Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions

The paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter in are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large in, for epsilon > 0 and increases for large in, for epsilon < 0.

Efficient simulation of unitary operators by combining two numerical algorithms: An NMR simulation of the mirror-inversion propagator of an XY spin chain

Precise experimental implementation of unitary operators is one of the most important tasks for quantum information processing. Numerical optimization techniques are widely used to find optimized control fields to realize a desired unitary operator. However, finding high-fidelity control pulses to realize an arbitrary unitary operator in larger spin systems is still a difficult task. In this work, we demonstrate that a combination of the GRAPE algorithm, which is a numerical pulse optimization technique, and a unitary operator decomposition algorithm Ajoy et al., Phys. Rev. A 85, 030303 (2012)] can realize unitary operators with high experimental fidelity. This is illustrated by simulating the mirror-inversion propagator of an XY spin chain in a five-spin dipolar coupled nuclear spin system. Further, this simulation has been used to demonstrate the transfer of entangled states from one end of the spin chain to the other end.

Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows

Single fluid schemes that rely on an interface function for phase identification in multicomponent compressible flows are widely used to study hydrodynamic flow phenomena in several diverse applications. Simulations based on standard numerical implementation of these schemes suffer from an artificial increase in the width of the interface function owing to the numerical dissipation introduced by an upwind discretization of the governing equations. In addition, monotonicity requirements which ensure that the sharp interface function remains bounded at all times necessitate use of low-order accurate discretization strategies. This results in a significant reduction in accuracy along with a loss of intricate flow features. In this paper we develop a nonlinear transformation based interface capturing method which achieves superior accuracy without compromising the simplicity, computational efficiency and robustness of the original flow solver. A nonlinear map from the signed distance function to the sigmoid type interface function is used to effectively couple a standard single fluid shock and interface capturing scheme with a high-order accurate constrained level set reinitialization method in a way that allows for oscillation-free transport of the sharp material interface. Imposition of a maximum principle, which ensures that the multidimensional preconditioned interface capturing method does not produce new maxima or minima even in the extreme events of interface merger or breakup, allows for an explicit determination of the interface thickness in terms of the grid spacing. A narrow band method is formulated in order to localize computations pertinent to the preconditioned interface capturing method. Numerical tests in one dimension reveal a significant improvement in accuracy and convergence; in stark contrast to the conventional scheme, the proposed method retains its accuracy and convergence characteristics in a shifted reference frame. Results from the test cases in two dimensions show that the nonlinear transformation based interface capturing method outperforms both the conventional method and an interface capturing method without nonlinear transformation in resolving intricate flow features such as sheet jetting in the shock-induced cavity collapse. The ability of the proposed method in accounting for the gravitational and surface tension forces besides compressibility is demonstrated through a model fully three-dimensional problem concerning droplet splash and formation of a crownlike feature. (C) 2014 Elsevier Inc. All rights reserved.

Numerical modeling of the non-isothermal liquid droplet impact on a hot solid substrate

The heat transfer from a solid phase to an impinging non-isothermal liquid droplet is studied numerically. A new approach based on an arbitrary Lagrangian-Eulerian (ALE) finite element method for solving the incompressible Navier Stokes equations in the liquid and the energy equation within the solid and the liquid is presented. The novelty of the method consists in using the ALE-formulation also in the solid phase to guarantee matching grids along the liquid solid interface. Moreover, a new technique is developed to compute the heat flux without differentiating the numerical solution. The free surface and the liquid solid interface of the droplet are represented by a moving mesh which can handle jumps in the material parameter and a temperature dependent surface tension. Further, the application of the Laplace-Beltrami operator technique for the curvature approximation allows a natural inclusion of the contact angle. Numerical simulation for varying Reynold, Weber, Peclet and Biot numbers are performed to demonstrate the capabilities of the new approach. (C) 2014 Elsevier Ltd. All rights reserved.

