Modeling crack micro-branching using finite elements with embedded strong discontinuities

The emphasis of this work lies in the development of a numerical method which is capable of representing the complex physical phenomena arising in the case of crack branching in brittle materials. In particular, the formation of crack micro-branches needs to be accounted for when it comes to the prediction of the propagation pattern of crack macro-branches which will ultimately lead to the failure of the material. This is achieved by numerically modeling the failure zones within the individual finite elements based on the concept of the embedded finite element method, where all the information with regard to the geometry of the failure zone is stored locally on the element level leading to a very efficient methodology capable of discretely resolving the failure zone. The main feature of the current work is the redundancy of the branching criterion based on crack tip velocity and that both, micro- as well as macro-branches can be modeled. Whether a micro-crack develops into a macro-crack solely depends on the local state of the material as it is outlined based on the application of the proposed numerical scheme on a rectangular block with a pre-existing notch set under tension. A comparison of the oscillatory behavior of the obtained crack tip velocity every time a micro-crack develops with experimental results from the literature is provided. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New three-dimensional finite elements with embedded strong discontinuities to model solids at failure

This work focuses on the development of new finite elements which can capture strong discontinuities in three-dimensional failure problems. The displacement jumps in the solid are approximated by a linear interpolation obtained by enforcing a new class of enhanced separation modes to exactly be satisfied by the formulation. Efforts are also put towards the development of a proper crack propagation tracking algorithm needed for the complicated crack surfaces appearing in realistic 3D failure simulations, based on a combination of the global tracking algorithm and the marching cubes algorithm. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical approach for the simulation of ductile fracture with embedded discontinuities

In the present study, a computational approach for the numerical simulation of ductile fracture within the framework of the finite element method is proposed. In the developed macroscopic formulation, the inelastic behavior in the bulk of the material is described by the finite elasto‐plastic material model proposed in [4]. The failure process is modeled by introducing discontinuities when a special local fracture criterion is satisfied. The discontinuities are incorporated via special triangular finite elements with embedded interfaces following the line of [2]. Finally, the numerical procedure is evaluated for a twodimensional representative test problem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational formulation of the material failure process in solids by embedded discontinuities model

This work presents a variational formulation of the material failure process, idealized as strain or displacement discontinuities, by weak, strong, or discrete embedded discontinuities into a continuum. It is shown that the solution of the proposed variational formulation may be approximated by different types of finite elements with embedded discontinuities. The developed displacement approximation of a finite element split by the discontinuity leads to a symmetric stiffness matrix, which considers not only the continuity of tractions but also the rigid body relative motions of the portions in which the element is split. The variational formulation of a continuum with more than one discontinuity in its interior is developed. It is shown that this formulation may lead to finite elements with embedded discontinuities that can be classified as displacement, force, mixed, and hybrid models. To show the effectiveness of the proposed formulation, the classical example of a bar under tension is solved using one and 2D finite element approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A unifying formulation of the discrete and continuum approximations for embedded discontinuities

A methodology for the numerical implementation of embedded discontinuities into the finite element method is developed. This is applicable for the discrete and continuum approximations of discontinuities. The variational formulation of the problem of a solid with discontinuities is established for both approximations, yielding the equations used in this methodology. Three sets of equations are obtained by applying this methodology; all are suitable to be numerically implemented. To show the application potential of this method, the numerical simulation of the formation and propagation of a discontinuity in a concrete specimen is carried out and the results are compared with those from the physical experiment, demonstrating the adequacy of the methodology and its corresponding implementations to model discontinuities. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A catastrophic queueing model with delayed action

A queueing model with catastrophes and delayed action is studied in this paper. This delayed action could be in the form of protecting or removing all the customers that are in the system based on the outcome of two random clocks which are simultaneously activated upon the occurrence of a catastrophic event. Assuming the customers to arrive according to a versatile Markovian point process to a single server system, the service times to be of phase type, and all other underlying random variables to be exponentially distributed, we use matrix-analytic methods to study the delayed catastrophic model in steady-state. Needed expressions for the number in the system as well as the waiting time distributions are derived along with a discussion on some special cases of this model. Detailed illustrative examples are presented.     

Simulation of crack–propagation using embedded discontinuities

The transition from microscale damage phenomena to crack initiation and growth at the macroscale is an important mechanism which constrains the lifetime of concrete structures. Analysing crack growth using the finite element method without enhancement of the shape functions is possible only by continuously updating the corresponding meshes, which constitutes a significant computational effort. But even then the results can be substantially mesh–dependent and hard to interpret. The extended Finite Element Method (XFEM) uses additional discontinuous shape–functions and is one possibility to overcome these problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A delayed SIRS epidemic model with pulse vaccination

A delayed SIRS epidemic model with pulse vaccination and saturated contact rate is investigated. By using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the system. Further, by using the comparison theorem, we prove that under the condition that R₀ < 1 the infection-free periodic solution is globally attractive, and that under the condition that R′ > 1 the disease is uniformly persistent, which means that after some period of time the disease will become endemic.     

