Nonlinear Feedback in a Five-Dimensional Lorenz Model

2014 ◽  
Vol 71 (5) ◽  
pp. 1701-1723 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Dissipative Term ◽  
Higher Dimensional ◽  
The Impact ◽  
Underlying Processes

Abstract In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term

2015 ◽  
Vol 22 (6) ◽  
pp. 749-764 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the streamfunction is referred to as a secondary streamfunction mode, while the two additional modes, which appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): "If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado." The implications of this and previous work, as well as future work, are also discussed.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

2018 ◽  
Vol 28 (06) ◽  
pp. 1850072 ◽  
Author(s):  
Sara Faghih-Naini ◽  
Bo-Wen Shen
Keyword(s):  
Periodic Solution ◽  
Feedback Loop ◽  
Three Dimensional ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Oscillatory Solutions ◽  
Nonlinear Temperature

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

A Sociocultural Perspective on Negotiating Digital Identities in a Community of Learners

2013 ◽  
pp. 210-224 ◽  
Author(s):  
Anna Peachey ◽  
Greg Withnail
Keyword(s):  
Social Interaction ◽  
Identity Development ◽  
Three Dimensional ◽  
Digital Identity ◽  
Sociocultural Perspective ◽  
Regular Feature ◽  
Virtual World Environments ◽  
The Impact ◽  
Digital Identities

Three dimensional virtual world environments are becoming an increasingly regular feature of the education landscape, providing the opportunity for richly graphical augmented and immersive learning activities. Those who participate in these experiences must mediate through an avatar, negotiating and managing the complexities of this new variation of digital identity alongside their more familiar identity as learner and/or teacher/facilitator. This chapter describes some key moments in the construction of digital identities as a lecturer and a student in the Open University’s community in Second LifeTM. The authors explore experiences in relation to the impact of trust and consistency from a sociocultural perspective, privileging the role of social interaction and context where meaning is socially produced and situationally interpreted, concluding that social interaction is pivotal to any meaningful identity development that takes place. The chapter ends with thoughts for future issues surrounding digital identity in relation to lifelong learning.

Role of Through Silicon Via in 3D Integration: Impact on Delay and Power

2020 ◽  
pp. 2150051
Author(s):  
Shivangi Chandrakar ◽  
Deepika Gupta ◽  
Manoj Kumar Majumder
Keyword(s):  
Power Dissipation ◽  
Three Dimensional ◽  
Propagation Delay ◽  
Model Parameters ◽  
3D Integration ◽  
Coupling Behavior ◽  
Power Delay Product ◽  
Silicon Vias ◽  
The Impact

The metal–semiconductor (MES)-based through silicon vias (TSV) has provided attractive solutions over conventional metal–insulator–semiconductor (MIS) TSVs in recent three-dimensional (3D) integration. This paper aims a comprehensive performance analysis of MIS and MES structures considering different TSV shapes such as cylindrical, tapered, annular, and square. At 32[Formula: see text]nm technology, a CMOS-based coupled driver-via-load (DVL) setup is introduced wherein each via is represented an equivalent RLGC model of MIS- and MES-based TSV shapes. The proposed electrical model accurately considers the impact of micro bump and inter-metal dielectric (IMD) effects at 32[Formula: see text]nm technology as per the fabrication house. A 3D electromagnetic (EM) structural wave simulation is performed to validate the RLGC model parameters of different TSV structures for an operating frequency of up to 20[Formula: see text]GHz. The proposed DVL setup is used to analyze the propagation delay, power dissipation, and dynamic crosstalk for different MIS- and MES-based TSV shapes. A significant improvement in the cross-coupling behavior can be obtained using the MES-based tapered TSV compared to the other MIS structures. Additionally, the power delay product (PDP) of the tapered MES is reduced by 92.4% compared to the conventional MIS-based cylindrical TSV.

scholarly journals The Role of Nanoparticle Shapes and Structures in Material Characterisation of Polyvinyl Alcohol (PVA) Bionanocomposite Films

Polymers ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 264 ◽  
Author(s):  
Mohanad Mousa ◽  
Yu Dong
Keyword(s):  
Polyvinyl Alcohol ◽  
Three Dimensional ◽  
Halloysite Nanotubes ◽  
Mechanical And Thermal Properties ◽  
Cloisite 30B ◽  
Bionanocomposite Films ◽  
The Impact ◽  
Nanoparticle Shapes ◽  
Selection Of

