Time-marching Technique Research Articles

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

Efficient high-order radial basis-function-based differential quadrature-finite volume method for incompressible flows on unstructured grids.

This paper presents an efficient high-order radial basis-function-based differential quadrature-finite volume method for incompressible flows on unstructured grids. In this method, a high-order polynomial based on the Taylor series expansion is applied within each control cell to approximate the solution. The derivatives in the Taylor series expansion are approximated by the mesh-free radial basis-function-based differential quadrature method. The recently proposed lattice Boltzmann flux solver is applied to simultaneously evaluate the inviscid and viscous fluxes at the cell interface by the local solution of the lattice Boltzmann equation. In the present high-order method, a premultiplied coefficient matrix appears in the time-dependent term, reflecting the implicit nature. The implicit time-marching techniques, i.e., the lower-upper symmetric Gauss-Seidel and the explicit first stage, singly diagonally implicit Runge-Kutta schemes, are incorporated to efficiently solve the resultant ordinary differential equations. Several numerical examples are tested to validate the accuracy, efficiency, and robustness of the present method on unstructured grids. Compared with the k-exact method, the present method enjoys higher accuracy and better computational efficiency.

Elastodynamic wave propagation modelling in geological structures considering fully-adaptive explicit time-marching procedures

In this paper, two fully-adaptive explicit time-marching procedures are discussed for the time-domain solution of wave propagation in elastic media. In these procedures, the time integration parameters of the methods are adaptively locally evaluated, following the properties of the discretized model and the values of the calculated responses, engendering more accurate solution procedures. In addition, automated domain decomposition and sub-cycling procedures are also performed, providing more efficient analyses. In this case, a domain decomposition procedure divides the domain of the model into different time-marching subdomains according to the properties of the discretized problem and the stability limit of the adopted time-marching technique, so that a different time-step value is applied to each subdomain, leading to more efficient (and yet stable) explicit time-domain computations. As it is indicated in this work, the adoption of multi-time-steps/sub-cycling splitting procedures further improves the accuracy of the discussed adaptive time-marching formulations. At the end of the paper, benchmark analyses are performed to illustrate the effectiveness of the proposed techniques, followed by synthetic case analyses, with degrees of complexity equivalent to that of real geophysical applications in the OIL & GAS industry, demonstrating the robustness of the proposed explicit approaches.

Two Efficient Time-Marching Explicit Procedures Considering Spatially/Temporally-Defined Adaptive Time-Integrators

In this paper, two explicit time-marching techniques are discussed for the solution of hyperbolic models, which are based on adaptively computed parameters. In both these techniques, time integrators are locally and automatically evaluated, taking into account the properties of the spatially/temporally discretized model and the evolution of the computed responses. Thus, very versatile solution techniques are enabled, which allows computing highly accurate responses. Here, the so-called adaptive [Formula: see text] method is formulated based on the elements of the adopted spatial discretization (elemental formulation), whereas the so-called adaptive [Formula: see text] method is formulated based on the degrees of freedom of the discretized model (nodal formulation). In this context, each adaptive procedure can be better applied according to the specific features of the focused spatial discretization technique. At the end of the paper, numerical results are presented, illustrating the excellent performance of the discussed adaptive formulations.

A weak form temporal quadrature element formulation for linear structural dynamics

PurposeVarious time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.Design/methodology/approachA variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.FindingsThe weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.Originality/valueThe weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Three novel truly-explicit time-marching procedures considering adaptive dissipation control

In this paper, three novel truly-explicit time-marching techniques are proposed for the solution of hyperbolic models. In these techniques, locally computed time-integration parameters are applied, which are defined taking into account the properties of the spatially/temporally discretised model and the evolution of the computed responses. These adaptive time integrators allow introducing enhanced numerical damping into the analysis, reducing spurious non-physical oscillations that occur due to the excitation of spatially unresolved modes. The proposed adaptive techniques are formulated based on second-, third-, and fourth-order accurate time-marching approaches, providing solution algorithms with different complexities. At the end of the paper, numerical results are presented, illustrating the good performance of the proposed procedures.

A multi-level explicit time-marching procedure for structural dynamics and wave propagation models

In this paper, a truly explicit and truly self-starting time-marching procedure is proposed for structural dynamics and wave propagation analyses. The novel technique can be subdivided into three different levels, providing distinct time-marching formulations that can be applied separately or considering a mixed adaptive approach. In this context, at levels 1 or 2, the new technique provides simpler and more accurate/stable formulations than their equivalent counterparts in standard time-marching techniques. At level 3, the novel approach enables a mixed self-adjustable space/time co-joint solution procedure, providing a highly effective time integration technique for wave propagation analyses. Numerical results are presented along the manuscript, illustrating the excellent performance of the proposed multi-level time-marching technique.

An enhanced explicit time-marching technique for wave propagation analysis considering adaptive time integrators

In this work, a novel family of explicit time marching techniques is proposed for wave propagation analyses. The main features of the new approach are: (i) the family is based only on single-step displacement–velocity relations, being very simple to implement; (ii) as usual in explicit analyses, it requires no system of equations to be dealt with, being very efficient; (iii) it is truly self-starting; (iv) it is second-order accurate; (v) it enables advanced controllable algorithmic dissipation in the higher modes; (vi) it is locally defined and based on adaptive time integrators, allowing enhanced accuracy to be obtained; (vii) it is function of the adopted temporal and spatial discretizations, as well as of the properties of the model and computed responses, establishing a link between these features; (viii) it is entirely automated, requiring no expertise from the user; (ix) it stands as an adaptive non-iterative approach, further improving its efficiency; (x) it is conditionally stable and its stability limit is twice that of the Central Difference Method; (xi) the stability limit of the method remains unaltered, despite the introduction of dissipation. The proposed technique is very versatile and effective, providing the main positive attributes that are requested from an efficacious explicit time marching procedure. Numerical results are presented at the end of the manuscript, illustrating the excellent performance of the novel methodology.

An explicit direct FEM–BEM coupling procedure for nonlinear dynamics

In this work, an explicit direct FEM–BEM coupling procedure is proposed for nonlinear dynamics. In this procedure, each subdomain of the model is analysed separately taking into account proper interface conditions, allowing optimized solver procedures to be applied within each subdomain. In addition, the solution procedure is carried out directly (i.e., without any iterative process), since an explicit time-marching technique is employed within the FEM subdomains. In this context, a locally stabilized central difference approach is utilized, ensuring stability for the FEM analyses, as well as enhanced accuracy. Thus, a very efficient and versatile coupled solution takes place, allowing the direct and independent solution of the subdomains of the model, without regarding critical time-steps limitations within the explicit subdomains. A multi-level time-step algorithm is also considered here, enabling different time-steps to be applied to each subdomain of the model, further improving the flexibility of the coupled analyses. At the end of the paper, numerical applications are presented, illustrating the effectiveness and versatility of the proposed methodology.

The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials

• Developing a combined KDC-GFDM scheme for long-time dynamic simulations. • Proposing a domain-decomposition GFDM for 3D anisotropic composite materials. • Introducing a new distance criterion for the adaptive selection of nodes in the GFDM. • Extremely large time step-sizes can be used in temporal discretization . In this paper we investigate the application of the generalized finite difference method (GFDM) to three-dimensional (3D) transient heat conduction in anisotropic composite (layered) materials. In our computations, the Krylov deferred correction (KDC) method, a pseudo-spectral type time-marching technique, is introduced to perform temporal discretization in time-domain. The KDC method allows discretizing the temporal direction using relatively large time-steps, making the method very promising for dynamic simulations, particularly when high precision is desired. A multi-domain GFDM scheme is also employed where the composite material considered is decomposed into several sub-domains and, in each sub-domain, the solution is approximated by using the GFDM expansion. On the sub-domain interface, compatibility of temperatures and normal heat fluxes is imposed. The method is tested on several benchmark numerical examples and its relative merits and disadvantages are discussed.

Layerwise/FSDT comparative studies on nonlinear transient response of SMA hybrid composite beam

A comprehensive study is provided to compare the results of layer-wise theory (LWT) with one of the renowned equivalent single layer beam theories (ESLs) known as First-Order shear deformation beam theory (FOBT) on the study of transient nonlinear dynamic response of an SMA reinforced composite beam. Considering von Karman strain–displacement formulations, the governing equations of forced vibrations under transient impulsive loads are formulated by LWT and ESL methods. In order to simulate the constitutive behavior of SMA wires, the 3D SMA model developed by Panico and Brinson is adopted and reduced to a 1D model which enables one to reproduce pseudo-elastic (PE), martensite transformation/orientation and in particular ferro-elastic (FE) effects. General transient transverse loading is considered and the governing equations are discretized by combined method of differential–integral quadratures (DQ-IQ). Based on Newmark time marching technique, an incremental solution algorithm is developed for the problem in which nonlinear governing equations are solved by Newton–Raphson method. The results are assessed by comparing with available literature. Moreover, the FOBT and layerwise solutions are compared with each other and the effects of layup geometry and boundary conditions upon the composite beam responds are explored. The influence of volume fraction of SMA-wire reinforcements and the FE/PE behaviors of the wires upon the diverging results of FOBT/LWT analyses are studied in detail.

A nonlinear study on structural damping of SMA hybrid composite beam

A more realistic analysis of forced and transient vibrations for a composite beam with SMA wires is provided by considering the following details: Low and high temperature vibrations are simulated by using the Panico–Brinson model which replicates the pseudo-elastic (PE) and ferro-elastic (FE) hysteresis types of martensite transformations. Besides first-order shear deformation beam theory (FOBT) and large deflection von Karman strain–displacement correlations are used to obtain more accurate estimations of stress field which consequently can result in better estimations of SMA wires hysteresis loops and their decaying characteristics. The governing equations of forced vibration in a beam under transient dynamical loading is developed and discretized by using the multiple method of differential-integral quadrature (DQ-IQ). Incremental time-domain solution of the problem is obtained by Newmark time marching technique. Nonlinear space-domain governing equations are solved by Newton-Raphson method. The results are assessed by comparing with available literature. Considering different types of boundary conditions, the influence of SMA layer position, the hysteresis behavior of pseudo-elastic and ferro-cycles, the pre-straining of SMA wires, the geometrical nonlinearities and the amplitudes of the impulsive loads are studied in detail.

Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems

In this paper we first present stability and error analysis of the fully discrete numerical schemes for general dissipative systems, in which the implicit Runge–Kutta (IRK) method is adopted for time integration. Under suitable conditions on the IRK time stepping method that we refer as the total stability, a priori error estimates can be simultaneously obtained. Then we apply such time-marching techniques and analysis framework to one-dimensional time-dependent nonlocal diffusion problems, together with the discontinuous Galerkin method being used for spatial discretization. Unconditional stability of approximations of both primal and auxiliary variables and the priori error estimates for the corresponding fully discrete systems are proved, and the results indicate the schemes are asymptotically compatible. In addition, long time asymptotic behavior of the approximate solutions is also investigated. Various numerical experiments are finally performed to verify the theoretical results.

A locally defined time-marching technique for structural dynamics

In this work, a new time marching procedure is proposed for structural dynamics analyses. In this novel technique, time integration parameters are locally defined and different values may be attributed to each structural element of the model. In addition, the time integrators are evaluated according to the properties of the elements, and the user may select in which structural elements numerical dissipation will be introduced. Since the integration parameters are locally defined as function of the structural element itself, the time marching technique adapts according to the model, providing enhanced accuracy. The method is very simple to implement and it stands as an efficient, direct, single-step procedure. It is second order accurate, unconditionally stable, truly self-starting and it allows highly controllable algorithm dissipation in the higher modes. Numerical results are presented along the paper, illustrating the good performance of the new technique.

Fully reliable error control for first-order evolutionary problems

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of solution approximation for evolutionary models, i.e., a time-marching technique and a space–time approach. The first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space–time approach. It is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method.The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.

An effective adaptive time domain formulation to analyse acoustic–elastodynamic coupled models

This work is concerned with the development of an effective technique to model the propagation of interacting acoustic and elastic waves. Here, spatial discretization by finite elements is adopted, and uncoupled analyses of acoustic and elastodynamic subdomains are considered, with the interaction between the different subdomains of the model being accomplished by interface forces. Adaptive explicit and implicit time-marching techniques are employed, in which the time integrators are locally computed, assuming different values along the spatial and temporal discretizations, as the solution evolves. The proposed solution algorithm is very efficient, accurate, flexible and easy to implement, standing as a very attractive approach. At the end of the paper, numerical results are presented, illustrating the good performance and potentialities of the new methodology.

