crack initiation and growth to establish the limits of repeated repair and guide the design of durable composites. This work is supported in part by an Army Research Office grant on fracture of self-healing composites.

Under this research area, we pursue the following:

Coupled-field modeling: Develop mathematical models of material degradation and healing that couple deformation, species diffusion, and heat transfer.
Fracture and damage mechanics: Develop phase-field formulations to predict crack initiation, propagation, and interactions with degradation and healing.
Numerical methods: Construct accurate formulations that preserve nonnegative concentrations and applicable maximum principles while resolving disparate material, spatial, and temporal scales.
Healing prediction and optimization: Quantify and optimize the recovery of mechanical performance in fiber-reinforced composites over repeated fracture–healing cycles.
Scalable simulation and inversion: Combine high-performance computing and machine learning to address large-scale engineering problems and identify material parameters from measured responses.
Experimental validation: Assess model predictions against targeted measurements of degradation, fracture, and healing through collaboration with experimental researchers.

Selected publications

M. K. Mudunuru*, and K. B. Nakshatrala, "A framework for coupled deformation-diffusion analysis with application to degradation / healing," International Journal for Numerical Methods in Engineering, 89: 1144-1170, 2012. [Journal link] [arXiv link]
C. Xu*, M. K. Mudunuru*, and K. B. Nakshatrala, “Material degradation due to moisture and temperature. Part 1: Mathematical model, analysis, and analytical solutions," Continuum Mechanics and Thermodynamics, 28:1847-1885, 2016. [Journal link] [arXiv link]
A. D. Snyder, Z. J. Phillips, J. S. Turicek, C. E. Diesendruck, K. B. Nakshatrala, and J. F. Patrick, "Prolonged in situ self-healing in structural composites via thermo-reversible entanglement," Nature Communications, 13: 6511, 2022. [Journal link] [Editors' highlights]
J. S. Turicek, Z. J. Phillips, K. B. Nakshatrala, and J. F. Patrick, "Self-healing for the long haul: In situ automation delivers century-scale fracture recovery in structural composites," Proceedings of the National Academy of Sciences (PNAS), 123 (2) e2523447123, 2026. [Journal link]

Realizing in situ self-healing in fiber-reinforced polymer composites via thermal remending.

In collaboration with Prof. Jason Patrick (NCSU).

Subsurface Modeling: Flow Through Porous Media

Background and Motivation. Flow and reactive transport in heterogeneous (deformable or rigid) porous media manifest in many important technological endeavors, such as geological carbon dioxide sequestration, seepage of contaminants, long-term storage of nuclear waste, and geothermal energy. Because of the inaccessibility of the subsurface, modeling plays a crucial role in making these technologies successful. But surface problems are notorious: (1) the presence of numerous coupled processes, (2) the existence of disparate spatial and temporal scales, (3) strong anisotropy, heterogeneity, and uncertainties in subsurface properties, (4) ubiquity of inverse problems, and (5) the large-scale nature of practical problems. The researchers at CAML work on both theoretical and computational fronts to advance the modeling capabilities of flow and reactive-transport in deformable (and fracturable) porous media.

Delving into the Depths: Harnessing Geothermal Energy from Beneath the Surface

In collaboration with Dr. Mudunuru (PNNL).

We pursue the following research activities:

On the theoretical front:
1. Develop predictive models for multi-component and multiphase flows in heterogeneous porous media leveraging the theory of interacting continua (TIC)
2. Incorporate and study the effect of mechanical deformation of the porous solid on the flow and transport
3. Provide critical reviews of current models used in the studies on flow through porous media
On the computational front:
1. Develop novel numerical (stabilized) formulations for models arising from TIC
2. Perform Verification, Validation, and Uncertainty Quantification (VV-UQ) studies
3. Develop high-performance and machine-learning tools
Under a recent project from EMSL (PNNL) (project link):
1. Leverage high-resolution pore-scale visualization to construct pore structure and understand concomitant processes
2. Devise multi-scale strategies and upscale soil structural information

Selected publications

K. B. Nakshatrala, H. S. H. Joodat*, and R. Ballarini, "Modeling flow in porous media with double porosity/permeability: Mathematical model, properties, and analytical solutions," Journal of Applied Mechanics, 85: 081009, 2018. [Journal link] [arXiv link]
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. [Journal link] [arXiv link] (Melsoh award was given to the student based on the research reported in this paper)
M. S. Joshaghani*, H. S. H. Joodat*, and K. B. Nakshatrala, “A stabilized mixed discontinuous Galerkin formulation for double porosity / permeability model," Computer Methods in Applied Mechanics and Engineering, 352: 508-560, 2019. [Journal link] [arXiv link]
N. V. Jagtap*, M. K. Mudunuru, and K. B. Nakshatrala, "A deep learning modeling framework to capture mixing patterns in reactive-transport systems," Communications in Computational Physics, 31: 188-223, 2022. [Journal link] [arXiv link]

Computational Method Development & Scientific Machine Learning

Background and Motivation. Computational predictions inform engineering design and scientific discovery, but their reliability depends on the methods used to obtain them. Strong coupling, material heterogeneity, and disparate scales can produce numerical instabilities, unphysical solutions, and excessive computational cost. Reliable simulation therefore requires formulations that preserve essential physical and mathematical properties while remaining practical for complex problems. These challenges motivate our research in computational method development, spanning finite element and lattice Boltzmann methods, physics-informed neural networks, and scientific machine learning. We seek to understand the strengths and limitations of these approaches and develop accurate, robust, and scalable methods for prediction, design, and inverse analysis.

