Analyzing Bilinear Systems Using a New Hybrid Symbolic-Numeric Computational Method

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Meng-Hsuan Tien ◽  
Kiran D'Souza
Keyword(s):  
Linear Response ◽  
Nonlinear Response ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Bilinear Systems ◽  
Computational Method ◽  
Transition Time ◽  
Time Interval ◽  
Beam Model ◽  
Nonlinear Characteristics

In this paper, an efficient and accurate computational method for determining responses of high-dimensional bilinear systems is developed. Predicting the dynamics of bilinear systems is computationally challenging since the piecewise-linear nonlinearity induced by contact eliminates the use of efficient linear analysis techniques. The new method, which is referred to as the hybrid symbolic-numeric computational (HSNC) method, is based on the idea that the entire nonlinear response of a bilinear system can be constructed by combining linear responses in each time interval where the system behaves linearly. The linear response in each time interval can be symbolically expressed in terms of the initial conditions. The transition time where the system switches from one linear state to the other and the displacement and velocity at the instant of transition are solved using a numerical scheme. The entire nonlinear response can then be obtained by joining each piece of the linear response together at the transition time points. The HSNC method is based on using linear features to obtain large computational savings. Both the transient and steady-state response of bilinear systems can be computed using the HSNC method. Thus, nonlinear characteristics, such as subharmonic motion, bifurcation, chaos, and multistability, can be efficiently analyzed using the HSNC method. The HSNC method is demonstrated on a single degree-of-freedom (DOF) system and a cracked cantilever beam model, and the nonlinear characteristics of these systems are examined.

A Hybrid Symbolic-Numeric Computational Method for Analysis of Bilinear Systems

2018 ◽  
Author(s):  
Meng-Hsuan Tien ◽  
Kiran D’Souza
Keyword(s):  
Linear Response ◽  
Nonlinear Response ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Bilinear Systems ◽  
Computational Method ◽  
Transition Time ◽  
Time Interval ◽  
Beam Model ◽  
Nonlinear Characteristics

In this paper, an efficient and accurate computational method for determining responses of high-dimensional bilinear systems is developed. Predicting the dynamics of bilinear systems is computationally challenging since the piecewise-linear nonlinearity induced by contact eliminates the use of efficient linear analysis techniques. The new method, which is referred to as the hybrid symbolic-numeric computational (HSNC) method, is based on the idea that the entire nonlinear response of a bilinear system can be constructed by combining linear responses in each time interval where the system behaves linearly. The linear response in each time interval can be symbolically expressed in terms of the initial conditions. The transition time where the system switches from one linear state to the other and the displacement and velocity at the instant of transition are solved using a numerical scheme. The entire nonlinear response can then be obtained by joining each piece of the linear response together at the transition time points. The HSNC method is based on using linear features to obtain large computational savings. Both the transient and steady-state response of bilinear systems can be computed using the HSNC method. Thus, nonlinear characteristics, such as subharmonic motion, bifurcation, chaos, and multistability, can be efficiently analyzed using the HSNC method. The HSNC method is demonstrated on a single degree of freedom system and a cracked cantilever beam model, and the nonlinear characteristics of these systems are examined.

Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations

2001 ◽  
Vol 446 ◽  
pp. 133-171 ◽  
Author(s):  
VICTOR I. SHRIRA ◽  
IGOR A. SAZONOV
Keyword(s):  
Boundary Layer ◽  
Shear Flows ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Initial Perturbation ◽  
Time Interval ◽  
Two Dimensional ◽  
Leading Order ◽  
Perturbation Field ◽  
Long Wave

The work, being the first in a series concerned with the evolution of small perturbations in shear flows, studies the linear initial-value problem for inviscid spatially harmonic perturbations of two-dimensional shear flows of boundary-layer type without inflection points. Of main interest are the perturbations of wavelengths 2π/k long compared to the boundary-layer thickness H, kH = ε [Lt ] 1. By means of an asymptotic expansion, based on the smallness of ε, we show that for a generic initial perturbation there is a long time interval of duration ∼ ε−3 ln(1/ε), where the perturbation representing an aggregate of continuous spectrum modes of the Rayleigh equation behaves as if it were a single discrete spectrum mode having no singularity to the leading order. Following Briggs et al. (1970), who introduced the concept of decaying wave-like perturbations due to the presence of the ‘Landau pole’ into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contribution to the entire perturbation field evolves for different field characteristics. We find that over O(ε−3 ln(1/ε)) time interval, the quasi-mode dominates the velocity field. In particular, over this interval the share of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O(ε4) (O(ε4) for the Blasius flow). The mode is weakly decaying, as exp(−ε3t). At larger times the quasi-mode ceases to dominate in the perturbation field and the perturbation decay law switches to the classical t−2. By definition, the quasi-modes are singular in a critical layer; however, we show that in our context their singularity does not appear in the leading order. From the physical viewpoint, the presence of a small jump in the higher orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Since we have established that the decay rate of the quasi-modes sharply increases with the increase of the wavenumber, one of the major conjectures of the analysis is that the long-wave components prevail in the large-time asymptotics of a wide class of initial perturbations, not necessarily the predominantly long-wave perturbations. Thus, the explicit expressions derived in the long-wave approximation describe the asymptotics of a much wider class of initial conditions than might have been anticipated. The concept of quasi-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics.