A non-local heat transport model in solids with discontinuities via Lagrangian particle method

A new Lagrangian non-local diffusion model, designed as a meshless particle simulation method, is proposed to predict the thermal response of solids involving discontinuous. The main idea is to understand the heat transfer process of solids via a Lagrangian treatment of a particle-discretized system, which provides a more general physical representation of the heat transfer process of solids. In contrast to the traditional differential model, the proposed mathematical model is expressed via an integral form, which can easily handle the case involving discontinuities. The spatial convergence studies show that the proposed model converges to its associated non-local solution while increasing the number of particles in a fixed cut-off radius; and it can reach the local exact solution when the cut-off radius is close to zero. The numerical examples not only verify that the proposed diffusion model converges to the continuum heat conduction model, but also show the capability of its application to heat transfer problems involving discontinuities. In particular, the computational performance of the proposed non-local thermal model is also demonstrated its computational performance in the case with practical crack propagation in a two-dimensional plate. In addition, the specific comparison between the proposed method and the well-developed peridynamics approach for heat conduction problems is carefully described with rigor via mathematical proofs and numerical results. Specifically, the proposed model is shown to be a variation of the corresponding peridynamic-type thermal diffusion model. This model not only has a formulation consistent with that of peridynamics in solid mechanics but also shows a better performance in the case with sharp corners than that of the peridynamic diffusion in [1].     

A delayed marine bacteriophage infection model

A marine bacteriophage infections model with stage structure is studied. Necessary and sufficient conditions for the extinction and permanence of the system are obtained, which enrich and improve the corresponding results given by S.A. Gourley and Y. Kuang [A delay reaction–diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math. 65 (2005) 550–566].     

A queueing model of delayed product differentiation

We consider a production system in which a supplier produces semi-finished items on a make-to-stock basis for a manufacturer that will customize the items on a make-to-order basis. The proportion of total processing time undertaken by the supplier determines how suitable the semi-finished items will be to meet customer demand. The manufacturer wishes to determine the optimal point of differentiation (the proportion of processing completed by the supplier) and its optimal semi-finished goods buffer size. We use matrix geometric methods to evaluate various performance measures for this system, and then, with enumeration techniques, obtain optimal solutions. We find that delayed product differentiation is attractive when the manufacturer can balance the costs of customer order fulfillment delay with the costs associated with unsuitable items.     

Modeling quasi-static crack growth with the embedded finite element method on multiple levels

The current work presents the multilevel approach of the embedded finite element method which is obtained by combining features of the method of domain decomposition with those of the standard embedded finite element method. The conventional requirement of fine mesh in a possible failure zone is rendered unnecessary with the new approach thereby reducing the computational expense. In addition, it is also possible to stop a propagating crack-tip in the middle of a finite element. In this approach, the finite elements at the failure-prone zone where cracks or shear bands, referred to as strong discontinuities which represent jumps in the displacement field, can form and propagate based on some failure criterion are treated as separate sub-boundary value problems which are adaptively discretized during the run time into a number of sub-elements and subjected to a kinematic constraint on their boundary. Each sub-element becomes equally capable of developing a strong discontinuity depending upon its state of stress. A linear displacement based constraint is applied initially which is modified accordingly as soon as a strong discontinuity propagates through the boundary of the main finite element. At the local equilibrium, the coupling between the quantities at two different levels of discretization is obtained by matching the virtual energies due to admissible variations of the main finite element and its constituent sub-elements. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A delayed epidemic model with stage-structure and pulses for pest management strategy

From a biological pest management standpoint, epidemic diseases models have become important tools in control of pest populations. This paper deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive, provided that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than another critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its attractivity. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.     

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

A delayed chemostat model with general nonmonotone response functions and differential removal rates

A chemostat model with general nonmonotone response functions is considered. The nutrient conversion process involves time delay. We show that under certain conditions, when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, the competitive exclusion principle holds. In the case of insignificant death rates, the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times (see [H.M. El-Owaidy, M. Ismail, Asymptotic behavior of the chemostat model with delayed response in growth, Chaos Solitons Fractals 13 (2002) 787-795; H.M. El-Owaidy, A.A. Moniem, Asymptotic behavior of a chemostat model with delayed response growth, Appl. Math. Comput. 147 (2004) 147-161; S. Yuan, M. Han, Z. Ma, Competition in the chemostat: convergence of a model with delayed response in growth, Chaos Solitons Fractals 17 (2003) 659-667]). However, the proofs are all incorrect. In this paper, we provide a correct proof that also applies in the case of differential death rates. In addition, we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium, thus showing the outcome can be initial condition dependent. Moreover, we show that when the species specific death rates are included, damped oscillations may occur even when there is no delay. Thus, the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments.     

A delayed SIR epidemic model with saturation incidence and a constant infectious period

In this paper, an SIR epidemic model with saturation incidence and a time delay describing a constant infectious period is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. When the basic reproduction number is greater than unity, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained to estimate the eventual lower bound of the fraction of infectious individuals. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global attractiveness of the endemic equilibrium. Numerical simulations are carried out to illustrate the main results.     

Admissible functions with multiple discontinuities

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ _i=1 ⁿ f(x+iα asn → α is resolved.     

Approximating scattered data with discontinuities

A method is presented for approximating scattered data by a function defined on a regular two-dimensional grid. It is required that the approximation is discontinuous across given curves in the parameter domain known as faults. The method has three phases: regularisation, local approximation and extrapolation. The main emphasis is put on the extrapolation which is based on a matrix equation which minimises second order differences. By approximating each fault by a set of line segments parallel with one of the axes, it is simple to introduce natural boundary conditions across the faults. The resulting approximation has, as expected, discontinuities across faults and is smooth elsewhere. The method is stable even for large data sets.This research was supported by the Royal Norwegian Council for Scientific and Industrial Research.