Voronoi‐based peridynamics and cracking analysis with adaptive refinement

The most common technique for the numerical implementation of peridynamic theory is the uniform discretization together with constant horizon. However, unlike the nonuniform discretization and varying horizons, it is not a natural and intrinsic component of the adaptive refinement analysis and multiscale modeling. Besides, it encounters discretization difficulty in analyzing irregular structures. Therefore, to analyze problems with nonuniform discretization and varying horizons and solve the resulting problems of ghost forces and spurious wave reflection, the dual‐horizon peridynamics based on uniform discretization is extended to the nonuniform discretization based on Voronoi diagrams, for which we call it Voronoi‐based peridynamics. We redefine the damage definition as well. Next, an adaptive refinement analysis method based on the proposed Voronoi‐based peridynamics and its numerical implementation is introduced. Finally, the traditional bond‐based peridynamics and the Voronoi‐based peridynamics with or without refinement are used to simulate 4 benchmark problems. The examples of 2‐D quasi‐static elastic deformation, 2‐D wave propagation, 2‐D dynamic crack growth, and 3‐D simulation of the Kalthoff‐Winkler experiment demonstrate the efficiency and effectivity of the proposed Voronoi‐based peridynamics. Further, the adaptive refinement analysis can be used to obtain reasonable crack path and crack propagation speed with reduced computational burden.     

A two‐dimensional ordinary,state‐based peridynamic model for linearly elastic solids

Peridynamics is a non‐local mechanics theory that uses integral equations to include discontinuities directly in the constitutive equations. A three‐dimensional, state‐based peridynamics model has been developed previously for linearly elastic solids with a customizable Poisson's ratio. For plane stress and plane strain conditions, however, a two‐dimensional model is more efficient computationally. Here, such a two‐dimensional state‐based peridynamics model is presented. For verification, a 2D rectangular plate with a round hole in the middle is simulated under constant tensile stress. Dynamic relaxation and energy minimization methods are used to find the steady‐state solution. The model shows m‐convergence and δ‐convergence behaviors when m increases and δ decreases. Simulation results show a close quantitative matching of the displacement and stress obtained from the 2D peridynamics and a finite element model used for comparison. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of the plastic zone near the crack tips under the uniaxial tension using ordinary state‐based peridynamics

In this paper, the plastic model of ordinary state‐based peridynamics is established. The size and shape of plastic zone around crack tips with the different inclination angles are simulated using ordinary state‐based peridynamics. Comparison of the size and shape of plastic zone around the crack tips obtained from peridynamic solution and analytic solution is made. It is found that the relative errors between the analytical and peridynamic solution are very little. Therefore, it is feasible to predict the plastic zone around crack tips using ordinary state‐based peridynamics.     

Generalization of the ordinary state-based peridynamic model for isotropic linear viscoelasticity

This paper presents a generalization of the original ordinary state-based peridynamic model for isotropic linear viscoelasticity. The viscoelastic material response is represented using the thermodynamically acceptable Prony series approach. It can feature as many Prony terms as required and accounts for viscoelastic spherical and deviatoric components. The model was derived from an equivalence between peridynamic viscoelastic parameters and those appearing in classical continuum mechanics, by equating the free energy densities expressed in both frameworks. The model was simplified to a uni-dimensional expression and implemented to simulate a creep-recovery test. This implementation was finally validated by comparing peridynamic predictions to those predicted from classical continuum mechanics. An exact correspondence between peridynamics and the classical continuum approach was shown when the peridynamic horizon becomes small, meaning peridynamics tends toward classical continuum mechanics. This work provides a clear and direct means to researchers dealing with viscoelastic phenomena to tackle their problem within the peridynamic framework.     

Convergence,adaptive refinement,and scaling in 1D peridynamics

We introduce here adaptive refinement algorithms for the non‐local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48 :175–209) as a reformulation of classical elasticity for discontinuities and long‐range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes. Copyright © 2008 John Wiley & Sons, Ltd.     

A posteriori error approximation in EFG method

Recently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated. An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.     

