Time Integration Algorithm Research Articles

An integrated enabling technology interfacing multiple space/time methods/algorithms/domains with model reduction for first-order systems

PurposeThe purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented.Design/methodology/approachThe GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept.FindingsThe present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency.Originality/valueThe novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations.

An inner-outer subcycling algorithm for parallel cardiac electrophysiology simulations.

This paper explores cardiac electrophysiological simulations of the monodomain equations and introduces a novel subcycling time integration algorithm to exploit the structure of the ionic model. The aim of this work is to improve upon the efficiency of parallel cardiac monodomain simulations by using our subcycling algorithm in the computation of the ionic model to handle the local sharp changes of the solution. This will reduce the turnaround time for the simulation of basic cardiac electrical function on both idealized and patient-specific geometry. Numerical experiments show that the proposed approach is accurate and also has close to linear parallel scalability on a computer with more than 1000 processor cores. Ultimately, the reduction in simulation time can be beneficial in clinical applications, where multiple simulations are often required to tune a model to match clinical measurements.

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

Time integration algorithm of velocity frequency-weighted root mean square for ship comfort index

Time integration algorithm of velocity frequency-weighted root mean square for ship comfort index

Time integration algorithm of velocity frequency-weighted root mean square for ship comfort index

Time integration algorithm of velocity frequency-weighted root mean square for ship comfort index

Dynamic deflection behaviour of plant fibre reinforced laminated composite plate structure using simulation model

In this work, a finite element model has been developed for the transient deflection analysis of plant fibre reinforced laminated composite flat panel using commercial finite element software platform. The plate model has been discretised using eight noded SHELL 281 serendipity element in ANSYS parametric design language (APDL) platform and the responses in time domain are obtained using Newmark time integration algorithm. The accuracy and performance of the present simulation model have been checked via reference literature. Further, the model is extended to analyse the influence of geometrical parameters and support condition on time dependent responses of plant fibre reinforced composite plate structure.

Numerical analysis of time-dependent negative skin friction on pile in soft soils

The generation of negative skin friction (NSF) induces the additional axial force on the pile, which exerts a detrimental load rather than a beneficial one. However, the effect of soil creep on the long-term development of NSF has remained poorly understood. The study uses numerical analysis to investigate this effect. First, an elasto-viscoplastic model with an enhanced time integration algorithm is successfully implemented into a finite element code. Next, the two-dimensional axisymmetric pile-soil interaction model is established and properly calibrated with a known field case. Finally, parametric studies are conducted to examine different degrees of creep effect on the variation in NSF and neutral plane (NP) within the primary and secondary consolidation periods. According to the findings, a high creep coefficient of the soil results in an increase in NSF and descending trend of the NP. The creep induced delay of NSF is observed attributed to the increase in excess pore pressure during the early stage of consolidation. The NP position varies drastically at the commencement of consolidation when taking creep into consideration. Lastly, the study proposes an exponential prediction model to reflect the time dependence of the location of the NP.

A study of impact induced stress due low velocity impact on laminated composite skewed hypar shell roof

The doubly ruled hypar shell, being one of the most feasible shell roof configurations, enjoys industrial preference for covering large column free areas. This class of shells is unique as the only curvature here is the cross curvature and these do not admit easy closed form solution particularly when the boundary conditions are complicated. Laminated composite, an innate choice to different industrial sectors for its huge specific strength and stiffness, good weathering resistance, is now being extensively employed in civil engineering. However the low transverse shear strength of the composite shell impelled the researchers to study the response of the same under the action of impact loads. In the present study, a finite element code is functioned to investigate the impact induced stress history generated in simply supported laminated composite hypar shell for different impact velocities of a spherical solid striker. Contact behavior is described by modified Hertzian contact law where as time dependent equations are solved using Newmark’s time integration algorithm in present analysis. A practicing engineer often has to select a particular shell option consisting stacking sequence among a number of possibilities. The present paper discusses the behaviour of composite hypars under low velocity impact and different stresses induced due to the same from engineering standpoint to propose a selection guide line among the available options and to proposes values of practical parameters like the equivalent static loads and dynamic magnification factors for designing such shells through static simplification of the problems.

