Time Integration Algorithm Research Articles

New Explicit Integration Algorithms with Controllable Numerical Dissipation for Structural Dynamics

This paper presents a new family of explicit time integration algorithms with controllable numerical dissipation for structural dynamic problems by utilizing the discrete control theory. Firstly, the equilibrium equation of the implicit Yu-[Formula: see text] algorithm is adopted, and the recursive formulas of velocity and displacement for the explicit CR algorithm are used in the algorithms. Then, the transfer function and characteristic equation of the algorithms with integration coefficients are obtained by the [Formula: see text] transformation. Furthermore, their integration coefficients are derived according to the poles condition. It was indicated that the proposed algorithms possess the advantages of second-order accuracy, self-starting, and unconditional stability for linear systems and nonlinear systems with softening stiffness. The numerical dissipation of the algorithms is controlled by the spectral radius at infinity [Formula: see text]. It was also shown that the proposed algorithms have the same poles as the Yu-[Formula: see text] algorithm, and thus the same numerical properties. Compared with the implicit Yu-[Formula: see text] algorithm, the proposed algorithms are explicit in terms of both the displacement and velocity formulas. Finally, the effectiveness of the proposed algorithms in reducing the undesired participation of higher modes for solving the dynamic responses of linear and nonlinear systems has been demonstrated.

A novel and simple a posteriori error estimator for LMS methods under the umbrella of GSSSS framework: Adaptive time stepping in second-order dynamical systems

A novel general purpose a posteriori error estimator that is agnostic to selection of time integration schemes arising under the umbrella of ”Generalized Single Step Single Solve” (GSSSS) framework and family of algorithms is proposed to foster adaptive time stepping; it encompasses the entire class of LMS methods for second order dynamical systems. Unlike several error estimators that have been applied to a limited selection of known time integration methods found in the literature, the proposed estimator offers a significant deviation and difference — that is, it is totally unconstrained from dependence upon algorithmic parameters and is general purpose. Consequently, it can be used for a wide class of numerically dissipative and non-dissipative algorithms for the general class of LMS methods without any modifications; the GSSSS family of algorithms encompasses most existing algorithms developed over the past 50 years or so as subsets, in addition to several new design developments and optimal algorithms arising from this framework. In addition, the proposed estimator is shown to possess the same order of convergence and error constant as the exact local error, which demonstrates its accuracy. The applicability of the proposed estimator to several but selected existing time integration algorithms including the well known schemes like the Newmark method, HHT- α, Classical midpoint rule and in addition, new algorithms and designs as well is demonstrated with single and multi-degree of freedom, linear and nonlinear dynamical problems. In addition, an adaptive time stepping procedure is employed to further demonstrate efficient implementation for multi-degree of freedom, linear and nonlinear dynamical systems.

Dynamic force reconstruction techniques from incomplete measurements

We present two time-domain algorithms derived from the popular Newmark’s time integration algorithm, to reconstruct the input force excitation from the vibration measurements. While the first method uses a direct formulation of the time integration algorithm, the second one uses a regularization tool to solve the associated inverse problem. A modified regularization technique is employed to improve the accuracy of the identification problem. Numerical studies with limited measurements, with measurement noise, multiple load identification etc., are presented to investigate the practical applicability of the two methods. Studies have clearly favored the inverse algorithm with modified regularization technique. Finally modeling errors associated with force reconstruction are investigated and a technique is proposed to identify the system parameters simultaneously while reconstructing the input dynamic force, using a recently developed quantum particle swarm optimization algorithm developed by the authors. The numerical simulation studies presented clearly indicate that the force, as well as the system parameters, can be identified accurately using the proposed algorithm.

A novel model order reduction framework via staggered reduced basis space-time finite elements in linear first order transient systems

A novel model order reduction framework for space and time domain discretizations is proposed. Iterative convergence of a Galerkin approximation in space and a Least Squares Petrov Galerkin approximation in time is obtained through a staggered reduced basis method in space-time. In every iteration, one of the two domains (space or time) is refined; and the other is reduced and a posteriori error indicators in space and time are used to drive the convergence iterations. Numerical results for 2D heat transfer and convection-diffusion problems demonstrate the significant computational efficiency of the proposed methodology. Comparisons of wall-clock times and solution accuracy with traditional time integration algorithms has been presented to validate the efficacy of the proposed framework and demonstrate computational savings of an order of magnitude.

