A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions

In this work we develop a dynamically adaptive sparse grids (SG) method for quasi-optimal interpolation of multidimensional analytic functions defined over a product of one dimensional bounded domains. The goal of such approach is to construct an interpolant in space that corresponds to the "best $M$-terms" based on sharp a priori estimate of polynomial coefficients. In the past, SG methods have been successful in achieving this, with a traditional construction that relies on the solution to a Knapsack problem: only the most profitable hierarchical surpluses are added to the SG. However, this approach requires additional sharp estimates related to the size of the analytic region and the norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we present an iterative SG procedure that adaptively refines an estimate of the region and accounts for the effects of the Lebesgue constant. Our approach does not require any a priori knowledge of the analyticity or operator norm, is easily generalized to both affine and non-affine analytic functions, and can be applied to sparse grids build from one dimensional rules with arbitrary growth of the number of nodes. In several numerical examples, we utilize our dynamically adaptive SG to interpolate quantities of interest related to the solutions of parametrized elliptic and hyperbolic PDEs, and compare the performance of our quasi-optimal interpolant to several alternative SG schemes

Investigation of small punch creep testing

Assessing the damage level of in-service components and obtaining material properties for welded structures exposed to creep is essential for the safe operating of power generation industry. Standard creep testing techniques require relatively large volumes of material for the machining of testing samples. For that reason they are not usually suitable for obtaining creep properties of in-service structures. It has been found that significant amount of the failures in welds exposed to elevated temperatures occur in an area formed due to the complex thermal and cooling cycles during the welding process. Because of this a different approach is needed for the derivation of creep properties from small amounts of metal. The small punch creep testing method is considered to be a, potentially, powerful technique for obtaining creep and creep rupture properties of in-service welded components. However, relating small punch creep test data to the corresponding uniaxial creep data has not proved to be simple and a straightforward approach is required. The small punch creep testing method is highly complex and involves interactions between a number of non-linear processes. The deformed shapes that are produced from such tests are related to the punch and specimen dimensions and to the elastic, plastic, and creep behaviour of the test material, under contact and large deformation conditions, at elevated temperature. Owing to its complex nature, it is difficult to interpret small punch creep test data in relation to the corresponding uniaxial creep behaviour of the material. One of the aims of this research is to identify the important characteristics of the creep deformation results from 'localized' deformations and from the 'overall' deformation of the specimen. For this purpose, the results of approximate analytical methods, experimental tests and detailed finite element analyses, of small punch tests, have been obtained. It is shown that the regions of the uniaxial creep test curves dominated by primary, secondary and tertiary creep are not those that are immediately apparent from the displacement versus time records produced during a small punch test. On the basis of the interpretation of the finite element results presented, a method based on the reference stress approach is proposed for interpreting the result of small punch experimental test data and relating it to the corresponding uniaxial creep data. Another aim of this study is to investigate the effect of friction between the sample and the punch as well as the effects of the basic dimensions, on the small punch creep testing data

Investigation of small punch creep testing

Assessing the damage level of in-service components and obtaining material properties for welded structures exposed to creep is essential for the safe operating of power generation industry. Standard creep testing techniques require relatively large volumes of material for the machining of testing samples. For that reason they are not usually suitable for obtaining creep properties of in-service structures. It has been found that significant amount of the failures in welds exposed to elevated temperatures occur in an area formed due to the complex thermal and cooling cycles during the welding process. Because of this a different approach is needed for the derivation of creep properties from small amounts of metal. The small punch creep testing method is considered to be a, potentially, powerful technique for obtaining creep and creep rupture properties of in-service welded components. However, relating small punch creep test data to the corresponding uniaxial creep data has not proved to be simple and a straightforward approach is required. The small punch creep testing method is highly complex and involves interactions between a number of non-linear processes. The deformed shapes that are produced from such tests are related to the punch and specimen dimensions and to the elastic, plastic, and creep behaviour of the test material, under contact and large deformation conditions, at elevated temperature. Owing to its complex nature, it is difficult to interpret small punch creep test data in relation to the corresponding uniaxial creep behaviour of the material. One of the aims of this research is to identify the important characteristics of the creep deformation results from 'localized' deformations and from the 'overall' deformation of the specimen. For this purpose, the results of approximate analytical methods, experimental tests and detailed finite element analyses, of small punch tests, have been obtained. It is shown that the regions of the uniaxial creep test curves dominated by primary, secondary and tertiary creep are not those that are immediately apparent from the displacement versus time records produced during a small punch test. On the basis of the interpretation of the finite element results presented, a method based on the reference stress approach is proposed for interpreting the result of small punch experimental test data and relating it to the corresponding uniaxial creep data. Another aim of this study is to investigate the effect of friction between the sample and the punch as well as the effects of the basic dimensions, on the small punch creep testing data

Submonolayer growth with decorated island edges

We study the dynamics of island nucleation in the presence of adsorbates using kinetic Monte Carlo simulations of a two-species growth model. Adatoms (A-atoms) and impurities (B-atoms) are codeposited, diffuse and aggregate subject to attractive AA- and AB-interactions. Activated exchange of adatoms with impurities is identified as the key process to maintain decoration of island edges by impurities during growth. While the presence of impurities strongly increases the island density, a change in the scaling of island density with flux, predicted by a rate equation theory for attachment-limited growth [D. Kandel, Phys. Rev. Lett. 78, 499 (1997)], is not observed. We argue that, within the present model, even completely covered island edges do not provide efficient barriers to attachment.Comment: 7 pages, 2 postscript figure

Sparse-grid Discontinuous Galerkin Methods for the Vlasov-Poisson-Lenard-Bernstein Model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov-Poisson-Lenard-Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD