A plastic nonlocal damage model

In the present paper, a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials. A new isotropic damage model, which is able to describe the behavior of a wide class of cementitious materials, is presented. A regularization technique, based on the introduction of the damage Laplacian in the damage limit function, is adopted to overcome the analytical and computational problems induced by the softening constitutive law. A Drucker–Prager type of plastic limit function is proposed considering isotropic hardening. A numerical procedure, based on an implicit `backward-Euler' technique for the time integration of the plastic and damage evolution equations, is presented. To solve each nonlinear step, a predictor–corrector iterative method is developed within the splitting method. In particular, the damage evolution is determined solving a constrained minimization problem of a convex functional. The proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials.     

Unified elasto-plastic associated and non-associated constitutive model and its engineering applications

A unified elasto-plastic associated and non-associated constitutive model, which is based on the unified theory, is proposed in this paper. This unified constitutive model can be easily used and has been implemented in UEPP (unified elasto-plastic finite element program). This new elasto-plastic constitutive model can be widely used in engineering.     

Predictor–corrector Obreshkov pairs

The combination of predictor–corrector (PEC) pairs of Adams methods can be generalized to high derivative methods using Obreshkov quadrature formulae. It is convenient to construct predictor–corrector pairs using a combination of explicit (Adams–Bashforth for traditional PEC methods) and implicit (Adams–Moulton for traditional PEC methods) forms of the methods. This paper will focus on one special case of a fourth order method consisting of a two-step predictor followed by a one-step corrector, each using second derivative formulae. There is always a choice in predictor–corrector pairs of the so-called mode of the method and we will consider both PEC and PECE modes. The Nordsieck representation of Adams methods, as developed by C. W. Gear and others, adapts well to the multiderivative situation and will be used to make variable stepsize convenient. In the first part of the paper we explain the basic approximations used in the predictor–corrector formula. Those can be written in terms of Obreshkov quadrature. Next section we discuss the equations in terms of Nordsieck vectors. This provides an opportunity to extend the Gear Nordsieck factorization to achieve a variable stepsize formulation. Numerical tests with the new method are also discussed. The paper will present Prothero–Robinson and Kepler problem to illustrate the power of the approach.     

Numerical analysis of anisotropic ductile continuum damage

The paper deals with the numerical analysis of large elastic–plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. Estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic (damage–plastic) predictor followed by an elastic corrector step. The finite element method is used to approximate the linearized variational problem. Furthermore, identification of material parameters is discussed. Numerical simulation of the elastic–plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation.     

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.     

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.     

Euler’s predictor–corrector technique for solving the depth-integrated shallow water equations

Euler’s predictor–corrector technique combined with finite analysis method is applied to solve 2D advection–diffusion shallow water equations. In this algorithm the momentum equations are calculated by the finite analysis method based on a single mesh, while the continuity equation is solved by Euler’s predictor–corrector technique. To verify the performance of this approach, the simulation of tidal flow in the Huangpu estuary is carried out. The numerical results are found to be consistent with the field results, implying that this proposed method is effective and applicable.     

Coupled viscoelastic–viscoplastic modeling of homogeneous and isotropic polymers: Numerical algorithm and analytical solutions

A coupled viscoelastic–viscoplastic (VE–VP) model is implemented and studied. The total strain is the sum of VE and VP parts, and the Cauchy stress is given by a linear VE model as a Boltzmann integral of the history of VE strains. The proposed computational algorithm features fully implicit integration, return mapping based on a two-step VE predictor/VP corrector strategy, and a consistent tangent operator. The algorithm is applied to J₂ VP coupled with VE. Very compact expressions are obtained which are form-identical to classical elasto-viscoplasticity (EVP) provided that the constant linear elastic shear and bulk moduli are replaced with incremental relaxation moduli which are appropriate functions of the time increment. Two different integration methods to obtain the incremental moduli are proposed and assessed. Closed-form solutions for uniaxial tension and simple shear are developed, based on an original solution method for integro-differential equations. The analytical results enable to illustrate the constitutive model and provide unambiguous benchmarks for numerical algorithms. Model predictions are compared with experimental data and reasonable correlation is obtained.     

