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This article presents and analyzes a $p^{th}$-degree immersed finite element (IFE) method for elliptic interface problems with nonhomogeneous jump conditions. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE piecewise polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements. This $p^{th}$-degree IFE method is proved to converge optimally under mesh refinement. In addition, this article addresses the stability of this IFE method and has established upper bounds for its condition numbers which are optimal with respect to the mesh size but suboptimal with respect to the contrast of the discontinuous coefficient.

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babu\v{s}ka and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions.

In this work we propose a stochastic model for estimating the occurrence of crack initiations on the surface of metallic specimens in fatigue problems that can be applied to a general class of geometries. The stochastic model is based on spatial Poisson processes with intensity function that combines stress-life (S-N) curves with averaged effective stress, $\sigma_{{\mathrm{eff}}}^{\Delta} (\mathbf{x})$, which is computed after solving numerically the linear elasticity equations on the specimen domains using finite element methods. Here, $\Delta$ is a parameter that characterizes the size of the neighbors covering the domain boundary. The averaged effective stress, parameterized by $\Delta$, maps the stress tensor to a scalar field upon the specimen domain. Data from fatigue experiments on notched and unnotched sheet specimens of 75S-T6 aluminum alloys are used to calibrate the model parameters for the individual data sets and for their combination. Bayesian and classical approaches are applied to estimate the survival-probability function for any specimen tested under a prescribed fatigue experimental setup. Our proposed model can predict the initiation of cracks in specimens made from the same material with new geometries.

The Generalized Finite Element Method (GFEM) is an extension of the Finite Element Method (FEM), where the standard finite element space is augmented with a space of non-polynomial functions, called the enrichment space. The functions in the enrichment space mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully applied to a wide range of problems. However, it often suffers from bad conditioning, i.e., its conditioning may not be robust with respect to the mesh and in fact, the conditioning could be much worse than that of the standard FEM. In this paper, we present a numerical study that shows that if the "angle" between the finite element space and the enrichment space is bounded away from 0, uniformly with respect to the mesh, then the GFEM is stable, i.e., the conditioning of GFEM is not worse than that of the standard FEM. A GFEM with this property is called a Stable GFEM (SGFEM). The last part of the paper is devoted to the derivation of a robust iterative solver exploiting this angle condition. It is shown that the required "wall-clock" time is greatly reduced compared to popular GFEMs used in the literature.

In this work, we present a statistical treatment of stress-life (S-N) data drawn from a collection of records of fatigue experiments that were performed on 75S-T6 aluminum alloys. Our main objective is to predict the fatigue life of materials by providing a systematic approach to model calibration, model selection and model ranking with reference to S-N data. To this purpose, we consider fatigue-limit models and random fatigue-limit models that are specially designed to allow the treatment of the run-outs (right-censored data). We first fit the models to the data by maximum likelihood methods and estimate the quantiles of the life distribution of the alloy specimen. To assess the robustness of the estimation of the quantile functions, we obtain bootstrap confidence bands by stratified resampling with respect to the cycle ratio. We then compare and rank the models by classical measures of fit based on information criteria. We also consider a Bayesian approach that provides, under the prior distribution of the model parameters selected by the user, their simulation-based posterior distributions. We implement and apply Bayesian model comparison methods, such as Bayes factor ranking and predictive information criteria based on cross-validation techniques under various a priori scenarios.

We study mathematical and computational models for computing the deformation of fiber-reinforced cross-plied laminates due to external forces. This requires an understanding of both micro-structural effects and different sources of uncertainty in the problem. We first show that the uncertainties in the problem are of both statistical (aleatoric) and systematic (epistemic) types and that current multiscale stochastic models, such as stationary random fields, which are based on precise probability theory, are not capable of correctly characterizing uncertainty in fiber composites. Next, we motivate the applicability of models based on imprecise uncertainty theory and present a novel fuzzy-stochastic model, which can more accurately describe uncertainties in fiber composites. The new model is constructed by combining stochastic fields and fuzzy variables through a simple calibration-validation approach. Finally, we construct a global-local multiscale algorithm for efficiently computing output quantities of interest. The method aims at approximating required quantities, such as displacements and stresses, in regions of relatively small size, e.g. hot spots or zones. The algorithm uses the concept of representative volume elements and computes a global solution to construct a local approximation that captures the microscale features of the solution. The results are based on and backed by real experimental data.

The paper analyses the convergence of the $p$-version of the FEM when the solution is piecewise analytic function. It focuses on pointwise convergence of the gradient. It shows that at boundary the rate is different than inside the element and there is a Gibbs phenomenon in the neighborhood of the point where the solution is not analytic. The major result is a conjecture written in the form of a theorem. The conjecture is based on careful numerical computations. The known theoretical results are stated.

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in $L^\infty(\Omega)$. This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in \cite{107}, and elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by partitioning the computational domain $\Omega$ into to a collection of preselected subsets $\omega_{i},i=1,2,..m$ and constructing finite dimensional approximation spaces $\Psi_{i}$ over each subset using local information. The notion of the Kolmogorov $n$-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom $N_{i}$ in the energy norm over $\omega_i$. The local spaces $% \Psi_{i}$ are used within the GFEM scheme to produce a finite dimensional subspace $S^N$ of $H^{1}(\Omega)$ which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., super-algebraicly) with respect to the degrees of freedom $N$. When length scales "`separate" and the microstructure is sufficiently fine with respect to the length scale of the domain $\omega_i$ it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime., Comment: 30 pages, 6 figures, updated references, sections 3 and 4 typos corrected, minor text revision, results unchanged

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\Omega$, using the Generalized Finite Element Method. For smooth, bounded domains $\Omega$, we obtain that the GFEM--approximation $u_S$ satisfies $\|u - u_S\|_{H\sp{1}(A)} \le C h^{\gamma}\|u\|_{H\sp{1-k}(\Omega)}$, where $h$ is the typical size of the ``elements'' defining the GFEM--space $S$ and $\gamma \ge 0 $ is such that the local approximation spaces contain all polynomials of degree $k + \gamma + 1$. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz \cite{NitscheSchatz72} and, especially, \cite{NitscheSchatz74}. It turns out that, in addition to the usual ``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev spaces $H\sp{-l}$, $l \ge 0$, which are defined by duality and contain the distributional boundary data., Comment: 23 pages