Numerical calculation of the tangent stiffness matrix in materials modeling

The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Accelerating column generation for variable sized bin-packing problems

In this paper, we study different strategies to stabilize and accelerate the column generation method, when it is applied specifically to the variable sized bin-packing problem, or to its cutting stock counterpart, the multiple length cutting stock problem. Many of the algorithms for these problems discussed in the literature rely on column generation, processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables. In the sequel, we introduce new dual-optimal inequalities, and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables. Two algorithms based on aggregation are proposed. The first one relies on a row aggregated LP, while the second one solves iteratively sequences of doubly aggregated models. Working with these approximations, in the various stages of an iterative solution process, has proven to be an effective way of achieving faster convergence.     

Multiaxial constitutive modeling of aircraft engine materials

Based on a newly developed theory (Lu and Weng, Acta Mech., in press) the high temperature behavior of an aircraft engine material is studied under combined stress state. Both monotonic and cyclic deformations are examined to uncover its stress-strain response, as well as its cyclic hardening and strain-ratchetting characteristics. Under a biaxial loading it is disclosed that tensile cyclic hardening is greatly magnified with a superimposed lateral tension, whereas the strain-ratchetting process is led to an enhanced, unsettling state with a superimposed lateral compression. The biaxial transient and steady-state creep strains have also been calculated. The results suggest that while a superimposed lateral tension will inhibit the creep deformation, a lateral compression can greatly promote the inelastic flow. To reflect the practical service conditions of an aircraft engine, the theory is further applied to examine the effect of loading frequency on the development of inelastic strains under concurrent thermal and mechanical loading. It is found that a more frequently flying aircraft will have a greater accumulation of creep strains and, consequently, a greater possibility of material damage in its engine components over the same total flying time.     

Mathematical modeling of current generation by electromagnetic waves in plasma

Translated from Programmnoe Oborudovanie i Voprosy Prinyatiya Reshenii, pp. 206–211, 1989.     

Minimal generation of ideals of algebraic varieties at points with multilinear tangent cones

Let \(A\) be the local ring, at a singular isolated point \(P\) of an affine irreducible algebraic variety \(V\), with regular normalization. Let \(\mathfrak p\) be the prime ideal of \(A\) corresponding to \(V\). In this paper we study the minimal number of generators of \(\mathfrak p\), when the projectivized tangent cone of \(V\) at \(P\) is multilinear (that is union of linear varieties) and has maximal rank.

     

Recent advances in the numerical modeling of constitutive relations

In this paper we present an overview of the recent developments in the area of numerical and finite element modeling of nonlinear constitutive relations. The paper discusses elastic, hyperelastic, elastoplastic and anisotropic plastic material models. In the hyperelastic model an emphasis is given to the method by which the incompressibility constraint is applied. A systematic and general procedure for the numerical treatment of hyperelastic model is presented. In the elastoplastic model both infinitesimal and large strain cases are discussed. Various concerns and implications in extending infinitesimal theories into large strain case are pointed out. In the anisotropic elastoplastic case, emphasis is given to the practicality of proposed theories and its feasible and economical use in the finite element environment.     

A viscoelastic constitutive modeling of rubber-like materials with the Payne effect

This paper aims to present a viscoelastic constitutive model of rubber-like materials, which can capture the Payne effect under dynamic cyclic loadings. The Payne effect is induced by a damage process of bond rupture inside the rubber-like materials, which leads to the storage and loss moduli changing with the dynamic strain amplitude. A viscoelastic constitutive relation is established based on the nonequilibrium thermodynamics for the rubber-like materials by constructing the Helmholtz free energy as the superposition of a hyperelastic model and a convolution viscous model. The neo-Hookean hyperelastic model and the convolution viscous model in terms of the Prony series are then employed in a modification that the material parameters concerned are treated as internal variables and can be identified through a simple but effective approach. At last, the Payne effect is effectively predicted in a good agreement with experimental results.     

Approximation of curves by fairness splines with tangent conditions

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper.     

Data generation for composite-based structural equation modeling methods

Advances in Data Analysis and Classification - Examining the efficacy of composite-based structural equation modeling (SEM) features prominently in research. However, studies analyzing the efficacy...     

Hot deformation behavior and constitutive modeling of VCN200 low alloy steel

The hot working behavior of VCN200 medium carbon low alloy steel was analyzed by performing hot compression tests in temperature range of 850–1150 °C and at strain rates of 0.001–1 s⁻¹. Flow curves were typical of dynamic recrystallization during hot working over temperature range of 900–1150 °C and strain rates of 0.001–1 s⁻¹. However, at lower temperatures no indication of flow softening was observed. The constitutive analysis using the hyperbolic sine function was performed and the value of apparent activation energy for the hot deformation determined to be about 435 kJ/mol. The flow curves up to the peak were successfully modeled using a dynamic recovery model. All the factors in this model were defined in terms of the Zener–Hollomon parameter. A modified form of Avrami equation was used to estimate the fractional softening due to the dynamic recrystallization for any given strain in flow curve. Using this model, the flow curves were successfully predicted and generalized to different deformation conditions.     

