Direct Time Integration Research Articles

The Numerical Identification of Basins of Attraction for the Vibration Response of the Rigid Rotor with Squeeze Film Dampers

The article deals with the computational study of the rigid rotor coupled with squeeze film dampers. Various techniques such as the method of computation of the synchronous response with a circular centred orbit, the harmonic balance method, and the direct time integration method are used to analyse the nonlinear behaviour of the rotor system. The results indicate that the rotor system can exhibit both a synchronous circular response with a large orbit radius and a nonsynchronous response with a quasiperiodic character. However, both responses are undesirable in rotating machinery and should be avoided. The new results are presented to provide insight into the impact of initial conditions on the vibration response via basins of attraction. The simulations show that: (i) the basins of attraction are more sensitive to the choice of the initial velocities than displacements, (ii) the basins of attraction are noticeably dependent on the rotor speed in the region of a nonsynchronous response, and (iii) the border between the basins of attraction can be smooth or without a clear structure. The research brings clear conditions defined by parameters such as the dimensionless SFD constant, unbalance, and rotational speed for the suppression of undesirable nonlinear phenomena. The results suggest that the damper can effectively improve the vibration response of high-speed rotating machinery, but its design must be chosen appropriately.

Fast Spin‐Up of Geochemical Tracers in Ocean Circulation and Climate Models

AbstractOcean geochemical tracers such as radiocarbon, protactinium and thorium isotopes, and noble gases are widely used to constrain a range of physical and biogeochemical processes in the ocean. However, their routine simulation in global ocean circulation and climate models is hindered by the computational expense of integrating them to a steady state. Here, a new approach to this long‐standing “spin‐up” problem is introduced to efficiently compute equilibrium distributions of such tracers in seasonally‐forced models. Based on “Anderson Acceleration,” a sequence acceleration technique developed in the 1960s to solve nonlinear integral equations, the new method is entirely “black box” and offers significant speed‐up over conventional direct time integration. Moreover, it requires no preconditioning, ensures tracer conservation and is fully consistent with the numerical time‐stepping scheme of the underlying model. It thus circumvents some of the drawbacks of other schemes such as matrix‐free Newton Krylov that have been proposed to address this problem. An implementation specifically tailored for the batch HPC systems on which ocean and climate models are typically run is described, and the method illustrated by applying it to a variety of geochemical tracer problems. The new method, which provides speed‐ups by over an order of magnitude, should make simulations of such tracers more feasible and enable their inclusion in climate change assessments such as IPCC.

The Meshfree Radial Point Interpolation Method (RPIM) for Wave Propagation Dynamics in Non-Homogeneous Media

This work presents a novel simulation approach to couple the meshfree radial point interpolation method (RPIM) with the implicit direct time integration method for the transient analysis of wave propagation dynamics in non-homogeneous media. In this approach, the RPIM is adopted for the discretization of the overall space domain, while the discretization of the time domain is completed by employing the efficient Bathe time stepping scheme. The dispersion analysis demonstrates that, in wave analysis, the amount of numerical dispersion error resulting from the discretization in the space domain can be suppressed at a very low level when the employed nodal support domain of the interpolation function is adequately large. Meanwhile, it is also mathematically shown that the amount of numerical error resulting from the time domain discretization is actually a monotonically decreasing function of the non-dimensional time domain discretization interval. Consequently, the present simulation approach is capable of effectively handling the transient analysis of wave propagation dynamics in non-homogeneous media, and the disparate waves with different speeds can be solved concurrently with very high computation accuracy. This numerical feature makes the present simulation approach more suitable for complicated wave analysis than the traditional finite element approach because the waves with disparate speeds always cannot be concurrently solved accurately. Several numerical tests are given to check the performance of the present simulation approach for the analysis of wave propagation dynamics in non-homogeneous media.

Post-buckling analysis of shell-like structures using an implicit dynamic time integration scheme

PurposeThis paper aims to investigate post-buckling responses of shell-like structures using an implicit conservative-decaying time integration dynamic scheme.Design/methodology/approachIn this work, the authors have proposed the use of a four-node quadrilateral flat shell finite element with drilling rotational degree of freedom within the framework of an updated Lagrangian formulation mutually with an implicit conservative-dissipative time integration dynamic scheme.FindingsSeveral numerical simulations were considered to evaluate the accuracy, robustness, stability and the capacity of the considered time integration scheme to dissipate numerical noise in the presence of high frequencies. The obtained results illustrate a very satisfying performance of the implicit conservative-dissipative direct time integration scheme conjointly with the quadrilateral flat shell finite element with drilling rotation.Originality/valueThe authors have investigated the potential of the implicit dynamic scheme to deal with unstable branches after limit points in the non-linear post-buckling response of shell structures with no need for structural damping. The capability of the studied algorithm to study buckling and post-buckling behaviour of thin shell structures is illustrated through several numerical examples.

