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.

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]

Modeling Degradation/Healing in Composite Materials

Background and Motivation. Every material degrades over time. However, some natural material systems (e.g., tissue, granular materials) are known to heal over time. The behavior and durability of materials and structures are significantly influenced by factors such as temperature fluctuations, moisture levels, exposure to reactive or inert chemical agents, radiation, and the presence of electrical and magnetic fields. Fatigue should not be underestimated in this context. Recent advancements have led researchers to achieve in situ self-healing capabilities even in synthetic materials, particularly in challenging material systems like fiber-reinforced polymer composites. At CAML, we actively investigate various degradation mechanisms and explore self-healing in composite materials.

Under this research area, we pursue the following:

Develop coupled-field mathematical models for degradation/healing of materials
Develop accurate numerical formulations respecting the physical constraints (e.g., non-negative constraint) and mathematical principles (e.g., maximum principles) and resolving disparate material, spatial, and temporal scales
Using mathematical and modeling tools, predict, and optimize self-healing in fiber-reinforced composites
Formulate material design algorithms using topology optimization by accounting for degradation and understand the resulting designs
Leverage high-performance computing and machine learning tools to solve large-scale practical problems and inverse problems
Validate models using state-of-the-art experiments

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]

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

In collaboration with Prof. Jason Patrick (NCSU).

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).

Structure-Preserving Numerical Formulations

At CAML, we work on fundamental numerical analysis and computational mechanics aspects. We develop accurate numerical algorithms for problems that frequently arise in multiple branches of engineering. Some related topics we pursue are as follows:

Maximum principles & the non-negative constraint: Transport equations satisfy the so-called maximum principles. Many numerical formulations, in general, do not satisfy maximum principles and physical restrictions (e.g., non-negativity of concentration). Stated differently, these formulations produce unphysical negative solutions for transport problems, especially under anisotropic medium properties. Researchers at CAML develop numerical formulations that satisfy the discrete version of maximum principles and meet physical constraints such as non-negativity.
Stabilized and mixed numerical formulations: The Galerkin weak formulation—the vanilla flavor finite element formulation—does not necessarily produce accurate solutions when odd derivatives in the model dominate (e.g., advection-dominated problems) or when one has to discretize multi-fields in a model (e.g., velocity and pressure fields in Stokes equations). At CAML, we develop stabilized and mixed formulations to overcome such numerical pathologies.
Transient domain decomposition methods: Over the last few decades, many domain decomposition (DD) methods have been developed for static problems. However, developing DD methods for transient problems is still in its infancy and is currently an active area of research. At CAML, we develop novel domain decomposition methods for first- and second-order transient systems that enable (i) different time steps in different subdomains, (ii) different time integrators in different subdomains, (iii) can handle multiple subdomains and cross points, and (iv) applicable even for nonlinear problems. We will also implement the proposed DD methods for large-scale problems and study their parallel performance.

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.

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