Non-iterative methods for dynamic analysis of nonlinear velocity-dependent problems

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 unusual amplitude growth property and its remedy for structure-dependent integration methods

In molecular dynamics simulations, the limited time step size has been a barrier to simulating long-time behaviors. Implicit time integration methods allow markedly larger time steps than the standard explicit time method, although they have major drawbacks such as overheads solving linear systems and instability of Newton iterations. To overcome these issues, we propose a semi-implicit time integration scheme, the semi-implicit Hessian correction (SimHec) scheme, for overdamped Langevin dynamics. The method focuses on the Hessian matrices of bonded and nonbonded interactions, where components with large negative Hessian eigenvalues are cut off in the linear approximation of momentum equations to avoid instability. The narrow band Hessian matrix enables an efficient parallelized linear solution with an overlapping approximation. We tested SimHec for the interdomain fluctuations in adenylate kinase and the powerstroke transition of myosin II using a coarse-grained protein model. SimHec reproduced the same dynamics as the explicit method, although the transition dynamics tended to be accelerated and fluctuations in bonded potentials were slightly reduced. These deviations were corrected using a hybrid method, SimHec-H, which adds explicit time steps after the semi-implicit time step. The proposed scheme allowed us to use time steps 50-200 times larger than those in explicit time integration, which resulted in a speedup factor of 7-30 taking the overhead into account.

A new implicit dynamic finite element analysis procedure with damping included

To analyze a multibody system, this paper proposes an implicit numerical integration method for joint coordinates subsystem synthesis method. To verify the proposed method, a multibody model for an unmanned robot vehicle, which consists of six identical independent suspension systems, is developed. The symbolic method is applied to compute the system Jacobian matrix for the implicit integration method. The proposed method is also verified by performing rough terrain run-over simulation in comparison with the conventional implicit integration method. In addition, to evaluate the efficiency of the proposed method, the CPU time obtained by using this method is compared with that obtained by using the conventional implicit method.

A GPU-based numerical manifold method for modeling the formation of the excavation damaged zone in deep rock tunnels

Describes simulations of impact forging processes. Uses the explicit time integration finite element method, which is based on direct time integration of equation of motion, to compute the deformation of the workpiece and the dies. Uses the program developed to simulate the copper blow test performed on a 350,000J counter‐blow hammer. The calculated result reveals a good agreement in the final deformed configurations between the experiment and the explicit simulation. In order to compare this with the explicit method, the implicit time integration rigid‐plastic finite element program considering the inertia effect is also applied to the copper blow test simulation. As a result of the copper blow test simulation using the explicit program and the implicit program, finds that the calculated results have good agreements in available plastic deformation energy, forging load and equivalent plastic strain distribution. Finally, applies the developed program to simulations of multi‐blow forging processes. Presents three major findings from the multi‐blow forging simulations: (1) the continuous analysis technique used for the multi‐blow forging simulations works well; (2) the blow efficiency and the forging load generated by blow operations can be analysed efficiently and simulated results coincide with previous experimental and analytical ones; (3) the geometrical configuration of the workpiece is closely related to blow efficiency.

This paper presents a computational strategy that combines a novel rate-independent phenomenological model with an explicit time integration method to efficiently perform nonlinear dynamic analyses of non-stiffening hysteretic mechanical systems. The novel rate-independent model, developed by specializing a general class of uniaxial phenomenological models, has an algebraic nature, is based on a set of only three parameters having a clear mechanical significance, and can be easily implemented in a computer program. The adopted explicit structure-dependent time integration method, belonging to the Chang’s family of explicit methods, is unconditionally stable for all non-stiffening hysteretic mechanical systems, has a second-order accuracy, does not suffer from numerical damping, and displays a small relative period error for small time step. Furthermore, it does not require iterative procedures and, consequently, does not suffer from convergence issues. Numerical accuracy and computational efficiency of the proposed procedure are assessed by performing several nonlinear time history analyses on hysteretic mechanical systems and comparing the results with those obtained by employing a conventional strategy based on the celebrated Bouc–Wen model, or its modified version, and the widely used Newmark’s constant average acceleration method.

A novel decoupling dynamic method with third-order accuracy and controllable dissipation

A novel sub-step explicit time integration method based on cubic B-spline interpolation for linear and nonlinear dynamics

Dialectic mechanics was introduced as an approximative modeling alternative to the classic Newtonian formulation of mechanics. It allows for additional freedom in placing a systems eigenvalues to facilitate simulation of systems, that are not suitable for most integration methods, when modeled according to the classic approach. The original idea of dialectic mechanics enables the suppression of high frequencies, but may still yield very stiff systems unsuitable for explicit integration methods. An additional term is added to enable real-time simulation with explicit methods. The goal of this paper is an analysis of the resulting equations and a comparison to the classic Newtonian formulation, aiming for an understanding of which applications most benefit from using dialectic mechanics.

A novel family of controllable, dissipative structure-dependent integration methods is derived from an eigen-based theory, where the concept of the eigenmode can give a solid theoretical basis for the feasibility of this type of integration methods. In fact, the concepts of eigen-decomposition and modal superposition are involved in solving a multiple degree of freedom system. The total solution of a coupled equation of motion consists of each modal solution of the uncoupled equation of motion. Hence, an eigen-dependent integration method is proposed to solve each modal equation of motion and an approximate solution can be yielded via modal superposition with only the first few modes of interest for inertial problems. All the eigen-dependent integration methods combine to form a structure-dependent integration method. Some key assumptions and new techniques are combined to successfully develop this family of integration methods. In addition, this family of integration methods can be either explicitly or implicitly implemented. Except for stability property, both explicit and implicit implementations have almost the same numerical properties. An explicit implementation is more computationally efficient than for an implicit implementation since it can combine unconditional stability and explicit formulation simultaneously. As a result, an explicit implementation is preferred over an implicit implementation. This family of integration methods can have the same numerical properties as those of the WBZ-α method for linear elastic systems. Besides, its stability and accuracy performance for solving nonlinear systems is also almost the same as those of the WBZ-α method. It is evident from numerical experiments that an explicit implementation of this family of integration methods can save many computational efforts when compared to conventional implicit methods, such as the WBZ-α method.

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

A proof of stability is developed for an explicit multi-time step integration method of the second order differential equations which result from a semidiscretization of the equations of structural dynamics. The proof is applicable to an algorithm that partitions the mesh into subdomains according to nodal groups which are updated with different time steps. The stability of the algorithm is demonstrated by showing that the eigenvalues of the amplification matrices lie within the unit circle and that a pseudo-energy remains constant. Bounds on the stable time steps for the nodal partitions are developed in terms of element frequencies.

This study investigates the full penetration simulation of piles from the ground surface, focusing on frictional contact modeling without mesh distortion. To overcome issues related to mesh distortion and improve solution convergence, the Arbitrary Lagrangian–Eulerian (ALE) adaptive mesh technique was implemented within both explicit and implicit time integration schemes. The numerical model was validated against field experiments conducted at Bothkennar, Scotland, using the Imperial College instrumented displacement pile (ICP) in soft clay, where the soil behavior was effectively represented using the modified Cam-Clay model and the Mohr–Coulomb model. The primary objectives of this study are to evaluate the ALE method performance in handling mesh distortion; analyze the effects of soil–pile interface friction, pile dimensions, and various dilation angles on pile resistance; and compare the effectiveness of explicit and implicit time integration schemes in terms of stability, computational efficiency, and solution accuracy. The ALE method effectively modeled pile penetration in Bothkennar clay, validating the numerical model against field experiments. Comparative analysis revealed the explicit time integration method as more robust and computationally efficient, particularly for complex soil–pile interactions with higher friction coefficients.