The progress of high performance computing platforms is dramatic, and most of the simulations carried out on these platforms result in improvements on one level, yet expose shortcomings of current CFD packages. Therefore, hardware-aware design and optimizations are crucial towards exploiting modern computing resources. This thesis proposes optimizations aimed at accelerating numerical simulations, which are illus- trated in OpenFOAM solvers. A hybrid MPI and GPGPU parallel conjugate gradient linear solver has been designed and implemented to solve the sparse linear algebraic kernel that derives from two CFD solver: icoFoam, which is an incompressible flow solver, and laplacianFoam, which solves the Poisson equation, for e.g., thermal dif- fusion. A load-balancing step is applied using heterogeneous decomposition, which decomposes the computations taking into account the performance of each comput- ing device and seeking to minimize communication. In addition, we implemented the recently developed pipeline conjugate gradient as an algorithmic improvement, and parallelized it using MPI, GPGPU, and a hybrid technique. While many questions of ultimately attainable per node performance and multi-node scaling remain, the ex- perimental results show that the hybrid implementation of both solvers significantly outperforms state-of-the-art implementations of a widely used open source package.

Centered finite difference is the basic method in this paper for spatial discretization. In general, except the schemes that will be used adjacent tothe boundary points, centered finite difference schemes will be used on the main mesh points. Depending on the requirement of order of accuracy and optimization, different multi-point stencil schemes will be built in Matlab in the form of matrix. As a result, solving PDEs is actually operating matrices in Matlab. Standard schemes and optimized schemeswill be tested with 1D linear convection equation before applying them to the solvers. In 2D-pulse case, the rectangular domain will be transformed into a wavy domain and as a result Jacobian transformation method will betested. Results from different schemes will be compared with the analytical solution in two dimensional pulse case.

This thesis describes the development, verification, and validation of a three dimensional time domain thermoacoustic solver. The purpose of the solver is to predict the frequencies, modeshapes, linear growth rates, and limit cycle amplitudes for combustion instability modes in gas turbine combustion chambers. The linearized Euler equations with nonlinear heat release source terms are solved using the finite volume method. The treatment of mean density gradients was found to be vital to the success of frequency and modeshape predictions due to the sharp density gradients that occur across deflagration waves. In order to treat mean density gradients with physical fidelity, a non-conservative finite volume method based on the wave propagation approach to the Riemann problem is applied. For modelling unsteady heat release, user input flexibility is maximized using a virtual class hierarchy within the OpenFOAM C++ library. Unsteady heat release based on time lag models are demonstrated. The solver gives accurate solutions compared with analytical methods for one-dimensional cases involving mean density gradients, cross-sectional area changes, uniform mean flow, arbitrary impedance boundary conditions, and unsteady heat release in a one-dimensional Rijke tube. The solver predicted resonant frequencies within 1% of the analytical solution for these verification cases, with the dominant component of the error coming from the finite time interval over which the simulation is performed. The linear iii growth rates predicted by the solver for the Rijke tube verification were within 5% of the theoretical values, provided that numerical dissipation effects were controlled. Finally, the solver is then used to predict the frequencies and limit cycle amplitudes for two lab scale experiments in which detailed acoustics data are available for comparison. For experiments at the University of Melbourne, an empirical flame describing function was provided. The present simulation code predicted a limit cycle of 0.21 times the mean pressure, which was in close agreement with the estimate of 0.25 from the experimental data. The experiments at Purdue University do not yet have an empirical flame model, so a general vortex-shedding model is proposed on physical grounds. It is shown that the coefficients of the model can be tuned to match the limit cycle amplitude of the 2L mode from the experiment with the same accuracy as the Melbourne case. The code did not predict the excitation of the 4L mode, therefore it is concluded that the vortex-shedding model is not sufficient and must be supplemented with additional heat release models to capture the entirety of the physics for this experiment.

This thesis presents the development of an automatic mesh generator, named WoodFrameMesh, using object oriented C++. The program developed is capable of generating complete finite element models of wooden houses incorporating frames, linear links, springs, nodal loads and restraints at the desired locations. The finite element mesh generated by the program may be triangular or quadrilateral. The triangular mesh can be generated over any arbitrary domain with multiple openings and line constraints. The program implements the advancing front method for triangulation as discussed by Lee and Hobbs. The difference is made by implementing the algorithm using object oriented concepts and the extensive use of the powerful C++ Standard Template Library (STL). Quadrilateral mesh generation is limited to simple quadrilateral domains with no openings or constraint lines. A simple structured technique is implemented to generate the quadrilateral mesh. The amount of time spent in manual generation of the complete finite element model of wooden houses has been considerably reduced by automating the modeling process. Overall, the use of object oriented design has facilitated the code development and has provided a platform for further additions. The program relies on the use of STL as it provides dynamic data structures, algorithms for storage, searching, sorting, etc. Efficiency of the program is improved by the use of the in-built features in STL instead of developing new code. Analysis of the finite element models generated by the automatic mesh generator is performed using SAP 2000 and WoodFrameSolver. WoodFrameSolver is a finite element analysis engine for WoodFrameMesh, which was developed at Virginia Tech by a group of graduate students (including the author) and professors as a separate project. A chapter discussing the WoodFrameSolver architecture, its extensibility features and its verification is also presented in this thesis. The solver performance and accuracy are similar to those of SAP 2000, which was chosen as the benchmark for testing the analysis results. / Master of Science