An incremental minimization principle suitable for the analysis of low cycle fatigue in metals: A coupled ductile–brittle damage model

The present paper is concerned with a novel variational constitutive update suitable for the analysis of low cycle fatigue in metals. The underlying constitutive model originally advocated in [1] accounts for plastic deformation as well as for damage accumulation. The latter is captured by a combination of two constitutive models. While the first of those is associated with ductile damage, the second material law is related to a quasi-brittle response. The complex overall model falls into the range of so-called generalized standard materials and thus, it is thermodynamically consistent. However, since the evolution equations are non-associative, it does not show an obvious variational structure. By enforcing the flow rule as well as the evolution equations through a suitable parameterization, a minimization principle can be derived nevertheless. Discretized in time, this principle is employed for developing an effective numerical implementation. Since the mechanical subproblems corresponding to ductile damage and that of quasi-brittle damage are uncoupled, an efficient staggered scheme can be elaborated. Within both steps, Newton’s method is applied. While the evolution of the quasi-brittle damage requires only the computation of a one-dimensional optimization problem, the ductile damage model is defined by a numerically more expensive tensor-valued variable. For further increasing the numerical performance of the respective minimization principle, a closed-form solution for the inverse of the Hessian matrix is derived. By numerically analyzing the prediction of mesocrack initiation in low-cycle fatigue simulations, the performance of the resulting algorithm is demonstrated.     

Parallel block pc methods with rkn-type correctors and adams-type predictors ∗

This paper describes the construction of block predictor - corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor - corrector method not only at step point, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-accurate predictions using Adams-type formulas. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictions using last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor - corrector methods are comparable with those of a conventional predictor - corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor -corrector methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the best parallel and sequential methods from the literature.     

An adaptative augmented Lagrangian method for three-dimensional multimaterial flows

The direct numerical simulation of incompressible multimaterial flows, based on predictor/corrector and volume of fluid (VOF) approaches is presented. An original adaptative augmented Lagrangian method is proposed to solve the predictor solution, satisfying at the same time the conservation equations as well as the incompressibility constraint. This algorithm is based on an Uzawa optimisation technique. The corrector solution is obtained with a projection method on a divergence free subspace. Several examples of two- and three-dimensional flows are proposed to illustrate the ability of the method to deal with unsteady, multimaterial problems.     

Artificial neural network training utilizing the smooth variable structure filter estimation strategy

A multilayered neural network is a multi-input, multi-output nonlinear system in which network weights can be trained by using parameter estimation algorithms. In this paper, a novel training method is proposed. This method is based on the relatively new smooth variable structure filter (SVSF) and is formulated for feed-forward multilayer perceptron training. The SVSF is a state and parameter estimation that is based on the sliding mode concept and works in a predictor–corrector fashion. The SVSF training performance is tested on three benchmark pattern classification problems. Furthermore, a study is presented comparing the popular back-propagation method, the extended Kalman filter, and the SVSF.     

Dynamic response simulation for reinforced concrete slabs

Present investigation comprises development of a new finite element numerical formulation for nonlinear transient dynamic analysis of reinforced concrete slab structures. Depending on many experimental data, new material constitutive relationships for concrete material have been formulated. A regression analysis of available experimental data in the SPSS-statistical program has been employed for formulating the proposed material finite element models, and the appropriateness of the models are confirmed through the histograms and measured indices of determination. Concrete slab structures were analyzed using eight-node serendipity degenerated plate elements. The constitutive models of the nonlinear materials are introduced to take into account the nonlinear stress–strain relationships of concrete. For studying the stress profile of the concrete slab through its thickness, a layered approach is adopted. Elastic perfectly plastic and strain hardening plasticity approaches have been employed to model the compressive behavior of concrete. Assumptions for strain rate effect were included in dynamic analysis by supposing the dynamic yield function as a function of the strain rate, in addition to be the total plastic strain. The yield condition is formulated in terms of the first two stress invariants. Geometrical nonlinearity was considered in analysis as a mathematical model based on the total lagrangian approach taking into account Von Karman assumptions. Implicit Newmark with corrector–predictor algorithm was used for time integration solution of the equation of the motion for slab structures. An incremental and iterative procedure is adopted to trace the entire response of the structure; a displacement convergence criterion is adopted in the present study. A computer program coded in FORTRAN has been developed and used for the dynamic analysis of reinforced concrete slabs. The numerical results show good agreement with other published studies’ results which include deflections.     