An improved implicit time integration algorithm: The generalized composite time integration algorithm

The weighted residual method is employed to develop one- and two-step time integration schemes. Newly developed time integration schemes are combined to obtain a new second-order accurate implicit time integration algorithm whose computational structure is similar to the Bathe method (Bathe and Noh, 2012). The newly developed algorithm can control algorithmic dissipation in the high frequency limit through the optimized weighting parameters. It contains only one free parameter, and always provides an identical effective stiffness matrix to the first and second sub-steps in linear analyses, which is not provided in the algorithm proposed by Kim and Reddy (2016). Various nonlinear test problems are used to investigate performance of the new algorithm in nonlinear analyses.

Portable implementation model for CFD simulations. Application to hybrid CPU/GPU supercomputers

ABSTRACTNowadays, high performance computing (HPC) systems experience a disruptive moment with a variety of novel architectures and frameworks, without any clarity of which one is going to prevail. In this context, the portability of codes across different architectures is of major importance. This paper presents a portable implementation model based on an algebraic operational approach for direct numerical simulation (DNS) and large eddy simulation (LES) of incompressible turbulent flows using unstructured hybrid meshes. The strategy proposed consists in representing the whole time-integration algorithm using only three basic algebraic operations: sparse matrix–vector product, a linear combination of vectors and dot product. The main idea is based on decomposing the nonlinear operators into a concatenation of two SpMV operations. This provides high modularity and portability. An exhaustive analysis of the proposed implementation for hybrid CPU/GPU supercomputers has been conducted with tests using up to 128 GPUs. The main objective consists in understanding the challenges of implementing CFD codes on new architectures.

A composite collocation method with low-period elongation for structural dynamics problems

This paper presents a novel time integration algorithm for solving linear structural dynamic problems in the framework of the high-order collocation method. When two Gauss points in the integration interval are selected as collocation points, both an A-stable algorithm with third order accuracy and a non-dissipative algorithm with fourth order accuracy can be derived from a second order collocation polynomial. The only difference is that the former obtains a numerical solution at the middle point of the time interval, while the latter has a solution at the end of the interval. A new composite method is established through applying these two algorithms alternately, which combines the advantages of the numerical dissipation property of the third order algorithm and the high-order accuracy of the fourth order algorithm. The usage frequency of the two algorithms during the whole step-by-step integration procedure is an important parameter affecting the numerical dissipation, which is investigated in this study. As the algebraic equations systems solved by the two algorithms are exactly same, no extra computation effort is introduced.

Performance analysis of coupled quasi-steady state air flow calculation and dynamic simulation of hygrothermal transport inside porous materials

The hygrothermal simulation model DELPHIN performs a transient simulation of energy and moisture transport in porous building materials and air gaps. The governing partial differential equations are solved with an advanced error control time integration algorithm. In comparison to the time scales of heat and moisture transport, air flow can be treated sufficiently well as steady-state process. The pressure and flow field needs to be updated whenever ambient air pressures change significantly or free convection is induced due to temperature or humidity variations. Within DELPHIN the air flow pattern is computed with a Darcy-type steady-state flow calculation.The calculation of the flow field is a time-consuming task and should only be done when necessary. A central question that impacts both physical results and performance is the update criteria for the quasi-steady flow field. Indeed, if the flow field is updated in with every change of temperature or moisture, we observe in some cases significant performance losses. For all simulation cases we use an error controlled time integration method.In the article we identify free convection inside an air gap as a performance critical simulation scenario. Dominating convective transport is responsible for divergence of the fully coupled simulation. Reduction of update frequency stabilizes simulation run without loss of calculation efficiency. These results may be generalized for less critical simulation cases.

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

SummaryIn this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.

A multi-order method for predicting stability of a multi-delay milling system considering helix angle and run-out effects

In this paper, a multi-delay milling system considering helix angle and run-out effects is firstly established. An exponential cutting force model is used to model the interaction between a work-piece and a cutting tool, and a new approach is presented for accurately calibrating exponential cutting force coefficients and cutter run-out parameters. Furthermore, based on an implicit multi-step Adams formula and an improved precise time-integration algorithm, a novel stability prediction method is proposed to predict the stability of the system. The involved time delay term and periodic coefficient term are integrated asa comprehensive state term in the integral response which is approximated by the Adams formula. Then, a Floquet transition matrix with an arbitrary-order form is constructed by using a series of matrix multiplication, and the stability of the system is determined by the Floquet theory. Compared to classical semi-discretization methods and full-discretization methods, the developed method shows a good performance in convergence, efficiency, accuracy, and multi-order complexity. A series of cutting tests is further carried out to validate the practicability and effectiveness of the proposed method. The results show that the calibration process needs a time of less than 5min, and the stability prediction method is effective.

A novel time integration formulation for nonlinear dynamic analysis

To solve the linear and nonlinear dynamic equation of motion, a family of explicit higher-order time integration algorithms is presented. The differential transformation method in conjunction with the state–space concept is used. In this study, four members of this family are considered thoroughly. To validate the performance of the new algorithms, comparison studies are accomplished with the well-known time integration schemes. By solving some linear and nonlinear dynamic problems, the merits of the authors' formulations will be clearly demonstrated.

Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method

SummaryNumerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids. To investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method, the implicit time integration schemes, ie, the Newmark, the HHT‐α, and the WBZ‐α methods, and the explicit time integration algorithms, ie, the central difference, the Zhai's, and Chung‐Lee methods, are implemented. Their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading, impact analysis of an elastic rod with an initial velocity, and excavation analysis of jointed rock masses, respectively. Parametric studies using different time steps are conducted for different time integration algorithms, and the convergence efficiency of the open‐close iterations for the contact problems is also investigated. It is proved that the Hilber‐Hughes‐Taylor‐α (HHT‐α), Wood‐Bossak‐Zienkiewicz‐α (WBZ‐α), Zhai's, and Chung‐Lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts. It is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy.

Modeling and assessment of partially debonded piezoelectric sensor in smart composite laminates

As substantive components of intelligent structures, piezoelectric sensors need to perform their intended function without any spuriousness. In this work, a mathematical model has been developed to characterize the performance degradation of a piezoelectric sensor due to its debonding from a smart composite laminate. The electromechanically coupled governing equation of motion is developed by incorporating improved layerwise theory and higher electric potential field along with the finite element method and Extended Hamilton's principle. The problem governing equation is solved by Newmark time-integration algorithm to assess the performance of a piezoelectric sensor in the presence of partial debonding at the edge and inner side of the sensor. The partially debonded piezoelectric sensor is investigated in frequency domain via power spectral density of the sensor output. The degradative performance of the partially debonded sensor is generalized for random loadings via basic signal statistics. The numerical results show that the developed model recover the presence and extent of partial debonding between the piezoelectric sensor and the host laminate.

A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics

For the development of a new family of implicit higher-order time integration algorithms, mixed formulations that include three time-dependent variables (i.e., the displacement, velocity, and acceleration vectors) are developed. Equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables in the mixed formulations, and the time finite element method and the modified weighted-residual method are applied to the velocity–displacement and velocity–acceleration relations of the mixed formulations. Weight parameters are introduced and optimized to achieve preferable attributes of the time integration algorithms. Specific problems of structural dynamics are used in the numerical examples to discuss some fundamental limitations of the well-known second-order accurate algorithms as well as to demonstrate advantages of using the developed higher-order algorithms.

