Particle Finite Element Simulations of Cutting Processes in Manufacturing

The cutting of metals is an important process in manufacturing and challenges established methods in the field of computational mechanics. The particle finite element method (PFEM) combines the benefits of particle based methods and the standard finite element method (FEM) to account for large deformations and separation of material. In cutting simulations the workpiece is realised as a set of particles, whose boundary is detected by the α-shape method. After the boundary detection, the particles are meshed with finite elements. Since metals show a plastic behavior under large deformations, a suitable material model needs to be considered. Numerical examples show the effect of the choice of the parameter α on the cutting force. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelación numérica de deslizamientos de ladera en embalses mediante el Método de Partículas y Elementos Finitos (PFEM)

The paper presents the results of the application of the Particle Finite Element Method (PFEM) to the analysis of landslides in reservoirs. This is a complex phenomenon, because of the interaction between the landslide, the still water in the reservoir and the dam. PFEM combines a Lagrangian approach with the solution of the governing equations of the problem using the FEM. A mesh connecting the initial set of particles (nodes) is re-generated in every time step. Some validation cases are presented, in which PFEM results are compared with experimental data. More complex calculations have been made over the actual geometry of reservoirs taken from the cartographic information of the sites. In these cases the wave generation, its propagation and dam overtopping are reproduced. Finally, Lituya bay rock slide in which 90 × 10⁶ tons of rocks fell on the bay, generating a huge wave that caused a maximum run-up of 524 m on the opposite shore, has been simulated in 3D. The results show that PFEM is a useful tool in risk assessment related with landslides in reservoirs as it gives a good approximation to the potential affections, thus allowing the appropriate design of protection measures.     

Computational Excavation Analysis of a Single Cutting Tool using a Hypoplastic Constitutive Model

We propose a computational model for the numerical analysis of the dynamic interaction between a single excavation tool and the surrounding soil. An incremental non-linear (hypoplastic) constitutive model is employed to capture the complex response of soft soils. Large displacements and deformations are handled by an Updated Lagrangian formulation, the particle finite element method (PFEM). (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polynomial shape functions on the logarithmic space: the LogFE method

We extend the Logarithmic finite element method, a novel finite element approach for solving boundary-value problems proposed in [1], to a complete set of degrees of freedom, i.e. translational and rotational degrees of freedom in three dimensions. In contrast to the standard Ritz-Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function. Unlike existing formulations based on Lie groups, they may include polynomial functions of arbitrary degree. The method focuses on reducing the low-frequency components in the error, while minimizing spurious high-frequency deformations, a characteristic that is particularly advantageous in the context of a multigrid algorithm, in which the method may be used to construct an approximation for the coarse grid. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodicity and unbordered segments of words

A nonempty word β is said to be a border of a word α if and only if α = λβ = βρ for some nonempty words λ and ρ. For an arbitrary (possibly infinite) sequence α the expression # α denotes the (possibly infinite) supremum of the set of all |β| for β an unbordered finite segment of α.     

On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes

We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.This research was supported in part by the Advanced Technology Program of the State of Texas, in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, and in part by the Air Force Office of Scientific Research (AFSC) under Contract F49620-89-C-0044.     

On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Hypermaps versus bipartite maps

A hypermap was defined by R. Cori to be a pair of permutations σ and α on a finite set B, such that the group generated by σ and α is transitive on B. The genus of a hypermap was defined according to a formula of A. Jacques for the genus of a pair of permutations. This paper presents a one-to-one correspondence between the set of hypermaps of a given genus and the set of 2-colored bipartite maps of the same genus.     

Shape sensitivity analysis in the extended finite element method

The contribution is concerned with the derivation and implementation of shape sensitivity analysis in the context of the extended finite element method (xfem). Here, the displacement approximation is enhanced in order to model strong and weak discontinuities along the interfaces and cracks. Consequently, sensitivity analysis must be accomplished for the extended formulation based on a sub-integration technique on sub-domains of the cut elements. This cumbersome and error-prone work can be circumvent using a standard treatment of sub-elements on an appropriately defined sub-mesh. These modifications guarantee exact analytical sensitivities even for distorted finite element domains. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoothed Particle ElectroMagnetics: A mesh-free solver for transients

In this paper an advanced mesh-free particle method for electromagnetic transient analysis, is presented. The aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid-based numerical methods. The basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain. The mesh-free smoothed particle hydrodynamics method has been adopted to obtain numerical solution of time domain Maxwell's curl equations. An explicit finite difference scheme has been employed for time integration. Details about the numerical treatment of electromagnetic vector fields components are discussed. Two case studies in one and in two dimensions are reported. In order to validate the new proposed methodology, named as Smoothed Particle ElectroMagnetics, a comparison with the standard finite difference time domain method results is performed. The intrinsic adaptive capability of the proposed method, has been exploited by introducing irregular particles distribution.     

