Numerical Integration Algorithm Research Articles

Different Numerical Time Integration Schemes for Elastoplasticity Coupled to Anisotropic Damage

Much effort has been put in the development of proper continuum damage mechanics models, in which damage is either represented as a scalar, vectorial or tensorial quantity. In this work the anisotropic damage theory of Lemaitre et al. (2000), which describes damage as a second order tensor, is utilized. Two numerical time integration algorithms, namely a fully implicit and a partially explicit scheme, are compared by means of finite element computations of a plate with a circular hole. The convergence behavior of the two algorithms is studied and compared regarding the number of time steps.

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.

An internal-state-variable based viscoelastic-plastic model for polymers

In this study, a new viscoelastic-plastic constitutive model which has been formulated by utilizing the formalism of stress-like internal state variables is introduced. The developed constitutive equation allows for a good description of the inelastic material response of polymeric materials over a wide range of strain rates. An algorithm for numerical integration of the model equations has been derived. The FE implementation of the constitutive equation is widely discussed and the results of solving several exemplary problems are presented.

APPLICATION OF NUMERICAL INTEGRATION IN TECHNICAL PRACTICE

In this contribution we present the methodology of the solution of ordinary differential equation by the Runge – Kutta numerical method of fourth order. Analysed function is continuous and it has derivatives at every point. Records of the centre of the gravity velocities of agricultural technological vehicle are function values of the derivatives. Algorithm of numerical integration was implemented by the help of programming language C# in MS Visual Studio Pro 2010 developing environment. Trajectory of the centre of the gravity movement in three-dimensional space is the result of the listed algorithm.

A Comparison of Explicit Semi-Analytical Numerical Integration Methods for Solving Stiff ODE Systems

In this study, a comparison among three semi-analytical numerical integration algorithms for solving stiff ODE systems is presented. The algorithms are based on Differential Transform Method (DTM) which are Multiple-Step DTM (MsDTM), Enhanced MsDTM (E-MsDTM) and MsDTM with PadA© approximants (MsDTM-P). These methods can be classified as explicit one step semi-analytical numerical integration methods. The error and stability analysis of each method is presented. New important relationships among the methods are introduced. To demonstrate our results, a comparison of the accuracy, stability and computational efficiency of the methods is presented through solving some linear and nonlinear problems arising in applied science and engineering.

Dynamic response of sleepers in a track with uneven rail irregularities using a 3D vehicle–track model with sleeper beams

Employing a 3D model of the vehicle–track interaction, dynamic responses of sleepers in railway tracks are determined due to the effect of uneven irregularities (cross-levels) in left/right rails. An integrated mathematical model of the vehicle–track system is used to calculate the dynamic behavior of sleepers in the railway track. The general methodology of the dynamic analysis is presented by extracting the mathematical equations and by introducing the coupling process in the wheel–rail interface. Besides the vertical (bounce) and pitch motions, roll degrees of freedom of the components are taken into account. Except rails and sleepers, other components of the vehicle–track system are simulated with the rigid-body elements, linked to each other within the resilient elements (springs and dashpots). The rails and sleepers are modeled with the beam elements, making the model more detailed in the track subsystem. A numerical integration algorithm based on Newmark’s method is developed, capable of solving the dynamic interaction of the entire system in time domain. Applying various irregularity profiles from the field measurement data as the source of excitation, the dynamic response of sleepers in track structure is treated in the longitudinal and traverse directions. Dynamic displacements under sleepers are compared with the analytical solution of the beam on elastic foundation (static response) to highlight the dynamic amplifications. The results of analysis show that the profile unevenness in left/right rails plays an important role in dynamic stability of the sleepers in railway track, particularly when the severity of the profile unevenness was increased.

Comparisons of three numerical integration algorithms in shake table-actuator hybrid testing under proportional errors

Shake Table-Actuator Hybrid (STAH) testing is a recently developed dynamic test method. A STAH test is in a certain sense a dynamic time history analysis of a structure under seismic loading, except that some nonlinear elements are tested in the laboratory. Therefore the numerical integration algorithm is significant for the accuracy of the structural seismic response. In order to compare the effects of different numerical integration algorithms, the stabilities and accuracies of STAH tests with Central Difference Method, Linear Acceleration Method and Chang Method are studied. In this paper, the equations of STAH tests with three different numerical integration algorithms are derived and then the numerical simulation platforms based on these algorithms are developed using the software MATLAB. Errors in seismic tests are inevitable, and proportional errors are the worst of all. Therefore the effects of proportional errors on the stability and accuracy of STAH testing are investigated. The effects of the three different numerical integration algorithms are also discussed and compared theoretically and numerical simulation platforms are used to verify the theoretical results. It turns out that the stability of Chang Method is better while the accuracy of Linear Acceleration Method is better. Therefore, it is suggested that different numerical integration algorithms should be adopted according to different situations.

