Research Areas

Researchers at CAML are interested in a wide variety of research topics. The research is spread across various disciplines: solid mechanics, fluid mechanics, heat transfer, and high-performance computing. CAML approaches any research problem they pursue on five fronts:

• 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 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 tools to tackle large-scale practical problems, and

  5. Validate the models using experiments by collaborating with experimental groups.

Currently, the following research topics are actively pursued.

A detailed modeling of flow and reactive transport in heterogeneous (deformable or rigid) porous media is vital for addressing many important problems like carbon-dioxide sequestration, enhanced oil recovery, improved efficiency in fuel cells, seepage of contaminants, and long-term storage of nuclear waste. Although existing mathematical models, software packages, and numerical formulations address some aspects of these problems, it is widely acknowledged that much more work needs to be done (in both modeling aspects and numerical techniques) to be able to perform reliable and predictive simulations. The researchers at CAML work on both theoretical and computational fronts to existing modeling capabilities of flow and reactive-transport in deformable porous media.

• Some theoretical aspects of interest include:

  • New and better models for multi-component and multiphase flows in heterogeneous porous media

  • Critical reviews of current models used in the studies of flow through porous media (e.g., models based on the theory of interacting continua, porosity models)

  • Modeling the dependence of viscosity on the pressure, and its effect on the flow and the deformation of the porous solid

  • Effect of the deformation of the porous solid on the flow and transport

• Some computational aspects of interest include:

  • Novel stabilized, mixed, and/or least-squares formulations for flow and transport equations

  • Multi-scale methods for capturing heterogeneity

  • Pore-scale visualization (e.g., high-resolution visualization of flow patterns in 2D and 3D) and digital imaging and construction of pore-structure

  • New and robust solvers, coupling algorithms, and adaptive mesh refinement

  • Parallel computing related to porous media

  • Uncertainty quantification and stochastic modeling

• Some specific engineering and technological applications of this research are

  • Enhanced oil recovery (EOR)

  • Geological carbon-dioxide sequestration

  • Modeling seepage of contaminants

The behavior of materials/structures is strongly affected by a number of factors, which include temperature, moisture, exposure to reactive or inert chemical species, radiation, and electrical and magnetic fields. Fatigue is another major factor that affects the behavior and serviceability of structures. The researchers at CAML work on computational, experimental and theoretical techniques for modeling degradation of materials/structures with a particular emphasis on environmental factors and fatigue. Some specific topics of interest are as follows:

• Constitutive modeling: New and better constitutive models for degradation of materials under harsh environment; constitutive modeling of fatigue; effect of anisotropy and heterogeneity on degradation and fatigue; and critical reviews of existing constitutive models.

• Computational techniques: New computational techniques to simulate various degradation mechanisms, predictive simulation capabilities for studying fatigue; and multiscale modeling of degradation of materials and fatigue.

• Design procedures: Current design requirements / procedures for fatigue; critical review on the current design procedures for designing structures against harsh environment; and suggestions for new design procedures and philosophies.

• Mathematical approaches: Application of mathematical techniques or results (from Analysis, Partial Differential Equations, Dynamical Systems, Statistics, Probability) to address problems on degradation of materials; and stochastic modeling of fatigue.

• Stabilized formulations: flow, transport & structural problems: At CAML, researchers develop stabilized numerical formulations for problems arising in solids, fluids and heat transfer. We shall employ various techniques, which include least-squares finite element method (LSFEM), variational multiscale (VMS) formalism. The researchers at CAML have been successful in developing stabilized one-field (primal) or two-field (mixed) formulations for Darcy flow, non-Darcy flows (modified Darcy and Darcy-Forchheimer equations), nearly incompressible linearized elasticity, advection-diffusion equation, Stokes flow, incompressible Navier-Stokes. We are currently developing novel numerical formulations for more complicated mathematical models arising in engineering.

