Explicit Runge–Kutta methods for the integration of rate-type constitutive equations

Modern constitutive models have the potential to improve the quality of engineering calculations involving non-linear anisotropic materials. The adoption of complex models in practice, however, depends on the availability of reliable and accurate solution methods for the stress point integration problem. This paper presents a modular implementation of explicit Runge–Kutta methods with error control, that is suitable for use, without change, with any rate-type constitutive model. The paper also shows how the complications caused by the algebraic constraint of conventional plasticity are resolved through a simple subloading modification. With this modification any rate-independent model can be implemented without difficulty, using the integration module as an accurate and robust standard procedure. The effectiveness and efficiency of the method are demonstrated through a comparative evaluation of second and fifth-order formulas, applied to a complex constitutive model for natural clay, full details of which are given. This work was undertaken with the financial support of the UK Engineering and Physical Sciences Research Council: Grant no. GR/S84897/01.     

Postintegration methods for numerical integration of non-linear dynamic equations of motion

The paper presents a family of methods with postintegration, for numerical integration of non-linear dynamic equations of motion of discrete systems. Both explicit and implicit algorithms of these methods are conditionally stable, but can give solutions of accuracy of a few orders higher than the known methods with preintegration, corresponding to them. A general analysis of stability and accuracy of single-step methods is carried out. The postintegration methods have been tested on a single-degree-of-freedom non-linear system.     

An integration scheme for Prandtl-Reuss elastoplastic constitutive equations

We investigate the Generalized Midpoint Rule for the time integration of elastoplastic constitutive equations for pressure-independent yield criteria. The incremental equations are divided into one scalar hydrostatic pressure/dilation rate equation, and a stress deviator/strain rate deviator tensorial equation, the solution of which reduces to one single scalar equation in the plastic multiplier. The existence and uniqueness of an incremental solution is discussed. The pressure/deviator decomposition is the basis for reduced integration of the pressure term in the Principle of Virtual Work, in order to avoid locking and spurious pressure oscillations. It is also shown that an optimal choice of the parameter of the Midpoint Rule can be computed by reference to the analytical solution of the equations assuming no work hardening. A benchmark test shows that this choice allows increased time steps. This formulation is applied to two classical problems: bulging of a tube under internal pressure and tension test on a notched specimen, and a comparison with the analytical solution is performed. Finally, the hypothesis which sustains these formulations of elastoplasticity (constant strain rate during an increment) is discussed with reference to elastic unloading and residual stress computation.     

Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis

An improved algorithm is presented for integrating rate constitutive equations in large-deformation analysis. The algorithm is shown to be ‘objective’ with respect to large rotation increments.     

钢筋混凝土微平面本构模型及其算法研究

阐述文献[1]的第二部分。第一部分在Ba ant Z.P.等人提出的混凝土微平面模型的基础上引入钢筋的影响提出了一个适用于配筋混凝土的微平面动态本构模型。混凝土模型采用能够反映各种复杂受力行为并被试验充分验证的微平面模型,钢筋采用Cowper-Symonds型率相关的双线性模型。该模型可用于钢筋混凝土的静力和动力显式分析。这一部分将给出模型的显式算法和试验验证。     

On the stability of explicit methods for the numerical integration of the equations of motion in finite element methods

This paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes. As an example, the two-dimensional wave propagation in a semi-infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method. Solution of the example has been compared with that of the Timoshenko theory.     

Elasto-viscoplastic constitutive equations for rock-type materials (finite deformation)

In the present paper the elasto-viscoplastic frame indifferent constitutive equations for rock and rock-type materials are proposed within the frame-work of large deformations. The constitutive equations are written with respect to an arbitrary configuration at time t as reference configurations. Our model describe the dilatancy or compressibility and creep for rock-type materials as well as the existence of the irreversible part of strain even when the applied stresses are relatively small. Constitutive hypotheses are based on the experimental evidences which reveal the complex behaviour or rock-type materials. All the constitutive functions and moduli involved in the model can be determined from a complete set of experimental data.     

An implicit integration algorithm with a projection method for viscoplastic constitutive equations

Robinson's viscoplastic model, a representative of the so-called overstress models, is integrated by use of the generalized midpoint rule. The solution of the non-linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with Newton's method. Consistent tangent moduli are calculated and the quadratic convergence of the global Newton equilibrium iteration is shown. The time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration. Various numerical examples are considered to demonstrate the efficiency of the methods.     