Benchmark values for molecular two-electron integrals arising from the Dirac equation.

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.

Implementation of HHT algorithm for numerical integration of multibody dynamics with holonomic constraints

The Hilber–Hughes–Taylor (HHT) algorithm is implemented in this paper to solve the dynamic problem of flexible multibody system with holonomic constraints. The incremental formulas of governing equations of motion, a set of differential algebraic equations of index 3, are presented and integrated direct by current approach. The parameterization of rotation has been paid a special attention, which makes the formula of HHT algorithm much more complicated because of the nonlinearity of rotations. The associated nonlinear algebraic equations are solved with the application of simplified Newton’s iteration to simplify the linearization of finite rotation. Two options are available for solving the incremental linear equations, one is the direct method based on LU decomposition and the other GMRES-based matrix-free method. Based on the estimated truncation error, the time adaptive scheme is designed to enhance the integrating speed. Numerical simulations prove the modified HHT algorithm behavior as the second-order accuracy with numerical damping. The application of conformal rotation vector improves the robustness of the current approach which makes it possible to integral rotation of arbitrary magnitude.

Синтез разностных алгоритмов управления цифровыми следящими электроприводами мобильных роботов Е-операторным методом

The mathematical Е-operating method is intended for transformation of the of differential equations system to the tracking electric drive mathematical model equations system. In contrast to the known algorithms for numerical integration, for example, in the Matlab-Simulink, E-method allows to synthesize of the transfer function of tracking electric drives for both sequential and parallel differential algorithms for working in real time on the basis of both software and hardware shifting registers. It is important to note that these algorithms cannot be implemented using Matlab-Simulink. Parallel algorithm provides greater performance than serial algorithm. Parallel algorithm can be implemented "conveyor" method of using multiple coprocessors at the same time. Method is an alternative of the means available in Matlab-Simulink programme for description and analysis of the tracking electric drive model. In the first part of the article the synthesis algorithm using traditional means of differential calculus and Laplace transforms describes. In the second part of the article a solution of the same problem with the use of the Е-operating method presents. The results obtained can be used in mechatronics, robotics, automatic control theory, in the practice of building looking for tracking electric drive.

Modelling and Simulation of Water Distribution Systems with Quantised State System Methods

The work in this paper describes a study of quantised state systems in order to formulate a new framework within which water distribution systems can be modelled and simulated. In contrast to the classic time-slicing simulators, depending on the numerical integration algorithms, the quantisation of system states would allow accounting for the discontinuities exhibited by control elements in a more efficient manner, and thereby, offer a significant increase in speed of the simulation of water network models. The proposed approach is evaluated on a case study and compared against the results obtained from the Epanet2 simulator and OpenModelica.

TFA-based Homogenization for Composites Subjected to Coupled Damage-friction Effects

The present paper deals with the formulation of a new homogenization procedure for the determination of the overall response of composites subjected to nonlinear phenomena due to the development and growth of cracks. An interface model accounting for damage, unilateral contact and friction is described. Then, a homogenization technique, based on the Transformation Field Analysis, is developed. A numerical algorithm for the time integration of the evolutive problem is formulated within a predictor- corrector procedure for solving the time step nonlinear problem. A numerical application is presented to assess the effectiveness of the proposed approach.

B-Methods for the Numerical Solution of Evolution Problems with Blow-Up Solutions Part I: Variation of the Constant

In the last two decades, the field of geometric numerical integration and structure-preserving algorithms has focused on the design of numerical methods that preserve properties of Hamiltonian systems, evolution problems on manifolds, and problems with highly oscillatory solutions. In this paper, we show that a different geometric property, namely, the blow up of solutions in finite time, can also be taken into account in the numerical integrator, giving rise to geometric methods we call B-methods. We give a first systematic approach for deriving such methods for scalar and systems of semi- and quasi-linear parabolic and hyperbolic partial differential equations. We show both analytically and numerically that B-methods have substantially better approximation properties than standard numerical integrators as the solution approaches the blow-up time.

Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method

We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.

The high-order Euler method and the spin–orbit model.

We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

A two-surface hardening plasticity model based on non-associated flow rule for anisotropic metals subjected to cyclic loading

In this paper a phenomenological material model for simulation of sheet metal forming processes was introduced. This model is able to describe the anisotropic behavior of sheet metals in both yield stresses and plastic strain ratios (r-values) by using the non-associated flow rule and quadratic yield and potential functions. Additionally, to reproduce an accurate prediction of cyclic plastic deformation phenomena, a two-surface mixed isotropic-nonlinear kinematic hardening model was combined with the quadratic non-associated anisotropic formulation. This mixed kinematic hardening model is amongst the most sophisticated models with acceptable degree of complexity and minimum requirement of experimental tests for material coefficients. The main advantage of the model over the more complex nonquadratic yield and potential functions along with associated or non-associated flow rules is its simplicity and computational efficiency. The plasticity foundation of the model was introduced and then a general return mapping algorithm for numerical stress integration of the constitutive model was developed in order to implement it into a finite element code. Finally, the model was used to simulate both forming stage and subsequent springback of a deep drawing problem. The results showed that the model can accurately predict springback as well as earing phenomenon of the stamped part.

An improved numerical integration algorithm for elastoplastic constitutive equations

A simplified methodology is proposed for elastoplastic calculations which holds for associative models. It is based on the representation of the elastoplastic model based on a rotation of the principal stresses and the fact that, in such system of coordinates, the direction that minimizes the square of a distance between a trial stress and the plastic surface has the same direction as the plastic deformation evolution. Such an approach allows for the elastoplastic calculation of complex models to be simpler and more efficient computationally. The proposed methodology is verified by the application to the elastoplastic model of Sandler–DiMaggio.

Modeling concrete like materials under sever dynamic pressures

The behavior of concrete, as well as that of all cohesive-frictional materials, is characterized by a strong dependence on hydrostatic pressure and strain rate sensitivity. The stress–strain response is nonlinear and, depending on the loading path, it can show compaction and dilatancy. Due to its internal constitution, the numerical simulation of this type of materials can be done with a great variety of approaches which are mainly differentiated by the modeling scales and the approximation methods used. Appropriate formulation and calibration of the models are necessary to accurately reproduce the physical phenomena involved in the simulated processes.A general elastic–viscoplastic model for concrete like materials under high strain rate dynamic loads that produce high confinement pressures, like those present in blast actions, is presented in this paper. The model developed is of phenomenological type and it is formulated within the framework of continuum thermodynamics for irreversible processes with internal variables and small strains. Non-associated plastic flow, cap function and hardening functions that depend on the loading path are proposed. The transition between compaction and dilatancy processes observed in triaxial compression experimental tests can be properly reproduced with the non-associated flow consideration and the proposed hardening functions. The cap function allows the appropriate simulation of the material volumetric response for high confinement pressures. The model formulation is general and it can be used for other cohesive-frictional materials.The numerical integration algorithm is implemented in a 2D finite element dynamic program that allows solving nonlinear problems of solid mechanics in small strains.For the model validation, static and dynamic tests with different loading paths and confinement levels for many types of concretes and mortars are analyzed. Numerical results are compared with experimental results obtaining a good description of the main characteristics of this type of materials response under high strain rate pressures.

Алгоритм численного интегрирования потенциально-потоковых уравнений в сосредоточенных параметрах с контролем корректности приближенного решения

Алгоритм численного интегрирования потенциально-потоковых уравнений в сосредоточенных параметрах с контролем корректности приближенного решения

On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

SUMMARYA simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity. Central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants. The set of bases can be regarded equivalently as a local cylindrical coordinate system in the three‐dimensional coaxial tensor subspace, namely, the principal space. The base tensors are given in the global coordinate system. Similar to the principal space approach, the proposed method reduces the problem dimension from six to three. In contrast to the conventional approach, the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed. With the proposed technique, the matrices, which need to be inverted during iteration, take a simple form for the great majority of constitutive equations in use. The tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it. Consequently, its closed form is derived in an extremely simple manner. Finally, numerical examples demonstrate the high quality performances of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.