• Coupling algorithms for fluid-structure interaction: Fluid-structure interaction is a classical example of a multiscale problem. The time scales of the structure are different from that of the fluid. Some examples of fluid-structure include: blood flow in deformable arteries, flow past a tall (flexible) building, flow through deformable porous media. All these problems are difficult to solve are the governing equations are coupled and nonlinear. One has to resort to numerical solutions as one cannot find analytical solutions in general. An important ingredient of a fluid-structure simulator is coupling algorithm that couples individual analyses (in this case, fluid and structure) to give the coupled response. Researchers CAML develop stable and accurate coupling algorithms. Both staggered and monolithic schemes are being considered, and stability analysis is also being performed. The application areas are: flow through deformable porous media, and blood flow in deformable arteries.

• Maximum principles & the non-negative constraint: Some elliptic and parabolic partial differential equations (PDEs) satisfy the so-called (continuous) maximum principle. For example, the Laplace's equation (which is a second-order elliptic PDE) and transient heat equation (which is a second-order parabolic PDE). The maximum principle says that (under certain assumptions on regularity of the domain and forcing function) the maximum can occur only on the boundary or in the initial condition. Under some conditions, it can be shown that the non-negative constraint is a consequence of maximum principles. The discrete version of the (continuous) maximum principle is commonly refer to as the discrete maximum principle (DMP). Many numerical formulations, in general, do not satisfy maximum principles. One has to impose restrictions on the mesh or material properties. For example, under the Galerkin formulation for the Poisson's equation, an arbitrary computational grid will not satisfy maximum principles. A sufficient condition for a triangular mesh to satisfy the DMP for the Poisson's equation is Delaunay mesh. Even mixed formulations and many popular stabilized formulations (in general) do not satisfy the DMP. A significant amount of research has been done on steady-state scalar diffusion equation. Many researchers have derived necessary, and sufficient conditions for the scalar diffusion equation. Now the current research is to develop novel numerical formulations to satisfy the DMP unconditionally for tensorial diffusion equation, time dependent problems (parabolic PDEs), and advective-diffusive-reactive systems. Researchers at CAML have developed a novel optimization-based method for tensorial diffusion equation to satisfy the DMP unconditionally for the lowest-order Raviart-Thomas (RT0), and variational multiscale mixed formulations.

• Transient domain decomposition methods: Over last couple of decades many domain decomposition (DD) methods have been developed for static problems. But developing DD methods for transient problems is still in its infancy, and is currently an active area of research. To this end, we have developed a novel domain decomposition for first-order transient systems that enables (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 have also developed four variants of domain decomposition methods for first-order linear transient systems based on trapezoidal family of time integrators, and also assessed their stability. In future, researchers at CAML will develop novel DD methods for second-order (linear and nonlinear) transient systems, perform mathematical stability analysis of the proposed DD methods for both first- and second-order transient systems in the nonlinear regime, and implement the proposed DD methods for large scale problems, and study their parallel performance.

A part of this research is in collaboration with Dr. Justin Chang (my former student and currently a postdoctoral research associate at Rice University) and Dr. Matthew Knepley (Department of Computer Science and Engineering, University at Buffalo).

Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to determine whether a computational framework is performing efficiently or is scalable. We develop various strategies for better understanding the performance of any parallel computational frameworks for solving PDEs. Important performance issues that negatively impact time-to-solution are taken into account, and we develop performance spectrum analysis strategies that can enhance one's understanding of critical aforementioned performance issues. In particular, we develop performance spectrum, which collectively assess the performance on three levels: intensity (which measure hardware efficiency), rate (which measure algorithmic efficiency) and time to solution (which measure optimal performance). We examine commonly used finite element simulation packages and software and apply the performance spectrum to quickly analyze the performance and scalability across various hardware platforms, software implementations, and numerical discretizations. The performance spectrum will be particularly useful for understanding hardware performance in a massively parallel computing environment.