On the higher-order stroboscopic method

The stroboscopic method is a semi-analytical method for the solution of the equations of a perturbed dynamical system depending on several slow variables and one fast variable. It is shown how this method can be generalised to higher order approximations. An important application has been made in satellite dynamics.

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Zusammenfassung Die stroboskopische Methode ist eine semi-analytische Methode zur Lösung der Gleichungen eines gestörten dynamischen Systems welches von mehreren langsamen Variablen abhängt und einer schnellen Variablen. Es wird gezeigt, daß sich die Methode auf Approximationen höherer Ordnung verallgemeinern läßt. Eine wichtige Anwendung findet sie in der Satellitendynamik.

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Dedicated to Professor Eduard Stiefel     

Modeling of microstructural pattern formation in crystal plasticity

The mechanical behavior of most materials is dictated by a present or emergent underlying microstructure which is a direct result of different, even competing physical mechanisms occurring at lower length scales. In this work, energetic microstructure interaction via different non-convex contributions to the free energy in metals is modeled. For this purpose rate dependent gradient extended crystal plasticity models at the glide-system level are formulated. The non-convex energy serves as the driving force for the emergent microstructure. The competition between the kinetics and the relaxation of the free energy is an essential feature of the model. Non-convexity naturally arises in finite-deformation single-slip crystal plasticity and the results of the gradient model for this case are compared with an effective laminate model based on energy relaxation. Similarities as well as essential differences are observed and explained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence of higher-order orthogonal iteration

The higher-order orthogonal iteration (HOOI) has been popularly used for finding a best low-multilinear rank approximation of a tensor. However, its convergence is still an open question. In this paper, we first analyse a greedy HOOI, which updates each factor matrix by selecting from the best candidates one that is closest to the current iterate. Assuming the existence of a block-nondegenerate cluster point, we establish its global iterate sequence convergence through the so-called Kurdyka–?ojasiewicz property. In addition, we show that if the starting point is sufficiently close to any block-nondegenerate globally optimal solution, the greedy HOOI produces an iterate sequence convergent to a globally optimal solution. Relating the iterate sequence by the original HOOI to that by the greedy HOOI, we then show that the original HOOI has global convergence on the multilinear subspace sequence and thus positively address the open question.     

On the fractional counterpart of the higher-order equations

In this work, we study the solutions of some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be expressed by means of the transition laws of stable subordinators and their inverse processes. In particular, we establish connections between fractional and higher-order equations.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

On physics-based crystal plasticity models: Application to virtual material testing of sheet metals

It is possible to pursue a multi-scale modeling approach for sheet forming simulations by applying the concept of virtual material testing to determine the yield surface from the microstructure of a given material. Full-field simulations with phenomenological crystal plasticity models are widely used for this kind of investigations. However, recent developments focus on incorporating physical quantities like dislocation density into these models. In this work, a dislocation density based crystal plasticity model is used to investigate the plastic anisotropy of the deep drawing steel DC04. In particular, we focus on the prediction of R-values, which can be used to calibrate macroscopic plasticity models. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On higher-order proto-differentiability of perturbation maps

This paper is concerned with higher-order sensitivity analysis in parametric vector optimization problems. Firstly, higher-order proto-differentiability of a set-valued mapping from one Euclidean space to another is defined. Then, we prove that the perturbation map/the proper perturbation map/the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable under some suitable qualification conditions.

     

Simulation of micro-cutting considering finite deformation crystal plasticity

Micro-machining processes on metalic microstructures are influenced by the crystal structure, i. e. the grain orientation. Furthermore, the chip formation underlies large deformations. To perform finite element simulations of micro-cutting processes, a large deformation material model is necessary in order to model the hyperelastic and finite plastic material behaviour. In the case of cp-titanium material with hcp-crystal structure the anisotropic behaviour must be considered by an appropriate set of slip planes and slip directions. In the present work the impact of the grain orientation on the plastic deformation is demonstrated by means of finite element simulations of a finite deformation single slip crystal plasticity model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometrically nonlinear models in crystal plasticity and the limit of rigid elasticity

It is shown that a standard model in crystal plasticity for which only rigid body motions have finite elastic energy can be approximated in the sense of Γ-convergence by models with more general elastic potentials. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

 We consider the elliptic operator P(D)+V in ℝ ^d , d≥2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2ℓ with a homogeneous convex symbol P(ξ), and V is a real periodic function in L ^∞(ℝ ^d ). We show that the number of gaps in the spectrum of P(D)+V is finite if 4ℓ>d+1. If in addition, V is smooth and the convex hypersurface {ξℝ ^d :P(ξ)=1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8ℓ>d+3 and 9≥d≥2, or 4ℓ>d−3 and d≥10. Received: 10 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 35J10 Research supported in part by NSF Grant DMS-9732894.     

On the higher-order differential-geometric objects of a multidimensional three-web

This survey is a study of the geometry of a multidimensional three-web defined by its higher-order differential-geometric objects. Relations are obtained that connect the fundamental tensors of akth-order differential neighborhood. Using these relations we solve the problem of determining whether theg-structure defined by a multidimensional hexagonal three-web is closed. The latter turns out to be a closedg-structure of class 4. This result is then generalized. To be specific, analytic conditions are found for ag-structure of arbitrary order to be closed. These, as shown in the article, can be interpreted algebraically using a certain number of identities in one variable which are satisfied in the coordinate loops of the web. A classification of identities of this type is given.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 101–154, 1987.     

A coupled phase-field model for ductile fracture in crystal plasticity

The objective of this work is to present a simplified, nonetheless representative first stage of a phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional well- stablished elasto-plastic crystal plasticity model and a well-known phase-field model (PFM) modified to predict ductile fracture. A two-dymensional initial boundary-value problem of ductile fracture is introduced considering a single crystal Nickel-base superalloy material. the model is implemented into the finite element context subjected to a one-dimensional tension test (displacement-controlled). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Hermite-Hadamard-type characterizations of higher-order differential inequalities

Let $I$ be an open interval of $\mathbb{R}$ and $f: I\to \mathbb{R}$. It is well-known that $f$ is convex in $I$ if and only if, for all $x,y\in I$ with $x相似文献   

On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation

     

An algorithm for massively parallel dislocation dynamics simulations of small scale plasticity

Accurate modeling of dislocation motion in bounded bodies is essential for the goal of obtaining desired properties, for example electronic or optical, of many microelectronic devices. At present, we lack high fidelity computer codes for such modeling that efficiently utilize modern parallel computer architectures. In contrast, many dislocation simulation codes are available for periodic or infinite bodies. In principle, these codes can be extended to allow for dislocation modeling in finite bodies. However, such extension may involve an additional solver to be employed, coupled with a dislocation simulation code. We present an algorithm for development of parallel dislocation simulation capability for bounded bodies based on such coupling. Subsequently, we analyze the performance of the algorithm for a demanding dislocation dynamics model problem.     

On the eigenvalues of certain canonical higher-order ordinary differential operators

We consider the operator of taking the 2pth derivative of a function with zero boundary conditions for the function and its first p−1 derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for p=1 and p=3.