Relative stability analysis for robustness design of real-time hybrid simulation

Stability is a prerequisite of a successful real-time hybrid simulation (RTHS), which depends on the time-integration algorithm, the delay-compensation method, the loading system dynamics (mostly characterized as a time delay), and the nonlinearity of the structures being tested. The existing stability analysis methods generally provide a qualitative judgement based on one or some of the factors above, without considering all these factors to estimate quantitative stability margin. However, it is critical to design a robust RTHS system with enough stability margin (i.e., robust RTHS system) to ensure the success of the actual test with inescapable uncertainties. To this end, a relative stability analysis method for robustness design of RTHS systems is proposed. In this method, the critical gain of the stiffness of an experimental substructure is used to quantify the stability limit, and the critical delay is used to quantitatively represent the sensitivity of an RTHS system to the phase discrepancy at the interface. The RTHS system with a nonlinear experimental substructure is modeled by a discrete transfer function in an incremental form, wherein the time-integration algorithm, the delay-compensation method, the loading system dynamics, and the nonlinearities of the experimental substructure are considered. The critical delay and gain are obtained from the discrete transfer function. Two relative stability criteria are proposed based on the critical delay and the critical gain of an RTHS system to facilitate the evaluation of the system's stability performance and the design of a robust RTHS system. To verify the critical delay and gain estimated using the proposed method, numerical simulation of RTHS systems was carried out first. Then RTHS experiments of single-degree-of-freedom models with experimental substructures consisting of a linear spring and a nonlinear stiffness hardening specimen were conducted. The consistent results of the numerical simulation and the physical RTHS tests with those derived from the proposed relative stability analysis method demonstrate its effectiveness and accuracy in determining the stability properties of RTHS systems, which can be further utilized in designing more robust RTHS testing systems.

A Study on the Vortex Induced Vibration of a Cylindrical Structure with Surface Bulges

Inspired by the cactus in nature, a cactus-like cross-sectional structure was proposed to achieve the VIV suppression. The VIV of the elastically mounted cylinder was realized based on the ANSYS Fluent and User Defined Function (UDF). The dynamic motion of the cylinder was solved by the single-step time integration algorithms Newmark-β method. The in-house code was first validated by studying the 2DOF VIV of a circular cylinder with small mass ratio over the range U*=2~13, and the results agree well with the published literature. Then, the performance of surface bulge on VIV suppression was studied and four different coverage ratios (CR) were considered, i.e., 0%, 20%, 33%, and 40%. The VIV of a bulged cylinder can be effectively suppressed. CR20 performs the best in VIV suppression and the suppression efficiency in streamwise and transverse direction are 44.6% and 63.1%, respectively. The mechanism of surface bulge on the VIV suppression is the shift of separation point of the shear layer and vortices form between the surface bulges.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Scaled Boundary Perfectly Matched Layer (SBPML): A novel 3D time-domain artificial boundary method for wave problem in general-shaped and heterogeneous infinite domain

Artificial boundary method is widely used in the numerical modeling of unbounded wave problem. However, the accurate modeling of truncated infinite domain with general geometry and heterogeneous materials is still a challenging task, especially for direct time-domain analysis in three dimensions (3D). In this paper, a novel 3D time-domain artificial boundary method, called Scaled Boundary Perfectly Matched Layer (SBPML), is proposed. This method is a generalization of the Perfectly Matched Layer (PML) based on a scaled boundary coordinates transformation inspired by the Scaled Boundary Finite Element Method (SBFEM), which is capable of using artificial boundary of general geometry (not necessarily convex) and considering plane physical surfaces and interfaces extending to infinity in the truncated infinite domain. Local scaled boundary coordinates are firstly introduced on the element-level into the truncated infinite domain to describe general geometry properties of the infinite domain. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. The spatial discretization of the SBPML domain produces semi-discrete mixed displacement–stress​ unsplit-field formulations of third orders in time. The order of the obtained formulation can be reduced by one, enabling a seamless coupling with the standard displacement-based finite element formulation of the interior domain. The coupled system can be solved by an explicit time integration algorithm efficiently. The validation of the SBPML is demonstrated through several benchmark tests, including wave problems in unbounded domains with general geometries and heterogeneous material properties. Furthermore, the application of the SBPML in dynamic soil–structure interaction (SSI) is demonstrated using two engineering problems, including an impact analysis of soft rock-nuclear island system and a vibration analysis of soil-lined tunnel system.