Enumeration Schemes and, more importantly, their automatic generation

The notion of Enumeration Scheme is introduced and applied to the problem of counting permutations with forbidden patterns. Most importantly, the process is completely automated in a software package, WILF, accompanying this article.Supported in part by the NSF.     

Extremal loading of soft fibrous tissues: multi-scale mechanics and constitutive modeling

In the current contribution, we present a multi-scale constitutive model capturing macroscopic inelastic effects (like stress softening and permanent set) in soft tissues under cyclic loading. Soft biological tissues can be described as a biological composite material. The extracellular matrix is hereby reinforced by collagen fibers which themself are an assembly of collagen fibrils embedded in a proteoglycan (PG) rich matrix. Micro-damage induced by cyclic loading is treated by an interaction scenario between the fibrils and the PGs. At the low strain regime PGs promote sliding between fibrils [1] which leads to the yielding of statistical distributed overlapping segments. The breakage of the PG-bridges is defined by a decreasing PG-density. Due to the accumulated damage of the PG connections at high tissue strains, the strains at the fibril level increases. This finally drives the over-stretching of the fibrils, which is associated with a permanent rupture of the hydrogen bonds inside of the tropocollagen molecules [2]. The so obtained model is in line with recent experimental findings [1, 2] and was additionally validated against experimental data available in literature. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric characterizations of C1 manifolds in Euclidean spaces by tangent cones

A remarkable and elementary fact that a locally compact set $F$ of Euclidean space is a smooth manifold if (and only if) the lower and upper paratangent cones to $F$ coincide at every point, is proved. The celebrated von Neumann’s result (1929) that a locally compact subgroup of the general linear group is a smooth manifold, is a straightforward application.     

A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences

In this paper, which may be considered a sequel to a recent article by Eric Rowland and Reem Yassawi, we present yet another approach for the automatic generation of automata (and an extension that we call congruence linear schemes) for the fast (log-time) determination of congruence properties, modulo small (and not so small!) prime powers, for a wide class of combinatorial sequences. Even more interesting than the new results that could be obtained is the illustrated methodology, that of designing ‘meta-algorithms’ that enable the computer to develop algorithms, that it (or another computer) can then proceed to use to actually prove (potentially!) infinitely many new results. This paper is accompanied by a Maple package, AutoSquared, and numerous sample input and output files, that readers can use as templates for generating their own, thereby proving many new ‘theorems’ about congruence properties of many famous (and, of course, obscure) combinatorial sequences.     

Local approximation of uniformly regular Carnot-Carathéodory quasispaces by their tangent cones

We prove a local approximation theorem for the Carnot-Carathéodory quasimetrics on uniformly regular (equiregular) Carnot-Carathéodory spaces. Using this theorem, we study convergence of the Carnot-Carathéodory quasispaces to their tangent cones. In particular, we prove a Mitchell type theorem on convergence of an equiregular Carnot-Carathéodory quasispace with distinguished point to its tangent cone.     

Accelerating autonomous learning by using heuristic selection of actions

This paper investigates how to make improved action selection for online policy learning in robotic scenarios using reinforcement learning (RL) algorithms. Since finding control policies using any RL algorithm can be very time consuming, we propose to combine RL algorithms with heuristic functions for selecting promising actions during the learning process. With this aim, we investigate the use of heuristics for increasing the rate of convergence of RL algorithms and contribute with a new learning algorithm, Heuristically Accelerated Q-learning (HAQL), which incorporates heuristics for action selection to the Q-Learning algorithm. Experimental results on robot navigation show that the use of even very simple heuristic functions results in significant performance enhancement of the learning rate.     

Microscopic modeling and control logic for incident-responsive automatic vehicle movements in single-automated-lane highway systems

Automatic response to lane-blocking incidents is a critical issue in the field of automated highway systems (AHS). Accordingly, this paper presents a microscopic vehicular control methodology for automatic-control (AC) vehicular movements in response to lane-blocking incidents in the AHS environment. The embedded traffic control logic is based on the basic safety requirements for automatic-control lane traffic maneuvers responding to lane-blocking incidents in the single-automated-lane AHS environment. Accordingly, respective automated vehicular control models are proposed to deal with AC vehicles moving in three corresponding sequential phases, i.e., (1) AC platoon approaching the incident site from the blocked lane, (2) mandatory lane changing and mixed car following in the adjacent lane, and (3) AC platoon reforming downstream from the incident site in the blocked automated lane. Using a microscopic simulation model which embeds these proposed models, preliminary tests are conducted to investigate the relative performance of the proposed method in various traffic flow and control scenarios. The resulting numerical results, including simplified sensitivity analyses, indicate that the proposed microscopic traffic control logic permits regulating automatic-control vehicular movements in response to the effects of lane-blocking incidents on traffic flows either in control-free lanes or in the automatic-control lanes. Implications of the results and some findings are discussed for further research.     

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.     

On the thermodynamics of fluids defined by implicit constitutive relations

In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.