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When a shock interacts with a bubble having a different density than the environment or medium, the interaction causes compression and deformation of the bubble and generation of a vortex pair. Later, secondary vortices appear causing enhanced mixing. The enhanced mixing induced by the shock bubble interactions is particularly of interest in supersonic combustion and detonation. The Wavelet Adaptive Multi-resolution Representation (WAMR) method is particularly suitable for challenging continuum physics problems like shock bubble interaction, which has strong multi-scale character. This method provides an efficient strategy to create a dynamically adaptive spatial grid and to obtain a verified solution. Since the wavelet amplitude provides a first-hand estimate of the local error at each point, the method is able to efficiently capture a wide spectrum of spatial scales by dynamically changing the adaptive grid. Highly resolved computations are done only in the regions where abrupt transition occurs. In this work a detailed investigation of Shock Bubble Interaction (SBI) is carried out using shocks having Mach numbers from 1.2 to 3 for helium, nitrogen and krypton bubbles. Simulations carried out using WAMR method were used to analyze the effects of Mach number and density contrast on the shape, location and velocity of the bubble as well as vorticity and pressure in the flow field.

Computational Science and Engineering (CSE) applications often exhibit the pattern of adaptive mesh applications. Adaptive mesh algorithm starts with a coarse base-level grid structure covering entire computational domain. As the computation intensified, individual grid points are tagged for refinement. Such tagged grid points are dynamically overlayed with finer grid points. Similarly if the level of refinement in a cell is greater than required, all such regions are replaced with coarser grids. These refinements proceed recursively. We have developed an object-oriented framework enabling time-stepped adaptive mesh application developers to convert their sequential applications to MPI applications in few easy steps. We present in this thesis our positive experience converting such application using our framework. In addition to the MPI support, framework does the grid expansion/contraction and load balancing making the application developer’s life easier.

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This dissertation develops a complete framework for fully discrete adjoint sensitivity analysis and inverse problem solutions, in the context of time dependent, adaptive mesh, and adaptive step models. The discrete framework addresses all the necessary ingredients of a state–of–the–art adaptive inverse solution algorithm: adaptive mesh and time step refinement, solution grid transfer operators, a priori and a posteriori error analysis and estimation, and discrete adjoints for sensitivity analysis of flux–limited numerical algorithms. / Ph. D.

Adaptive mesh refinement schemes are used to find accurate low-dimensional approximating spaces when solving elliptic PDEs with Galerkin finite element methods. For nonlinear PDEs, solving the nonlinear problem with Newton's method requires an initial guess of the solution on a refined space, which can be found by interpolating the solution from a previous refinement. Improving the accuracy of the representation of the converged solution computed on a coarse mesh for use as an initial guess on the refined mesh may reduce the number of Newton iterations required for convergence. In this thesis, we present an algorithm to compute an orthogonal L^2 projection between two dimensional finite element spaces constructed from a triangulation of the domain. Furthermore, we present numerical studies that investigate the efficiency of using this algorithm to solve various nonlinear elliptic boundary value problems. / Master of Science

<p> This thesis details work done in the development of the phase field model which allows simulation of elasticity with diffuse interfaces and the extension of a thin interface analysis developed by previous authors to study non-dilute ideal alloys. These models are coupled with a new finite difference adaptive mesh algorithm to efficiently simulate a variety of physical systems. The finite difference adaptive mesh algorithm is shown to be at worse 4-5 times faster than an equivalent finite element method on a per node basis. In addition to this increase in speed for explicit solvers in the code, an iterative solver used to compute elastic fields is found to converge in O(N) time for a dynamically growing precipitate, where N is the number of nodes on the adaptive mesh. A previous phase field formulation is extended such as to make possible the study of non-ideal binary alloys with complex phase diagrams. A phase field model is also derived for a free energy that incorporates an elastic free energy and is used to investigate the competitive development of solid state structures in which the kinetic transfer rate of atoms from the parent phase to the precipitate phase is large. This results in the growth of solid state dendrites. The morphological effects of competing surface anisotropy and anisotropy in the elastic modulus tensor is analyzed. It is shown that the transition from surfaceenergy driven dendrites to elastically driven dendrites depends on the magnitudes of the surface energy anisotropy coefficient (E4 ) and the anisotropy of the elastic tensor (β) as well as on the super saturation of the particle and therefore to a specific Mullins-Sekerka onset radius. The transition point of this competitive process is predicted from these three controlling parameters. </p> / Thesis / Doctor of Philosophy (PhD)

