Introducing a modified gradient vector method for optimization of accident prediction non-linear functions

In this paper, a new optimization method has been proposed for accident prediction non-linear models. This has been achieved by eliminating the Hessian matrix from the equation of optimal pace length in the gradient vector method. One advantage is that it is independent of the starting point in optimization processes and it provides convergence at the highest top as well. This method has been tested on an accident prediction model and its preference over the gradient vector method has been proven.     

Methacrylate monolithic stationary phases for gradient elution separations in microfluidic devices

Methacrylate monolithic stationary phases were produced in fused-silica chips by UV initiation. Poly(butyl methacrylate-co-ethylene dimethacrylate) (BMA) and poly(lauryl methacrylate-co-ethylene dimethacrylate) (LMA) monoliths containing 30, 35 and 40% monomers were evaluated for the separation of peptides under gradient conditions. The peak capacity was used as an objective tool for the evaluation of the separation performance. LMA monoliths of the highest density gave the highest peak capacities (≈40) in gradients of 15 min and all LMA monoliths gave higher peak capacities than the BMA monoliths with the same percentage of monomers. Increasing the gradient duration to 30 min did not increase the peak capacity significantly. However, running fast (5 min) gradients provides moderate peak capacities (≈20) in a short time. Due to the system dead volume of 1 μL and the low bed volume of the chip, early eluting peptides migrated over a significant part of the column during the dwell time under isocratic conditions. It was shown that this could explain an increased band broadening on the monolithic stationary phase materials used. The effect is stronger with BMA monoliths, which partly explains the inferior performance of this material with respect to peak capacity. The configuration of the connections on the chip appeared to be critical when fast analyses were performed at pressures above 20 bar.     

Separation of poly-3-hydroxyvalerate-co-3-hydroxybutyrate through gradient polymer elution chromatography

Polyhydroxyalkanoates are biodegradable polyesters produced by bacteria that can have a wide distribution in molecular weight, composition of monomers, and functionalities. This large distribution often leads to unpredictable physical properties making commercial applications challenging. To improve polymer homogeneity and obtain samples with a clear set of physical characteristics, poly-3-hydroxyvalerate-co-3-hydroxybutyrate copolymers were fractionated using gradient polymer elution chromatography (GPEC) as opposed to extensively used bulk fractionation. Separation was achieved using a reversed-phase column with chloroform and ethanol as the solvent and non-solvent, respectively. A separation was also conducted on a normal-phase column to compare elution patterns between columns of varied polarity. The fractions were analyzed using Size Exclusion Chromatography (SEC) and NMR to determine the percentage of 3-hydroxyvalerate in the copolymer as well as its molecular weight. It was found that as the percentage of "good" solvent was increased in the mobile phase, the polymers eluted with decreasing percentage of 3-hydroxyvalerate and increasing molecular weight which indicates the importance of precipitation/redissolution in the separation. The elution pattern of the polymer remained unchanged when using both a normal- and reversed-phase column which also illustrates the dominance of precipitation/redissolution in GPEC of polyhydroxyalkanoates. As such, GPEC is shown to be an excellent choice to provide polyhydroxyalkanoate samples with a narrower distribution in composition than the original bulk copolymer sample.     

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time-space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.     

User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide non asymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.     

Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions

This paper deals with the field enhancement due to insertion of a bow-tie structure of perfectly conducting inclusions into the two-dimensional space with a given field. The field enhancement is represented by the gradient blow-up of a solution to the conductivity problem. The bow-tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles. The domains are parts of cones near the vertices which are nearly touching to each other. We construct functions explicitly which characterize the field enhancement. As consequences, we derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners. The estimates show in quantitatively precise way that the field is enhanced beyond the corner singularities due to the interaction between two inclusions, and the blow-up rate is much higher than the one for the case of inclusions with smooth boundaries.     

求解一类二阶段有补偿问题的对偶梯度法

利用对偶理论，本文给出了求解一类具有简单补偿的非线性二阶段问题的新对偶梯度法．在假设目标函数为可分连续可微凸函数的条件下，在每一选代步可将原二阶段有补偿问题转化为几个一维凸规划问题，大大简化了问题的求解．所给算法简单易行，文中还证明了该算法的全局收敛性．     

Kink band propagation in layered structures

Confined layered structures in layer-parallel compression exhibit kink band deformation that has an associated severe initial instability. However, once the first kink band forms the system restabilizes and the bands then propagate in two mechanisms: band broadening and band progression. Previous work on triggering the initial instability and band broadening is now extended to include band progression. A new model for this propagation mechanism that successfully accounts for the modulating restabilizing response is presented and quantitative comparisons with the physical experiments yield excellent results.