Time domain modelling of aeroelastic bridge decks: a comparative study and an application

This paper has the purpose of describing, developing and comparing the formulae of aeroelastic forces in the time domain for the aerodynamic study of bridges. In particular, it considers the ‘quasi‐steady’ formulation and the formulation ‘derived from the extension of aeroelastic derivatives to the time domain’ with the use of the recursive expression for the memory term. Both formulae are then applied to the analysis of ‘stress ribbon’ pedestrian bridges, aerodynamically similar to a thin airfoil immersed in a fluid. The thin airfoil theory is described and then extended to general coverage of bridge dynamics in order to identify the essential aeroelastic coefficients and derivatives. Different formulae are also compared with simplified theory of the aeroelastic issue frequency domain. Finally, this comparison, the obtained results and all input data are recorded and can be used as a possible benchmark for other theoretical formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit dual-based approach to couple peridynamics with classical continuum mechanics

A general local/nonlocal implicit coupling technique called the dual-based approach is proposed to couple peridynamics (PD) with classical continuum mechanics. In the present method, physical information is transmitted mutually from local to nonlocal regions through the coupling elements; no transition region is introduced. For different mesh discretizations, two coupling methods are achieved with simplicity and effectivity. To obtain the stiffness matrix of the coupled model, without loss of generality, the implicit dual-horizon ordinary state-based peridynamic model is proposed, in which the linearization of dual-horizon ordinary state-based PD is derived and the dual assembly algorithm of the peridynamic stiffness matrix is developed. It will be seen that the implicit dual-based coupling approach provides a new implicit coupling method that is easy to implement and makes full use of the internal connection between PD and classical continuum mechanics. Several numerical examples involving static crack propagation are investigated, and the satisfactory results show both quantitative and qualitative agreement with either the analytic solution or the available experiment.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.     

On continuous,discontinuous, mixed,and primal hybrid finite element methods for second‐order elliptic problems

Finite element formulations for second‐order elliptic problems, including the classic H¹‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L²‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.     

Elastoplastic stability analysis of shells using the physically stabilized finite element SHB8PS

In this paper, we present the formulation of the SHB8PS element, its implementation into the incremental, non‐linear and implicit calculation code Stanlax‐INCA and examples of applications which demonstrate its efficiency. This element is an 8‐node, three‐dimensional cube with a preferential direction called the thickness. Therefore, it can be used to represent thin structures while, at the same time, correctly taking into account phenomena throughout the thickness thanks to the use of a numerical integration with five Gauss points in that direction. This element is subintegrated and thus requires a stabilization mechanism in order to control the hourglass modes. The stabilization technique used is based on the works by Belytschko and Bindeman, which apply an ‘assumed strain method’. The main advantage of this element is the adaptivity of its stabilization term, which is made variable with the elastoplastic evolution throughout the thickness. Copyright © 2003 John Wiley & Sons, Ltd.     

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Computer‐aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce ‘mega‐sized’, arbitrary‐shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis (FEA) and reduced engineering manpower requirement for final quality assurance. A method, denoted as c‐type, has been proposed by constructively defining a finite element space whereby the above hurdles may be overcome with a minimal number of hyper‐sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B‐splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter‐element continuity on demand unifies modelling and finite element method. The c‐type method may introduce additional control, namely, an inter‐element continuity condition to the existing h‐type and p‐type methods. Adaptation of the c‐type method to existing commercial and general‐purpose computer programs based on a conventional displacement‐based finite element method is straightforward. The c‐type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non‐linear formulation for the two‐dimensional curved beams/arches is presented. Several beam problems ranging from truly three‐dimensional tortuous linear curved beams to geometrically extremely non‐linear two‐dimensional arches are solved to establish numerical efficiency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non‐linearity and to treat general material non‐linearity. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

Discounting risks in the far future

Risks to life and health in the future must be discounted in quantitative risk analysis. Yet, risks in the distant future become trivialized if any reasonable constant interest rate is used. Our responsibility toward future generations rules out such drastic discounting. A solution to this problem is proposed here, resting on the ethical principle that our duty with respect to saving lives is the same to all generations, whether in the near or far future. It is shown that when a choice between prospects involving different risks has a financing horizon T, then ordinary principles of discounting apply up to this time T, while no further discounting is justifiable after T. The principle implies that risk events beyond the financing horizon should be valued as if they occurred at the financing horizon.     

