Michael Griebel, Alexander Hullmann and Oeter Oswald Optimal scaling parameters for sparse grid discretizations Numerical Linear Algebra with Applications, 22(1): 76-100 2015 http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable non-linear preconditioner.
Stefanie Heyden, Bo Li, Kerstin Weinberg, Sergio Conti and Michael Ortiz A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity J. Mech. Phys. Solids, 74: 175-195 2015 http://dx.doi.org/10.1016/j.jmps.2014.08.005
Abstract: We derive and numerically verify scaling laws for the macroscopic fracture energy of poly- mers undergoing crazing from a micromechanical model of damage. The model posits a local energy density that generalizes the classical network theory of polymers so as to account for chain failure and a nonlocal regularization based on strain-gradient elasticity. We specifically consider periodic deformations of a slab subject to prescribed opening dis- placements on its surfaces. Based on the growth properties of the energy densities, scaling relations for the local and nonlocal energies and for the specific fracture energy are derived. We present finite-element calculations that bear out the heuristic scaling relations.
Aicke Hinrichs, Lev Markhasin, Jens Oettershagen and Tino Ullrich Optimal quasi-Monte Carlo rules on higher order digital nets for the numerical integration of multivariate periodic functions 2015 http://arxiv.org/pdf/1501.01800v1
Brandon Runnels, Irene Beyerlein, Sergio Conti and Michael Ortiz A relaxation method for the energy and morphology of grain boundaries and interfaces J. Mech. Phys. Solids 2015 http://dx.doi.org/10.1016/j.jmps.2015.11.007
2014
Landry Fokoua Djodom, Sergio Conti and Michael Ortiz Optimal Scaling in Solids undergoing Ductile Fracture by Void Sheet Formation Arch. Ration. Mech. Anal., 212(1): 331-357 2014 http://dx.doi.org/10.1007/s00205-013-0687-8
Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.
Landry Fokoua Djodom, Sergio Conti and Michael Ortiz Optimal scaling laws for ductile fracture derived from strain-gradient microplasticity J. Mech. Phys. Solids, 62: 295-311 2014 http://dx.doi.org/10.1016/j.jmps.2013.11.002
Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.
Michael Griebel and Jens Oettershagen Dimension-adaptive sparse grid quadrature for integrals with boundary singularities In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 109-136.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_5
Aicke Hinrichs and Jens Oettershagen Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives 2014 http://arxiv.org/pdf/1409.5894v1
Shin-ichi Ohta and Karl-Theodor Sturm Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds Adv. Math., 252: 429-448 2014 http://dx.doi.org/10.1016/j.aim.2013.10.018
0
P.L. Ferrari and A. Occelli Time-time covariance for last passage percolation in half-space preprint:arXiv:2204.06782 0 https://arxiv.org/abs/2204.06782
Abstract: This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we have to overcome the difficulty that the point-to-point model in half-space with generic start and end points is not known.