Imposing displacements in implicit direct time integration & a patch test

We consider the transient analysis of finite element models when an implicit direct time integration scheme is used and focus on the accuracy of the response solution when time varying nodal displacements are imposed. The solutions of such problems are abundantly sought in engineering practice. We analyze the performance of the widely used Trapezoidal Rule, a specific case of Newmark time integration, and the performance of the Bathe method. The theoretical results show that the Trapezoidal Rule is unstable and this instability becomes more pronounced as the time step size for solution is decreased, whereas the Bathe method is stable. We illustrate these theoretical findings in the transient analyses of some finite element models. These investigations lead us to define a “patch test” useful for the general evaluation of direct time integration schemes.

Optimization of Seismic Base Isolation System Using a Fuzzy Reinforced Swarm Intelligence

The current study deals with the optimization of seismic isolation system parameters applied in base-isolated buildings. In this regard, to consider the more realistic condition, both near-fault and far-fault earthquake ground motions are taken into account by simultaneously imposing six different acceleration records during the direct time integration analysis of the structure. The minimization of the system's peak acceleration ratio is defined as the objective function of the optimization model. To get feasible results, the determining parameters of the isolator are restricted. Such that, while the isolation system period and damping ratio are kept within a predefined range, the maximum displacement of the system is monitored and limited during the optimization process. To solve the proposed optimization model, an enhanced version of the Deferential Evolution method so-called Fuzzy Differential Evolution incorporated Virtual Mutant (FDEVM) is employed. The FDEVM method applies a fuzzy decision-making mechanism to adopt its search behavior based on the governing conditions of the current problem. It also uses a new mutant vector called virtual mutant to contribute information of all population based on each agent's quality. So, the FDEVM works as a self-adaptive and parameter-free search algorithm. The acquired results show that the FDEVM works properly to find the optimal values for the dynamic parameters of the seismic base isolation system.

A unified scheme for nonlinear dynamic direct time integration methods: a comparative study on the application of multi-point methods

A unified scheme for nonlinear dynamic direct time integration methods: a comparative study on the application of multi-point methods

Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave.

Nonlinear nonplanar dynamics of porous functionally graded pipes conveying fluid

This paper aims to investigate the three-dimensional (3-D) nonlinear dynamics of a porous functionally graded (FG) pipe subjected to lateral harmonic excitation. Material properties of the porous pipe are graded across the radius in a power-law distribution form. Based on the Euler–Bernoulli beam theory, the nonlinear equations of motion are derived employing the Hamilton’s principal to achieve third-order accuracy with the fluid-related loads associated with the bending–torsional vibration. The new nonlinear model, consisting of three strongly coupled nonlinear partial differential equations, is discretized into second-order ordinary differential equations via Galerkin method. Subsequently, the pseudo-arclength continuation technique together with a direct time-integration method are employed to perform nonlinear static and dynamic responses of this gyroscopic system. Numerical parametric investigations are conducted to assess the significant effects of different parameters on the resonant dynamic behavior of system, with special focus on the modal interaction that could lead to multiple coexisting solutions including planar and nonplanar motions. Moreover, the nonplanar resonance behavior is revealed that the influences of structural symmetries and symmetry-breaking effects on bifurcations and instabilities in comparison to the previously classical planar resonance behavior.

Bounds for rotating convection at infinite Prandtl number from semidefinite programs

The technique of semidefinite programming is used to replace direct numerical simulations and time integrations of turbulent flows with an optimization problem which is easier to solve but which returns only upper bounds to the time averages the direct numerical simulation would have produced. This idea is applied to rotating Rayleigh-B\'enard convection at infinite Prandtl number. The obtained bounds are not sharp but they reproduce the qualitative behavior of the results known from direct simulations and experiments.

