scholarly journals A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021 ◽  
Author(s):  
Stefan Hante ◽  
Denise Tumiotto ◽  
Martin Arnold
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Equations Of Motion ◽  
Second Order ◽  
The Body ◽  
Benchmark Problems ◽  
Beam Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

scholarly journals Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1652
Author(s):  
Dong-Sin Shih ◽  
Gour-Tsyh Yeh
Keyword(s):  
Stokes Equations ◽  
Field Application ◽  
Benchmark Problems ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Cross Sectional ◽  
Inertia Effects ◽  
Time Step Size ◽  
Saint Venant Equations

One-dimensional (1D) Saint-Venant equations, which originated from the Navier–Stokes equations, are usually applied to express the transient stream flow. The governing equation is based on the mass continuity and momentum equivalence. Its momentum equation, partially comprising the inertia, pressure, gravity, and friction-induced momentum loss terms, can be expressed as kinematic wave (KIW), diffusion wave (DIW), and fully dynamic wave (DYW) flow. In this study, the method of characteristics (MOCs) is used for solving the diagonalized Saint-Venant equations. A computer model, CAMP1DF, including KIW, DIW, and DYW approximations, is developed. Benchmark problems from MacDonald et al. (1997) are examined to study the accuracy of the CAMP1DF model. The simulations revealed that CAMP1DF can simulate almost identical results that are valid for various fluvial conditions. The proposed scheme that not only allows a large time step size but also solves half of the simultaneous algebraic equations. Simulations of accuracy and efficiency are both improved. Based on the physical relevance, the simulations clearly showed that the DYW approximation has the best performance, whereas the KIW approximation results in the largest errors. Moreover, the field non-prismatic case of the Zhuoshui River in central Taiwan is studied. The simulations indicate that the DYW approach does not ensure achievement of a better simulation result than the other two approximations. The investigated cross-sectional geometries play an important role in stream routing. Because of the consideration of the acceleration terms, the simulated hydrograph of a DYW reveals more physical characteristics, particularly regarding the raising and recession of limbs. Note that the KIW does not require assignment of a downstream boundary condition, making it more convenient for field application.

Horizontally Explicit and Vertically Implicit (HEVI) Time Discretization Scheme for a Discontinuous Galerkin Nonhydrostatic Model

2015 ◽  
Vol 143 (3) ◽  
pp. 972-990 ◽  
Author(s):  
Lei Bao ◽  
Robert Klöfkorn ◽  
Ramachandran D. Nair
Keyword(s):  
Discontinuous Galerkin ◽  
Strong Stability ◽  
Euler System ◽  
Second Order ◽  
Spatial Discretization ◽  
Step Size ◽  
Time Step ◽  
Exact Integration ◽  
Time Step Size ◽  
Propagating Waves

Abstract A two-dimensional nonhydrostatic (NH) atmospheric model based on the compressible Euler system has been developed in the (x, z) Cartesian domain. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method with exact integration. The orography is handled by the terrain-following height-based coordinate system. The time integration uses the horizontally explicit and vertically implicit (HEVI) time-splitting scheme, which is introduced to address the stringent restriction on the explicit time step size due to a high aspect ratio between the horizontal (x) and vertical (z) spatial discretization. The HEVI scheme is generally based on the Strang-type operator-split approach, where the horizontally propagating waves in the Euler system are solved explicitly while the vertically propagating waves are treated implicitly. As a consequence, the HEVI scheme relaxes the maximum allowed time step to be mainly determined by the horizontal grid spacing. The accuracy of the HEVI scheme is rigorously compared against that of the explicit strong stability-preserving (SSP) Runge–Kutta (RK) scheme using several NH benchmark test cases. The HEVI scheme shows a second-order temporal convergence, as expected. The results of the HEVI scheme are qualitatively comparable to those of the SSP-RK3 scheme. Moreover, the HEVI DG formulation can also be seamlessly extended to account for the second-order diffusion as in the case of the standard SSP-RK DG formulation. In the presence of orography, the HEVI scheme produces high quality results, which are visually identical to those produced by the SSP-RK3 scheme.

scholarly journals A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

scholarly journals Multi-Degree of Freedom Propeller Force Models Based on a Neural Network and Regression

2020 ◽  
Vol 8 (2) ◽  
pp. 89 ◽  
Author(s):  
Bradford Knight ◽  
Kevin Maki
Keyword(s):  
Neural Network ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Open Water ◽  
Small Time ◽  
Training Data ◽  
Step Size ◽  
Time Step ◽  
Neural Network Approach ◽  
Time Step Size

Accurate and efficient prediction of the forces on a propeller is critical for analyzing a maneuvering vessel with numerical methods. CFD methods like RANS, LES, or DES can accurately predict the propeller forces, but are computationally expensive due to the need for added mesh discretization around the propeller as well as the requisite small time-step size. One way of mitigating the expense of modeling a maneuvering vessel with CFD is to apply the propeller force as a body force term in the Navier–Stokes equations and to apply the force to the equations of motion. The applied propeller force should be determined with minimal expense and good accuracy. This paper examines and compares nonlinear regression and neural network predictions of the thrust, torque, and side force of a propeller both in open water and in the behind condition. The methods are trained and tested with RANS CFD simulations. The neural network approach is shown to be more accurate and requires less training data than the regression technique.