A compromise programming method using multibounds formulation and dual approach for multicriteria structural optimization

To enhance the reliability and efficiency of the multicriteria optimization procedure, a compromise programming method using multibounds formulation (MBF) is proposed in this paper. This method can be used either to obtain the Pareto optimum set or to find the ‘best’ Pareto optimum solution in the sense of having the minimum distance to the utopia point. By introducing a set of artificial design variables, it is shown that a simplified and easy‐to‐use formulation can be established for practical applications. Particularly, this formulation is well adapted to the efficient dual solution approach due to the convexity of objective function. Theoretically, based on the Kuhn–Tucker optimality conditions, demonstrations show that the new formulation is equivalent to its original form and thus retains the basic properties of the latter. Numerical examples will be solved to show the capacity of this method. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical analysis of passive scalar transport in turbulent shear flows

Abstract

The development of the formation and vortex pairing process in a two‐dimensional shear flow and the associated passive scalar (mass concentration or energy) transport process was numerically simulated by using the Vortex‐in‐Cell (VIC) Method combined with the Upwind Finite Difference Method. The visualized temporal distributions of passive scalars resemble the vortex structures and the turbulent passive scalar fluxes showed a definite connection with the occurrence of entrainment during the formation and pairing interaction of large‐scale vortex structures. The profiles of spatial‐averaged passive scalar ø, turbulent passive scalar fluxes, u'ø’ and v'ø’, turbulent diffusivity of mean‐squared scalar fluctuation, v'ø‘ ², mean‐squared turbulent passive scalar fluctuation, √ø‘ ², skewness, and flatness factor of the probability density function of scalar fluctuation ø’ at three different times are calculated. With the lateral dimension scaled by the momentum thickness and the velocity scaled by the velocity difference across the shear layer, these profiles were shown to be self‐preserved. The probability density function of turbulent scalar fluctuation was found to be asymmetric and double‐peaked.     

Hybrid hyper‐reduced modeling for contact mechanics problems

The model reduction of mechanical problems involving contact remains an important issue in computational solid mechanics. In this article, we propose an extension of the hyper‐reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation. As the potential contact zone is naturally reduced through the reduced mesh involved in hyper‐reduced equations, the dual reduced basis is chosen as the restriction of the dual full‐order model basis. We then obtain a hybrid hyper‐reduced model combining empirical modes for primal variables with finite element approximation for dual variables. If necessary, the inf‐sup condition of this hybrid saddle‐point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper‐reduced/full‐order model strategy. This way, a better approximation on the potential contact zone is further obtained. A posttreatment dedicated to the reconstruction of the contact forces on the whole domain is introduced. In order to optimize the offline construction of the primal reduced basis, an efficient error indicator is coupled to a greedy sampling algorithm. The proposed hybrid hyper‐reduction strategy is successfully applied to a 1‐dimensional static obstacle problem with a 2‐dimensional parameter space and to a 3‐dimensional contact problem between two linearly elastic bodies. The numerical results show the efficiency of the reduction technique, especially the good approximation of the contact forces compared with other methods.     

Efficient computation of order and mode of corner singularities in 3D‐elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non‐smooth domains in three‐dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin–Petrov finite element method. A quadratic eigenvalue problem ( P +λ Q +λ² R ) u = 0 is obtained, with explicitly analytically defined matrices P , Q , R . Moreover, the three matrices are found to have optimal structure, so that P , R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ?e(λ)∈(?0.5,1.0) (no eigenpairs can be ‘lost’) as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill‐in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd.     

Least square-finite element for elasto-static problems. Use of ‘reduced’ integration

A least square based finite element algorithm is developed for some elasto-static problems. In the formulation both stresses and displacements appear as simultaneous variables. In two dimensional (plane) analysis, parabolie isoparametric elements are used. Considerable improvement of performance is obtained with a numerical integration based on 2 × 2 Gauss point distribution over more accurate integration schemes. Reasons for this are presented. The formulation is extended in the section ‘General least square formulation’ to beams and plates with a similar success of ‘reduced’ integration.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.