A detailed numerical study of NOx kinetics in low calorific value H-2/CO syngas flames

The present study provides an extensive and detailed numerical analysis of NO chemical kinetics in low calorific value H-2/CO syngas flames utilizing predictions by five chemical kinetic mechanisms available out of which four deal with H-2/CO while the fifth mechanism (GRI 3.0) additionally accounts for hydrocarbon chemistry. Comparison of predicted axial NO profiles in premixed flat flames with measurements at 1 bar, 3.05 bar and 9.15 bar shows considerably large quantitative differences among the various mechanisms. However, at each pressure, the quantitative reaction path diagrams show similar NO formation pathways for most of the mechanisms. Interestingly, in counterflow diffusion flames, the quantitative reaction path diagrams and sensitivity analyses using the various mechanisms reveal major differences in the NO formation pathways and reaction rates of important reactions. The NNH and N2O intermediate pathways are found to be the major contributors for NO formation in all the reaction mechanisms except GRI 3.0 in syngas diffusion flames. The GRI 3.0 mechanism is observed to predict prompt NO pathway as the major contributing pathway to NO formation. This is attributed to prediction of a large concentration of CH radical by the GRI 3.0 as opposed to a relatively negligible value predicted by all other mechanisms. Also, the back-conversion of NNH into N2O at lower pressures (2-4 bar) was uniquely observed for one of the five mechanisms. The net reaction rates and peak flame temperatures are used to correlate and explain the differences observed in the peak NO] at different pressures. This study identifies key reactions needing assessment and also highlights the need for experimental data in syngas diffusion flames in order to assess and optimize H-2/CO and nitrogen chemistry. Copyright (C) 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.

NUMERICAL SIMULATIONS OF LIQUID JET BREAK UP IN A CROSSFLOW

Atomization is the process of disintegration of a liquid jet into ligaments and subsequently into smaller droplets. A liquid jet injected from a circular orifice into cross flow of air undergoes atomization primarily due to the interaction of the two phases rather than an intrinsic break up. Direct numerical simulation of this process resolving the finest droplets is computationally very expensive and impractical. In the present study, we resort to multiscale modelling to reduce the computational cost. The primary break up of the liquid jet is simulated using Gerris, an open source code, which employs Volume-of-Fluid (VOF) algorithm. The smallest droplets formed during primary atomization are modeled as Lagrangian particles. This one-way coupling approach is validated with the help of the simple test case of tracking a particle in a Taylor-Green vortex. The temporal evolution of the liquid jet forming the spray is captured and the flattening of the cylindrical liquid column prior to breakup is observed. The size distribution of the resultant droplets is presented at different distances downstream from the location of injection and their spatial evolution is analyzed.

Interphase anisotropy effects on lamellar eutectics: A numerical study

In directional solidification of binary eutectics, it is often observed that two-phase lamellar growth patterns grow tilted with respect to the direction z of the imposed temperature gradient. This crystallographic effect depends on the orientation of the two crystal phases alpha and beta with respect to z. Recently, an approximate theory was formulated that predicts the lamellar tilt angle as a function of the anisotropy of the free energy of the solid(alpha)-solid(beta) interphase boundary. We use two different numerical methods-phase field (PF) and dynamic boundary integral (BI)-to simulate the growth of steady periodic patterns in two dimensions as a function of the angle theta(R) between z and a reference crystallographic axis for a fixed relative orientation of alpha and beta crystals, that is, for a given anisotropy function (Wulff plot) of the interphase boundary. For Wulff plots without unstable interphase-boundary orientations, the two simulation methods are in excellent agreement with each other and confirm the general validity of the previously proposed theory. In addition, a crystallographic ``locking'' of the lamellae onto a facet plane is well reproduced in the simulations. When unstable orientations are present in the Wulff plot, it is expected that two distinct values of the tilt angle can appear for the same crystal orientation over a finite theta(R) range. This bistable behavior, which has been observed experimentally, is well reproduced by BI simulations but not by the PF model. Possible reasons for this discrepancy are discussed.

Adaptive executions of hyperbolic block-structured AMR applications on GPU systems

A block-structured adaptive mesh refinement (AMR) technique has been used to obtain numerical solutions for many scientific applications. Some block-structured AMR approaches have focused on forming patches of non-uniform sizes where the size of a patch can be tuned to the geometry of a region of interest. In this paper, we develop strategies for adaptive execution of block-structured AMR applications on GPUs, for hyperbolic directionally split solvers. While effective hybrid execution strategies exist for applications with uniform patches, our work considers efficient execution of non-uniform patches with different workloads. Our techniques include bin-packing work units to load balance GPU computations, adaptive asynchronism between CPU and GPU executions using a knapsack formulation, and scheduling communications for multi-GPU executions. Our experiments with synthetic and real data, for single-GPU and multi-GPU executions, on Tesla S1070 and Fermi C2070 clusters, show that our strategies result in up to a 3.23 speedup in performance over existing strategies.