Reduced-Order Model for the Nonlinear Dynamics of Cables

Formulation of governing equations for an elastic cable have a long and dated history. A unified framework to detect the dynamics of such systems is detailed, justified and assessed numerically. Modal analyses are performed in the Frenet basis which parts the motion into the local frame and accounts accurately for the system's physics. In this article, a methodology to produce arbitrary reduced-order models for cable nonlinear dynamics is provided. The results obtained from the latter via direct time integration are challenged numerically via a comparison with results of nonlinear finite element method.

Nonlinear vibratory energy exchanges in a meta-cell

Meta-materials are utilized for bringing passive attenuation solution especially in the acoustics and vibration domains. Generally speaking, they are composed of an array or a chain of meta-cells. Here the vibratory energy exchanges between particles of a nonlinear meta-cell are studied. The meta-cell is composed of an outer mass which houses an inner mass with a compound nonlinear restoring forcing term. It is globally non-smooth, containing pure cubic and piece-wise linear terms, which constitutes a new type of nonlinearity for such mass-in-mass cells. The complexified form of system equations is treated by the multiple time scale method to find out its fast and slow dynamics, leading to determination of the slow invariant manifold as well as the singular and equilibrium points of the system. The compound nonlinear restoring forcing function of the inner mass makes the global geometry of the slow invariant manifold to be different from those of systems with pure cubic nonlinearities, for example, including four singular lines and two distinct unstable zones. Amplitude dependency of the frequency of such a nonlinear system in the conservative form is represented by the backbone curves. Furthermore, detected equilibrium points for different external forcing amplitudes are represented by three-dimensional frequency response curves. Finally, analytical predictions are confronted with numerical results obtained by direct time integration of the system equations. An application of such system can be the passive control of main systems via embedding another nonlinear oscillator inside it. Moreover, such system can also be extended in the form of an array to create meta-materials.

An engineering perspective on the numerics of quasi-periodic oscillations

The phenomenon of quasi-periodicity in deterministic dynamical systems describes stationary solutions, which neither exhibit a finite period length nor are chaotic. Recently, an increasing demand for robust numerical methods is driven by applied dynamics and industrial applications. In this context, direct time integration proves to be impractical due to extensive integration intervals. Therefore, in a first step, this contribution aims on giving an application oriented survey of the basic theory as well as alternative concepts. In the following, the focus is set on the direct computation of invariant manifolds (surfaces) on which quasi-periodic solutions evolve. This approach offers a unique framework from which classical methods (e.g., the multi-harmonic-balance) can be systematically deduced and mutual similarities between different methods may be revealed. This contribution starts with a brief summary of related mathematical basics, which is followed by an overview of available methods. Subsequently, the computation of invariant manifolds by means of solving a partial differential equation is emphasized. These PDEs may be formulated using different parametrization strategies. Here, the concept of hyper-time parametrization is particularly interesting, since it is a promising starting point for the development of numerical schemes with general applicability in engineering problems. In order to solve the underlying PDE, various methods may be used. The implementation of a Fourier-Galerkin method as well as a finite difference method is presented and compared on the basis of computational results of the van-der-Pol equation (with and without forcing). Moreover, it is demonstrated that both methods apply to periodic as well as quasi-periodic solutions alike. In order to exemplify the practical use, these methods are applied to a generic rotordynamic model problem.

Dynamical analysis of sliding connections with mesh independent roughness by a total Lagrangian FEM

Sliding connections are present in several applications on the mechanics, civil and aerospace industries. A framework consisting on an accurate and stable formulation to describe the dynamics of flexible systems with sliding connections is developed. The total Lagrangian positional approach of the Finite Element Method is employed using 2D solid and frame elements to discretize bodies and connections. This allows a wide range of applications, particularly the local modelling of joints. The proposed formulation includes roughness along sliding paths independent from the finite element geometry discretization. Following variational principles, Lagrange multipliers are used to impose sliding constraints on the equations of motion. A direct time integration is performed by the generalized-α method and its stability in the present finite deformation context is evaluated. The resulting nonlinear equations are solved by the Newton-Raphson method. Examples are presented where the proposed framework is evaluated regarding its dynamical behavior and to solve practical scenarios for which sliding modelling is a necessity.

