Publications

Shadid, John N.

Abstract not provided.

Sala, Marzio S.; Shadid, John N.; Tuminaro, Raymond S.

Abstract not provided.

Sala, Marzio S.; Shadid, John N.; Tuminaro, Raymond S.

Abstract not provided.

Proposed for publication in Computation Methods in Applied Mechanics and Engineering.

Shadid, John N.; Salinger, Andrew G.; Pawlowski, Roger P.; Lin, Paul L.; Hennigan, Gary L.; Tuminaro, Raymond S.; Lehoucq, Richard B.

The solution of the governing steady transport equations for momentum, heat and mass transfer in fluids undergoing non-equilibrium chemical reactions can be extremely challenging. The difficulties arise from both the complexity of the nonlinear solution behavior as well as the nonlinear, coupled, non-symmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this paper, we briefly review progress on developing a stabilized finite element ( FE) capability for numerical solution of these challenging problems. The discussion considers the stabilized FE formulation for the low Mach number Navier-Stokes equations with heat and mass transport with non-equilibrium chemical reactions, and the solution methods necessary for detailed analysis of these complex systems. The solution algorithms include robust nonlinear and linear solution schemes, parameter continuation methods, and linear stability analysis techniques. Our discussion considers computational efficiency, scalability, and some implementation issues of the solution methods. Computational results are presented for a CFD benchmark problem as well as for a number of large-scale, 2D and 3D, engineering transport/reaction applications.

Proposed for publication in International Journal for Numerical Methods in Engineering.

Sala, Marzio S.; Shadid, John N.; Tuminaro, Raymond S.

This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton-Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behavior of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non-zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two-level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large-scale distributed-memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two- and three-level preconditioners are demonstrated to be scalable to 1024 processors.

Shadid, John N.; Tuminaro, Raymond S.; Sala, Marzio S.

Abstract not provided.

Lin, Paul L.; Sala, Marzio S.; Shadid, John N.; Tuminaro, Raymond S.

Abstract not provided.

Proposed for publication in Journal of Computational Physics.

Tuminaro, Raymond S.; Shadid, John N.

Two heuristic strategies intended to enhance the performance of the generalized global basis (GGB) method [H. Waisman, J. Fish, R.S. Tuminaro, J. Shadid, The Generalized Global Basis (GGB) method, International Journal for Numerical Methods in Engineering 61(8), 1243-1269] applied to nonlinear systems are presented. The standard GGB accelerates a multigrid scheme by an additional coarse grid correction that filters out slowly converging modes. This correction requires a potentially costly eigen calculation. This paper considers reusing previously computed eigenspace information. The GGB? scheme enriches the prolongation operator with new eigenvectors while the modified method (MGGB) selectively reuses the same prolongation. Both methods use the criteria of principal angles between subspaces spanned between the previous and current prolongation operators. Numerical examples clearly indicate significant time savings in particular for the MGGB scheme.

Proposed for publication in the Journal of Fluid Mechanics.

Salinger, Andrew G.; Shadid, John N.

Abstract not provided.

Proposed for publication in Applied Numerical Mathematics.

Shadid, John N.; Romero, L.A.

Abstract not provided.

Bochev, Pavel B.; Christon, Mark A.; Collis, Samuel S.; Lehoucq, Richard B.; Shadid, John N.; Slepoy, Alexander S.

Existing approaches in multiscale science and engineering have evolved from a range of ideas and solutions that are reflective of their original problem domains. As a result, research in multiscale science has followed widely diverse and disjoint paths, which presents a barrier to cross pollination of ideas and application of methods outside their application domains. The status of the research environment calls for an abstract mathematical framework that can provide a common language to formulate and analyze multiscale problems across a range of scientific and engineering disciplines. In such a framework, critical common issues arising in multiscale problems can be identified, explored and characterized in an abstract setting. This type of overarching approach would allow categorization and clarification of existing models and approximations in a landscape of seemingly disjoint, mutually exclusive and ad hoc methods. More importantly, such an approach can provide context for both the development of new techniques and their critical examination. As with any new mathematical framework, it is necessary to demonstrate its viability on problems of practical importance. At Sandia, lab-centric, prototype application problems in fluid mechanics, reacting flows, magnetohydrodynamics (MHD), shock hydrodynamics and materials science span an important subset of DOE Office of Science applications and form an ideal proving ground for new approaches in multiscale science.

Shadid, John N.; van Bloemen Waanders, Bart G.; Bartlett, Roscoe B.

Abstract not provided.

Proposed for publication in IJNME.

Tuminaro, Raymond S.; Shadid, John N.

Abstract not provided.

Proposed for publication in Journal of Computational Physics.

Shadid, John N.; Tuminaro, Raymond S.; Devine, Karen D.; Hennigan, Gary L.; Lin, Paul L.

Abstract not provided.

Proposed for publication in Journal of Computer Methods in Application and Mechanical Engineering.

Bochev, Pavel B.; Bochev, Pavel B.; Gunzburger, Max D.; Shadid, John N.

Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.

Proposed for publication in Computer Methods in Applied Mechanics and Engineering.

Bochev, Pavel B.; Bochev, Pavel B.; Gunzburger, Max D.; Shadid, John N.

Abstract not provided.

Means, Shawn A.; Rintoul, Mark D.; Shadid, John N.; Rintoul, Mark D.

We demonstrate two specific examples that show how our exiting capabilities in solving large systems of partial differential equations associated with transport/reaction systems can be easily applied to outstanding problems in computational biology. First, we examine a three-dimensional model for calcium wave propagation in a Xenopus Laevis frog egg and verify that a proposed model for the distribution of calcium release sites agrees with experimental results as a function of both space and time. Next, we create a model of the neuron's terminus based on experimental observations and show that the sodium-calcium exchanger is not the route of sodium's modulation of neurotransmitter release. These state-of-the-art simulations were performed on massively parallel platforms and required almost no modification of existing Sandia codes.

Salinger, Andrew G.; Pawlowski, Roger P.; Shadid, John N.; van Bloemen Waanders, Bart G.

Abstract not provided.