scholarly journals Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

2021 ◽  
Author(s):  
Hannes Uecker
Keyword(s):  
Pattern Formation ◽  
Degrees Of Freedom ◽  
Numerical Continuation ◽  
Nonlinear Pdes ◽  
Large Set ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Time Periodic Solutions ◽  
User Friendly ◽  
Parameter Dependent

AbstractNumerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

RELATIVISTIC NUCLEAR ENERGY DENSITY FUNCTIONALS

2010 ◽  
Vol 19 (04) ◽  
pp. 548-557 ◽  
Author(s):  
D. VRETENAR ◽  
T. NIKŠIĆ ◽  
P. RING
Keyword(s):  
Energy Density ◽  
Degrees Of Freedom ◽  
Nuclear Energy ◽  
Effective Interaction ◽  
Binding Energies ◽  
Large Set ◽  
Density Functionals ◽  
Heavy Nuclei ◽  
Interaction Terms ◽  
Range Dynamics

A class of relativistic nuclear energy density functionals is explored, in which only nucleon degrees of freedom are explicitly used in the construction of effective interaction terms. Short-distance correlations, as well as intermediate and long-range dynamics, are encoded in the nucleon-density dependence of the strength functionals of an effective interaction Lagrangian. The resulting phenomenological effective interaction, adjusted to experimental binding energies of a large set of axially deformed nuclei, together with a new separable pairing interaction adjusted to reproduce the pairing gap in nuclear matter calculated with the Gogny force, is applied in triaxial relativistic Hartree-Bogoliubov calculations of sequences of heavy nuclei: Th , U , Pu , Cm , Cf , Fm , and No .

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

APPLICATION OF THE FINITE STRIP METHOD TO MODELING OF VIBRATIONS OF COLLECTING ELECTRODES

2013 ◽  
Vol 13 (07) ◽  
pp. 1340001 ◽  
Author(s):  
IWONA ADAMIEC-WÓJCIK ◽  
ANDRZEJ NOWAK ◽  
STANISŁAW WOJCIECH
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Geometrical Parameters ◽  
Experimental Measurements ◽  
Newmark Method ◽  
Algebraic Equations ◽  
Finite Strip ◽  
Strip Method ◽  
Finite Strip Method ◽  
Large Length

The paper presents an application of the finite strip method to modeling of vibrations of the collecting electrodes, which are shells with large length (up to 16 m), width of 0.5 m and thickness of 0.002 m. The models and computer programs have been worked out and validated. Comparison of results obtained from numerical simulations and experimental measurements are presented and discussed. The equations of motion have been solved using methods for solution of sparse algebraic equations and Newmark method. The strip method has proved to be numerically effective. The programs enable us to carry out calculations for a system with several hundred thousands of degrees of freedom with time of analysis requiring thousand integration steps during less than 90 min on a PC computer. High numerical efficiency enables the geometrical parameters of the collecting electrodes to be selected in order to ensure large accelerations caused by a beater to be spread evenly over the surface of the electrodes. Conclusions concerning the influence of length of the collecting electrodes on the normal and tangentz accelerations are formulated.

scholarly journals Modelling of a Microslip Friction Damper Subjected to Translation and Rotation

1999 ◽  
Author(s):  
Gabor Csaba
Keyword(s):  
Surface Roughness ◽  
Degrees Of Freedom ◽  
Forced Response ◽  
Least Square ◽  
Contact Model ◽  
Contact Radius ◽  
Algebraic Equations ◽  
Friction Damper ◽  
Six Degrees Of Freedom ◽  
Winkler Elastic Foundation

This paper presents a friction interface model where one of the mating surfaces is curved. The model is based on a discretization of the Winkler elastic foundation model and is general in the sense that it allows for relative motion in all six degrees of freedom. The variables for the contact model are based on damper geometry and material data, except coefficient of friction and tangential stiffness coefficient, which have to be measured. Simulated and experimental hysteresis curves are presented. A model of a curved wedge damper has been developed using the contact model. An algorithm for solving forces and displacement when the damper is allowed to move in all six degrees of freedom has been presented. The governing algebraic equations are solved using a nonlinear least-square method routine in a commercial software package. Forced response of a beam-damper-beam test set-up has been simulated and compared with experiments. The results highlighted some effects which have not been modelled e.g. the actual contact area between damper and blade is influenced by surface roughness for low normal loads. It is assumed that this effect resulted in problems in getting agreement between experiments and analysis. The influence of surface roughness is assumed to be negligible when vibrations of real turbomachinery are considered. This is due to the fact that both normal and excitation force on the clamper are about ten times higher than what was used in the experiments and simulations in this paper. Variation of contact radius of the damper shows that a larger radius e.g. a flatter contact gives better damping and increases the resonance frequency. The disadvantage is that the alignment of the damper becomes more unreliable.

Sensitivity Analysis for the Operating Efficiency of an Axial Piston Pump

2015 ◽  
Author(s):  
Noah D. Manring ◽  
Viral S. Mehta ◽  
Jeff L. Kuehn ◽  
Bryan E. Nelson
Keyword(s):  
Power Efficiency ◽  
Design Parameters ◽  
Operating Efficiency ◽  
Parameter Study ◽  
Total Efficiency ◽  
Pump Efficiency ◽  
Large Set ◽  
Algebraic Equations ◽  
Pump Design ◽  
Axial Piston

Axial piston pumps of swash-plate type are extensively used in off-highway machines to convert rotating mechanical power into hydraulic power. Efficiency of such pumps is of considerable importance to hydraulic design engineers. Many researchers have tried to create mathematical models for describing pump efficiency. These models are typically a system of nonlinear algebraic equations dependent upon a total of four variables (pressure, speed, temperature, displacement) and a set of experimentally determined coefficients. Since these models are not of the a-priori type, they are not of much value to a design engineer who is trying to design an efficient pump. Others have tried to use physics based models and numerical programs to accurately predict the influence of component design on efficiency. Such programs are considerably slow to run and of not much use to a design engineer who needs to make quick decisions. Hence the objective of this paper is to understand the sensitivity of various design parameters on the total efficiency of the pump by conducting a dimensionless parameter study of a large set of pump design parameters. Using this method it will be shown that a small group of design parameters have the highest influence on the efficiency of these pumps.

Cluster state quantum computation for many-level systems

2007 ◽  
Vol 7 (3) ◽  
pp. 184-208
Author(s):  
W. Hall
Keyword(s):  
Quantum Computation ◽  
Degrees Of Freedom ◽  
Cluster State ◽  
Quantum Systems ◽  
Explicit Description ◽  
State Model ◽  
Mutually Unbiased Bases ◽  
Large Set ◽  
Adaptive Computation ◽  
Complete Framework

The cluster state model for quantum computation [Phys. Rev. Lett. \textbf{86}, 5188] outlines a scheme that allows one to use measurement on a large set of entangled quantum systems in what is known as a cluster state to undertake quantum computations. The model itself and many works dedicated to it involve using entangled qubits. In this paper we consider the issue of using entangled qudits instead. We present a complete framework for cluster state quantum computation using qudits, which not only contains the features of the original qubit model but also contains the new idea of adaptive computation: via a change in the classical computation that helps to correct the errors that are inherent in the model, the implemented quantum computation can be changed. This feature arises through the extra degrees of freedom that appear when using qudits. Finally, for prime dimensions, we give a very explicit description of the model, making use of mutually unbiased bases.

Numerical and experimental performance estimation for a ExoFing - 2 DOFs finger exoskeleton

Robotica ◽  
2021 ◽  
pp. 1-13
Author(s):  
G Carbone ◽  
M Ceccarelli ◽  
C. E. Capalbo ◽  
G Caroleo ◽  
C Morales-Cruz
Keyword(s):  
Degrees Of Freedom ◽  
Experimental Validation ◽  
Index Finger ◽  
Kinematic Model ◽  
Experimental Tests ◽  
Performance Estimation ◽  
Experimental Performance ◽  
Finger Motion ◽  
User Friendly ◽  
Operation Characteristics

Abstract This paper presents a numerical and experimental validation of ExoFing, a two-degrees-of-freedom finger mechanism exoskeleton. The main functionalities of this device are investigated by focusing on its kinematic model and by computing its main operation characteristics via numerical simulations. Experimental tests are designed and carried out for validating both the engineering feasibility and effectiveness of the ExoFing system aiming at achieving a human index finger motion assistance with cost-oriented and user-friendly features.

System-Level Modal Representation of Flexible Multibody Systems

2009 ◽  
Author(s):  
Gert H. K. Heirman ◽  
Wim Desmet
Keyword(s):  
Model Reduction ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Flexible Multibody Systems ◽  
System Level ◽  
Algebraic Equations ◽  
Flexible Multibody ◽  
Simulation Results ◽  
Model Equations ◽  
Model Reduction Technique

The presence of both differential and algebraic equations in the model equations, as well as the number of degrees of freedom needed to accurately represent flexibility, prohibit fast simulation of flexible multibody systems (e.g. real-time). In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of a fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The complexity of simulation of the reduced model equations is estimated. In a numerical experiment, simulation results for the original model equations are compared with simulation results for the model equations obtained after model reduction, showing a good match. The dominant sources of error of the proposed methodology are illustrated and explained.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

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