Shape memory and phase transitions for auxetic materials

We present a mathematical model describing the auxetic‐austenitic phase transition phenomenon by a second order shape memory phase transition. The typical properties of auxetic materials, as the negative Poisson ratio ν, are described by a function of the phase ?, called order parameter, which relates the phase transition with a change of the internal order structure of the material. In our model, the auxetic phase is represented by an order parameter ? = 1, which provides a negative Poisson's ratio, while the austenitic phase will be denoted by ? = 0. Copyright © 2013 John Wiley & Sons, Ltd.     

A projection method for phase field models in micromagnetics

Magnetic materials have been finding increasingly wider areas of application in industry and therefore, as indicated by the reviews [1], [2] and [3], there is an increased interest in the efficient modeling of such materials that have an inherent coupling between the magnetic and mechanical characteristics. A particular challenge in the modeling of such materials is the algorithmic preservation of the geometric constraint on the magnetization field, that remains constant in magnitude [4]. In earlier works, [5] and [6], we presented a phase field model within a geometrically exact incremental variational framework where the geometric property of the magnetization director is exactly preserved pointwise by nonlinear rotational updates at the nodes. In the current work however, we present an alternative approach that involves an operator split along with a projection step for the magnetization vector. This method provides significant advantages in terms of speed and ease of implementation at the cost of the maximum time step size used. The current work therefore presents comparative study of the the two methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of large domain Al foam permeability by Finite Difference methods

Classical methods to calculate permeability of porous media have been proposed mainly for high density (e.g. granular) materials. These methods present shortcomings in high porosity, i.e. high permeability media (e.g. metallic foams). While for dense materials permeability seems to be a function of bulk properties and occupancy averaged over the volume, for highly porous materials these parameters fail to predict it. Several authors have attacked the problem by solving the Navier-Stokes equations for the pressure and velocity of a liquid flowing through a small domain (Ω_s) of aluminium foam and by comparing the numerical results with experimental values (prediction error approx. 9%). In this article, we present calculations for much larger domains (Ω_L) using the Finite Difference (FD) method, solving also for the pressure and velocity of a viscous liquid flowing through the Packed Spheres scenario. The ratio Vol(Ω_L)/Vol(Ω_s) is around 10³. The comparison of our results with the Packed Spheres example yields a prediction error of 5% for the intrinsic permeability. Additionally, numerical permeability calculations have been performed for Al foam samples. Our geometric modelling of the porous domain stems from 3D X-ray tomography, yielding voxel information, which is particularly appropriate for FD. Ongoing work concerns the reduction in computing times of the FD method, consideration of other materials and fluids, and comparison with experimental data. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanics model for tortuosity and homogenized diffusion properties of porous materials with distributed micro-cracks

Continuum micromechanics [2,3] is used to model tortuosity within the context of ion transport to obtain the homogenized diffusivity of intact and micro-cracked porous materials. A novel cascade homogenization technique to model ion diffusion in porous materials is proposed. A REV^uc that represents the porous material is modelled as a spherical pore-space inclusion in a recursively updated matrix with an initial diffusivity without any tortuosity. A REV^c representing the micro-cracked material is modelled with prolate ellipsoidal inclusions representing micro-cracks embedded in a porous matrix [4]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of homogenized diffusion properties in micro-cracked porous materials

Methods of micromechanics [2, 3] are used to obtain homogenized diffusion properties for intact and micro-cracked porous materials. A two scale homogenization technique is proposed. A REV^uc that represents the porous microstructure is modelled as a spherical pore-space inclusion in a matrix, whose diffusion coefficient does not depend on the microstructure geometry but only on the porosity. For cracked porous materials, a REV^c is modelled with the diffusion coefficient for intact porous materials as the matrix phase with ellipsoidal inclusions representing micro-cracks. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie symmetry of the Landau-Lifshitz-Gilbert equation and exact linearization in the Minkowski space

In this paper we present a linear representation of the Landau-Lifshitz-Gilbert equation for describing the magnetization of ferromagnetic materials. According to Lie P \Bbb M³⁺¹,P {\Bbb M}^{3+1}, of which the projective proper orthochronous Lorentz group PSO ^o(3,1) left acts. By the Lie symmetry a group preserving scheme is developed, which improves the computational accuracy and efficiency.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

Fractal materials, beams, and fracture mechanics

Continuing in the vein of a recently developed generalization of continuum thermomechanics, in this paper we extend fracture mechanics and beam mechanics to materials described by fractional integrals involving D, d and R. By introducing a product measure instead of a Riesz measure, so as to ensure that the mechanical approach to continuum mechanics is consistent with the energetic approach, specific forms of continuum-type equations are derived. On this basis we study the energy aspects of fracture and, as an example, a Timoshenko beam made of a fractal material; the local form of elastodynamic equations of that beam is derived. In particular, we review the crack driving force G stemming from the Griffith fracture criterion in fractal media, considering either dead-load or fixed-grip conditions and the effects of ensemble averaging over random fractal materials.     