The Forestry Department at Mississippi State University has been contracted to design and test a novel beam to be used to create crossing platforms for cranes operating in muddy, swampy areas. To date, they have performed stress analyses on 30 beams, but their physical testing method requires costly amounts of material and man hours. It is theorized that the finite element method may be used as an alternative method of analysis in order to reduce costs. The focus of this study is to create models of tested beams using the finite element solver, ANSYS, and verify the accuracy of these models using the results of the Forestry Department’s physical testing.

No description available.

No description available.

Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.

<p> Systems of differential-algebraic equations (DAEs) arise in numerious applications, and there has been considerable research on solving DAE initial value problems (IVPs). Existing methods and software for solving DAEs usually handle at most index-three problems. However, DAE problems of index three and higher do arise, for example, in actuator dynamics, multi-stage processes, and optimization.</p> <p> We present the method of J. Pryce and N. Nedialkov for solving DAEs, which can be of high index, fully implicit, and contain derivatives of order higher than one. We solve such DAEs by expanding their solution in Taylor series (TS). To compute Taylor coefficients, we employ J. Pryce's structural analysis and automatic differentiation. Then we compute an approximate TS solution with appropriate stepsize and project this solution to satisfy the constraints (explicit and hidden) of the problem.</p> <p> This thesis discusses the algorithms involved in this method, including the algorithms for Taylor coefficients computation, consistent point projection, error estimation, stepsize control, and the overall integration process. The author has implemented a software package named HIDAETS (High-Index DAE by Taylor Series). In this thesis, we present the specification, design, implementation, and usage of HIDAETS. Numerical results on several high-index DAEs are reported. These results demonstrate that HIDAETS is efficient and accurate for solving IVP in DAEs.</p> / Thesis / Master of Science (MSc)

Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity analysis are to understand the robustness of the model with respect to perturbations, and to select the most important parameters for the model. MATLODE is a tool for sensitivity analysis of models described by ordinary differential equations (ODEs). MATLODE implements two distinct approaches for sensitivity analysis: direct (via the tangent linear model) and adjoint. Within each approach, four families of numerical methods are implemented, namely explicit Runge-Kutta, implicit Runge-Kutta, Rosenbrock, and single diagonally implicit Runge-Kutta. Each approach and family has its own strengths and weaknesses when applied to real world problems. MATLODE has a multitude of options that allows users to find the best approach for a wide range of initial value problems. In spite of the great importance of sensitivity analysis for models governed by differential equations, until this work there was no MATLAB ordinary differential equation sensitivity analysis toolbox publicly available. The two most popular sensitivity analysis packages, CVODES [8] and FATODE [10], are geared toward the high performance modeling space; however, no native MATLAB toolbox was available. MATLODE fills this need and offers sensitivity analysis capabilities in MATLAB, one of the most popular programming languages within scientific communities such as chemistry, biology, ecology, and oceanogra- phy. We expect that MATLODE will prove to be a useful tool for these communities to help facilitate their research and fill the gap between theory and practice. / Master of Science / Sensitivity analysis is the study of how small changes in a model?s input effect the model’s output. Sensitivity analysis provides tools to quantify the impact that small, discrete changes in input values have on the output. The objective of this research is to develop a MATLAB sensitivity analysis toolbox called MATLODE. This research is critical to a wide range of communities who need to optimize system behavior or predict outcomes based on a variety of initial conditions. For example, an analyst could build a model that reflects the performance of an automobile engine, where each part in the engine has a set of initial characteristics. The analyst can use sensitivity analysis to determine which part effects the engine’s overall performance the most (or the least), without physically building the engine and running a series of empirical tests. By employing sensitivity analysis, the analyst saves time and money, and since multiple tests can usually be run through the model in the time needed to run just one empirical test, the analyst is likely to gain deeper insight and design a better product. Prior to MATLODE, employing sensitivity analysis without significant knowledge of computational science was too cumbersome and essentially impractical for many of the communities who could benefit from its use. MATLODE bridges the gap between computational science and a variety of communities faced with understanding how small changes in a system’s input values effect the systems output; and by bridging that gap, MATLODE enables more large scale research initiatives than ever before.