Computaional Mathematics, Science and Engineering

Computational Mathematics, Science & Engineering



Poster Number: CMSE-01

Authors: Michael M. Crockatt, Andrew J. Christlieb, Cory D. Hauck

Title:  High Order Hybrid Methods for Radiative Transport


Abstract: A selection of high order hybrid schemes for solving linear, kinetic radiative transport problems are considered. The hybrid schemes are constructed by splitting the radiative flux into collisional and free-streaming components to which standard time integration techniques are applied. It is demonstrated that, compared to traditional approaches, such hybrid schemes have the potential to significantly reduce the computational cost required to produce accurate solutions to transport problems across a wide range of collisionality regimes.


This work was supported in part by Oak Ridge National Laboratory (ORNL) and Oak Ridge Associated Universities (ORAU) through the ORAU/ORNL High Performance Computing (HPC) Grant Program



Poster Number: CMSE-02

Authors: Gautham Dharuman, Michael S. Murillo

Title:  Effective Quantum Potentials for Large-scale Simulation of Atomic and Molecular Cluster Explosion by High Intensity Lasers


Abstract: Ionization of clusters using high intensity lasers has gained significant attention as a source for fusion energy (T. Ditmire et al., Nature 1999). The process can also serve as a high energy (~1MeV) ion beam source (T. Ditmire et al., Nature 1997). Improved design for the applications requires detailed simulations of the correlated quantum dynamics post ionization. Cluster sizes can range from ~10^2 - 10^3 atoms or molecules (Last and Jortner, PRL 2001) requiring a simulation involving ~10^2-10^5 electrons and ions. An existing high fidelity approach is time-dependent density functional theory, but its O(N^3) scaling for N particles makes it intractable for the large cluster sizes of interest. We present an alternative approach using a classical Hamiltonian framework employing momentum-dependent potentials (MDPs) to approximately capture the large-scale quantum dynamics (Dharuman et al., PRE 2016). These MDPs are empirical, therefore need to be trained on suitable properties of the system with the expectation of a good predictive capability for properties not included in the training. Using a specific MDP form we showed that training on ground state energies of some atoms translated reasonably well into prediction of their first and second ionization energies. But, that MDP form failed to capture the momentum transfer cross section (MTCS) for an electron-ion scattering. We are currently exploring new MDP forms that can adequately predict MTCS and suitable properties of atoms or molecules that constitute the clusters of interest.


This work was supported in part by AFOSR



Poster Number: CMSE-03

Authors: Connor Glosser, Jack Hamel, Shanker Balasubramaniam, Carlo Piermarocchi

Title:  Maxwell-Bloch Quantum Electrodynamics: A Full-wave Solution Strategy for Disordered Photonic Media


Abstract: Here we consider a disordered system of interacting quantum dots---nanoscale semiconductors with wide applicability in systems ranging from lasing to quantum computing to biological contrast imaging and next-generation displays. Much like atoms, individual quantum dots facilitate absorptive and emissive processes at specific frequencies over timescales independent of those in the incident radiation. These processes couple between dots due to the presence of electromagnetic fields, giving rise to emergent nonlinear behavior within the system. By treating quantum dots semiclassically within our simulation, we maintain the discrete dynamics inherent to quantum objects without resorting to cumbersome second quantization to describe electromagnetic fields (i.e. fields behave classically). This has the advantage of partitoning the simulation into two distinct parts, (1) determination of source polarizations through evolution of the differential optical Bloch equations, and (2) evaluation of radiation patterns through methods adapted from well-known computational electromagnetics techniques. We employ a highly-tuned predictor-corrector integration scheme to advance (1) in time and the subsequent polarizations serve as pointlike sources to electromagnetic integral equations (chosen to facilitate accurate point-to-point communication of fields without the computational overhead of a ``radiation grid''). The coupled solution of (1) and (2), then, gives a complete description of both the quantum and electromagnetic dynamics at each timestep, giving rise to nonlinear effects such as dynamical frequency shifts, strongly correllated quantum behaviors, and other optical phenomena.