NUMERICAL STUDIES OF DYNAMO ACTION IN A TURBULENT SHEAR FLOW. I.

We perform numerical experiments to study the shear dynamo problem where we look for the growth of a large-scale magnetic field due to non-helical stirring at small scales in a background linear shear flow in previously unexplored parameter regimes. We demonstrate the large-scale dynamo action in the limit where the fluid Reynolds number (Re) is below unity while the magnetic Reynolds number (Rm) is above unity; the exponential growth rate scales linearly with shear, which is consistent with earlier numerical works. The limit of low Re is particularly interesting, as seeing the dynamo action in this limit would provide enough motivation for further theoretical investigations, which may focus attention on this analytically more tractable limit of Re < 1 compared to the more formidable limit of Re > 1. We also perform simulations in the regimes where (i) both (Re, Rm) < 1, and (ii) Re > 1 and Rm < 1, and compute all of the components of the turbulent transport coefficients (alpha(ij) and alpha(ij)) using the test-field method. A reasonably good agreement is observed between our results and the results of earlier analytical works in similar parameter regimes.

Numerical study of solidification of A356 aluminum alloy flowing on an oblique plate with experimental validation

Molten A356 aluminum alloy flowing on an oblique plate is water cooled from underneath. The melt partially solidifies on plate wall with continuous formation of columnar dendrites. These dendrites are continuously sheared off into equiaxed/fragmented grains and carried away with the melt by producing semisolid slurry collected at plate exit. Melt pouring temperature provides required solidification whereas plate inclination enables necessary shear for producing slurry of desired solid fraction. A numerical model concerning transport equations of mass, momentum, energy and species is developed for predicting velocity, temperature, macrosegregation and solid fraction. The model uses FVM with phase change algorithm, VOF and variable viscosity. The model introduces solid phase movement with gravity effect as well. Effects of melt pouring temperature and plate inclination on hydrodynamic and thermo-solutal behaviors are studied subsequently. Slurry solid fractions at plate exit are 27%, 22%, 16%, and 10% for pouring temperatures of 620 degrees C, 625 degrees C, 630 degrees C, and 635 degrees C, respectively. And, are 27%, 25%, 22%, and 18% for plate inclinations of 30, 45, 60, and 75, respectively. Melt pouring temperature of 625 degrees C with plate inclination of 60 generates appropriate quality of slurry and is the optimum. Both numerical and experimental results are in good agreement with each other. (C) 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

An Optimizing Code Generator for a Class of Lattice-Boltzmann Computations

The Lattice-Boltzmann method (LBM), a promising new particle-based simulation technique for complex and multiscale fluid flows, has seen tremendous adoption in recent years in computational fluid dynamics. Even with a state-of-the-art LBM solver such as Palabos, a user has to still manually write the program using library-supplied primitives. We propose an automated code generator for a class of LBM computations with the objective to achieve high performance on modern architectures. Few studies have looked at time tiling for LBM codes. We exploit a key similarity between stencils and LBM to enable polyhedral optimizations and in turn time tiling for LBM. We also characterize the performance of LBM with the Roofline performance model. Experimental results for standard LBM simulations like Lid Driven Cavity, Flow Past Cylinder, and Poiseuille Flow show that our scheme consistently outperforms Palabos-on average by up to 3x while running on 16 cores of an Intel Xeon (Sandybridge). We also obtain an improvement of 2.47x on the SPEC LBM benchmark.

Experimental and numerical studies on protection of buried pipelines and underground utilities using geocells