Under this research area, we pursue the following:

Computational method development: Develop numerical formulations and algorithms for flow, transport, solid mechanics, and coupled multiphysics problems.
Structure-preserving formulations: Develop methods that respect conservation laws, nonnegative quantities, and applicable maximum principles.
Scientific machine learning: Develop physics-informed neural networks and adaptive learning strategies for solving governing equations and incorporating observational data.
Inverse problems: Identify material properties and model parameters from measured responses, and assess sensitivity to noise and data availability.
Mathematical analysis and verification: Establish stability and convergence, quantify numerical errors, and assess reliability using analytical solutions and benchmark problems.

Selected publications

S. Karimi*, and K. B. Nakshatrala, "On multi-time-step monolithic coupling algorithms for elastodynamics," Journal of Computational Physics, 273: 671-705, 2014. [Journal link] [arXiv link]
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. [Journal link] [arXiv link]
J. Chang*, and K. B. Nakshatrala, “Variational inequality approach to enforcing the non-negative constraint for advection-diffusion equations," Computer Methods in Applied Mechanics and Engineering, 320: 287-334, 2017. [Journal link] [arXiv link]
M. S. Joshaghani*, J. Chang, K. B. Nakshatrala, and M. G. Knepley, "Composable block solvers for the four-field double porosity / permeability model," Journal of Computational Physics, 386: 428-466, 2019. [Journal link] [arXiv link]
N. V. Jagtap*, M. K. Mudunuru, and K. B. Nakshatrala, "CoolPINNs: A physics-informed neural network modeling of active cooling in vascular systems," Applied Mathematical Modelling, 122: 265-287, 2023. [Journal link] [arXiv link]
V. S. Maduri* and K. B. Nakshatrala, "A machine learning-enhanced Hopf-Cole formulation for nonlinear gas flow in porous media," Computer Methods in Applied Mechanics and Engineering, 461: 119185, 2026. [Journal link] [arXiv link] [Code on GitHub]
V. S. Maduri* and K. B. Nakshatrala, “An adaptive machine learning framework for fluid flow in dual-network porous media,” accepted in Computer Methods in Applied Mechanics and Engineering, 2026. [Journal link]

Viscous fingering (Saffman-Taylor instability)

The figures below simulate viscous fingering in porous media and show the obtained concentration field, which, theoretically, should lie between zero and unity.

Classical numerical formulation produces negative values for the concentration field. Also, the solution violates the maximum principle, as the concentration from the formulation exceeds unity.

On the other hand, the formulation developed at CAML provides a physically meaningful concentration field -- the values lie between zero and unity.

Seismic Metamaterials

Background and Motivation. Periodic foundations/wave barriers capable of attenuating low-frequency waves (5 – 100 Hz) are now feasible. However, near-field earthquakes generate ultra-low-frequency pulses (0.01 – 5 Hz), rendering previous methods impractical due to their design limitations (e.g., the wave barrier size exceeding that of the superstructure). Thus, at CAML, in collaboration with Prof. Y. L. Mo and NCREE (Taiwan), we develop novel seismic metamaterials capable of attenuating ultra-low frequency waves besides low-frequency ones. Our approach is as follows:

Avail mathematical modeling, material design, and additive manufacturing in creating such seismic metamaterials
Leverage phononic (backscattering) and sonic (local resonance) ways of realizing band gaps to create novel seismic metamaterials
Understand and build a knowledge base of wave propagation in such seismic metamaterials
Perform lab-scale and field-scale studies to validate their performance

Selected publications

W. Witarto*, K. B. Nakshatrala, and Y. L. Mo, “Global sensitivity analysis of frequency band gaps in one-dimensional phononic crystals," Mechanics of Materials, 134: 38-53, 2019. [Journal link] [arXiv link]
H. W. Huang*, B. Zhang, J. Wang, F.-Y. Menq, K. B. Nakshatrala, Y. L. Mo, and K. H. Stokoe, "Experimental study on wave isolation performance of periodic barriers," Soil Dynamics and Earthquake Engineering, 144: 106602, 2021. [Journal link]
B. Zhang, H. W. Huang*, F. Menq, J. Wang, K. B. Nakshatrala, K. H. Stokoe, and Y. L. Mo, "Field experimental investigation on broadband vibration mitigation using metamaterial-based barrier-foundation system," Soil Dynamics and Earthquake Engineering, 155: 107167, 2022. [Journal link]

Field-scale testing of a periodic foundation -- a phononic-based seismic metamaterial -- using a thruster capable of generating seismic disturbances.

In collaboration with Prof. Yi-Lung Mo (UH).

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