Showing papers by "Michael Ortiz published in 2013"

Optimal Uncertainty Quantification

Abstract: We propose a rigorous framework for uncertainty quantification (UQ) in which the UQ objectives and its assumptions/information set are brought to the forefront. This framework, which we call optimal uncertainty quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop optimal concentration inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the nonpropagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained minitutorial on the basic concepts and issues of UQ.

Coarse-graining Kohn-Sham Density Functional Theory

Abstract: We present a real-space formulation for coarse-graining Kohn–Sham Density Functional Theory that significantly speeds up the analysis of material defects without appreciable loss of accuracy. The approximation scheme consists of two steps. First, we develop a linear-scaling method that enables the direct evaluation of the electron density without the need to evaluate individual orbitals. We achieve this by performing Gauss quadrature over the spectrum of the linearized Hamiltonian operator appearing in each iteration of the self-consistent field method. Building on the linear-scaling method, we introduce a spatial approximation scheme resulting in a coarse-grained Density Functional Theory. The spatial approximation is adapted so as to furnish fine resolution where necessary and to coarsen elsewhere. This coarse-graining step enables the analysis of defects at a fraction of the original computational cost, without any significant loss of accuracy. Furthermore, we show that the coarse-grained solutions are convergent with respect to the spatial approximation. We illustrate the scope, versatility, efficiency and accuracy of the scheme by means of selected examples.

Modeling fracture by material-point erosion

Abstract: The present work is concerned with the verification and validation of an implementation of the eigenfracture scheme of Schmidt et al. (SIAM J Multi-scale Model Simul 7:1237–1266, 2009) based on material-point erosion, which we refer to as eigenerosion. Eigenerosion is derived from the general eigenfracture scheme by restricting the eigendeformations in a binary sense: they can be either zero, in which case the local behavior is elastic; or they can be equal to the local displacement gradient, in which case the corresponding material neighborhood is failed, or eroded. When combined with a material-point spatial discretization, this scheme gives rise to material-point erosion, i. e., each material point can be either intact, in which case its behavior is elastic, or be completely failed—or eroded—and has no load bearing capacity. We verify the eigenerosion scheme through convergence studies for mode I fracture propagation in three-dimensional problems. By way of validation we apply the eigene-rosion scheme to the simulation of combined torsion- traction experiments in aluminum-oxide bars.

Finite element analysis of geometrically necessary dislocations in crystal plasticity

Abstract: We present a finite element method for the analysis of ductile crystals whose energy depends on the density of geometrically necessary dislocations (GNDs). We specifically focus on models in which the energy of the GNDs is assumed to be proportional to the total variation of the slip strains. In particular, the GND energy is homogeneous of degree one in the slip strains. Such models indeed arise from rigorous multiscale analysis as the macroscopic limit of discrete dislocation models or from phenomenological considerations such as a line-tension approximation for the dislocation self-energy. The incorporation of internal variable gradients into the free energy of the system renders the constitutive model non-local. We show that an equivalent free-energy functional, which does not depend on internal variable gradients, can be obtained by exploiting the variational definition of the total variation. The reformulation of the free energy comes at the expense of auxiliary variational problems, which can be efficiently solved using finite element approximations. The addition of surface terms in the formulation of the free energy results in additional boundary conditions for the internal variables. The proposed framework is verified by way of numerical convergence tests, and simulations of three-dimensional problems are presented to showcase its applicability. A performance analysis shows that the proposed framework solves strain-gradient plasticity problems in computing times of the order of local plasticity simulations, making it a promising tool for non-local crystal plasticity three-dimensional large-scale simulations.

A micromechanical model of distributed damage due to void growth in general materials and under general deformation histories

Abstract: We develop a multiscale model of ductile damage by void growth in general materials undergoing arbitrary deformations. The model is formulated in the spirit of multiscale finite element methods (FE 2), that is, the macroscopic behavior of the material is obtained by a simultaneous numerical evaluation of the response of a representative volume element. The representative microscopic model considered in this work consists of a space-filling assemblage of hollow spheres. Accordingly, we refer to the present model as the packed hollow sphere (PHS) model. A Ritz–Galerkin method based on spherical harmonics, specialized quadrature rules, and exact boundary conditions is employed to discretize individual voids at the microscale. This discretization results in material frame indifference, and it exactly preserves all material symmetries. The effective macroscopic behavior is then obtained by recourse to Hill's averaging theorems. The deformation and stress fields of the hollow spheres are globally kinematically and statically admissible regardless of material constitution and deformation history, which leads to exact solutions over the entire representative volume under static conditions. Excellent convergence and scalability properties of the PHS model are demonstrated through convergence analyses and examples of application. We also illustrate the broad range of material behaviors that are captured by the PHS model, including elastic and plastic cavitation and the formation of a vertex in the yield stress of porous metals at low triaxiality. This vertex allows ductile damage to occur under shear-dominated conditions, thus overcoming a well-known deficiency of Gurson's model.

A Γ-Convergence Analysis of the Quasicontinuum Method

Abstract: We present a $\Gamma$-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defect-free crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with $\Gamma$-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a ...

Optimal uncertainty quantification for legacy data observations of lipschitz functions

Abstract: We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

Optimal Uncertainty Quantification: Distributional Robustness Versus Bayesian Brittleness

Abstract: We discuss recent mathematical and computational results on uncertainty quantification (UQ) in the presence of uncertainty about the correct probabilistic and physical models. Such UQ problems can be formulated as constrained optimization problems with information acting as the constraints, with consequent optimal assessments of risk, and advantages for interdisciplinary communication and open science. We also report consequences of this point of view for the robustness of Bayesian methods under prior perturbation.

Automatically inf − sup compliant diamond-mixed finite elements for Kirchhoff plates

Abstract: We develop a mixed finite-element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment-equilibrium problem for the rotations is in direct analogy to the problem of incompressible two-dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf − sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment-equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf − sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf − sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems.

Coupled thermoelastic simulation of nanovoid cavitation by dislocation emission at finite temperature

Abstract: In this work we study the early onset of void growth by dislocation emission at finite temperature in single crystal of copper under uniaxial loading conditions using the HotQC method. The results provide a detailed characterization of the cavitation mechanism, including the geometry of the emitted dislocations, the dislocation reaction paths and attendant macroscopic quantities of interest such as the cavitation pressure. In addition, this work shows that as prismatic dislocation loops grow and move away from the void, the material surrounded by these loops is pushed away from the void surface, giving rise to a flux of material together with a heat flux through the crystal.

A Cohesive Model for Fatigue Crack

Abstract: We describe fatigue processes within the framework of cohesive theories offracture. Crack formation is due to the gradual separation of material surfaces resisted bycohesive tractions. The relationship between traction and opening displacement is governed byan irreversible law with unloading-reloading hysteresis. We assume that the unloadingreloadingresponse of the cohesive model degrades with the number of cycles and assume thereloading stiffness as damage variable. The fatigue behavior is embedded into surface-like finiteelements, compatible with a standard discretization of solid volumes. The potential fatiguecracks are identified by inter-element surfaces, initially coherent. When a fatigue initiationcriterion is satisfied, a self-adaptive remeshing procedure inserts a cohesive element.

Thermalization of rate-independent processes by

Abstract: We consider the eective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to \thermalize" the process is an interior-point entropic regularization of the Moreau{Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the eective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a dierent and non-trivial eective dissipation potential.