A novel family of explicit time marching techniques for structural dynamics and wave propagation models

Numerical Meth Engineering

2019

Summary In this article, a two‐stage explicit time integration method is presented for the numerical analysis of various dynamic problems described by second‐order ordinary differential equations in time. The newly proposed method have negligible damping and period errors in the important low‐frequency range and can introduce moderate numerical damping into the spurious high‐frequency range. The computational effort per step required in the new method is the same as the effort needed in the existing two‐stage explicit methods. The proposed method can provide fourth‐order accurate solutions for velocity independent problems and second‐order accurate solutions for velocity dependent problems, which is not possible in the existing methods. The proposed new method is expected to be an excellent tool for numerical analyses of various dynamic problems including structural dynamics.

“…The Soares method is summarized as

\begin{matrix} right {\dot{u}}_{t_{s} + Δ t} = & left {(M + \frac{Δ t}{2} C)}^{- 1} (M {\dot{u}}_{t_{s}} - \frac{Δ t}{2} C {\dot{u}}_{t_{s}} - K (Δ t u_{t_{s}} + \frac{Δ t^{2}}{2} {\dot{u}}_{t_{s}}) + Δ t (β_{1} q_{t_{s}} + β_{2} q_{t_{s} + Δ t})) \end{matrix}

\begin{matrix} right u_{t_{s} + Δ t} = & left {(M + \frac{Δ t}{2} C)}^{- 1} (- \frac{Δ t^{2}}{2} C {\dot{u}}_{t_{s} + Δ t} - K (β b_{1} b_{2} Δ)) \end{matrix}

…”

Section: A Brief Review Of the Various Methods Used In This Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

Numerical Meth Engineering

2019

“…As more sophisticated spatial finite element models [1][2][3][4] are developed constantly, demands for more accurate time integration methods are also increasing to take full advantage of improved spatial models in transient analyses. Recently, numerous implicit [5][6][7][8][9] and explicit [10][11][12][13][14][15][16] time integration methods have been introduced to effectively analyze challenging dynamic problems.…”

Section: Introductionmentioning

confidence: 99%

“…Soares developed an explicit time integration method based on the weighted residual method [13]. The Soares method can be used for the analysis of wave propagation problems, but it becomes an only first-order accurate implicit method in the presence of non-diagonal damping matrix.…”

Section: Introductionmentioning

confidence: 99%

Higher-order explicit time integration methods for numerical analyses of structural dynamics

Lat. Am. j. solids struct.

2019

In this article, third-and fourth-order accurate explicit time integration methods are developed for effective analyses of various linear and nonlinear dynamic problems stated by second-order ordinary differential equations in time. Two sets of the new methods are developed by employing the collocation approach in the time domain. To remedy some shortcomings of using the explicit Runge-Kutta methods for second-order ordinary differential equations in time, the new methods are designed to introduce small period and damping errors in the important low-frequency range. For linear cases, the explicitness of the new methods is not affected by the presence of non-diagonal damping matrix. For nonlinear cases, the new methods can handle velocity dependent problems explicitly without decreasing order of accuracy. The new methods do not have any undetermined algorithmic parameters. Improved numerical solutions are obtained when they are applied to various linear and nonlinear problems.

A simple explicit single step time integration algorithm for structural dynamics

Numerical Meth Engineering

2019

In this article, a new single-step explicit time integration method is developed based on the Newmark approximations for the analysis of various dynamic problems. The newly proposed method is second-order accurate and able to control numerical dissipation through the parameters of the Newmark approximations. Explicitness and order of accuracy of the proposed method are not affected in velocity-dependent problems. Illustrative linear and nonlinear examples are used to verify performances of the proposed method. KEYWORDS controllable numerical dissipation, explicit time integration method, impact and wave propagation, linear and nonlinear structural dynamics, pendulums Int J Numer Methods Eng. 2019;119:383-403.wileyonlinelibrary.com/journal/nme 384 KIM highly nonlinear problems, the higher-order methods are more suitable than the second-order methods. However, improved accuracy of the higher-order accurate methods is often accompanied by more complicated computations and increased computational efforts. For example, four stages are required to advance a step in the fourth-order accurate Runge-Kutta method, whereas only one stage is required in many of the second-order methods. Due to this reason, second-order accurate methods are broadly used for practical analyses. Among the classical explicit methods, the central difference (CD) 14 method and the leap-frog method 15 have simple and intuitive computational structures. Both methods do not have undetermined parameters, whereas algorithmic parameters should be properly determined in many of the recently developed explicit methods. The CD method is known as one of the most accurate nondissipative second-order explicit methods with the highest stability limit. Due to this reason, the CD method is often considered as a reference method for the study of accuracy and stability of explicit methods. The leap-frog and Verlet methods are also nondissipative and broadly used in various engineering and science research works. However, the aforementioned methods become only first-order accurate for velocity dependent problems when the velocity vector is explicitly treated.The objective of the current work is to propose a simple second-order explicit time integration algorithm with dissipation control capability. To verify the linear performance of the proposed explicit method, commonly used spectral analysis 16,17 is conducted by using the single-degree-of-freedom (SDOF) problem. Furthermore, linear and nonlinear test problems are solved by using the new and existing methods. The numerical results obtained from the various methods are compared with each other to investigate improved performances of the newly proposed method.