Double Arrays, Triple Arrays and Balanced Grids

Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids

Double Arrays, Triple Arrays, and Balanced Grids with \u3cem\u3ev\u3c/em\u3e = \u3cem\u3er\u3c/em\u3e + \u3cem\u3ec\u3c/em\u3e - 1

In Theorem 6.1 of McSorley et al. [3] it was shown that, when v = r+c−1, every triple array TA(v, k, λrr, λcc, k : r × c) is a balanced grid BG(v, k, k : r×c). Here we prove the converse of this Theorem. Our final result is: Let v = r +c−1. Then every triple array is a TA(v, k, c−k, r−k, k : r × c) and every balanced grid is a BG(v, k, k : r × c), and they are equivalent

Complete Enumeration and Properties of Binary Pseudo-Youden Designs PYD(9, 6, 6)

A binary pseudo -Youden design PYD(9, 6, 6) is a 6 × 6 array in which each cell contains one element from the set V = {1, 2, . . ., 9}, and each element from V occurs 4 times. Every row of the array contains distinct elements and every column contains distinct elements. The rows and columns, when taken together, are pairwise balanced and form a (9, 12, 8, 6, 5)-BIBD. In Preece (1968) and (1976) a total of 345 species of binary PYD(9, 6, 6) were found. Here we complete this enumeration and find 348 species of binary PYD(9, 6, 6). We give a complete set of invariants for these species based upon the numbers of intercalates and anti-intercalates that they contain; and discuss some of their properties. We also show that there are 696 non-isomorphic binary PYD(9, 6, 6), and give a complete set of invariants for these arrays

Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

In this paper we study Youden rectangles of small orders. We have enumerated all Youden rectangles for all small parameter values, excluding the almost square cases, in a large scale computer search. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values. We refer to these objects as \emph{near Youden rectangles}. For all our designs we calculate the size of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.Comment: 33 pages, 21 Table

Non-invasive multigrid for semi-structured grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid

Advanced Photovoltaic Devices Enabled by Lattice-Mismatched Epitaxy

Thin-film III-V semiconductor-based photovoltaic (PV) devices, whose light conversion efficiency is primarily limited by the minority carrier lifetimes, are commonly designed to minimize the formation of crystalline defects (threading dislocations or, in extreme cases, fractures) that can occur, in particular, due to a mismatch in lattice constants of the epitaxial substrate and of the active film. At the same time, heteroepitaxy using Si or metal foils instead of costly III-V substrates is a pathway to enabling low-cost thin-film III-V-based PV and associated devices, yet it requires to either use metamorphic buffers or lateral confinement either by substrate patterning or by growing high aspect ratio structures. Mismatched epitaxy can be used for high-efficiency durable III-V space PV systems by incorporation of properly engineered strained quantum confined structures into the solar cells that can enable bandgap engineering and enhanced radiation tolerance. One of the major topics covered in this work is optical and optoelectronic modeling and physics of the triple-junction solar cell featuring planar Si middle sub-cell and GaAs0.73P0.27 and InAs0.85P0.15 periodic nanowire (NW) top and bottom sub-cells, respectively. In particular, the dimensions of the NW arrays that would enable near-unity broad-band absorption for maximum generated current were identified. For the top cell, the planarized array dimensions corresponding to maximum generated current and current matching with the underlying Si sub-cell were found to be 350 nm for NW diameter and 450 – 500 nm for NW spacing. For the GaAs0.73P0.27, resonant coupling was the main factor driving the absorption, yet addressing the coupling of IR light in the transmission mode in the InAs0.85P0.15 nanoscale arrays was challenging and unique. Given the nature of the Si and bottom NW interface, the designs of high refractive index encapsulation materials and conformal reflectors were proposed to enable the use of thin NWs (300 – 400 nm) for sufficient IR absorption. A novel co-simulation tool combining RSoft DiffractMOD® and Sentaurus Device® was established and utilized to design the p-i-n 3D junction and thin conformal GaP passivation coating for maximum GaAs0.73P0.27 NW sub-cell efficiency (16.5%) mainly impacted by the carrier surface annihilation. Development of a highly efficient GaAs solar cell enhanced with InxGa1-xAs/GaAsyP1-y quantum wells (QWs) is also demonstrated as one of the key parts of the dissertation. The optimizations including design of GaAsP strain balancing that would support efficient thermal (here, 17 nm-thick GaAs0.90P0.10 for 9.2 nm-thick In0.10Ga0.90As QWs) and/or tunneling (4.9 nm-thick GaAs0.68P0.32) carrier escape out of the QW while maintaining a consistent morphology of the QW layers in extended QW superlattices were performed using the principles of strain energy minimization and by tuning the growth parameters. The fundamental open-circuit voltage (V¬oc) restraints in radiative and non-radiative recombination-limited regimes in the QW solar cells were studied for a variety of InxGa1-xAs compositions (x=6%, 8%, 10%, and 14%) and number of QWs using spectroscopic and dark current analysis and modeling. Additionally, the design and use of distributed Bragg reflectors for targeted up to 90% QW absorption enhancement is demonstrated resulting in an absolute QW solar cell efficiency increase by 0.4% due to nearly doubled current from the QWs and 0.1% enhancement relatively to the optically-thick baseline device with no QWs

An unstructured parallel least-squares spectral element solver for incompressible flow problems

The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows

An Unstructured Parallel Least-Squares Spectral Element Solver for Incompressible Flow Problems

The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows