Reflection error analysis for wave propagation problems solved by a heterogeneous asynchronous time integrator

Within the framework of dynamic calculations, the hybrid multi–time‐step method proposed by Gravouil and Combescure (GC method) has proven to be an efficient algorithm that enables the use of arbitrary time steps and Newmark time schemes in each subdomain. Nonetheless, when dealing with wave propagation problems, the amount of reflections at the interfaces between subdomains strongly depends on the choice of the time integrators and the time steps used for the simulation study. In this paper, we deal with both one‐ and two‐dimensional wave propagation problems (only the anti‐plane shear wave problem is considered for the two‐dimensional case) with the aim of deriving an analytical estimation of the numerical reflection coefficient at the interface between two linear elastic subdomains having their own time integrators and time scales. The model is approximated using the lowest‐order finite elements, whereas the propagation process is described using harmonic waves. The study is carried out on the explicit/implicit and explicit/explicit integrations using arbitrary time‐step ratios. The numerical reflection coefficient is then analyzed with emphasis on the effect of the time‐step ratio and the direction of incidence.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Regularization and perturbation technique to solve plasticity problems

This paper investigates new procedures to solve plasticity problems by using the asymptotic numerical method (ANM). As the elastic-plastic behavior involves two unilateral conditions, we replace these two conditions by regular functions depending upon the stress field and its time derivative which permits one to take into account elastic-plastic transition and elastic unloading. Several applications in structural plasticity problems are presented to assess the ability of the proposed algorithm.     

A FETI‐based domain decomposition technique for time‐dependent first‐order systems based on a DAE approach

We present a novel partitioned coupling algorithm to solve first‐order time‐dependent non‐linear problems (e.g. transient heat conduction). The spatial domain is partitioned into a set of totally disconnected subdomains. The continuity conditions at the interface are modeled using a dual Schur formulation where the Lagrange multipliers represent the interface fluxes (or the reaction forces) that are required to maintain the continuity conditions. The interface equations along with the subdomain equations lead to a system of differential algebraic equations (DAEs). For the resulting equations a numerical algorithm is developed, which includes choosing appropriate constraint stabilization techniques. The algorithm first solves for the interface Lagrange multipliers, which are subsequently used to advance the solution in the subdomains. The proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps (i.e. it allows subcycling) in each subdomain. This implies that existing software and numerical techniques can be used to solve each subdomain separately. The coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers. We present examples showing the feasibility of the proposed coupling algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

A time‐staggered partitioned coupling algorithm for transient heat conduction

We present a time‐staggered partitioned coupling algorithm for transient heat conduction finite element simulations. This algorithm divides a large structural mesh into a number of smaller subdomains, solves the individual subdomains separately and couples the solutions to obtain the response to the original problem. The proposed algorithm is a mixed multi‐timestep algorithm and enables arbitrary time integration schemes and meshes to be coupled with different timesteps in the various subdomains. In this procedure, the solution of each partition is separately evaluated over a system timestep after which the interfacial conditions are enforced making this a staggered algorithm that facilitates parallel computation. We present examples showing the feasibility of the coupling algorithm and discuss the merits in terms of convergence and stability. Copyright © 2009 John Wiley & Sons, Ltd.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics

In dynamics, domain decomposition methods (DDMs) enable one to use different spatial and temporal discretizations depending on the physical phenomenon being taken into account. Thus, DDMs provide the analyst with key tools for dealing with problems in which phenomena occur on different temporal and spatial scales. This paper focuses on a less intrusive variation of this type of method which enables the global (industrial) mesh to remain unchanged while the local problem is being refined in space and in time where needed. This property is particularly useful in the case of a local problem whose localization evolves rapidly with time, as is the case for delamination. The downside is that the technique is iterative. The method is presented in the context of linear explicit dynamics, but, as with domain decomposition, its extension to other integration schemes and to nonlinear problems should be possible.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

Computation of time and space derivatives in a CQM-based BEM formulation

This work proposes an approach for the numerical computation of time and space derivatives of the time-domain solution of scalar wave propagation problems by means of a boundary element method formulation. Here, this formulation employs the so-called convolution quadrature method. Non-homogeneous initial conditions are taken into account by means of a general procedure, known as initial condition pseudo-force procedure, which replaces the initial conditions by equivalent pseudo-forces. The boundary integral equation with initial conditions contribution is differentiated analytically and the quadrature weights of the standard formulation are transformed in order to compute time and space derivatives at interior points. Numerical examples are presented to show the efficiency of the implemented formulation.     

Novel partitioned time integration methods for DAE systems based on L‐stable linearly implicit algorithms

Real‐time applications based on the principle of Dynamic Substructuring require integration methods that can deal with constraints without exceeding an a priori fixed number of steps. For these applications, first we introduce novel partitioned algorithms able to solve DAEs arising from transient structural dynamics. In particular, the spatial domain is partitioned into a set of disconnected subdomains and continuity conditions of acceleration at the interface are modeled using a dual Schur formulation. Interface equations along with subdomain equations lead to a system of DAEs for which both staggered and parallel procedures are developed. Moreover under the framework of projection methods, also a parallel partitioned method is conceived. The proposed partitioned algorithms enable a Rosenbrock‐based linearly implicit LSRT2 method, to be strongly coupled with different time steps in each subdomain. Thus, user‐defined algorithmic damping and subcycling strategies are allowed. Secondly, the paper presents the convergence analysis of the novel schemes for linear single‐Degree‐of‐Freedom (DoF) systems. The algorithms are generally A‐stable and preserve the accuracy order as the original monolithic method. Successively, these results are validated via simulations on single‐ and three‐DoFs systems. Finally, the insight gained from previous analyses is confirmed by means of numerical experiments on a coupled spring–pendulum system. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations

A proof of stability is developed for an explicit multi-time step integration method of the second order differential equations which result from a semidiscretization of the equations of structural dynamics. The proof is applicable to an algorithm that partitions the mesh into subdomains according to nodal groups which are updated with different time steps. The stability of the algorithm is demonstrated by showing that the eigenvalues of the amplification matrices lie within the unit circle and that a pseudo-energy remains constant. Bounds on the stable time steps for the nodal partitions are developed in terms of element frequencies.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

On the numerical treatment of the delamination problem in laminated composites under cleavage loading

A method for the effective numerical treatment of the delamination problem in laminated composites under cleavage loading is herein proposed. The interlaminar interface mechanical behaviour is described by means of the so-called complete laws which are non-monotone and possibly multivalued force/ displacement laws including jumps (or in general, decreasing branches) corresponding to the discontinuous strength reduction. These complete laws that take into account the development of delamination phenomena in a quasistatic way are derived by non-convex energy functions, called delamination superpotentials which in turn, lead to the formulation of the principle of virtual work for the laminated composite in a hemivariational inequality form and to the generalisation of the principle of minimum potential energy as a substationarity principle. Applying an appropriate finite element discretisation scheme to the laminated composite, the respective discrete problem is formulated which describes the response of the structure taking into account the development of the delamination phenomenon. The numerical treatment of the latter problem is successfully performed by applying a new algorithm that approximates the nonmonotone law by a sequence of monotone ones. The performed numerical applications presented in the last part of the paper and several analogous numerical experiments exhibit very good convergence properties.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Compact schemes for anisotropic reaction–diffusion equations with adaptive time step

Many problems in biology and engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. These characteristics of the system imply the use of very fine meshes and small time steps in order to accurately capture the propagating wave avoiding the appearance of spurious oscillations in the wave front. This work develops a fourth‐order compact scheme for anisotropic reaction–diffusion equations with stiff reactive terms. As mentioned, the scheme accounts for the anisotropy of the media and incorporates an adaptive time step for handling the stiff reactive term. The high‐order scheme allows working with coarser meshes without compromising numerical accuracy rendering a more efficient numerical algorithm by reducing the total computation time and memory requirements. The order of convergence of the method has been demonstrated on an analytical solution with Neumann boundary conditions. The scheme has also been implemented for the solution of anisotropic electrophysiology problems. Anisotropic square samples of normal and ischemic cardiac tissue have been simulated by means of the monodomain model with the reactive term defined by Luo–Rudy II dynamics. The simulations proved the effectiveness of the method in handling anisotropic heterogeneous non‐linear reaction–diffusion problems. Bidimensional tests also indicate that the fourth‐order scheme requires meshes about 45% coarser than the standard second‐order method in order to achieve the same accuracy of the results. Copyright © 2009 John Wiley & Sons, Ltd.     

Sequencing of parts on single-stage multifunctional machining systems using a chaos-embedded simulated annealing algorithm

This paper deals with the objective of determining an optimal part sequence on a single-stage multifunctional machining system (SSMS) with a view to achieve the broad objectives of cost minimization and time minimization. SSMS has become a preferred alternative for manufacturers to use the resource efficiently, owing to the flexibility and process variety offered by it. This paper formulates a mathematical model that considers the minimization of both set-up cost and time simultaneously. The option of hiring an additional fixture has also been considered that enables the reduction in tool magazine replenishment and re-fixturing operations, which in turn offers economic advantage by way of reducing set-up cost. This study has proposed a new heuristic by modifying the simulated annealing concept to solve the underlying problem. The conventional simulated annealing search scheme is replaced by a chaotic search that takes into account the ergodic and stochastic properties of chaotic systems. In order to restrict the premature convergence and to diversify the search space, a modified perturbation scheme has been employed. The performance of the proposed algorithm was tested on a simulated case study adopted from the literature and the results obtained reveal the effectiveness and scalability of the proposed algorithm. The results establish that the proposed approach is effective and reactive to severe disturbances and must take place in the manufacturing environment.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.