Computational Differential Algebraic Equation Framework and Multi Spatial and Time Discretizations Preserving Consistent Second-Order Accuracy: Nonlinear Dynamics

Abstract We propose the novel design, development, and implementation of the well-known generalized single step single solve (GS4) family of algorithms into the differential algebraic equation framework which not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains but also additionally allows for the selection of different space discretized methods such as the finite element method and particle methods, and other spatial methods as well in a single analysis unlike existing state-of-the-art. For the first time, the user has the flexibility and robustness to embed different algorithms for time integration and different spatial methods for space discretization in a single analysis. In addition, the GS4 family enables a wide variety of choices of time integration methods in a single analysis and also ensures the second-order accuracy in time of all primary variables and Lagrange multipliers. This is not possible to date. However, the present framework provides the fusion of a wide variety of choices of time discretized methods and spatial methods and has the bandwidth and depth to engage in various types of research investigations as well and features for fine tuning of numerical simulations. It provides generality/versatility of the computational framework incorporating subdomains with different spatial and time integration algorithms with improved accuracy. The robustness and accuracy of the present work is not feasible in the current state of technology. Various numerical examples illustrate the significant capabilities and generality and effectiveness for general nonlinear dynamics.

A nonhydrostatic atmospheric dynamical core on cubed sphere using multi-moment finite-volume method

A nonhydrostatic dynamical core has been developed by using the multi-moment finite volume method that ensures the rigorous numerical conservation of mass in this study. To represent the spherical geometry free of the polar problems, the cubed-sphere grid is adopted. A fourth-order multi-moment discretization formulation is applied to solve the governing equations cast in the local curvilinear coordinates on each patch of the cubed sphere through a gnomonic projection. In the vertical direction, the height-based terrain-following coordinate is used to deal with the topography and a finite difference scheme, also assuring the conservation of mass, is adopted for the spatial discretization. The proposed dynamical core adopts the nonhydrostatic governing equations. To get around the CFL stability restriction imposed by the sound wave and the relatively small grid spacing in the vertical direction, the dimensional splitting time integration algorithm using the HEVI (horizontally-explicit and vertically-implicit) strategy is implemented by applying the IMEX (implicit-explicit) Runge-Kutta method. The proposed model was checked by the widely-used benchmark tests in this study. The numerical results show that the multi-moment model has the comparable solution quality in comparison with the existing advanced ones and the great practical potential as a numerical platform for development of the atmospheric general circulation models.

Nonlinear dynamic analysis of galloping of a single iced conductor with four degrees of freedom based on curved beam theory

Based on the nonlinear strain-displacement relationship of spatial curved beam theory, a finite element model for galloping analysis of iced transmission lines is established, which involves three translational degrees of freedom (DOF) and one rotational degree of freedom for each node. Considering the aerodynamic nonlinearity and the geometric nonlinearity of large amplitude motions of the iced transmission line, the nonlinear dynamic finite element equation is presented by using the virtual work principle. The equation is transformed into the sub-space according to the mode superposition method and a time integration algorithm is also performed in the sub-space. Then, the element independence and modal convergence are researched. Finally, the influence of aerodynamic forces on structural frequency is analyzed. The numerical results show that aerodynamic forces will greatly affect the torsional frequency of the transmission line, which accurately reflects the dynamic characteristics of transmission lines. Furthermore, the galloping calculation of the transmission line has reliable accuracy by using 4-DOF (per node) elements for the iced conductor. On the other hand, the model can predict the actual galloping response of the transmission line, which is convenient for the implementation of subsequent control design.

A truly self-starting composite isochronous integration analysis framework for first/second-order transient systems

Due to the lack of and limited work done for first/second-order systems useful for multidisciplinary problems, this paper proposes a new and novel composite isochronous integration ([i-Integration]) analysis framework to solve first/second-order transient systems via a single computational framework. The main contributions are summarized as follows: (1) The ρ∞-Bathe method is newly reformed to an identical truly self-starting representation with the absence of acceleration and is more efficient for practical applications; (2) The novel [i-Integration] process is applied to automatically generate a family of time integration algorithms for first/second-order transient systems, whilst preserving features of second-order time accuracy, unconditional stability, controllable numerical dissipation in high-frequency, and truly self-starting; (3) The newly proposed algorithms design framework provides a unified toolkit to solve first/second-order transient systems in a single analysis and hence is more effective for coupled problems and numerical analysis of fluid/heat transfer/structure-type transient simulations. Numerical examples encompassing a manufactured example, the C-F heat conduction model, and the thermal stress wave problem are demonstrated to validate the proposed algorithms.