Three different types of nanoparticles, 1D Cloisite 30B clay nanoplatelets, 2D halloysite nanotubes (HNTs), and 3D nanobamboo charcoals (NBCs) were employed to investigate the impact of nanoparticle shapes and structures on the material performance of polyvinyl alcohol (PVA) bionanocomposite films in terms of their mechanical and thermal properties, morphological structures, and nanomechanical behaviour. The overall results revealed the superior reinforcement efficiency of NBCs to Cloisite 30B clays and HNTs, owing to their typical porous structures to actively interact with PVA matrices in the combined formation of strong mechanical and hydrogen bondings. Three-dimensional NBCs also achieved better nanoparticle dispersibility when compared with 1D Cloisite 30B clays and 2D HNTs along with higher thermal stability, which was attributed to their larger interfacial regions when characterised for the nanomechanical behaviour of corresponding bionanocomposite films. Our study offers an insightful guidance to the appropriate selection of nanoparticles as effective reinforcements and the further sophisticated design of bionanocomposite materials.

The Impact of Example Modality and Physical Interactions on Design Creativity

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Christine A. Toh ◽  
Scarlett R. Miller
Keyword(s):  
Engineering Design ◽  
Three Dimensional ◽  
Idea Generation ◽  
Controlled Study ◽  
First Year ◽  
Design Creativity ◽  
Physical Interactions ◽  
2D And 3D ◽  
The Impact

Interacting with example products is an essential and widely practiced method in engineering design, yet little information exists on how the representation (pictorial or physical) or interaction a designer has with an example impacts design creativity. This is problematic because without this knowledge we do not understand how examples affect idea generation or how we can effectively modify or develop design methods to support example usage practices. In this paper, we report the results of a controlled study with first year engineering design students (N = 89) developed to investigate the impact of a designer's interaction with either a two-dimensional (2D) pictorial image or a three-dimensional (3D) product (through visual inspection or product dissection activities) and the resulting functional focus and creativity of the ideas developed. The results of this study reveal that participants who interacted with the physical example produced ideas that were less novel and less functionally focused than those who interacted with the 2D representation. Additionally, the results showed that participants who dissected the product produced a higher variety of ideas than those that visually inspected it. These results contribute to our understanding of the benefits and role of 2D and 3D designer-product interactions during idea development. We use these findings to develop recommendations for the use of designer-product interactions throughout the design process.

Modeling of the impact of rigid ellipsoidal blocks by means of an elastic-visco-plastic constitutive model 

2021 ◽  
Author(s):  
Giuseppe Dattola ◽  
Giovanni Battista Crosta ◽  
Claudio Giulio di Prisco
Keyword(s):  
Constitutive Model ◽  
Three Dimensional ◽  
Impact Angle ◽  
Material Point ◽  
Lumped Mass ◽  
Mountain Areas ◽  
Block Motion ◽  
The Impact ◽  
Constitutive Parameters

<p>Rockfall is one of the most common hazards in mountain areas causing severe damages to structures/infrastructures and, human lives. For this reason, numerous are the papers published in the last decades on this subject, both introducing reliable approaches to simulate the boulder trajectory and defining design methods for sheltering structures. As is well known, the most popular strategy to simulate the block trajectory and velocity is based on the lumped mass material point approach. This is capable of describing the block trajectory, before either its natural arrest or impact against an artificial/natural obstacle, by suitably considering its interaction with soil/rock materials, interaction always dynamic, very often highly dissipative and defined, according to its nature, as sliding, rolling or impact.</p><p>In this framework, this study focusses on impacts and, in particular, on the role of block geometry in affecting the block kinematic response. The problem is approached numerically; by modifying a previously conceived elastic-viscoplastic constitutive model, based on the macro-element concept. and capable of satisfactorily simulating impacts of spherical blocks.</p><p>The modified constitutive model relaxes the assumption of spherical block by assuming an ellipsoidal shape and by allowing for the boulder rotation. These two changes make the problem more complex but allow to model more realistically the impact. For the sake of simplicity, the results shown in this work consider the block motion to be planar, but the model already allows to include general three dimensional conditions.</p><p>In this work, the model is briefly outlined and the procedure for calibrating the model constitutive parameters described. Then, the results of an extensive parametric analysis, employing constitutive parameters calibrated on experimental data taken from the literature, are discussed. In particular, the role of (i) the inner block orientation, and (ii) the inner impact angle is considered in terms of both kinematic variables and restitution coefficients. Finally, interpolation functions to compute restitution coefficients, once both block shape and inner impact block orientation are known, are provided.</p>

