On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

Conservative integration of rigid body motion by quaternion parameters with implicit constraints

An angular momentum and energy‐conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four‐component conjugate momentum vector via Hamilton's equations. The introduction of an extended mass matrix leads to a symmetric set of eight state‐space equations of motion. The extra inertial parameter serves as a multiplier on the kinematic constraint, and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments. External loads enter these equations via the set of momentum equations. Initially, the normalization of the quaternion array is introduced via a Lagrange multiplier. However, this Lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential, and elimination of the Lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient. An algorithm is developed by forming a finite increment of the Hamiltonian. This procedure identifies the proper selection of increments and mean values, and leads to an algorithm with conservation of momentum and energy. Implementation, conservation properties, and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top. Copyright © 2012 John Wiley & Sons, Ltd.     

Local modal reduction in explicit dynamics with domain decomposition. Part 1: extension to subdomains undergoing finite rigid rotations

We present an extension of the dual Schur multidomain method with local linear modal reduction previously introduced by Gravouil, Combescure, Herry and Faucher to the case of modal reduction on geometrically non‐linear vibrating subdomains. This first part of a two‐part paper describes a new formalism, based on an original set of parameters, to represent a subdomain's finite rigid‐body motion. Special attention is paid to the stability issues with time integration using the central difference scheme. The method is validated on an academic example and its efficiency is demonstrated on a large‐scale example. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient and robust method for parameterized nonintrusive reduced-order modeling

A method of constructing parameterized nonintrusive reduced-order models (NIROMs) is given. The approach relies on a geometrical interpretation of NIROM, requires only a single layer of interpolation to be applied for both system state and parametric dependence of the model and is applicable to systems characterized by any number of parameters spanning arbitrary orders of magnitude. The method is applied to three representative test problems and evaluated in terms of accuracy and speed, showing good performance.     

3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D

An algorithm for transient dynamics of discrete element systems comprising a large number of irregular discrete elements in 3D is presented. The algorithm is a natural extension of contact detection, contact interaction and transient dynamics algorithms developed in recent years in the context of discrete element methods and also the combined finite‐discrete element method. It complements the existing algorithmic procedures enabling transient motion including finite rotations of irregular discrete elements in 3D space to be accurately integrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Meshfree and finite element nodal integration methods

Nodal integration can be applied to the Galerkin weak form to yield a particle‐type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided. Copyright © 2007 John Wiley & Sons, Ltd.     

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

A multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain-to-subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one-dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one-dimensional heterogeneous bar and in heterogeneous plain strain problems.     

Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies

We reformulate the traditional velocity based vector‐space Newmark algorithm for the rotational dynamics of rigid bodies, that is for the setting of the SO(3) Lie group. We show that the most naive re‐write of the vector space algorithm possesses the properties of symplecticity and (almost) momentum conservation. Thus, we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms, but which curiously does not seem to have been considered in the open literature so far. Copyright © 2005 John Wiley & Sons, Ltd.     

An explicit asynchronous contact algorithm for elastic body–rigid wall interaction

The use of multiple‐time‐step integrators can provide substantial computational savings over traditional one‐time‐step methods for the simulation of solid dynamics, while maintaining desirable properties, such as energy conservation. Contact phenomena generally require either the adoption of an implicit algorithm or the use of unacceptably small time steps to prevent large amount of numerical dissipation from being introduced. This paper introduces a new explicit dynamic contact algorithm that, by taking advantage of asynchronous time stepping, delivers an outstanding energy performance at a much more acceptable computational cost. We demonstrated the performance of the numerical method with several three‐dimensional examples. Copyright © 2011 John Wiley & Sons, Ltd.     

An explicit time integration technique for dynamic analyses

A simple explicit solution technique for problems in structural dynamics, based on a Modified Trapezoidal rule Method (MTM) approximation of the governing ordinary differential equations, is developed. The resulting conditionally stable explicit method (MTM) can be easily implemented and is extremely simple to use. Particular attention is focused herein on the concept of numerical stability of the proposed method for a free-vibrational response of a linear undamped Single-Degree-Of-Freedom system (SDOF). To examine the effectiveness, strengths, and limitations of MTM, error analyses for the natural period, the displacement, the velocity and the associated phase angle for a free undamped simple mass–spring system are derived and compared with Modified Euler Method (MEM) and the well-known Newmark Beta Method (NBM). Numerical examples for a SDOF system and a Multi-Degree-Of-Freedom (MDOF) system are presented to illustrate the strengths and the limitations of the proposed method.     

Finite rotations in dynamics of beams and implicit time-stepping schemes

We examine theoretical and computational aspects of three-dimensional finite rotations pertinent to the dynamics of beams. The model problem chosen for consideration is the Reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner. Special emphasis is placed on clarifying the geometry aspects, finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters. The latter is shown to play an important role in constructing the optimal implementation of a time-stepping scheme. © 1998 John Wiley & Sons, Ltd.     