This work was supported in part by NSF grant #1408115



Poster Number: CMSE-04

Authors: Stephen Hughey, Hasan Metin Aktulga, Shanker Balasubramaniam

Title:  Parallel Adaptive Fast Multipole Method for Electromagnetics


Abstract: Multiscale electromagnetic (EM) scattering and radiation problems find a wide range of applications in modern engineering design and analysis, from military aircraft to consumer electronics. Integral equation (IE) solutions to these problems are desirable due to their accuracy; however, method of moments discretization of IEs produces dense NxN matrix systems whose solution requires O(N^2) operations in an iterative solver. The fast multipole method (FMM) is often employed to reduce this cost to O(N log N). As the problem size becomes very large, parallelization of the FMM becomes critical.

The FMM achieves this speedup by splitting the simulation domain into a tree of cubes of equal size at each level and having them interact according to a particular set of rules. In the traditional FMM, the densest region of spatial discretization dictates the leaf box size. For multiscale problems, the spatial discretization may be highly non-uniform. Regions outside the densely-discretized region may be over-partitioned, resulting in degradation of the scaling of the FMM.

We describe and demonstrate a set of parallel algorithms that facilitate adaptation of the FMM tree to a non-uniform distribution while preserving the accuracy of the FMM. We define the necessary operators and present an efficient method for handling the cross-level interactions incurred by adapting the tree.



Poster Number: CMSE-05

Authors: Scott O'Connor, Zane Crawford, Shanker Balasubramaniam, John Verboncoeur

Title:  Stability Analysis of Dual FIeld Domain Decomposition


Abstract: Electromagnetic simulation tools are critical for many industrial applications. Time domain finite element methods are one class of methods to simulate transient Electric and Magnetic fields. A common approach to solve for both Electric and Magnetic fields in time is to use a leap frog method. In this approach, the electric field is solved at a time step, while the magnetic field is solved at the next half time step or vice versa. These methods have certain stability criterion that dictate various parameters of the simulation; e.g. time step size, mesh size or what range of frequencies can be simulated. One specific method, the Dual Field Domain Decomposition with an Element Level Decomposition (DFDD-ELD), provides a highly parallelizable framework. This work presents a stability analysis of the DFDD-ELD method along with improvements to the method.



Poster Number: CMSE-06

Authors: Jorge Suzuki, Mohsen Zayernouri

Title:  A Fractional-order Uniaxial Visco-elasto-plastic Model with Damage for Structural Analysis


Abstract: We incorporate a damage formulation to a fractional-order visco-elasto-plastic model considering uniaxial large strains, to describe material degradation. The fractional-order modeling takes into account memory effects for the plastic strains and damage, in order to determine the internal plastic variables, damage and stress. In the context of Lemaitre’s ductile damage theory, we introduce a scalar damage variable to describe the stress softening, along with a differential equation in time that describes the evolution of damage coupled with the plastic strains. The model uses two fractional-orders, respectively, beta_E, beta_K in (0,1), for visco-elasticity and visco-plasticity. A nonlinear system of equations results from the damage evolution and consistency condition, and is solved for the plastic strains and damage using Steffensen’s method and fractional-order time integration in the framework of a fractional-order return-mapping algorithm with damage. We test the model for convergence, prescribed monotonic and cyclic strains, and subsequently implement the model in a finite element space truss code to solve a two-bar and a star-dome problem, that account with snap-through instability and dynamic plasticity with high strain rates. The simulation results demonstrate the flexibility of the fractional-orders beta_E and beta_K when using the Caputo derivative to describe the rate-dependent hardening, softening and viscous dissipation of the model. We believe that such models have promising potential for complex constitutive laws with rate-dependent behavior with damage, such as engineering materials and biological tissues involving low-cycle fatigue, dynamic plasticity and creep damage.