This paper presents the results of the laboratory model tests and the numerical studies conducted on small diameter PVC pipes, buried in geocell reinforced sand beds. The aim of the study was to evaluate the suitability of the geocell reinforcement in protecting the underground utilities and buried pipelines. In addition to geocells, the efficacy of only geogrid and geocell with additional basal geogrid cases were also studied. A PVC (Poly Vinyl Chloride) pipe with external diameter 75 mm and thickness 1.4 mm was used in the experiments. The vehicle tire contact pressure was simulated by applying the pressure on the top of the bed with the help of a steel plate. Results suggest that the use of geocells with additional basal geogrid considerably reduces the deformation of the pipe as compared to other types of reinforcements. Further, the depth of placement of pipe was also varied between 1B to 2B (B is the width of loading plate) below the plate in the presence of geocell with additional basal geogrid. More than 50% reduction in the pressure and more than 40% reduction in the strain values were observed in the presence of reinforcements at different depths as compared to the unreinforced beds. Conversely, the performance of the subgrade soil was also found to be marginally influenced by the position of the pipe, even in the presence of the relatively stiff reinforcement system. Further, experimental results were validated with 3-dimensional numerical studies using FLAC(3D) (Fast Lagrangian Analysis of Continua in 3D). A good agreement in the measured pipe stain values were observed between the experimental and numerical studies. Numerical studies revealed that the geocells distribute the stresses in the lateral direction and thus reduce the pressure on the pipe. In addition, the results of the 1-g model tests were scaled up to the prototype case of the shallow buried pipeline below the pavement using the appropriate scaling laws. (C) 2015 Elsevier Ltd. All rights reserved.

Three-dimensional numerical analysis of geocell-reinforced soft clay beds by considering the actual geometry of geocell pockets

Due to its complex honeycomb structure, the numerical modeling of the geocell has always been a big challenge. Generally, the equivalent composite approach is used to model the geocells. In the equivalent composite approach, the geocellsoil composite is treated as the soil layer with improved strength and stiffness values. Though this approach is very simple, it is unrealistic to model the geocells as the soil layer. This paper presents a more realistic approach of modeling the geocells in three-dimensional (3D) framework by considering the actual curvature of the geocell pocket. A square footing resting on geocell reinforced soft clay bed was modeled using the ``fast Lagrangian analysis of continua in 3D'' (FLAC(3D)) finite difference package. Three different material models, namely modified Cam-clay, Mohr-Coulomb, and linear elastic were used to simulate the behaviour of foundation soil, infill soil and the geocell, respectively. It was found that the geocells distribute the load laterally to the wider area below the footing as compared to the unreinforced case. More than 50% reduction in the stress was observed in the clay bed in the presence of geocells. In addition to geocells, two other cases, namely, only geogrid and geocell with additional basal geogrid cases were also simulated. The numerical model was systematically validated with the results of the physical model tests. Using the validated numerical model, parametric studies were conducted to evaluate the influence of various geocell properties on the performance of reinforced clay beds.

Numerical study of laminar, standing hydraulic jumps in a planar geometry

We solve the two-dimensional, planar Navier-Stokes equations to simulate a laminar, standing hydraulic jump using a Volume-of-Fluid method. The geometry downstream of the jump has been designed to be similar to experimental conditions by including a pit at the edge of the platform over which liquid film flows. We obtain jumps with and without separation. Increasing the inlet Froude number pushes the jump downstream and makes the slope of the jump weaker, consistent with experimental observations of circular jumps, and decreasing the Reynolds number brings the jump upstream while making it steeper. We study the effect of the length of the domain and that of a downstream obstacle on the structure and location of the jump. The transient flow which leads to a final steady jump is described for the first time to our knowledge. In the moderate Reynolds number regime, we obtain steady undular jumps with a separated bubble underneath the first few undulations. Interestingly, surface tension leads to shortening of wavelength of these undulations. We show that the undulations can be explained using the inviscid theory of Benjamin and Lighthill (Proc. R. Soc. London, Ser. A, 1954). We hope this new finding will motivate experimental verification.

Co-Exploration of NLA Kernels and Specification of Compute Elements in Distributed Memory CGRAs

Coarse Grained Reconfigurable Architectures (CGRA) are emerging as embedded application processing units in computing platforms for Exascale computing. Such CGRAs are distributed memory multi- core compute elements on a chip that communicate over a Network-on-chip (NoC). Numerical Linear Algebra (NLA) kernels are key to several high performance computing applications. In this paper we propose a systematic methodology to obtain the specification of Compute Elements (CE) for such CGRAs. We analyze block Matrix Multiplication and block LU Decomposition algorithms in the context of a CGRA, and obtain theoretical bounds on communication requirements, and memory sizes for a CE. Support for high performance custom computations common to NLA kernels are met through custom function units (CFUs) in the CEs. We present results to justify the merits of such CFUs.