Developments in the Extended Finite Element Method and Algebraic Multigrid for Solid Mechanics Problems Involving Discontinuities

In this dissertation, some contributions related to computational modeling and solution of solid mechanics problems involving discontinuities are discussed. The main tool employed for discrete modeling of discontinuities is the extended finite element method and the primary solution method discussed is the algebraic multigrid. The extended finite element method has been shown to be effective for both weak and strong discontinuities. With respect to weak discontinuities, a new approach that couples the extended finite element method with Monte Carlo simulations with the goal of quantifying uncertainty in homogenization of material properties of random microstructures is presented. For accelerated solution of linear systems arising from problems involving cracks, several new methods involving the algebraic multigrid are presented. In the first approach, the Schur complement of the linear system arising from XFEM is used to develop a Hybrid-AMG method such that crack-conforming aggregates are formed. Another alternative approach involves transforming the original linear system into a modified system that is amenable for a direct application of algebraic multigrid. It is shown that if only Heaviside-enrichments are present, a simple transformation based on the phantom-node approach is available, which decouples the linear system along the discontinuities such that crack conforming aggregates are automatically generated via smoother aggregation algebraic multigrid. Various numerical examples are presented to verify the accuracy of the resulting solutions and the convergence properties of the proposed algorithms. The parallel scalability performance of the implementation are also discussed

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Analytical Rebridging Monte Carlo: Application to cis/trans Isomerization in Proline-Containing, Cyclic Peptides

We present a new method, the analytical rebridging scheme, for Monte Carlo simulation of proline-containing, cyclic peptides. The cis/trans isomerization is accommodated by allowing for two states of the amide bond. We apply our method to five peptides that have been previously characterized by NMR methods. Our simulations achieve effective equilibration and agree well with experimental data in all cases. We discuss the importance of effective equilibration and the role of bond flexibility and solvent effects on the predicted equilibrium properties.Comment: 29 pages, 8 PostScript figures, LaTeX source. to appear in J. Chem. Phys., 199

A fast GPU Monte Carlo Radiative Heat Transfer Implementation for Coupling with Direct Numerical Simulation

We implemented a fast Reciprocal Monte Carlo algorithm, to accurately solve radiative heat transfer in turbulent flows of non-grey participating media that can be coupled to fully resolved turbulent flows, namely to Direct Numerical Simulation (DNS). The spectrally varying absorption coefficient is treated in a narrow-band fashion with a correlated-k distribution. The implementation is verified with analytical solutions and validated with results from literature and line-by-line Monte Carlo computations. The method is implemented on GPU with a thorough attention to memory transfer and computational efficiency. The bottlenecks that dominate the computational expenses are addressed and several techniques are proposed to optimize the GPU execution. By implementing the proposed algorithmic accelerations, a speed-up of up to 3 orders of magnitude can be achieved, while maintaining the same accuracy

Adaptive System Identification using Markov Chain Monte Carlo

One of the major problems in adaptive filtering is the problem of system identification. It has been studied extensively due to its immense practical importance in a variety of fields. The underlying goal is to identify the impulse response of an unknown system. This is accomplished by placing a known system in parallel and feeding both systems with the same input. Due to initial disparity in their impulse responses, an error is generated between their outputs. This error is set to tune the impulse response of known system in a way that every change in impulse response reduces the magnitude of prospective error. This process is repeated until the error becomes negligible and the responses of both systems match. To specifically minimize the error, numerous adaptive algorithms are available. They are noteworthy either for their low computational complexity or high convergence speed. Recently, a method, known as Markov Chain Monte Carlo (MCMC), has gained much attention due to its remarkably low computational complexity. But despite this colossal advantage, properties of MCMC method have not been investigated for adaptive system identification problem. This article bridges this gap by providing a complete treatment of MCMC method in the aforementioned context