Efficient Three-Dimensional Model to Predict Time History of Structural Dynamics in Cold Rolling Mills

Abstract Introduced is a new physics-based three-dimensional (3D) mathematical model capable of efficiently predicting time histories of the nonlinear structural dynamics in cold rolling mills used to manufacture metal strips and sheets. The described model allows for the prediction of transient strip thickness profiles, contact force distributions, and roll-stack deformations due to dynamic disturbances. Formulation of the new 3D model is achieved through a combination of the highly efficient simplified-mixed finite element method with a Newmark-beta direct time integration approach to solve the system of differential equations that governs the motion of the roll-stack. In contrast to prior approaches to predict structural dynamics in cold rolling, the presented method abandons several simplifying assumptions and restrictions, including 1D or 2D linear lumped parameter analyses, vertical symmetry, continuous and constant contact between the rolls and strip, as well as the inability to model cluster-type mill configurations and accommodate typical profile/flatness control mechanisms used in industry. Following spatial and temporal convergence studies of the undamped step response, and validation of the damped step response, the new model is demonstrated for a 4-high mill equipped with both work-roll bending and work-roll crown, a 6-high mill with continuously variable crown (CVC) intermediate rolls, and finally a complex 20-high cluster mill. Solution times on a single computing processor for the damped 4-high and 20-high case studies are just 0.37 s and 3.38 s per time-step, respectively.

Dynamic fracture analysis using a high-accuracy manifold element modelling scheme

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.

Optimum topological bracing design of tall steel frames subjected to dynamic loading

In this paper, a frame structural topology optimization methodology for the generation of moment resisting braced frames for tall buildings, considering dynamic seismic loading, is presented. Real-world loading conditions along with standardized steel profiles of Euronorm are employed. In the literature, the studies that employ the ground structure topology optimization method are limited mostly to truss structures under static loading conditions. The dynamic response in the framework of structural topology optimization remains a challenging problem, especially when the ground structure method is used for deriving the structural system of tall buildings. The contribution of this study to the state of the art, relies on the implementation of the proposed methodology in the conceptual design phase of civil engineering frame structures, where both axial and flexural stiffness is considered. Direct time integration methods are used, to implement forced vibrations and real recorded earthquake data in the topology optimization procedure, resulting to novel layouts for lateral resisting systems of tall buildings. Aiming to aid the engineer to the final design phase, the response spectrum modal analysis is employed and the seismic loading is applied according to the Eurocode 8.

A simplified variant of the time finite element methods based on the shape functions of an axial finite bar

In the field of structural dynamics, the structural responses in the time domain are of major concern. There already exist many methods proposed previously including widely used direct time integration methods such as ones in the β-Newmark family, Houbolt’s method, and Runge-Kutta method. The time finite element methods (TFEM) that followed the well-posed variational statement for structural dynamics are found to bring about a superior accuracy even with large time steps (element sizes), when compared with the results from methods mentioned above. Some high-order time finite elements were derived with the procedure analogous to the conventional finite element methods. In the formulation of these time finite elements, the shape functions are like the ones for a (spatial) 2-order finite beam. In this article, a simplified variant for the TFEM is proposed where the shape functions similar to the ones for a (spatial) axial bar are used. The accuracy in the obtained results of some numerical examples is found to be comparable with the accuracy in the previous results.

Friction-induced traveling wave coupling in tuned bladed-disks

Flutter is a major constraint on modern turbomachines; as the designs move toward more slender, thinner, and loaded blades, they become more prone to experience high cycle fatigue problems. Dry friction, present at the root attachment for cantilever configurations, is one of the main sources of energy dissipation. It saturates the flutter vibration amplitude growth, producing a limit cycle oscillation whose amplitude depends on the balance between the energy injected and dissipated by the system. Both phenomena, flutter and friction, typically produce a small correction of the purely elastic response of the structure. A large number of elastic cycles is required to notice their effect, which appears as a slow modulation of the oscillation amplitude. Furthermore, even longer time scales appear when multiple traveling waves are aerodynamically unstable and exhibit similar growth rates. All these slow scales make the system time integration very stiff and CPU expensive, bringing some doubts about whether the final solutions are properly converged. In order to avoid these uncertainties, a numerical continuation procedure is applied to analyze the solutions that set in, their traveling wave content, their bifurcations and their stability. The system is modeled using an asymptotic reduced order model and the continuation results are validated against direct time integrations. New final states with multiple traveling wave content are found and analyzed. These solutions have not been obtained before for the case of microslip friction at the blade attachment; only solutions consisting of a single traveling wave have been reported in previous works.

Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.