Computational Aspects of Time Integration Procedures in Structural Dynamics—Part 1: Implementation

1978 ◽  
Vol 45 (3) ◽  
pp. 595-602 ◽  
Author(s):  
C. A. Felippa ◽  
K. C. Park
Keyword(s):  
Structural Dynamics ◽  
Error Propagation ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Effort ◽  
Computational Error ◽  
Unified Approach ◽  
Time Step ◽  
Computational Aspects ◽  
Specific Implementation

A unified approach for the implementation of direct time integration procedures in structural dynamics is presented. Two key performance assessment factors are considered, viz., computational effort and error propagation. It is shown that these factors are strongly affected by details in the reduction of the second-order equations of motion to a system of first-order equations, and by the computational path followed at each time step. Part 1 is primarily devoted to the study of the organization of the computational process. Specific implementation forms derivable from the unified approach are studied in detail, and rated accordingly. An analysis of the computational error propagation characteristics of these implementations is presented in Part 2.

Computational Aspects of Time Integration Procedures in Structural Dynamics—Part 2: Error Propagation

1978 ◽  
Vol 45 (3) ◽  
pp. 603-611 ◽  
Author(s):  
K. C. Park ◽  
C. A. Felippa
Keyword(s):  
Structural Dynamics ◽  
Error Propagation ◽  
Error Control ◽  
Time Integration ◽  
Trapezoidal Rule ◽  
Computational Error ◽  
Asymptotic Error ◽  
Average Acceleration ◽  
Computational Aspects ◽  
Acceleration Method

The propagation of computational error in the direct time integration of the equations of structural dynamics is investigated. Asymptotic error propagation equations corresponding to the computational paths presented in Part 1 are derived and verified by means of numerical experiments. It is shown that there exists an implementation form that achieves optimum error control when used in conjunction with one-derivative methods. No such form is found for two-derivative methods. A numerical beating phenomenon is observed for certain implementations of the average acceleration method and the trapezoidal rule, which from an error propagation standpoint, is highly undesirable.

A Dissipative Momentum-Conserving Time Integration Algorithm for Nonlinear Structural Dynamics

2021 ◽  
pp. 2150116
Author(s):  
Salvatore Lopez
Keyword(s):  
Angular Momentum ◽  
Structural Dynamics ◽  
Time Integration ◽  
Integration Scheme ◽  
Second Phase ◽  
Time Step ◽  
Integration Algorithm ◽  
Nonlinear Structural Dynamics ◽  
Nonlinear Structural ◽  
Solution Point

A second-order accurate single-step time integration method for nonlinear structural dynamics is developed. The method combines algorithmic dissipation of higher modes and conservation of linear and angular momentum and is composed of two phases. In the first phase, a solution point is computed by a basic integration scheme, the generalized-[Formula: see text] method being adopted due to its higher level of high-frequency dissipation. In the second phase, a correction is hypothesized as a linear combination of the solution in the basic step and the gradient of vector components of the incremental linear and angular momentum. By solving a system composed of six linear equations, the searched for corrected solution in the time step is then provided. The novelty in the presented integration scheme lies in the way of imposing the conservation of linear and angular momentum. In fact, this imposition is carried out as a correction of the computed solution point in the time step and not through an enlarged system of equations of motion. To perform tests on plane and spatial motion of three-dimensional structural models, a small strains — finite rotations corotational formulation is also described.

An efficient approach for dynamic analysis of a rotating beam using the discrete singular convolution

2016 ◽  
Vol 230 (20) ◽  
pp. 3642-3654 ◽  
Author(s):  
JunWei Chen ◽  
Ye Ding ◽  
Han Ding
Keyword(s):  
Dynamic Analysis ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
Efficient Approach ◽  
Discrete Singular Convolution ◽  
Dsc Method

This paper proposes an efficient approach for dynamic analysis of a rotating beam using the discrete singular convolution (DSC). By spatially discretizing the nonlinear equations of motion of the rotating beam using the DSC method, natural frequencies of the rotating beam are obtained. Numerical results show that the DSC method accurately captures not only the low-order but also the high-order frequencies of the beam rotating at a high angular velocity in very short time, compared with the classical finite element method. Moreover, by combining the DSC method and the differential quadrature method, the dynamic equations are reduced to a set of algebraic equations. Thus the dynamic response of the rotating beam is resolved accurately and efficiently with much less computational effort, and is able to be numerically stable for long-time integration.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Applicability of Attractor Control Techniques for a Particle Conveyed by a Propagating Wave

2004 ◽  
Vol 10 (7) ◽  
pp. 1057-1070 ◽  
Author(s):  
M. Ragulskis ◽  
K. Koizumi
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Integration ◽  
Algebraic Equations ◽  
Reverse Time ◽  
Time Step ◽  
Control Techniques ◽  
Potential Applicability ◽  
Nonlinear Features ◽  
Time Marching

The governing equations of motion describing the dynamics of a conveyed particle by a propagating surface wave are derived. Although the problem may look rather primitive, it holds considerable complications first of all due to the fact that the shape of the surface cannot be described explicitly Special forward and reverse time marching numerical techniques, incorporating the solution of nonlinear algebraic equations in every time step, are developed for time integration of derived differential equations. It is shown that the described system possesses numerous nonlinear features such as sensitivity to initial conditions. cocxistint, attractors. This fact builds the foundation for the potential applicability of attractor control techniques based on small external impulses.

scholarly journals Damping Acoustic Modes in Compressible Horizontally Explicit Vertically Implicit (HEVI) and Split-Explicit Time Integration Schemes

2018 ◽  
Vol 146 (6) ◽  
pp. 1911-1923 ◽  
Author(s):  
Joseph B. Klemp ◽  
William C. Skamarock ◽  
Soyoung Ha
Keyword(s):  
Acoustic Waves ◽  
Acoustic Noise ◽  
Equations Of Motion ◽  
Time Integration ◽  
Time Step ◽  
Explicit Time Integration ◽  
Physical Interest ◽  
Acoustic Modes ◽  
Integration Schemes ◽  
Time Integration Schemes

Although the equations of motion for a compressible atmosphere accommodate acoustic waves, these modes typically play an insignificant role in atmospheric processes of physical interest. In numerically integrating the compressible equations, it is often beneficial to filter these acoustic modes to control acoustic noise and prevent its artificial growth. Here, a new technique is proposed for filtering the 3D divergence that may damp acoustic modes more effectively than filters previously implemented in numerical modes using horizontally explicit vertically implicit (HEVI) and split-explicit time integration schemes. With this approach, a divergence damping term is added as a final adjustment to the horizontal velocity at the new time level after completing the vertically implicit portion of the time step. In this manner, the divergence used in the filter term has exactly the same numerical form as that used in the discrete pressure equation. Analysis of the dispersion equation for this form of the filter documents its stability characteristics and confirms that it effectively damps acoustic modes with little artificial influence on the amplitude or propagation of the gravity wave modes that are of physical interest. Some specific aspects of the implementation of the filter in the Model for Prediction Across Scales (MPAS) are discussed, and results are presented to illustrate some of the beneficial aspects of suppressing acoustic noise.

Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations

2011 ◽  
Author(s):  
Olivier Bru¨ls ◽  
Martin Arnold ◽  
Alberto Cardona
Keyword(s):  
Numerical Solution ◽  
Lie Group ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Cost ◽  
Differential Algebraic Equations ◽  
Rigid Body Dynamics ◽  
Time Step ◽  
Iteration Matrix

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix.

scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

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