Horizon Sizes Research Articles

An implicit non-ordinary state-based peridynamics with stabilised correspondence material model for finite deformation analysis

This paper is devoted to the development of a stabilised implicit non-ordinary state-based peridynamics approach. We propose a geometrically nonlinear implicit approach focusing on quasi-static analyses. Since the construction of the Jacobian matrix is the most time-consuming step in conducting this nonlinear analysis, we formulate an analytical expression based on the equation of motion of non-ordinary state-based peridynamics to ensure optimum convergence of the global residual force. The implicit formulation can adopt fairly large time increments, making it a good choice for analyses of finite deformation. Another important extension presented in this paper is the modification of the correspondence material model to remove zero-energy mode instabilities and reduce the spurious oscillations, as proposed by Silling (2017). The derivative of the additional stabilisation term with respect to displacement is included in the formulation of the Jacobian for the first time. Computational examples of 2D finite deformation problems with a stabilised correspondence model are presented. We assess the effectiveness of different values of the stabilisation parameter, G in terms of the particles’ spacings and horizon sizes for different problems. This allows the non-ordinary state-based peridynamics approach to model material behaviour with greater accuracy where correspondence materials have previously failed due to instabilities. In this paper, a damage model is also proposed, which provides for the first time an implicit approach for the static solution of crack propagation problems for non-ordinary state-based peridynamics. This paper lays the groundwork for non-ordinary state-based peridynamics to be used for a much greater variety of solid mechanics problems than is currently possible and at the same time satisfying the stability condition.

Derivation of dual-horizon state-based peridynamics formulation based on Euler–Lagrange equation

The numerical solution of peridynamics equations is usually done by using uniform spatial discretisation. Although implementation of uniform discretisation is straightforward, it can increase computational time significantly for certain problems. Instead, non-uniform discretisation can be utilised and different discretisation sizes can be used at different parts of the solution domain. Moreover, the peridynamic length scale parameter, horizon, can also vary throughout the solution domain. Such a scenario requires extra attention since conservation laws must be satisfied. To deal with these issues, dual-horizon peridynamics was introduced so that both non-uniform discretisation and variable horizon sizes can be utilised. In this study, dual-horizon peridynamics formulation is derived by using Euler–Lagrange equation for state-based peridynamics. Moreover, application of boundary conditions and determination of surface correction factors are also explained. Finally, the current formulation is verified by considering two benchmark problems including plate under tension and vibration of a plate.

Determination of horizon size in state-based peridynamics

Peridynamics is based on integro-differential equations and has a length scale parameter called horizon which gives peridynamics a non-local character. Currently, there are three main peridynamic formulations available in the literature including bond-based peridynamics, ordinary state-based peridynamics and non-ordinary state-based peridynamics. In this study, the optimum horizon size is determined for ordinary state-based peridynamics and non-ordinary state-based peridynamics formulations by using uniform and non-uniform discretisation under dynamic and static conditions. It is shown that the horizon sizes selected as optimum sizes for uniform discretisation can also be used for non-uniform discretisation without introducing significant error to the system. Moreover, a smaller horizon size can be selected for non-ordinary state-based formulation which can yield significant computational advantage. It is also shown that same horizon size can be used for both static and dynamic problems.

The Multi-Horizon Peridynamics

We present a novel refinement approach in peridynamics (PD). The proposed approach takes advantage of the PD flexibility in choosing the shape of the horizon by introducing multiple domains (with no intersections) to the nodes of the refinement zone. We will show that no ghost forces are needed when changing the horizon sizes in both subdomains. The approach is applied to both bond-based and state-based peridynamics and verified for a simple wave propagation refinement problem illustrating the efficiency of the method.

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.

Higher-order approximation to suppress the zero-energy mode in non-ordinary state-based peridynamics

This paper presents a numerical method to control the spurious deformation mode conventionally found in a non-ordinary state-based peridynamic formulation. The proposed approach introduces a higher-order approximation for a deformation gradient tensor in order to suppress oscillations from the zero-energy mode. The results are compared to those determined by other available methods. The findings of this study well demonstrate that the proposed method provides highly effective solutions for controlling zero-energy modes in peridynamic models. Contrasting other available zero-energy control methods, the proposed method remains robust for relatively large horizon sizes. The method is incorporated by implicit weight functions in peridynamics.

Mesh Sensitivity in Peridynamic Quasi-static Simulations

Peridynamics is a non-local formulation of continuum mechanics that does not rely on spatial derivatives, therefore peridynamics is well suited for crack and failure modeling. Body is discretized in a finite number of particles and each particle connects to other particles within a range called a material's horizon. In this study authors performed fifty glass-fiber coupon tensile and compression simulations with different horizon size and particle spacing combinations to see how they influence maximum displacement. Values from simulations are compared with values calculated using Hooke's law. The results show that the horizon size of three particle spacings gives the best results, which, support's the general view on the issue. However, simulations with horizon sizes ▪ times larger than particle spacing show similar accuracy.

