In this work, we propose to extend the Arlequin framework to couple particle and continuum models. Three different coupling strategies are investigated based on the L 2 norm, H 1 seminorm, and H 1 norm. The mathematical properties of the method are studied for a one-dimensional model of harmonic springs, with varying coefficients, coupled with a linear elastic bar, whose modulus is determined by simple homogenization. It is shown that the method is wellposed for the H 1 seminorm and H 1 norm coupling terms, for both the continuous and discrete formulations. In the case of L 2 coupling, it cannot be shown that the Babuška-Brezzi condition holds for the continuous formulation. Numerical examples are presented for the model problem that illustrate the approximation properties of the different coupling terms and the effect of mesh size.

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In this paper, we present an extension of goal-oriented error estimation and adaptation to the simulation of multi-scale problems of molecular statics. Computable error estimates for the quasicontinuum method are developed with respect to specific quantities of interest and an adaptive strategy based upon these estimates is proposed for error control. The theoretical results are illustrated on a nanoindentation problem in which the quantity of interest is the force acting on the indenter. The promising capability of such error estimates and adaptive procedure for the solution of multi-scale problems is demonstrated on numerical examples.

In the present study, a general dynamic data-driven application system (DDDAS) is developed for real-time monitoring of damage in composite materials using methods and models that account for uncertainty in experimental data, model parameters, and in the selection of the model itself. The methodology involves (i) data data from uniaxial tensile experiments conducted on a composite material; (ii) continuum damage mechanics based material constitutive models; (iii) a Bayesian framework for uncertainty quantification, calibration, validation, and selection of models; and (iv) general Bayesian filtering, as well as Kalman and extended Kalman filters. A software infrastructure is developed and implemented in order to integrate the various parts of the DDDAS. The outcomes of computational analyses using the experimental data prove the feasibility of the Bayesian-based methods for model calibration, validation, and selection. Moreover, using such DDDAS infrastructure for real-time monitoring of the damage and degradation in materials results in results in an improved prediction of failure in the system. ‡ It is with great pleasure that we contribute this study in honor of our dear colleague, friend, and visionary in the field of numerical methods in engineering, Professor Ted Belytschko. We thank him for his numerous contributions to this field, for his statesmanship, and for his leadership over the many years of development of the subject of computational engineering.highly nonlinear material damage theories of the type used in contemporary fatigue analysis, fracture mechanics, and structural mechanics. These typically involve material parameters that exhibit uncertainties. In order to provide information for real-time monitoring of damage, the dynamically collected data of uniaxial tensile experiments conducted on composite materials [8] are taken into consideration. Thus, the system itself must be calibrated and validated, and the inherent uncertainties in data must be factored into a statistical analysis for the validation of the full system. A Bayesian framework is also developed for defining, updating, and quantifying uncertainties in the model, the experimental data, and the target quantities of interest. This paper is structured as follows. A summary of some physical models for damage that are considered for adoption along with the finite element solution procedure is presented in Section 2. This is followed in Section 3, the development of a corresponding DDDAS. In Section 4, the experimental results used in the statistical analyses are presented. Bayesian methods for model calibration, validation, and selection with quantification of uncertainties are outlined in Section 5. Section 6 describes the developed and implemented software infrastructure in order to integrate the acquired experimental data along with the finite element solution of the continuum damage mechanics model in order to calibrate the model and compute model plausibilities that guide the selection of the models themselves. This is accomplished...

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Abstract. The progression of scientific computing resources has enabled the numerical approximation of mathematical models describing complex physical phenomena. A significant portion of researcher time is typically dedicated to the development of software to compute the numerical solutions. This work describes a flexible C++ software framework, built on the libMesh finite element library, designed to alleviate developer burden and provide easy access to modern computational algorithms, including quantity-of-interest-driven parallel adaptive mesh refinement on unstructured grids and adjoint-based sensitivities. Other software environments are highlighted and the current work motivated; in particular, the present work is an attempt to balance software infrastructure and user flexibility. The applicable class of problems and design of the software components is discussed in detail. Several examples demonstrate the effectiveness of the design, including applications that incorporate uncertainty. Current and planned developments are discussed.