L p – L q Estimates To The Solution Of The Cauchy Problem For Partial Differential Equations Describing Non-Simple Thermoelastic Materials

Theory of non-simple materials is different from that of simple materials because in it the first strain gradient is taken into consideration as the constitutive variable. The consequence of this fact, from mathematical point of view, is that the equation of motion consists either of higher order derivatives of displacement (four order derivatives) and some material parameters can depend not only on the temperature and the gradient of displacement but also on the second derivative of displacement. We consider the system of partial differential equations describing non-simple thermoelastic materials. This system consists of four scalar equations, three equations of motion and one of energy balance, describing the field of displacement and the temperature in an elastic body. Using the Fourier transform, we found the L ^p – L ^q time decay estimates of the solution of the Cauchy problem for the system of equation describing the non-simple thermoelastic materials, being important for proving the global-in-time solution of this problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effective conductivity of an infinitely interchangeable mixture

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n-point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.     

Operations planning test bed under rolling horizons,multiproduct, multiechelon,multiprocess for capacitated production planning modelling with strokes

One of the problems when conducting research in mathematical programming models for operations planning is having an adequate database of experiments that can be used to verify advances and developments with enough factors to understand different consequences. This paper presents a test bed generator and instances database for a rolling horizons analysis for multiechelon planning, multiproduct with alternatives processes, multistroke, multicapacity with different stochastic demand patterns to be used with a stroke-like bill of materials considering production costs, setup, storage and delays for operations management. From the analysis of the operations planning obtained from this test bed, it is concluded that a product structure with an alternative process obtains the lowest total cost and the highest service level. In addition, decreasing seasonal demand could present a lower total cost than constant demand, but would generate a worse service level. This test bed will allow researchers further investigation so as to verify improvements in forecast methods, rolling horizons parameters, employed software, etc.

     

Untangling Polygons and Graphs

Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C _n while keeping Ω(n ^2/3) vertices fixed. For any connected graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of G. One consequence is that every 3-connected planar graph has a drawing δ such that at most O((nlog n)^2/3) vertices are fixed in every untangling of δ.     

A Large-Deformation Phase-Field Approach for the Modeling of Magneto-Sensitive Elastomers

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magneto-mechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1, 7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

Variational Treatment and Stability Analysis of Coupled Electro-Mechanics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by a representative model problem. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Intervals in l-groups as L-algebras

L-algebras are related to algebraic logic and quantum structures. They were introduced by the first author [J. Algebra 320 (2008)], where a self-similar closure S(X) of any L-algebra X was employed to derive a criterion for X to be representable as an interval in a lattice-ordered group. In the present paper, this criterion is improved without using the embedding. It is shown that an L-algebra is representable as an interval in a lattice-ordered group if and only if it is semiregular with a smallest element and bijective negation. Any such L-algebra gives rise to a perfect dual with respect to the inverse of the negation. This is proved by a self-dual characterization of semiregularity.     

A necessary and sufficient condition for the total unimodularity of a matrix in terms of graph theory

We present a necessary and sufficient condition for an arbitrary matrix A to be totally unimodular. The matrix A is interpreted as the adjacency matrix of a bipartite graph G(A) The total unimodularity of A corresponds to non-existence of a cycle in G(A) which has an odd column valuation and which is equal to the induced subgraph, Some applications of the results are also discussed.     

Inference for the Number of Topics in the Latent Dirichlet Allocation Model via Bayesian Mixture Modeling

In latent Dirichlet allocation, the number of topics, T, is a hyperparameter of the model that must be specified before one can fit the model. The need to specify T in advance is restrictive. One way of dealing with this problem is to put a prior on T, but unfortunately the distribution on the latent variables of the model is then a mixture of distributions on spaces of different dimensions, and estimating this mixture distribution by Markov chain Monte Carlo is very difficult. We present a variant of the Metropolis–Hastings algorithm that can be used to estimate this mixture distribution, and in particular the posterior distribution of the number of topics. We evaluate our methodology on synthetic data and compare it with procedures that are currently used in the machine learning literature. We also give an illustration on two collections of articles from Wikipedia. Supplemental materials for this article are available online.     

Analysis of Micro- and Macro-Instability Phenomena in Computational Homogenization of Finite Electro-Statics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by representative model problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gradient methods for multiple state optimal design problems

We optimise a distribution of two isotropic materials α I and β I (α < β) occupying the given body in R ^d . The optimality is described by an integral functional (cost) depending on temperatures u ₁, . . . , u _m of the body obtained for different source terms f ₁, . . . ,f _m with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the Gateaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.