Thermal Conductivity of Nanostructure in Microelectronic Equipment and the Enhancement of Its Thermo-Physical Properties

To optimize the heat dissipation performance of microelectronic equipment, optical instrument, and optoelectronic devices, it’s necessary to explore the heat transfer mechanism of the nanostructure in them, however, due to the limitation of the research objects, there isn’t a uniform simulation method for the thermo-physical properties of such nanostructure. Therefore, this paper aims to study the thermo-physical properties of nanostructure in microelectronic equipment. At first, by structuring Si super lattice nanowires in a specific direction and then using the Ge atoms to replace the Si atoms in the nanowires, this paper built a model for the periodic Si/Ge super lattice nanowires. Then, this paper performed integral operation on the dynamic equations of the two types of atoms using the time integration algorithm to attain the trajectories of Si and Ge atoms in the nanowire structure of the microelectronic equipment, and discussed the influence of the size and geometrical shape of the cross section of nanowires on thermal conductivity. After that, this paper simulated the structure of the nano-scale SiC material, selected appropriate cut-off radius, and optimized the structure of the nanostructure model based on the Discover Minimization module. At last, the thermal conductivity of the nanostructure was calculated and its thermo-physical properties were analyzed.

Temporal stability of collocation, Petrov–Galerkin, and other non-symmetric methods in elastodynamics and an energy conserving time integration

Non-symmetric matrices may arise in the discretization of self-adjoint problems when a Petrov–Galerkin, collocation, or finite-volume method is employed. While these methods have been widely applied, in this paper it is shown that the use of these non-symmetric matrices is incompatable with the conservation of energy in elastodynamics. First, the consistency between the continuous forms of the momentum equation and the energy equation is examined. It is shown that the conservation of linear momentum is equivalent to conservation of energy provided the solution is sufficiently smooth. The semi-discrete counterparts are then analyzed, where it is demonstrated that they are also equivalent, but only conditionally: the mass and stiffness matrices must be symmetric. As a result, employing a non-symmetric method in elastodynamics may artificially generate or dissipate energy. The fully discrete forms with Newmark time integration are then examined where it is shown that unconditionally unstable algorithms may arise. An energy-conserving time integration algorithm is then proposed which provides stability in the solutions of non-symmetric systems. The collocation and finite-volume methods are employed in numerical examples to demonstrate stability issues and the effectiveness of the proposed time integration methodology.

A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

AbstractAn important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high‐frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first‐order overshooting components enter into several well‐known time integration algorithms which were previously thought to exhibit zero‐order overshooting in displacement. The Newmark method, Wilson‐ method, Bazzi‐ method, HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method are analyzed, as well as the generalized single‐step single‐solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.

Effective time-dependent behavior of three-phase polymer matrix smart composites

This study presents a mathematical framework to predict the effective time-dependent behavior of three-phase smart composites with typical 0-3, 1-3, and 2-2 connectivities. A composite is composed of magnetostrictive and piezoelectric reinforcements that both show nonlinear multiphysics coupling and a polymer matrix that exhibits viscoelastic behavior. For dealing with nonlinear and time-dependent problems, a tangent linearization is employed to linearize the nonlinear constituents and a time-integration algorithm is applied to numerically determine the viscoelastic response of the matrix. A unified constitutive equation can be subsequently formulated to cover various phase constitutive laws following by a simplified unit-cell micromechanics model to set up a composite constitutive relation. The presented formulation is validated by limited experimental data available in literatures. Numerical results show that inclusion of a polymer matrix in a smart composite causes magnetoelectric coupling to be creep due to strain-mediated coupling among these three phases. The established unit-cell micromechanics model can be further integrated into a finite element framework to analyze composite structures, which is a great merit in a variety of practical applications. It will do so by implementing the presented micromechanics method at every integration point.