scholarly journals Granular jets and hydraulic jumps on an inclined plane

2011 ◽  
Vol 675 ◽  
pp. 87-116 ◽  
Author(s):  
C. G. JOHNSON ◽  
J. M. N. T. GRAY
Keyword(s):  
Granular Material ◽  
Numerical Solutions ◽  
Slope Angle ◽  
Three Dimensional ◽  
Breaking Wave ◽  
Hydraulic Jumps ◽  
Inclined Plane ◽  
Steady Regime ◽  
Diverse Range ◽  
The Impact

A jet of granular material impinging on an inclined plane produces a diverse range of flows, from steady hydraulic jumps to periodic avalanches, self-channelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steady-state flow, in which the jet generates a closed teardrop-shaped hydraulic jump on the plane, enclosing a region of fast-moving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardrop-shaped, but exhibits a more complex ‘blunted’ shape with a steadily breaking wave. We explain these regimes by consideration of the supercritical or subcritical nature of the flow surrounding the shock. A model is developed in which the impact of the jet on the inclined plane is treated as an inviscid flow, which is then coupled to a depth-integrated model for the resulting thin granular avalanche on the inclined plane. Numerical simulations produce a flow regime diagram strikingly similar to that obtained in experiments, with the model correctly reproducing the regimes and their dependence on the jet velocity and slope angle. The size and shape of the steady experimental shocks and the location of sub- and supercritical flow regions are also both accurately predicted. We find that the physics underlying the rapid flow inside the shock is dominated by depth-averaged mass and momentum transport, with granular friction, pressure gradients and three-dimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a friction–gravity balance, and some flow features, such as a persistent indentation in the free surface, are not reproduced in the numerical solutions. On planes inclined at a shallow angle, the effect of stationary granular material becomes important in the flow evolution, and oscillatory and more general time-dependent flows are observed. The hysteretic transition between static and dynamic friction leads to two phenomena observed in the flows: unsteady avalanching behaviour, and the feedback from static grains on the flowing region, leading to levéed, self-channelised flows.

scholarly journals Improvisation as Responsible Innovation in Organizations

2021 ◽  
Vol 13 (4) ◽  
pp. 1597
Author(s):  
Milena Gojny-Zbierowska ◽  
Przemysław Zbierowski
Keyword(s):  
Empirical Evidence ◽  
Entrepreneurial Orientation ◽  
Positive Impact ◽  
Three Dimensional ◽  
Responsible Innovation ◽  
Senior Managers ◽  
Different Dimensions ◽  
The Impact ◽  
Responsible Manner

Improvisation might be seen as a method of responsible innovation in organizations, due to its potential to be more responsive and enable bottom-up initiative. Considering that improvising involves the ability to pivot we argue that enhancing entrepreneurial orientation of existing firms means that their entrepreneurial behaviors can be also displayed in more responsible manner. The paper aims at investigating the influence of improvisation on entrepreneurial orientation (EO). While intuitively improvisation is closely connected to EO, surprisingly, there is very little theoretical and empirical evidence on that relation. The paper closes that gap by empirically investigating the role that improvisation plays in enhancing EO. Building on empirical evidence on the role of improvising in individual entrepreneurship, we use Hmieleski and Corbett’s framework of improvisation as a three-dimensional construct (creativity and bricolage, ability to function and excel under pressure and in stress-filled environments, and spontaneity and persistence) and entrepreneurial orientation as a three-dimensional construct (innovativeness, proactiveness, and risk taking) to investigate the impact of improvisation on individual components of EO. Using the data from 567 senior managers from medium and large organizations we find that improvisation has moderate effect on entrepreneurial orientation. Importantly, different dimensions of improvisation shape components of EO in different way: Creativity and bricolage have positive impact on innovativeness and proactiveness and ability to function and excel under stress has impact on propensity to take risk. The study has implications for the theory of responsible innovation by highlighting the potential of improvising to generate more responsive and stakeholder-involving and, in consequence, more responsible innovation.

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