Adaptive Mesh Refinement (AMR) is a method which dynamically varies the spatio-temporal resolution of localized mesh regions in numerical simulations, based on the strength of the solution features. Due to high resolution discretization of localized regions of interests into rectangular mesh units called patches, AMR provides low cost of computations and high degree of accuracy. General purpose graphics processing units (GPGPUs) with their support for fine-grained parallelism, offer an attractive option for obtaining high performance for AMR applications. The data parallel computations of the finite difference schemes of AMR can be efficiently performed on GPGPUs. This research deals with challenges and develops techniques for efficient executions of AMR applications with uniform and non-uniform patches on GPUs. In the first part of the thesis, we optimize an AMR model with uniform patches. We have developed strategies for continuous online visualization of time evolving data for AMR applications executed on GPUs. In-situ visualization plays an important role for analyzing the time evolving characteristics of the domain structures. Continuous visualization of the output data for various time steps results in better study of the underlying domain and the model used for simulating the domain. We reorder the meshes for computations on the GPU based on the users input related to the subdomain that he wants to visualize. This makes the data available for visualization at a faster rate. We then perform asynchronous executions of the visualization steps and fix-up operations on the coarse meshes on the CPUs while the GPU advances the solution. By performing experiments on Tesla S1070 and Fermi C2070 clusters, we found that our strategies result in up to 60% improvement in response time and 16% improvement in the rate of visualization of frames over the existing strategy of performing fix-ups and visualization at the end of the time steps. The second part of the thesis deals with adaptive strategies for efficient execution of block structured AMR applications with non-uniform patches on GPUs. Most AMR approaches use patches of uniform sizes over regions of interests. Since this leads to over-refinement, some efforts have focused on forming patches of non-uniform dimensions to improve computational efficiency since the dimensions of a patch can be tuned to the geometry of a region of interest. 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 a geometric bin-packing method to load balance GPU computations and reduce thread idling, adaptive determination of amount of work to maximize asynchronism between CPU and GPU executions using a knapsack formulation, and scheduling communications for multi-GPU executions. We test our strategies for synthetic inputs as well as for traces from real applications. Our experiments on Tesla S1070 and Fermi C2070 clusters with both single-GPU and multi-GPU executions show that our strategies result in up to 69% improvement in performance over existing strategies. Our bin-packing based load balancing gives performance gains up to 39%, kernel optimizations give an improvement of up to 20%, and our strategies for adaptive asynchronism between CPU-GPU executions give performance improvements of up to 17% over default static asynchronous executions.

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Bayesian inference is commonly used to quantify and reduce modeling uncertainties in coastal ocean models by computing the posterior probability distribution function (pdf) of some uncertain quantities to be estimated conditioned on available observations. The posterior can be computed either directly, using a Markov Chain Monte Carlo (MCMC) approach, or by sequentially processing the data following a data assimilation (DA) approach. The advantage of data assimilation schemes over MCMC-type methods arises from the ability to algorithmically accommodate a large number of uncertain quantities without a significant increase in the computational requirements. However, only approximate estimates are generally obtained by this approach often due to restricted Gaussian prior and noise assumptions. This thesis aims to develop, implement and test novel efficient Bayesian inference techniques to quantify and reduce modeling and parameter uncertainties of coastal ocean models. Both state and parameter estimations will be addressed within the framework of a state of-the-art coastal ocean model, the Advanced Circulation (ADCIRC) model. The first part of the thesis proposes efficient Bayesian inference techniques for uncertainty quantification (UQ) and state-parameters estimation. Based on a realistic framework of observation system simulation experiments (OSSEs), an ensemble Kalman filter (EnKF) is first evaluated against a Polynomial Chaos (PC)-surrogate MCMC method under identical scenarios. After demonstrating the relevance of the EnKF for parameters estimation, an iterative EnKF is introduced and validated for the estimation of a spatially varying Manning’s n coefficients field. Karhunen-Lo`eve (KL) expansion is also tested for dimensionality reduction and conditioning of the parameter search space. To further enhance the performance of PC-MCMC for estimating spatially varying parameters, a coordinate transformation of a Gaussian process with parameterized prior covariance function is next incorporated into the Bayesian inference framework to account for the uncertainty in covariance model hyperparameters. The second part of the thesis focuses on the use of UQ and DA on adaptive mesh models. We developed new approaches combining EnKF and multiresolution analysis, and demonstrated significant reduction in the cost of data assimilation compared to the traditional EnKF implemented on a non-adaptive mesh.