A parameter to improve stability for a family of dissipative integration methods

There is a family of integration methods which has unconditional stability for linear elastic and stiffness softening-type systems; however, it becomes conditionally stable for stiffness hardening-type systems. Consequently, its applications are inconvenient or limited due to the conditional stability in stiffness hardening-type systems. This drawback can be overcome by introducing a free parameter into its formulation. The numerical properties of this family method are almost unaffected by this free parameter except that the stability property is improved. Thus, the method’s unconditional stability is successfully extended into stiffness hardening-type systems in addition to linear elastic and stiffness softening-type systems.     

An efficient method for nonlinear analysis of layered shells

Finite element formulation based on explicit through-thickness integration scheme assumes importance when applied to multilayered shells, as it is numerically accurate and computationally efficient. Explicit integration becomes possible on assuming the variation of the inverse Jacobian through the thickness. The element stiffness matrices are discussed for (i) large rotation, and (ii) small rotation. Relative efficiencies of the explicit through-thickness integration schemes are compared with that of the conventional formulation involving numerical integration in three directions in each layer and summation over the layers. The small rotation formulation assuming linear variation of the Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient. The geometric nonlinear behaviours of laminated composite cylindrical panels subjected to external pressure are discussed. The parameters considered are: number of layers, symmetric/antisymmetric, cross-ply/angleply, boundary conditions and central angle. The strength of shallow panels with longitudinal edges hinged and curved edges free is controlled by the limit point load, while for deep panels it is controlled by the bifurcation load. The boundary conditions have significant influence on load carrying capacities. A list of symbols is given at the end of the paper     

Accuracy of a composite implicit time integration scheme for structural dynamics

A comprehensive study of the two sub‐steps composite implicit time integration scheme for the structural dynamics is presented in this paper. A framework is proposed for the convergence accuracy analysis of the generalized composite scheme. The local truncation errors of the acceleration, velocity, and displacement are evaluated in a rigorous procedure. The presented and proved accuracy condition enables the displacement, velocity, and acceleration achieving second‐order accuracy simultaneously, which avoids the drawback that the acceleration accuracy may not reach second order. The different influences of numerical frequencies and time step on the accuracy of displacement, velocity, and acceleration are clarified. The numerical dissipation and dispersion and the initial magnitude errors are investigated physically, which measure the errors from the algorithmic amplification matrix's eigenvalues and eigenvectors, respectively. The load and physically undamped/damped cases are naturally accounted. An optimal algorithm‐Bathe composite method (Bathe and Baig, 2005; Bathe, 2007; Bathe and Noh, 2012) is revealed with unconditional stability, no overshooting in displacement, velocity, and acceleration, and excellent performance compared with many other algorithms. The proposed framework also can be used for accuracy analysis and design of other multi‐sub‐steps composite schemes and single‐step methods under physical damping and/or loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Explicit momentum‐conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

We reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel accurate and computationally efficient integration approach to viscoplastic constitutive model

A novel, accurate, and computationally efficient integration approach is developed to integrate small strain viscoplastic constitutive equations involving nonlinear coupled first-order ordinary differential equations. The developed integration scheme is achieved by a combination of the implicit backward Euler difference approximation and the implicit asymptotic integration. For the uniaxial loading case, the developed integration scheme produces accurate results irrespective of time steps. For the multiaxial loading case, the accuracy and computational efficiency of the developed integration scheme are better than those of either the implicit backward Euler difference approximation or the implicit asymptotic integration. The simplicity of the developed integration scheme is equivalent to that of the implicit backward Euler difference approximation since it also reduces the solution of integrated constitutive equations to the solution of a single nonlinear equation. The algorithm tangent constitutive matrix derived for the developed integration scheme is consistent with the integration algorithm and preserves the quadratic convergence of the Newton–Raphson method for global iterations.     

Multi‐model Arlequin method for transient structural dynamics with explicit time integration

This paper addresses the explicit time integration for solving multi‐model structural dynamics by the Arlequin method. Our study focuses on the stability of the central difference scheme in the Arlequin framework. Although the Arlequin coupling matrices can introduce a weak instability, the time integrator remains stable as long as the initial kinematic conditions of both models agree on the coupling zone. After showing that the Arlequin weights have an adverse impact on the critical time step, we present two approaches to circumvent this issue. Computational tests confirm that the two approaches effectively preserve a feasible critical time step and show the efficiency of the Arlequin method for structural explicit dynamic simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

An explicit time integration method for structural dynamics using septuple B‐spline functions

In this paper, an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple B‐spline interpolation polynomial functions. In this way, by use of septuple B‐splines, the authors have proceeded to solve the DE of motion governing a single DOF system, and later, the presented method has been generalized for a multiple DOF system. In the proposed method, a direct recursive formula for response of the system was formulated on the basis of septuple B‐spline interpolation approximation. In terms of the specific requirements of this proposed method, two initialization approaches are given for initial calculation. One is called direct initialization, and the other is indirect initialization. The stability analysis of the proposed method illustrates that, by use of adjustable parameters, a high‐frequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes. The computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples, and the results from the proposed method are compared with those from some of the existent numerical methods, such as the Newmark and Wilson‐ θ methods. The compared results show that the proposed method has high accuracy with low time consumption. Copyright © 2013 John Wiley & Sons, Ltd.