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1 
Adaptive Algorithms for Deterministic and Stochastic Differential EquationsMoon, KyoungSook January 2003 (has links)
No description available.

2 
Numerical Investigation of Shock Bubble Interaction using Wavelet Adaptive MultiResolution MethodDhopeshwar, Rahul 07 1900 (has links)
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 Multiresolution Representation (WAMR) method is particularly suitable for challenging continuum physics problems like shock bubble interaction, which has strong multiscale 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 firsthand 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.

3 
Adaptive Algorithms for Deterministic and Stochastic Differential EquationsMoon, KyoungSook January 2003 (has links)
No description available.

4 
An Adaptive Mesh MPI Framework for Iterative C++ ProgramsSilva, Karunamuni Charuka 23 March 2009 (has links)
Computational Science and Engineering (CSE) applications often exhibit the pattern of adaptive mesh applications. Adaptive mesh algorithm starts with a coarse baselevel 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 objectoriented framework enabling timestepped 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.

5 
Galerkin Projections Between Finite Element SpacesThompson, Ross Anthony 17 June 2015 (has links)
Adaptive mesh refinement schemes are used to find accurate lowdimensional 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

6 
Adjointbased spacetime adaptive solution algorithms for sensitivity analysis and inverse problemsAlexe, Mihai 14 April 2011 (has links)
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.

7 
Using PhaseField Modeling With Adaptive Mesh Refinement To Study ElastoPlastic Effects In Phase TransformationsGreenwood, Michael 11 1900 (has links)
<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 nondilute 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 45 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 nonideal 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
MullinsSekerka onset radius. The transition point of this competitive process
is predicted from these three controlling parameters. </p> / Thesis / Doctor of Philosophy (PhD)

8 
Efficient Execution Of AMR Computations On GPU SystemsRaghavan, Hari K 11 1900 (has links) (PDF)
Adaptive Mesh Refinement (AMR) is a method which dynamically varies the spatiotemporal 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 finegrained 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 nonuniform 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. Insitu 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 fixup 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 fixups 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 nonuniform patches on GPUs. Most AMR approaches use patches of uniform sizes over regions of interests. Since this leads to overrefinement, some efforts have focused on forming patches of nonuniform 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 nonuniform patches with different workloads. Our techniques include a geometric binpacking 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 multiGPU 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 singleGPU and multiGPU executions show that our strategies result in up to 69% improvement in performance over existing strategies. Our binpacking 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 CPUGPU executions give performance improvements of up to 17% over default static asynchronous executions.

9 
Integrated adaptive numerical methods for transient twophase flow in heterogeneous porous mediaChueh, ChihChe 26 January 2011 (has links)
Transient multiphase flow problems in porous media are ubiquitous in engineering and environmental systems and processes; examples include heat exchangers, reservoir simulation, environmental remediation, magma flow in the earth crust and water management in porous electrodes of PEM fuel cells. This thesis focuses on the development of accurate and computationally efficient numerical models to simulate such flows. The research challenges addressed in this work fall in two areas. For a numerical standpoint, conventional numerical methods including NewtonRaphson linearization and a simple upwind scheme do not always provide the required computational efficiency or sufficiently accurate resolution of the flow field. From a modelling perspective, closure schemes required in volumeaveraged formulations, such as the generalized Leverett J function for capillary pressure, are specific to certain media (e.g. lithologic media) and are not valid for fibrous porous media, which are of central interest in fuel cells.
This thesis presents a set of algorithms that are integrated efficiently to achieve computations that are more than two orders of magnitude faster compared to traditional techniques. The method uses an adaptive operator splitting method based on an a posteriori criterion to separate the flow from the transport equations which eliminates unnecessary and costly solution of the implicit pressurevelocity term at every time step; adaptive meshing to reduce the size of the discretized problem; efficient block preconditioned solver techniques for fast solution of the discrete equations; and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. The significant improvements in accuracy and efficiency of the approach is demosntrated using numerical experiments in 2D and 3D. The method is also extended to advectiondominated problems to specifically investigate twophase flow in heterogeneous porous media involving capillary transport. Both hydrophilic and hydrophobic media are considered, and insights relevant to fuel cell electrodes are discussed.

10 
Adaptive Solvers for HighDimensional PDE Problems on Clusters of Multicore ProcessorsGrandin, Magnus January 2014 (has links)
Accurate numerical solution of timedependent, highdimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massivescale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on largescale parallel computers. We utilize highorder discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (nonuniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBPSAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable. For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs. As an example application problem, we have considered the timedependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids. / eSSENCE

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