Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Range convexity,continuity, and strategy-proofness of voting schemes

A voting scheme assigns to each profile of alternatives reported byn individuals a compromise alternative. A voting scheme is strategy-proof if no individual is better off by lying, i.e., not reporting a best alternative. In this paper the main results concern the case where the set of alternatives is the Euclidean plane and the preferences are Euclidean. It is shown that for strategy-proof voting schemes continuity is equivalent to convexity of the range of the voting scheme. Using a result by Kim and Roush (1984), this leads to characterizations of surjective or unanimous, anonymous, strategy-proof voting schemes.Furthermore, the paper contains an extensive discussion of related results in the area.     

Topological Sorting of Finitely Near Sets

This paper introduces a natural approach in the evaluation of the nearness of sets in topological spaces. The objective is to classify levels of nearness of sets relative to each given set. The main result is a proximity measure of nearness for disjoint sets in an extremally disconnected topological space. Another result is that if A, B are nonempty semi-open sets such that \(A\ \delta \ B\), then \((\text{int }A)\ \delta \ (\text{int }B)\).     

Convergence theory for nonconvex stochastic programming with an application to mixed logit

Monte Carlo methods have extensively been used and studied in the area of stochastic programming. Their convergence properties typically consider global minimizers or first-order critical points of the sample average approximation (SAA) problems and minimizers of the true problem, and show that the former converge to the latter for increasing sample size. However, the assumption of global minimization essentially restricts the scope of these results to convex problems. We review and extend these results in two directions: we allow for local SAA minimizers of possibly nonconvex problems and prove, under suitable conditions, almost sure convergence of local second-order solutions of the SAA problem to second-order critical points of the true problem. We also apply this new theory to the estimation of mixed logit models for discrete choice analysis. New useful convergence properties are derived in this context, both for the constrained and unconstrained cases, and associated estimates of the simulation bias and variance are proposed. Research Fellow of the Belgian National Fund for Scientific Research     

On converse Lyapunov theorems for fluid network models

We consider the class of closed generic fluid network (GFN) models, which provides an abstract framework containing a wide variety of fluid networks. Within this framework a Lyapunov method for stability of GFN models was proposed by Ye and Chen. They proved that stability of a GFN model is equivalent to the existence of a functional on the set of paths that is decaying along paths. This result falls short of a converse Lyapunov theorem in that no state-dependent Lyapunov function is constructed. In this paper we construct state-dependent Lyapunov functions in contrast to path-wise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem. To resolve this problem we introduce the class of strict GFN models by forcing closed GFN models to satisfy a concatenation and a semicontinuity condition. For the class of strict GFN models we define a state-dependent Lyapunov function and show that a converse Lyapunov theorem holds. Finally, it is shown that common fluid network models, like general work-conserving and priority fluid network models as well as certain linear Skorokhod problems define strict GFN models.     

Chain homogeneous Souslin algebras

Assuming Jensen's principle ?⁺ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todor?evi?.     

Semi-crossed products of C1-algebras

Given a C¹-algebra $\text{U}$ and endomorphim α, there is an associated nonselfadjoint operator algebra $\text{Z}$⁺ X_α$\text{U}$, called the semi-crossed product of $\text{U}$ with α. If α is an automorphim, $\text{Z}$⁺ X_α$\text{U}$ can be identified with a subalgebra of the C¹-crossed product $\text{Z}$⁺ X_α$\text{U}$. If $\text{U}$ is commutative and α is an automorphim satisfying certain conditions, $\text{Z}$⁺ X_α$\text{U}$ is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C₀(S) given by α(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z⁺X_αC₀(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.     

Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

Journal of Theoretical Probability - We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha...