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

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.

Integration algorithm is a key to ensure the better application of the dynamic constitutive model. As for the bounding surface model based on swell–shrink rules in this paper, an explicit integration algorithm based on sub-stepping and error control scheme has be described in detail. For being adaptable to the constitutive model, several modifications have been made. The modified explicit algorithm includes two procedures: the first procedure is the second-order forward modified Euler scheme with sub-stepping and error control; the second procedure is the stress correction. However, due to the shrink rule of the bounding surface, the second procedure is not needed at the unloading phase. Finally, several numerical simulations have been conducted for assessing the performance of the explicit algorithm, which demonstrates that the corresponding algorithm is stable and accurate, and can simulate the mechanical characteristics of the clay pretty well.

We numerically calculate the quasinormal frequencies of the Klein-Gordon and Dirac fields propagating in a two-dimensional asymptotically anti-de Sitter black hole of the dilaton gravity theory. For the Klein-Gordon field we use the Horowitz-Hubeny method and the asymptotic iteration method for second order differential equations. For the Dirac field we first exploit the Horowitz-Hubeny method. As a second method, instead of using the asymptotic iteration method for second order differential equations, we propose to take as a basis its formulation for coupled systems of first order differential equations. For the two fields we find that the results that produce the two numerical methods are consistent. Furthermore for both fields we obtain that their quasinormal modes are stable and we compare their quasinormal frequencies to analyze whether their spectra are isospectral. Finally we discuss the main results.

Currently, the mechanical dynamics of soft tissue deformation is achieved by numerical time integrations such as the explicit or implicit integration; however, the explicit integration is stable only under a small time step, whereas the implicit integration is computationally expensive in spite of the accommodation of a large time step. This paper presents a cellular neural network method for stable simulation of soft tissue deformation dynamics. The non-rigid motion equation is formulated as a cellular neural network with local connectivity of cells, and thus the dynamics of soft tissue deformation is transformed into the neural dynamics of the cellular neural network. Results show that the proposed method can achieve good accuracy at a small time step. It still remains stable at a large time step, while maintaining the computational efficiency of the explicit integration. The proposed method can achieve stable soft tissue deformation with efficiency of explicit integration for surgical simulation.

The main application of explicit time integration algorithms for finite element simulations -as they are characterized by rather small time steps -are highly dynamic problems.An implementation of the element routines with a specific view on efficiency of the algorithms is essential, while the internal force vector has to be computed extremely often, compared to implicit algorithms.In this contribution, a specific implementation concept is presented, using automatic generation and optimization of code for a so-called 'Solid-Shell' formulation with quadratic shape functions in membrane direction and linear approximation in thickness direction.The efficiency of the numerical algorithms is shown and compared to otherwise programmed and compiled routines.

We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z)) dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient $\kappa_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlev\'e II equation. The leading coefficient and second leading coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the Painlev\'e II function.

This work exploits the recently developed telescopic projective numerical integration method for efficient parallel circuit simulation. Stable explicit numerical integration is achieved by adopting an explicit inner integrator (e.g. forward Euler) in a multi-level telescopic projective framework, thereby addressing the well-known stability limitation of many explicit numerical integration methods. In the presented approach, the effective time step of the entire multilevel integration is no longer limited by the smallest time constant of the circuit so as to safeguard stability. Rather, it is controlled solely by the accuracy requirement. This makes it possible to explore the natural parallelizability of such explicit integration method for parallel circuit simulation. We demonstrate the potential of the presented approach and its parallel implementation on multi-core machines with encouraging initial results.

An approach that increases the critical time step in explicit time integration is presented. By derivation as a variational integrator, symplecticity and momentum preservation are ensured. The linear response is assumed to be dominant and is integrated implicitly, while the nonlinear forces are explicitly integrated. MOLLY filters the high‐frequent portions of the nonlinear forces and increases the critical time step. An SDOF example and a structural example using modal reduction verify the improved stability. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A tendon-driven robot inspired from the bird neck is studied in this paper. The objective is to propose numerical integrations for this robot. The classical explicit and implicit Euler integration schemes can be used on this robot. When the elasticity in the tendons is not considered, these schemes are stable and give similar numerical integration results but the explicit implementation is faster. By considering the dynamics of the robot joints on one hand and the dynamics of the motors on the other hand, we propose two Euler integration schemes, explicit and implicit, that take into account the elasticity in the tendons. These schemes rely on the dynamics of the robot joints on the one hand, and on the dynamics of the motors on the other hand and it links them by the tendon elastic model. We observe that with the tendon elasticity model, the explicit integration becomes unstable, depending on the time step, while the implicit one stays stable. The numerical integration that uses this implicit scheme with tendon elasticity allows one to obtain more precise results, while the model of the system is more realistic. Moreover, the evolution of the tendon elongation during a desired motion can be obtained while they are computed in the integration scheme.

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

A class of nonlinear velocity-dependent problems must be solved iteratively for conventional integration methods since there exists no completely explicit integration method among them. A completely explicit structure-dependent integration method is developed in this work, and it can be applied to solve such problems non-iteratively. A completely explicit method for time integration is characterized by the adoption of explicit difference formulas for both the displacement and velocity increment. It can be derived from an eigen-based theory, and this theory can provide a fundamental basis for the feasibility of time integration. The new method generally has no serious disadvantages of weak instability and unusual overshoot in high-frequency steady-state responses that have been found in the current completely explicit structure-dependent integration methods. It is analytically and numerically verified that it can be used to solve nonlinear velocity-dependent problems without nonlinear iterations and can have a comparable accuracy in contrast to the solutions obtained from conventional integration methods with an iteration procedure. Its computational efficiency due to no nonlinear iterations is also affirmed by numerical examples.

Numerical calculations for a time-accurate solution of the equations of e uid dynamics often require a timestep constraint. One can reduce this constraint to an inequality relating the time step, the grid spacing, and some reference wave velocity. Historically, the literature in numerical analysis refers to this parametric cluster as the Courant number (nondimensional) and the condition for the linear case as the Courant ‐Friedrichs‐Lewy (CFL)condition. Classically, numerical analysis relies on linearization and von Neumann’ s useof Fourier series to derive the CFL condition. In practice, computational e uid dynamics mostly relies on rules of thumb and heuristic arguments to justify the equation that determines time-step size and numerical stability for complicated and nonlinear calculations. The approach proposed in this paper uses the second law of thermodynamics as a way of imposing a restriction on the time step, applied to linear and nonlinear equations and systems of equations like the equations of gas dynamics. Basically, by transforming the truncation error for the numerical formula approximating a conservation equation into an equation representing the balance of entropy, one can obtain an inequality that restricts the time step to satisfy the second law. The second law as developed extends its role by analogy for the simple linear advection equation, then a nonlinear equation, and e nally a system of equations representing the one-dimensional equations of gas dynamics. In each case results obtained agree with the classical approachforlinearequationsbutdifferinothers,indicatingthatthesecondlawhassignie cantimplicationsbeyond its role in thermodynamics. This work develops the topic only for explicit numerical algorithms with truncation errors no greater than second order. By conjecture one expects that the most general conclusions will hold for implicit and higher-order methods because of the universality of the second law and the concept of entropy.

Discussion of paper ‘Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm’ by Cheng Chen, James M. Ricles, Thomas M. Marullo and Oya Mercan, Earthquake Engineering and Structural Dynamics 2009; 38:23–44

We present a fast first order explicit time integration scheme for solving parabolic problems in mechanics via standard numerical methods in space using irregular grids, such as unstructured finite element meshes, or grids containing elements or cells of very different sizes. The new scheme extends one of the explicit FIC-Time (EFT) integration methods derived by the authors in [23] that allow considerable larger time steps than the forward-Euler (FE) scheme. The new EFT scheme overcomes the limitations in the time step size of explicit time integration schemes for irregular grids containing large and small elements. A variable time step is used for eliminating the oscillations near Dirichlet boundaries when large time steps are used. The advantages of the new EFT scheme versus the FE scheme are shown in one-, two- and three-dimensional transient heat conduction problems using irregular finite element grids.

A satisfied cloth simulation should be general, accurate, efficient and stable. Explicit, implicit and semi-implicit integration methods have contributed to large performance enhancements in the field of cloth simulation. In this paper, we propose a modified embedded Runge-Kutta method to solve the cloth simulation based on classical spring-mass model. Traditional embedded Runge-Kutta methods generally apply some optimized coefficients to solve ordinary differential equations of cloth simulation. Most of them tend to concentrate on the efficiency of the simulation process, and not the fidelity of the simulation result. We investigate and determine the extent to which the overall quality must be compromised in order for the stable conditions to be satisfied. The modified Runge-Kutta method proposed in our paper incorporates the fixed time step and adaptive time step in solving cloth motion equations to achieve a controllable error evaluation. Compared with the other Runge-Kutta methods, the proposed method has some advantages for cloth simulation: controllable error evaluation without extra computations, excellent efficiency, good stability and satisfied precision. Experiment demonstrates that this method results in simulation efficiency improvements and is considerable practicable.