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.

Multiscale Dynamic Modeling of Flexibility in Myosin V

2013 ◽  
Author(s):  
Mahdi Haghshenas-Jaryani ◽  
Alan Bowling
Keyword(s):  
Dynamic Modeling ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Physical Parameters ◽  
Myosin V ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Run Time

This paper presents a multiscale dynamic model for the simulation and analysis of flexibility in myosin V. A three dimensional (3D) flexible multibody model is developed to mechanically model the biological structure of myosin V. Experimental studies have shown that myosin’s neck domain can be considered as three pairs of tandem elements which can bend at junctures between them. Therefore, each neck is modeled by three rigid bodies connected by flexible spherical joints. One of the most important issues in dynamic modeling of micro-nanoscale sized biological structures, likes DNA and motor proteins, is the long simulation run time due to the disproportionality between physical parameters involved in their dynamics such as mass, drag coefficient, and stiffness. In order to address this issue, the mostly used models, based on the famous overdamped Langevin dynamics, omit the inertial terms in the equations of motion; that leads to a first order model which is inconsistent with the Newton’s second law. However, the proposed model uses the concept of the method of multiple scales (MMS) that brings all terms of the equations of motion into proportion with each other that helps to retain the inertia terms. This keeps consistency of the model with the physical laws and increases time step size of numerical integration from commonly used sub-femto seconds to sub-milli seconds. Therefore, simulation run time will be many orders of magnitude less than ones based on the other approaches. The simulation results obtained by the proposed multiscale model show more realistic dynamic behavior of myosin V in compared with other models.

Application of the Modified Rosenbrock Algorithm for Transient Rotordynamic Analysis

2012 ◽  
Author(s):  
Anand Srinivasan ◽  
Dhruv Kumar
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Single Step ◽  
Degree Of Freedom ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Forcing Function ◽  
Transient Simulations

It is well known that transient rotordynamic analyses involve numerical integration of the equations of motion in order to study the response of the system under an applied forcing function. A common problem that arises in such simulations is the choice of step-size that needs to be used to obtain numerically stable results. Traditional numerical integration techniques such as the Runge-Kutta algorithms not only require splitting up second order differential equations as two first order equations, but also necessitate multiple integrations at each time-step, thus increasing the solution time. The Newmark-beta and Wilson-theta algorithms are some of the prevalent methods that have been used for transient simulations in rotordynamics. However, those single-step methods are only conditionally stable, and require iterations to converge to a solution at each time step, thus making it pseudosingle-step. In the more recent years, a modified form of the Rosenbrock algorithm has been proposed as a numerically stable and true single-step mathematical formulation for the integration of structural dynamics problems. In this paper, the modified Rosenbrock algorithm has been applied to a transient start-up multi-degree-of-freedom rotordynamics problem. A constant time step-size algorithm has been used for the simulations, and results of the transient analysis have been presented. The fact that a multi-degree-of-freedom system can be solved without condensation of the higher order modes makes the superior numerical damping characteristics of the algorithm become evident.

scholarly journals A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Philku Lee ◽  
George V. Popescu ◽  
Seongjai Kim
Keyword(s):  
Numerical Solution ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Step Size ◽  
Time Step ◽  
Potential Oscillations ◽  
Time Step Size ◽  
Time Stepping ◽  
Spurious Oscillations

After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. However, the CN method may introduce spurious oscillations for nonsmooth data unless the time step size is sufficiently small. This article studies a nonoscillatory second-order time-stepping procedure for RD equations, called a variable-θmethod, as a perturbation of the CN method. In each time level, the new method detects points of potential oscillations to implicitly resolve the solution there. The proposed time-stepping procedure is nonoscillatory and of a second-order temporal accuracy. Various examples are given to show effectiveness of the method. The article also performs a sensitivity analysis for the numerical solution of biological pattern forming models to conclude that the numerical solution is much more sensitive to the spatial mesh resolution than the temporal one. As the spatial resolution becomes higher for an improved accuracy, the CN method may produce spurious oscillations, while the proposed method results in stable solutions.

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