Dual‐horizon peridynamics

SummaryIn this paper, we develop a dual‐horizon peridynamics (DH‐PD) formulation that naturally includes varying horizon sizes and completely solves the ‘ghost force’ issue. Therefore, the concept of dual horizon is introduced to consider the unbalanced interactions between the particles with different horizon sizes. The present formulation fulfills both the balances of linear momentum and angular momentum exactly. Neither the ‘partial stress tensor’ nor the ‘slice’ technique is needed to ameliorate the ghost force issue. We will show that the traditional peridynamics can be derived as a special case of the present DH‐PD. All three peridynamic formulations, namely, bond‐based, ordinary state‐based, and non‐ordinary state‐based peridynamics, can be implemented within the DH‐PD framework. Our DH‐PD formulation allows for h‐adaptivity and can be implemented in any existing peridynamics code with minimal changes. A simple adaptive refinement procedure is proposed, reducing the computational cost. Both two‐dimensional and three‐dimensional examples including the Kalthoff–Winkler experiment and plate with branching cracks are tested to demonstrate the capability of the method. Copyright © 2016 John Wiley & Sons, Ltd.

Wave dispersion analysis and simulation method for concrete SHPB test in peridynamics

The Split Hopkinson Pressure Bar (SHPB) technique is a widely used method for measuring mechanical properties of materials subjected to high-strain-rate loads, while it is difficult to simulate the whole testing process including high-rate deformation, local damage and failure of materials by the common numerical methods. In this paper, an improved numerical approach based on the non-local peridynamic (PD) theory is employed to study the elastic wave dispersion and propagation and the impact failure of concrete Brazilian discs in SHPB test. Meanwhile, an improved PMB (Prototype Microelastic Brittle) model and an implementation method of the contact-impact process are introduced. In PD, the nonlocal long-range force controls the numerical dispersion of wave through different material point sizes and horizon sizes. The numerical dispersion can result in a slight distortion of the wave speed and crack propagation speed, which is not conductive to failure analysis of solids. The improved PMB model can effectively lessen the numerical dispersion compared with the original PMB model. Furthermore, the PD simulation of concrete Brazilian disc SHPB test can reproduce damage accumulation and progressive failure of a specimen, and produce typical final failure pattern. The PD simulation method for SHPB test can be used to analyze the dynamic response of solids suffering impact load.

Switching Extensible FIR Filter Bank for Adaptive Horizon State Estimation With Application

Horizon size is an important parameter that affects the estimation performance of finite impulse response (FIR) filters. In this brief, we propose a novel adaptive horizon approach that aims to adapt the horizon size at each time point. The approach suggests providing state estimation using a bank of FIR filters called the switching extensible FIR filter bank (SEFFB), which consists of several FIR filters operating using different horizon sizes. The horizon sizes and the number of FIR filters in the SEFFB are adapted to changes in system characteristics using maximum likelihood. The SEFFB is applied to target tracking using a ground moving target indicator. A significant performance improvement is demonstrated using the SEFFB in comparison with a single FIR filter using constant optimal horizon size.

Selecting the kernel in a peridynamic formulation: A study for transient heat diffusion

The kernel in a peridynamic diffusion model represents the detailed interaction between points inside the nonlocal region around each material point. Several versions of the kernel function have been proposed. Although solutions associated with different kernels may all converge, under the appropriate discretization scheme, to the classical model when the horizon goes to zero, their convergence behavior varies. In this paper, we focus on the particular one-point Gauss quadrature method of spatial discretization of the peridynamic diffusion model and study the convergence properties of different kernels with respect to convergence to the classical, local, model for transient heat transfer equation in 1D, where exact representation of geometry is available. The one-point Gauss quadrature is the preferred method for discretizing peridynamic models because it leads to a meshfree model, well suited for problems with damage and fracture. We show the equivalency of two definitions for the peridynamic heat flux. We explain an apparent paradox and discuss a common pitfall in numerical approximations of nonlocal models and their convergence to local models. We also analyze the influence of two ways of imposing boundary conditions and that of the “skin effect” on the solution. We explain an interesting behavior of the peridynamic solutions for different horizon sizes, the crossing of m-convergence curves at the classical solution value that happens for one of the ways of implementing the classical boundary conditions. The results presented here provide practical guidance in selecting the appropriate peridynamic kernel that makes the one-point Gauss quadrature an “asymptotically compatible” scheme. These results are directly applicable to any diffusion-type model, including mass diffusion problems.

Weighted average extended FIR filter bank to manage the horizon size in nonlinear FIR filtering

In this paper, we propose a novel approach to manage the horizon size in nonlinear finite impulse response (FIR) filtering. The proposed approach is to perform state estimation through a bank of FIR filters called a weighted average extended FIR filter bank (WAEFFB). In the WAEFFB, the state estimate is obtained by weighting the average of multiple estimates from a bank of extended FIR filters that uses different horizon sizes. The horizon sizes used for the WAEFFB are adjusted constantly by maximizing the likelihood function. We show through simulations that the WAEFFB yields better results than the conventional approach that uses a constant (i.e., fixed) horizon size.