A New Method to Find the Forced Response of Nonlinear Systems with Dry Friction

2021 ◽  
Author(s):  
Gregory L. Altamirano ◽  
Meng-Hsuan Tien ◽  
Kiran D'Souza
Keyword(s):  
Dry Friction ◽  
Coulomb Friction ◽  
Nonlinear Response ◽  
Piecewise Linear ◽  
Time Integration ◽  
Nonlinear Behavior ◽  
Forced Response ◽  
New Method ◽  
Analysis Tools ◽  
Compatibility Constraint

Abstract Coulomb friction has an influence on the behavior of numerous mechanical systems. Coulomb friction systems or dry friction systems are nonlinear in nature. This nonlinear behavior requires complex and time demanding analysis tools to capture the dynamics of these systems. Recently, efforts have been made to develop efficient analysis tools able to approximate the forced response of systems with dry friction. The objective of this paper is to introduce a methodology that assists in these efforts. In this method, the piecewise-linear nonlinear response is separated into individual linear responses that are coupled together through compatibility constraint equations. The new method is demonstrated on a number of systems of varying complexity. The results obtained by the new method are validated through the comparison with results obtained by time integration. The computational savings of the new method is also discussed.

scholarly journals Analysis of periodically switched nonlinear circuits and nonlinear oversampled sigma-delta modulators

2021 ◽  
Author(s):  
Quan Li
Keyword(s):  
Initial Conditions ◽  
Nonlinear Circuits ◽  
Brute Force ◽  
Nonlinear Characteristics ◽  
Sigma Delta ◽  
Sigma Delta Modulators ◽  
Newton Raphson ◽  
Step Algorithm ◽  
Time Domain Response ◽  
Volterra Functional Series

This thesis proposes a new method for time domain response and sensitivity analysis of periodically switched nohnlinear circuits and nonlinear oversampled sigma-delta modulators. Using Volterra functional series and interpolating Fourier series, it extends the sampled datasimulation and two-step algorithm of linear circuits to periodically switched nonlinear circuits. The method can handle the inconsistent initial conditions encountered at switching instants. It is applied to the analysis of nonlinear oversampled sigma-delta modulators. The efficiency and accuracy of the proposed method are assessed using SPICE, brute-force, and the law of charge conservation on periodically switched nonlinear circuits, and nonlinear oversampled sigma-delta modulators. The method does not require the costly Newton-Raphson iterations. It is most efficient for analysis of circuits with mildly nonlinear characteristics and simulation over a long period of time.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

2021 ◽  
pp. 15-38
Author(s):  
D.Y. Ivanov ◽  
Keyword(s):  
Fourier Series ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Initial Conditions ◽  
Fourier Method ◽  
Time Interval ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Element Method

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

Solving linear systems from dynamical energy analysis - using and reusing preconditioners

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell
Keyword(s):  
Wave Equations ◽  
Energy Analysis ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Computational Method ◽  
Computational Time ◽  
Shell Elements ◽  
Finite Dimensional ◽  
Large Matrix

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

scholarly journals A Heuristic Approach for Precipitation Data Assimilation: Characterization Using OSSEs

2019 ◽  
Vol 147 (9) ◽  
pp. 3445-3466 ◽  
Author(s):  
Andrés A. Pérez Hortal ◽  
Isztar Zawadzki ◽  
M. K. Yau
Keyword(s):  
Data Assimilation ◽  
Initial Conditions ◽  
Grid Point ◽  
Wrf Model ◽  
Ensemble Member ◽  
Absolute Difference ◽  
Time Interval ◽  
State Variables ◽  
Vertical Column ◽  
A New Technique

Abstract We introduce a new technique for the assimilation of precipitation observations, the localized ensemble mosaic assimilation (LEMA). The method constructs an analysis by selecting, for each vertical column in the model, the ensemble member with precipitation at the ground that is locally closest to the observed values. The proximity between the modeled and observed precipitation is determined by the mean absolute difference of precipitation intensity, converted to reflectivity and measured over a spatiotemporal window centered at each grid point of the model. The underlying hypothesis of the approach is that the ensemble members that are locally closer to the observed precipitation are more probable to be closer to the “truth” in the state variables than the other members. The initial conditions for the new forecast are obtained by nudging the background states toward the mosaic of the closest ensemble members (analysis) over a 30 min time interval, reducing the impacts of the imbalances at the boundaries between the different selected members. The potential of the method is studied using observing system simulation experiments (OSSEs) employing a small ensemble of 20 members. The ensemble is produced by the WRF Model, run at a horizontal grid spacing of 20 km. The experiments lend support to the validity of the hypothesis and allow the determination of the optimal parameters for the approach. In the context of OSSE, this new data assimilation technique is able to produce forecasts with considerable and long-lived error reductions in the fields of precipitation, temperature, humidity, and wind.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Optimal Control against the Human Papillomavirus: Protection versus Eradication of the Infection

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Fernando Saldaña ◽  
Andrei Korobeinikov ◽  
Ignacio Barradas
Keyword(s):  
Optimal Control ◽  
Human Papillomavirus ◽  
Control Problem ◽  
Initial Conditions ◽  
Fixed Time ◽  
Hpv Infection ◽  
Control Problems ◽  
Time Interval ◽  
Optimal Controls ◽  
Total Prevalence

We investigate the optimal vaccination and screening strategies to minimize human papillomavirus (HPV) associated morbidity and the interventions cost. We propose a two-sex compartmental model of HPV-infection with time-dependent controls (vaccination of adolescents, adults, and screening) which can act simultaneously. We formulate optimal control problems complementing our model with two different objective functionals. The first functional corresponds to the protection of the vulnerable group and the control problem consists of minimizing the cumulative level of infected females over a fixed time interval. The second functional aims to eliminate the infection, and, thus, the control problem consists of minimizing the total prevalence at the end of the time interval. We prove the existence of solutions for the control problems, characterize the optimal controls, and carry out numerical simulations using various initial conditions. The results and properties and drawbacks of the model are discussed.

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