A new formulation and analysis of a collapsing bubble with different content in a liquid induced during acoustic cavitation

Purpose – The purpose of this paper is to present numerical investigation of the gas/vapor bubble dynamics under the influence of an ultrasonic field to give a more comprehensive understanding of the phenomenon and present new results Design/methodology/approach – In order to formulate the mathematical model, a set of governing equations for the gas inside the bubble and the liquid surrounding it are used. All hydrodynamics forces acting on the bubble are considered in the typical solution. The systems of equations required to be solved consist of ordinary and partial differential equations, which are both nonlinear and time dependent equations. A fourth order Runge-Kutta method is applied to solve the ordinary differential equations. On the other hand, the finite difference method is employed to solve the partial differential equations and a time-marching technique is applied. Findings – The numerical model which is developed in the current study permits a correct prediction of the bubble behavior and its characteristics in an acoustic field generated at this occasion. Originality/value – Previous studies considering numerical simulations of an acoustic bubble were performed based on the polytropic approximation or pressure uniformity models of the contents inside the bubble. In this study, an enhanced numerical model is developed to study the acoustic cavitation phenomenon and the enhancement concerns taking into account both the pressure and temperature gradients inside the bubble as well as heat transfer through the bubble surface into account which is very important to obtain the temperature of the liquid surrounding the bubble surface.

Stabilized Times Schemes for High Accurate Finite Differences Solutions of Nonlinear Parabolic Equations

The Residual Smoothing Scheme (RSS) have been introduced in Averbuch et al. (A fast and accurate multiscale scheme for parabolic equations, unpublished) as a backward Euler's method with a simplified implicit part for the solution of parabolic problems. RSS have stability properties comparable to those of semi-implicit schemes while giving possibilities for reducing the computational cost. A similar approach was introduced independently in Costa (Time marching techniques for the nonlinear Galerkin method, 1998), Costa et al. (SIAM J Sci Comput 23(1):46---65, 2001) but from the Fourier point of view. We present here a unified framework for these schemes and propose practical implementations and extensions of the RSS schemes for the long time simulation of nonlinear parabolic problems when discretized by using high order finite differences compact schemes. Stability results are presented in the linear and the nonlinear case. Numerical simulations of 2D incompressible Navier---Stokes equations are given for illustrating the robustness of the method.

An efficient staggered time-marching procedure for porodynamics

The present work proposes a novel time-marching procedure, which is designed for the dynamic simulation of the coupled two phase Biot equations, considering the displacement–pressure (u−p) formulation. The new approach provides unconditional stability and excellent efficiency. Due to the staggered character of the temporal discretization, where equilibrium is established at time t for the u group of governing equations and at time t+Δt for the p group of governing equations, the resulting systems of equations have an improved format and can be factorized/solved at significantly less computational effort when compared to standard approaches, such as the generalized Newmark scheme. In addition, the resulting new systems of equations are characterized by considerably lower condition numbers, reducing correlated numerical issues. Unconditional stability is achieved by a consistent manipulation of the mass and compressibility matrix of the model, which causes virtually no computational overhead. As a byproduct of the altered compressibility matrix, the method is capable of simulating incompressible and impermeable phenomena without the need for any further feats. The newly developed time-marching technique is implemented and tested considering different benchmark problems. For comparison, the commonly used Newmark scheme is also applied and both direct (LU decomposition) and iterative (GMRES) linear solvers are employed, in order to assess the efficiency of the new technique.

Parallel implicit anisotropic block-based adaptive mesh refinement finite-volume scheme for the study of fully resolved oblique shock wave reflections

A new parallel, fully implicit, anisotropic block-based adaptive mesh refinement (AMR) finite-volume scheme is proposed, described and demonstrated for the prediction of laminar, compressible, viscous flows associated with unsteady oblique shock reflection processes. The proposed finite-volume method provides numerical solutions to the Navier–Stokes equations governing the flow of polytropic gases in an accurate and efficient manner on two-dimensional, body-fitted, multi-block meshes consisting of quadrilateral computational cells. The combination of the anisotropic AMR and parallel implicit time-marching techniques adopted is shown to readily facilitate the simulation of challenging and complex shock interaction problems, as represented by the time-accurate predictions of unsteady oblique shock reflection configurations with fully resolved internal shock structures.