一类拟线性退化抛物型方程的初边值问题

本文讨论下面方程的初边值问题。光滑有界区域，且ｐ＞２．本文用正则化方法证明了当ｐ＞α＋２时广义解的全局存在性，对ｐ＜α＋２的情形，证明当初值ｕ＿ｏ（ｘ）属于一稳定集Ｗ时广义解的全局存在性，而当ｕ＿ｏ（ｘ）充分大时广义解只局部存在并在有限时间内爆破。最后用共轭算子法得到了广义解的唯一性定理。     

Convolution type operators on cones and their finite sections

This paper is concerned with finite sections of convolution type operators defined on cones, whose symbol is the Fourier transform of an integrable function on ?². The algebra of these finite sections satisfies a set of axioms (standard model) that ensures some asymptotic properties like the convergence of the condition numbers, singular values, ε‐pseudospectrum and also gives a relation between the singular values of an approximation sequence and the kernel dimensions of a set of associated operators. This approach furnishes a method to determine whether a Fredholm convolution operator on a cone is invertible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of Metropolis-type algorithms for a large canonical ensemble

In this paper, we study the convergence of Metropolis-type algorithms used in modeling statistical systems with a fluctuating number of particles located in a finite volume. We justify the use of Metropolis algorithms for a particular class of such statistical systems. We prove a theorem on the geometric ergodicity of the Markov process modeling the behavior of an ensemble with a fluctuating number of particles in a finite volume whose interaction is described by a potential bounded below and decreasing according to the law r ^?3?α, α ≥ 0, as r → 0.     

Strategies for the Computation of Configurational Forces in Dissipative Media

Configurational forces can be interpreted as driving forces on material inhomogeneities such as crack tips. In dissipative media the total configurational force on an inhomogeneity consists of an elastic contribution and a contribution due to the dissipative processes in the material. For the computation of discrete configurational forces acting at the nodes of a finite element mesh, the elastic and dissipative contributions must be evaluated at integration point level. While the evaluation of the elastic contribution is straightforward, the evaluation of the dissipative part is faced with certain difficulties. This is because gradients of internal variables are necessary in order to compute the dissipative part of the configurational force. For the sake of efficiency, these internal variables are usually treated as local history data at integration point level in finite element (FE) implementations. Thus, the history data needs to be projected to the nodes of the FE mesh in order to compute the gradients by means of shape function interpolations of nodal data as it is standard practice. However, this is a rather cumbersome method which does not easily integrate into standard finite element frameworks. An alternative approach which facilitates the computation of gradients of local history data is investigated in this work. This approach is based on the definition of subelements within the elements of the FE mesh and allows for a straightforward integration of the configurational force computation into standard finite element software. The suitability and the numerical accuracy of different projection approaches and the subelement technique are discussed and analyzed exemplarily within the context of a crystal plasticity model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetries of the Julia sets of Newton’s method for multiple root

The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained.     

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Abstract

We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.     

A simple heuristic for the p-centre problem

We describe a simple heuristic for determining the p-centre of a finite set of weighted points in an arbitrary metric space. The computational effort is O(np) for an n-point set. We show that the ratio of the objective function value of the heuristic solution to that of the optimum is bounded by min(3, 1 + α), where α is the maximum weight divided by the minimum weight of points in the set.     

Solving Diffusion Problems on Rough Surfaces with a Hierarchical Multiscale FEM

Diffusion on rough surfaces is a basic problem for many applications in engineering and the sciences. Solving these problems with a standard finite element method is often difficult or even impossible, due to the computational work and the amount of memory needed to triangulate the whole surface with a mesh which resolves its oscillations. We discuss in this paper a hierarchical Finite Element Method of “heterogeneous multiscale” type, which only needs to resolve the surface's fine scale on small sampling domains within a macro triangulation of the underlying smooth surface. This method converges, for periodic surface roughness and sufficiently small amplitude, at a robust (i.e. scale independent) rate, to the homogenized solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Test-Based Variable Selection via Cross-Validation

Abstract

Test-based variable selection algorithms in regression often are based on sequential comparison of test statistics to cutoff values. A predetermined a level typically is used to determine the cutoffs based on an assumed probability distribution for the test statistic. For example, backward elimination or forward stepwise involve comparisons of test statistics to prespecified t or F cutoffs in Gaussian linear regression, while a likelihood ratio. Wald, or score statistic, is typically used with standard normal or chi square cutoffs in nonlinear settings. Although such algorithms enjoy widespread use, their statistical properties are not well understood, either theoretically or empirically. Two inherent problems with these methods are that (1) as in classical hypothesis testing, the value of α is arbitrary, while (2) unlike hypothesis testing, there is no simple analog of type I error rate corresponding to application of the entire algorithm to a data set. In this article we propose a new method, backward elimination via cross-validation (BECV), for test-based variable selection in regression. It is implemented by first finding the empirical p value α*, which minimizes a cross-validation estimate of squared prediction error, then selecting the model by running backward elimination on the entire data set using α* as the nominal p value for each test. We present results of an extensive computer simulation to evaluate BECV and compare its performance to standard backward elimination and forward stepwise selection.