Effective Higher-Order Time Integration Algorithms for the Analysis of Linear Structural Dynamics

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

An efficient finite differences method for the computation of compressible, subsonic, unsteady flows past airfoils and panels

A finite differences scheme is proposed in this work to compute in the time domain the compressible, subsonic, unsteady flow past an aerodynamic airfoil using the linearized potential theory. It improves and extends the original method proposed in this journal by Hariharan, Ping and Scott [1] by considering: (i) a non-uniform mesh, (ii) an implicit time integration algorithm, (iii) a vectorized implementation and (iv) the coupled airfoil dynamics and fluid dynamic loads. First, we have formulated the method for cases in which the airfoil motion is given. The scheme has been tested on well known problems in unsteady aerodynamics —such as the response to a sudden change of the angle of attack and to a harmonic motion of the airfoil— and has been proved to be more accurate and efficient than other finite differences and vortex-lattice methods found in the literature. Secondly, we have coupled our method to the equations governing the airfoil dynamics in order to numerically solve problems where the airfoil motion is unknown a priori as happens, for example, in the cases of the flutter and the divergence of a typical section of a wing or of a flexible panel. Apparently, this is the first self-consistent and easy-to-implement numerical analysis in the time domain of the compressible, linearized coupled dynamics of the (generally flexible) airfoil–fluid system carried out in the literature. The results for the particular case of a rigid airfoil show excellent agreement with those reported by other authors, whereas those obtained for the case of a cantilevered flexible airfoil in compressible flow seem to be original or, at least, not well-known.

An improved time integration scheme based on uniform cubic B-splines and its application in structural dynamics

A time integration algorithm for structural dynamic analysis is proposed by uniform cubic B-spline functions. The proposed algorithm is successfully used to solve the dynamic response of a single degree of freedom (SDOF) system, and then is generalized for a multiple-degree of freedom (MDOF) system. Stability analysis shows that, with an adjustable algorithmic parameter, the proposed method can achieve both conditional and unconditional stabilities. Validity of the method is shown with four numerical simulations. Comparison between the proposed method and other methods shows that the proposed method possesses high computation accuracy and desirable computation efficiency.

Comparison and assessment of time integration algorithms for nonlinear vibration systems

A corrected explicit method of double time steps (CEMDTS) was introduced to accurately simulate nonlinear vibration problems in engineering. The CEMDTS, the leapfrog central difference method, the Newmark method, the generalized-α method and the precise integration method were implemented in typical nonlinear examples for the purpose of comparison. Both conservative and non-conservative systems were examined. The results show that it isn’t unconditionally stable for the precise integration method, the Newmark method and the generalized-α method in nonlinear systems. The stable interval of the three methods is less than that of the CEMDTS. When a small time step (Δt≤T min/20) is employed, the precise integration method is endowed with the highest accuracy while the CEMDTS possesses the smallest computation effort. However, the CEMDTS becomes the most accurate one when the time step is large (Δt≥0.3T min) or the system is strongly nonlinear. Therefore, the CEMDTS is more applicable to the nonlinear vibration systems.

Continuous chip formation in metal cutting processes using the Particle Finite Element Method (PFEM)

This paper presents a study on the metal cutting simulation with a particular numerical technique, the Particle Finite Element Method (PFEM) with a new modified time integration algorithm and incorporating a contact algorithm capability . The goal is to reproduce the formation of continuous chip in orthogonal machining. The paper tells how metal cutting processes can be modelled with the PFEM and which new tools have been developed to provide the proper capabilities for a successful modelling. The developed method allows for the treatment of large deformations and heat conduction, workpiece-tool contact including friction effects as well as the full thermo-mechanical coupling for contact. The difficulties associated with the distortion of the mesh in areas with high deformation are solved introducing new improvements in the continuous Delaunay triangulation of the particles. The employment of adaptative insertion and removal of particles at every new updated configuration improves the mesh quality allowing for resolution of finer-scale features of the solution. The performance of the method is studied with a set of different two-dimensional tests of orthogonal machining. The examples consider, from the most simple case to the most complex case, different assumptions for the cutting conditions and different material properties. The results have been compared with experimental tests showing a good competitiveness of the PFEM in comparison with other available simulation tools.

On generation of lumped mass matrices in partition of unity based methods

SummaryThe partition of unity based methods, such as the extended finite element method and the numerical manifold method, are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials. In the dynamic analysis of cracked bodies using an explicit time integration algorithm, as a result, huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions. Observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force, we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses, where we inversely utilize the ‘from local to global’ skill. The proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix, but has more excellent dynamic property than the latter. Meanwhile, the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes — a very excellent dynamic property. Copyright © 2017 John Wiley & Sons, Ltd.