Browsing by Subject "Transfer matrix method"

Abundant and widely available solar energy is one possible solution to the increasing demands for clean energy. The Thermodynamics and Sustainable Energy Laboratory (T-SEL) in Duke University has been dedicated to investigating methods to harness solar energy. Hybrid Solar System (HSS) is one of the promising methods to use solar energy, as it absorbs sunlight to produce hydrogen, which then can electrically power equipment through fuel cells. Hydrogen is produced through a biofuel reforming process, which occurs at a high temperature (over 700℃ for methane). Methods to design a high-temperature solar selective coating are investigated in this thesis.

The solar irradiance spectrum was assumed to be the same as Air Mass (AM) 1.5. A transfer-matrix method was adopted in this work to calculate the optical properties of the NREL #6, a design of nine-layer solar selective coating. Based on the design of NREL #6 coating, Differential Evolution (DE) algorithm was introduced to optimize this design. Two objective functions were considered: selectivity-oriented function and efficiency-oriented function, yielding the design of Revision #1 and Revision #2 respectively. The results showed a high selectivity (around 13) with low efficiency (66.6%) in Revision #1 and a high efficiency (82.6%) with moderate selectivity (around 9) in Revision #2.

In statistical physics, the exact partition function of simple (quasi)-one-dimensional models can be obtained from the numerical transfer matrix (TM) method. This method involves solving for the leading eigenvalues of a matrix representing all possible interactions between the states that a unit of the system can take. Because the size of this matrix grows exponentially with the number of those units, the TM method is ideally suited for models that have a finite state space and short-range interactions. Its success nevertheless relies on the use of efficient iterative eigensolvers and on leveraging system symmetry, whenever possible.

By careful finite-size extrapolation of sufficiently large systems, the TM method can also be used to examine two-dimensional models. A particularly interesting series of such systems are Ising models modified with next-nearest-neighbor frustration, which recapitulate the formation of equilibrium modulated phases in systems as varied as magnetic alloys, lipid surfactants, and cell morphogenesis. In these models, frustration results in large mixing times for Markov chain Monte Carlo simulations, but the TM approach sidesteps this slowdown and thus provides a putatively well-controlled computational scheme. The effectiveness of TM approach for these models, however, had previously been obfuscated by the limited range of system sizes computationally available for the numerical analysis. In this thesis, I build on the sparse matrix decomposition and take advantage of the structure and symmetry of the TM to develop optimized algorithms for the method, and thereby overcome the computational challenge. The resulting algorithm is implemented in various canonical frustrated next-nearest-neighbor Ising models, aiming to solve long-standing physical problems in these models. The approach provides benchmark results for related statistical physics models. It could also inspire the development of adapted eigensolver for similarly structured matrices.