Solving a finite horizon EPQ problem with backorders

This paper presents an alternative approach to solve a finite horizon production lot sizing model with backorders using Cauchy-Bunyakovsky-Schwarz Inequality. The optimal batch size is derived from a sequence number of batches. We prove that a constant batch size policy with one fill rate is better than the variable batch sizes with variable fill rates. Finally, a practical approach is proposed to find the optimal solutions for a discrete planning horizon and discrete batch sizes.

Coarse-graining of bubbling geometries and the fuzzball conjecture

In the Lin-Lunin-Maldacena (LLM) bubbling geometries, we compute the entropies of black holes and estimate their horizon sizes from the fuzzball conjecture, based on coarse-graining on the gravity side. The differences of black hole microstates cannot be seen by classical observations. Conversely, by counting the possible deformations of the geometry which are not classically detectable, we can calculate the entropy. We carry out this method on the black holes of the LLM bubbling geometries, such as the superstar, and obtain the same result as was derived by coarse-graining directly on the conformal field theory (CFT) (fermion) side. Second, by application of this method, we can estimate the horizon sizes of those black holes, based on the fuzzball conjecture. The Bekenstein-Hawking entropy computed from this horizon agrees with that microscopic entropy above. This result supports the fuzzball conjecture.

CHARGED COSMIC STRING NUCLEATION IN DE SITTER SPACE

We investigate aspects of the nucleation of charged circular cosmic strings in de Sitter space. By including the self-gravity for a single circular string loop we obtain the classical potential energy barrier and compute the quantum mechanical tunneling probability to horizon sizes in the semiclassical approximation. We also discuss the classical evolution of a pair of oppositely charged circular strings after their nucleation.

Inhomogeneous initial conditions for inflation.

We address the question whether the inflationary paradigm frees cosmology from the worry about initial conditions. That is, we study the influence of initial inhomogeneities on the inflationary epoch. We derive a simple set of approximate equations and use it to illustrate some features of deviations from homogeneity. However, the approximation methods are limited and in order to explore the full effect of initial inhomogeneities we turn to numerical calculations. The picture that emerges from these calculations is that inhomogeneities can prohibit ``new'' inflation from taking place. The crucial feature necessary for ``chaotic'' inflation is a sufficient high average field over a region of several horizon sizes.

Inhomogeneity and the onset of inflation.

We study numerically the onset of inflation under inhomogeneous initial conditions for various inflationary scenarios and various initial parameters. We solve the full Einstein equations for the gravitational field coupled to scalar-field and radiation-field sources, under the assumptions of spherical symmetry and a closed-universe background. We find that a large initial inhomogeneity suppresses the onset of inflation, and that the scalar field must have values suitable for inflation over a region of several horizon sizes for the inflationary phase to begin.

Sequential medical trials involving paired data

SUMMARY A general approach to sequential decision-theoretic problems is illustrated with Anscombe's formulation of the problem of comparing two treatments in a medical trial. The Bayes sequential procedure for a continuous time verion of the problem is explicitly determined. Suboptimal procedures are proposed and evaluated; asymptotic results and numerical descriptions are provided. The continuous time version is found to provide accurate approximations even for clinical trials involving relatively small horizon sizes. A natural formulation for many statistical problems is one combining Bayesian, sequential and decision-theoretic aspects. For deciding the sign of a normal mean, Chernoff (1961, 1965a, 1965b), Breakwell & Chernoff (1964) and Bather (1962) replaced sums of successive observations by a continuous time Wiener process. The heat equation plays a prominent role in the resulting analysis, since optimal procedures may be characterized in terms of solutions of free boundary problems involving the heat equation. It would seem that the formidable appearance of some of the analysis involved in this approach has distracted potential users from taking advantage of other aspects which are easy or routine and which contribute clarity, asymptotic results with important theoretical implications, and numerical descriptions of sequential procedures and the resulting risks. Recent work by Lai, Levin, Robbins & Siegmund (1980) and by Begg & Mehta (1979) on a model for sequential medical trials proposed by Anscombe (1963) has prompted us to investigate the same problem using the above approach. In this paper we give the optimal procedure and show how it compares with suboptimal procedures. Secondly we use Anscombe's model to illustrate how such results may be obtained for similar problems. The key feature is the asymptotic resemblance of the problem to an optimal stopping problem for a Wiener process. Since many sequential decision-theoretic problems can be framed in terms of efficient scores which are asymptotically normal, the generality of the approach should be apparent. The formulation we refer to as Anscombe's model is the following. There is a horizon of N patients to be treated by one of two available treatments. In the initial experimental phase, n pairs of patients are treated sequentially, with different treatments randomly assigned to the patients in each pair. The differences, Xi, in the values of a continuous outcome variable for the ith pair are assumed to be independently and normally