NICE—An explicit numerical scheme for efficient integration of nonlinear constitutive equations

The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage.     

A higher-order alternating group explicit scheme for the diffusion equation

In this paper, a higher-order alternating group explicit scheme for the diffusion equation is developed. The scheme has fourth-order truncation error approximately. The numerical simulations show that the new scheme can provide more accurate solutions. A discussion on the numerical stability of the scheme is also included.     

A second-order explicit scheme for the numerical solution of a fox-rabies model

A second-order, unconditionally-stable, finite-difference scheme is developed for the numerical solution of the SI model of fox-rabies dynamics. The local stability of the scheme, by direct inspection of the eigenvalues dependent on the time step size and on two parameters, is shown to be unconditionally stable.     

One-step explicit methods for the numerical integration of perturbed oscillators

In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

Comparison of models for finite plasticity: A numerical study

We introduce a general framework for the numerical approximation of finite multiplicative plasticity. The method is based on a fully implicit discretization in time which results in an iteratively evaluated stress response; the arising nonlinear problem is then solved by a Newton method where the linear subproblems are solved with a parallel multigrid method. The procedure is applied to models with different elastic free energy functionals and a plastic flow rule of von Mises type. In addition these models are compared to a recently derived frame indifferent approximation of finite multiplicative plasticity valid for small elastic strains which leads to linear balance equations. Rate independent and rate dependent realizations of the former models are considered. We demonstrate by various 3D simulations that the choice of the elastic free energy is not essential (for material parameters representative for metals) and that the new model gives the same response quantitatively and qualitatively as the standard models.     

An explicit structure‐preserving numerical scheme for EPDiff

We present a new structure‐preserving numerical scheme for solving the Euler‐Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non‐linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure‐preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre‐factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave‐like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally‐supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.     

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.     

Object-oriented numerical integration—a template scheme for FEM and BEM applications

An object-oriented numerical integration template implementation is presented on the basis of the C++ programming language. Aiming its straightforward application in finite and boundary element methods, the design supports integrand objects of scalar, vector or matrix types, so that a single programming statement is able to integrate element matrices and vectors. The integrand can contain singularities like the ones typically found in boundary element methods, allowing the evaluation of both regular and singular integrals under the same programming structure. The use of the proposed design is illustrated through some elementary applications as well as finite element and boundary element code excerpts.     

On the numerical integration of elasto-plastic constitutive model structures for nonproportional cyclic loading

The classical Runge-Kutta method with Gill coefficients, a non-iterative Adams predictorcorrector method, an Euler's method with automatic step-size control, an iterative Adams predictorcorrector method with automatic step-size control and Gear's method are the numerical solution algorithms considered in this study. Their computational accuracy and efficiency are evaluated for two cases of axial-torsional loading with transient, nonproportional, cyclic plasticity. The constitive equations implemented include a modified classical single-surface theory, a two-surface theory and a unified creep-plasticity or state variable theory.     

Implicit numerical integration of a three-invariant,isotropic/kinematic hardening cap plasticity model for geomaterials

The mechanical constitutive behavior of geomaterials is quite complex, involving pressure-sensitive yielding, differences in strength in triaxial extension vs. compression, the Bauschinger effect, dependence on porosity, and other factors. Capturing these behaviors necessitates the use of fairly complicated and expensive non-linear material models. For elastically isotropic materials, such models usually involve three-invariant plasticity formulations. Spectral decomposition has been used to increase the efficiency of numerical simulation for such models for the isotropically hardening case. We modify the spectral decomposition technique to models that include kinematic hardening. Finally, we perform some numerical simulations to demonstrate quadratic convergence.     

The unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models: Revisited

The unconditionally positive-convergent implicit scheme for two-equation turbulence models, originally developed by Mor-Yossef and Levy, is revisited. A compact, simple, and uniform reformulation of the method for the use of both structured and unstructured grid based flow solvers is presented. An analytical proof of the scheme revision is given showing that positivity of the turbulence model solutions and convergence of the turbulence model equations are guaranteed for any time step. Numerical experiments are conducted, simulating two test cases of three-dimensional complex flow fields using structured and hybrid unstructured grids. To demonstrate the overall scheme’s robustness, it is applied to non-linear k-ω and non-linear k- turbulence models. Results from the numerical simulations show that the scheme exhibits very good convergence characteristics, is robust, and it always preserves the positivity of the turbulence model dependent variables, even for an infinite time step.     

A numerical study of plasticity models and finite element types

The available experimental results for the cyclic loading of a simply supported circular plate using a rigid circular punch is utilized to compare the numerical results obtained from computations performed using the four most popular work hardening plasticity models, namely, the isotropic, kinematic, mechanical sublayer and the Mroz models. The mechanical sublayer model appears to be the most efficient in this group. Using the mechanical sublayer model the computational efficiencies of the 3, 4 and 8-node finite elements are studied. The 8-node element emerges as the most suitable element type for the present problem.     