A family of integration algorithms for constitutive equations in finite deformation elasto-viscoplasticity

A two parameter family of incrementally objective integration schemes is proposed for the analysis of a broad range of unified rate-dependent viscoplastic constitutive models in large deformation problems. A similar scheme can be applied to rate-independent solids as well. These algorithms are a generalization of the mid-point integration rule. Full linearization of the principle of virtual work is performed in an updated Lagrangian framework together with a calculation of the consistent linearized moduli. Some details of the finite element implementation are given for plane strain and axisymmetric problems. The method is compared with other objective integration schemes and is tested with several examples where large strains and rotations occur.     

An implicit time integration scheme for inelastic constitutive equations with internal state variables

In a broad class of inelastic constitutive models for the deformation of metals the inelastic strain rates are functions of the current state of stress and internal state variables only. All known models are in some regions of application mathematically stiff and therefore difficult to integrate. The unconditionally stable implicit Euler rule is used for integration. It leads to a system of highly nonlinear algebraic equations which have to be solved by an iterative process. The general Newton-Raphson method, which converges under very broad conditions, requires repeated solution of the finite element system and is infeasible for large inelastic problems. But for the inelastic strains and internal state variables the Jacobian can be computed analytically and therefore the NRI can be used. For the stresses the Jacobian cannot be computed analytically and therefore the accelerated Jacobi iteration is used. A new method for computing the relaxation parameter is introduced which increases the rate of convergence significantly. The new algorithm is applied on Hart's model. A comparison with prior computations using an approximation is made.     

Method of approximating series for numerical integration of systems of differential equations

A method of numerical integration of systems of differential equations is proposed that can be used for equations that describe processes occurring in every field of physics, namely, fluid mechanics, nuclear physics, solid-state physics, etc. The APPROX program package, which implements the method of approximating series, makes it possible to write programs for computations in no more than 2–3 hours and reduces the calculation time by 1–2 orders in comparison with finite-difference methods.Kharkov Polytechnic Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 2, pp. 238–244, February, 1994.     

Two high-order numerical methods for solving boundary-value nonlinear ordinary differential equations

Two numerical methods for solving two-point boundary-value problems associated with systems of first-order nonlinear ordinary differential equations are described. The first method, which is based on Lobatto quadrature, requires four internal function evaluations for each subinterval. It does not need derivatives and is of order h⁷, where h is the space chop. The second method, which is similar to the first but is based on Lobatto–Hermite quadrature, makes the additional use of derivatives to achieve O(h⁹) accuracy. Results of computational experiments comparing these methods with other known methods are given.     

A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics

A comprehensive study of A-stable linear two-step time integration methods for structural dynamics analysis is presented in this paper. An optimal A-stable linear two-step (OALTS) time integration method is revealed with desirable performance on low-frequency accuracy and high-frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second-order equations of structural dynamics; is implicit, A-stable, and second-order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.     

A family of noniterative integration methods with desired numerical dissipation

A new family of unconditionally stable integration methods for structural dynamics has been developed, which possesses the favorable numerical dissipation properties that can be continuously controlled. In particular, it can have zero damping. This numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes, whereas the low frequency modes are almost unaffected. The most important improvement of this family method is that it involves no nonlinear iterations for each time step, and thus it is very computationally efficient when compared with a general second‐order accurate integration method, such as the constant average acceleration method. Copyright © 2014 John Wiley & Sons, Ltd.     

The direct numerical integration of linear matrix differential equations using pade approximations

A class of approximations to the matrix linear differential equation is presented. The approximations range, in accuracy, from the simplest forward difference scheme to the exact solution. The infinite series defining the exponential matrix is used to ascertain the accuracy of the various approximations. A clear distinction is made between approximations to the system equations and the forcing function, with the forcing term being represented by a piecewise linear function. Special application is given to the equations arising in structural dynamics of the form For these structural dynamic equations, the measure of the energy of the system is used to analyse the stability of the numerical approximations.     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

A survey of parallel numerical methods for initial value problems for ordinary differential equations

The parallel solution of initial value problems for ordinary differential equations has become an active area of research. Recent developments in this area are surveyed with particular emphasis on traditional forward-step methods that offer the potential for effective small-scale parallelism on existing machines.<>