A continuum model of jointed rock masses based on micromechanics and its integration algorithm

The paper is devoted to proposing a constitutive model based on micromechanics. The joints in rock masses are treated as penny-shaped inclusion in solid but not through structural planes by considering joint density, closure effect, joint geometry. The mechanical behavior of the joints is represented by an elasto-plastic constitutive law. Mori-Tanaka method is used to derive the relationship between the joint deformations and macroscopic strains. The incremental stress-strain relationship of rock masses is ...     

On a statistical damage constitutive model for rock materials

A statistical damage constitutive model for rocks with strain softening behaviour is put forward, using the theory of continuous damage mechanics together with statistical mesoscopic strength theory based on maximum entropy distribution. In statistical constitutive modelling, the probabilistic distribution of mesoscopic element strength in rocks is a critical piece of information, which was conventionally described by various empirical distributions among which the Weibull distribution was extensively used. In this paper, an alternative and universal distribution, i.e. maximum entropy distribution, is proposed to describe the statistical property of mesoscopic strength in rocks. By incorporating the entropy distribution information into a damage variable, the paper derives a new constitutive model, which is verified by conventional triaxial experiments and dynamic experiments on rock specimens. To illustrate suitability and flexibility, the new model is compared to the well-established Weibull-distribution-based models.     

A P-Stable Linear Multistep Method For Solving General Third Order Ordinary Differential Equations

A P-stable linear multistep method for solving general third order initial value problems of ordinary differential equations without first reducing the problems into a system of first order equations is considered. The approach for the development of this method is essentially based on collocation of the differential system generated from a basis function. A predictor for the evaluation of $y_{n+k}$ for an odd $k\ge 3$ in the main method is also proposed. The two resulting methods, the corrector and the predictor are P-stable for $k = 3$ . These as a block are tested on a number of problems to show their efficiency. When the methods (the corrector and the predictor) are evaluated at $x = x_{n+3}$ identical schemes are obtained as special cases of the methods, while the set of first and second derivatives obtained from the corrector are different from those obtained from the predictor.     

Estimation with lossy measurements: jump estimators for jump systems

In this paper, we consider estimation with lossy measurements. This problem can arise when measurements are communicated over wireless channels. We model the plant/measurement loss process as a Markovian jump linear system. While the time-varying Kalman estimator (TVKE) is known to be optimal, we introduce a simpler design, termed a jump linear estimator (JLE), to cope with losses. A JLE has predictor/corrector form, but at each time selects a corrector gain from a finite set of precalculated gains. The motivation for the JLE is twofold. The computational burden of the JLE is less than that of the TVKE and the estimation errors expected when using JLE provide an upper bound for those expected when using TVKE. We then introduce a special class of JLE, termed finite loss history estimators (FLHE), which uses a canonical gain selection logic. A notion of optimality for the FLHE is defined and an optimal synthesis method is given. The proposed design method is compared to TVKE in a simulation study.     

Elasto-plastic analysis of axisymmetric structures subject to arbitrary loads by hybrid-stress finite elements

A semi-analytic finite element method in conjuction with a hybrid-stress functional based on the initial stress approach is presented for elasto-plastic analysis. A three-dimensional solid element based on hybrid-stress model is also implemented for the elasto-plastic analysis. A procedure to compute the non-linear effects in terms of Fourier series in the hybrid-stress model is described. The accuracy and efficiency of the semi-analytic method is evaluated via numerical examples by comparing the solution with a full 3-D solution. The semi-analytic method is observed to be a viable alternative to 3-D analysis in elasto-plastic analysis of axisymmetric structures subject to arbitrary loads.     

Computer generated elasto-plastic design data for pressure loaded circular plates

Two finite-difference computer programs for the non-axisymmetric elasto-plastic large deflection analysis of circular plates are described. One of the programs uses a constitutive model based on the Ilyushin full-section yield criterion, whereas the other uses the von Mises layered yield criterion. The programs have been used to model the full-range response including multi-stage unloading, of uniformly loaded circular plates with various boundary conditions for a range of slendernesses. From the computed results design charts of first yield over-pressure versus maximum and residual centre deflection have been compiled. A numerical example of the use of these charts in the design of an end closure to a cylindrical pressure vessel is presented. Finally, sample charts for ‘pre-dished’ plates are presented as an illustration of the further potential of these programs for design data generation.