A new model-dependent time integration scheme with effective numerical damping for dynamic analysis

This paper presents a new semi-explicit dissipative model-dependent time integration algorithm for solving structural dynamics problems. Motivated by the superior properties of the composite time-stepping scheme, the proposed method is designed, so that it fully inherits the numerical characteristics of its parent algorithm, namely the Bathe method. The algorithm design procedure is carried out by assuming unknown integration parameters for the proposed method. Afterwards, by time discretization of an SDOF model equation, the unknown parameters can be obtained explicitly by solving nonlinear system of equations. Some numerical examples are analyzed by the presented technique and comparisons are also made with two other dissipative model-dependent time integration algorithms as well as the Bathe method. Results demonstrate that the suggested technique can effectively damp out the spurious oscillations of the high-frequency modes, while the other schemes exhibit significant overshoot in the calculated responses. Furthermore, it is also observed that numerical results of the presented method totally coincide with the parent algorithm. While the Bathe method subdivides each time increment into two sub-steps, the proposed algorithm is single-step, non-iterative and does not involve any time-step subdividing.

     

On a constitutive driver as a useful tool in soil plasticity

A mathematical basis for the development of Constitutive Drivers in soil plasticity has recently been proposed by the authors. A Constitutive Driver is here understood as a computer program, containing a number of selected constitutive models, in which different laboratory and field tests can be simulated and model parameters optimised. As a pilot study of the mathematical concept, a Constitutive Driver for soils, in the form of a PC-program, has been developed. The paper discusses this particular program, i.e. its structure, the mathematical basis, included soil models and some application examples, to give an idea of how a general and user-friendly Constitutive Driver can be designed. Such a program can be used for practical, research and educational purposes. In fact, it is believed that so many important applications for Constitutive Drivers exist that it would be beneficial if such programs were easily accessible as complementary programs in commercial software.     

A smoothing scheme for a minimum weight problem in structural plasticity

We revisit an improtant and challenging class of minimum weight structural plasticity problems, the feature of which is the presence of complementarity constraints. Such relations mathematically express the perpendicularity of two sign-constrained vectors and mechanically describe an inherent property of plasticity. The optimization problem in point is referred to in the mathematical programming literature as a Mathematical Program with Equilibrium Constraints (MPEC). Due to its intrinsic complexity, MPECs are computationally very hard to solve. In this paper, we adopt recent ideas, proposed by mathematical programmers, on smoothing to develop a simple scheme for reformulating and solving our minimum weight problem as a standard nonlinear program. Simple examples concerning truss-like structures are also presented for illustrative purposes.     

Whole-sentence exponential language models: a vehicle for linguistic-statistical integration

We introduce an exponential language model which models a whole sentence or utterance as a single unit. By avoiding the chain rule, the model treats each sentence as a “bag of features", where features are arbitrary computable properties of the sentence. The new model is computationally more efficient, and more naturally suited to modeling global sentential phenomena, than the conditional exponential (e.g. maximum entropy) models proposed to date. Using the model is straightforward. Training the model requires sampling from an exponential distribution. We describe the challenge of applying Monte Carlo Markov Chain and other sampling techniques to natural language, and discuss smoothing and step-size selection. We then present a novel procedure for feature selection, which exploits discrepancies between the existing model and the training corpus. We demonstrate our ideas by constructing and analysing competitive models in the Switchboard and Broadcast News domains, incorporating lexical and syntactic information.     

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.     

An adaptive numerical integration code for a chain of transputers

We consider the calculation of ∫_a^bƒ(x)dx, on a linear chain of transputers of arbitrary length, in which each transputer calculates an approximation to this integral on a subinterval of equal length using an adaptive method based on the ten point Gaussain rule and the twenty-one point Kronrod extension. Extensive numerical testing on a chain of 32 T800s shows that if a large number of function evaluations are needed then linear speed-ups are achievable with any number of transputers.     

An explicit scheme for the solution of the filtration problems

Different models of compressible fluid filtration are considered. Unlike the classical system of equations, the continuity equation is modified with allowance for the minimum scale of space averaging and for the internal relaxation time of the system. Three-level explicit finite difference schemes are proposed that are convenient for high-performance parallel implementation. The transition from the parabolic to the hyperbolic system of equations makes the stability requirements for them less stringent than for the two-level schemes.     

Coefficients of Euler-Maclaurin formulas for numerical integration

Recurrent formulas for Euler-Maclaurin coefficients are obtained based on trapezoid, midpoint, and Simpson’s quadrature rules. Asymptotics of coefficients of each rule are found. Expressions for the first ten coefficients are given in the form of rational fractions.