The small punch creep testing method is highly complex and involves interactions between a number of nonlinear processes. The deformed shapes which 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. Due to its complex nature, it is difficult to interpret the small punch test creep data in relation to the corresponding uniaxial creep behaviour of the material. One of the aims of this paper is to identify the important characteristics of the creep deformation resulting from "localised" deformations and from the "overall" deformation of the specimen. Following this, the results of approximate analytical and detailed finite element analyses of small punch tests are investigated. It is shown that the regions of the uniaxial creep test curves dominated by primary, secondary and tertiary creep, are not those which 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 a reference stress approach is proposed for interpreting the results of small punch test experimental data. Future work planned for the interpretation of small punch tests data is briefly addressed.

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.

Calibration of the Energy Exascale Earth System Model (E3SM), land model (ELMv0) is challenging because of its model complexity, strong model nonlinearity, and significant computational requirements. Therefore, only a limited number of simulations can be allowed in any attempt to find a near‐optimal solution within an affordable time. The goal of this study is to calibrate some of the ELMv0 parameters to improve model projection of carbon fluxes. We propose a computationally efficient global optimization procedure using sparse‐grid based surrogates. We first use advanced sparse grid (SG) interpolation to construct a surrogate system of the ELMv0, and then calibrate the surrogate model in the optimization process. As the surrogate model is a polynomial whose evaluation is fast, it can be efficiently evaluated a sufficiently large number of times in the optimization, which facilitates the global search. We calibrate eight parameters against five years of net ecosystem exchange, total leaf area index, and latent heat flux data from the U.S. Missouri Ozark flux tower. The calibrated model is then used for predicting the three variables in the following 4 years. The results indicate that an accurate surrogate model can be created for the ELMv0 with a relatively small number of SG points, i.e., a few ELMv0 simulations that can be fully parallel. And, the application of the optimized parameters leads to a better model performance and a higher predictive capability than the default parameter values in the ELMv0.

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White noise is a very common way to account for randomness in the inputs to partial differential equations, especially in cases where little is know about those inputs. On the other hand, pink noise, or more generally, colored noise having a power spectrum that decays as 1/f α , where f denotes the frequency and α ∈ (0, 2] has been found to accurately model many natural, social, economic, and other phenomena. Our goal in this paper is to study, in the context of simple linear and nonlinear two-point boundary-value problems, the effects of modeling random inputs as 1/f α random fields, including the white noise (α = 0), pink noise (α = 1), and brown noise (α = 2) cases. We show how such random fields can be approximated so that they can be used in computer simulations. We then show that the solutions of the differential equations exhibit a strong dependence on α, indicating that further examination of how randomness in partial differential equations is modeled and simulated is warranted.