The Gibbs-Wilbraham phenomenon: An episode in fourier analysis

Hewitt, Edwin; Hewitt, Robert E.

doi:10.1007/bf00330404

Cited by 300 publications

(177 citation statements)

References 31 publications

Supporting

Mentioning

172

Contrasting

Unclassified

Order By: Relevance

“…In other areas of science, the collapse of biological populations was well known from the time of Lotka (1925) andof Volterra (1928). In mathematics, the discontinuities shown by some function were examined already in the mid-nineteenth century in terms of the ''Gibbs-Willbraham'' phenomenon (Hewitt and Hewitt 1979). This series of studies gave rise of the use of the term ''overshoot'' in control theory and in electronics, as well as that of ''feedback.''…”

mentioning

confidence: 99%

Jay Wright Forrester (1918–2016): His Contribution to the Concept of Overshoot in Socioeconomic Systems

Bardi

2016

Biophys Econ Resour Qual

View full text Add to dashboard Cite

The recent death of Jay Wright Forrester has generated several obituaries in the press describing his many achievements. However, often his involvement in the somewhat controversial issue of «World Modeling» and of «The Limits to Growth» has gone unmentioned or only barely mentioned [see, e.g., the article that appeared in The New York Times (Hafner in Jay W. Forrester dies at 98; a pioneer in computer models, 2016)]. In this commentary, I aim at providing a brief discussion of one of Forrester's main scientific achievements: the introduction of the concept of overshoot and collapse as applied to socioeconomic systems.

show abstract

mentioning

confidence: 99%

Jay Wright Forrester (1918–2016): His Contribution to the Concept of Overshoot in Socioeconomic Systems

Bardi

2016

Biophys Econ Resour Qual

View full text Add to dashboard Cite

show abstract

“…The negligible thickness of shock waves compared with the volume of the simulation implies that the FFT of this discontinuous distribution might display numerical artifacts. In particular, the presence of strong discrete signals surrounded by null values could act as flat window functions, which can lead to oscillations in Fourier space around the signal wavelength (Hewitt and Hewitt 1979).…”

Section: Appendix A: Fft Methods For the Shocks Distributionmentioning

confidence: 99%

On the interplay between cosmological shock waves and their environment

Martin-Alvarez

Planelles

Quilis

2017

Astrophys Space Sci

View full text Add to dashboard Cite

Cosmological shock waves are tracers of the thermal history of the structures in the Universe. They play a crucial role in redistributing the energy within the cosmic structures and are also amongst the main ingredients of galaxy and galaxy cluster formation. Understanding this important function requires a proper description of the interplay between shocks and the different environments where they can be found. In this paper, an Adaptive Mesh Refinement (AMR) Eulerian cosmological simulation is analysed by means of a shock-finding algorithm that allows to generate shock wave maps. Based on the population of dark matter halos and on the distribution of density contrast in the simulation, we classify the shocks in five different environments. These range from galaxy clusters to voids. The shock distribution function and the shocks power spectrum are studied for these environments dynamics. We find that shock waves on different environments undergo different formation and evolution processes, showing as well different characteristics. We identify three different phases of formation, evolution and dissipation of these shock waves, and an intricate migration between distinct environments and scales. Shock waves initially form at external, low density regions and are merged and amplified through the collapse of structures. Shock waves and cosmic structures follow a parallel evolution. Later on, shocks start to detach from them and dissipate. We also find that most of the power that shock waves dissipate is found at scales of k ∼ 0.5 Mpc −1 , with a secondary peak at k ∼ 8 Mpc −1 . The evolution of the shocks power spectrum confirms that shock waves evolution is coupled and conditioned by their environment.

show abstract

“…The standard spectral scheme described above, in the presence of discontinuities in τ(X) and F(X), exhibits numerical ringing artifacts associated with the Gibbs phenomenon for truncated Fourier series (Gibbs, 1898(Gibbs, , 1899Hewitt and Hewitt, 1979). Inspired by Willot et al (2014), Berbenni et al (2014) and Lebensohn and Needleman (2016), we use discrete (modified) spectral differentiation with a finite difference-based scheme (Müller, 1996).…”

Section: Constitutive Model: Finite-strain Crystal Plasticity In Magnmentioning

confidence: 99%

Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium

Vidyasagar

Tutcuoglu

Kochmann

2018

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

Complex microstructural patterns arise as energy-minimizers in systems having non-convex energy landscapes such as those associated with phase transformations, deformation twinning, or finite-strain crystal plasticity. The prediction of such patterns at the microscale along with the resulting, effective material response at the macroscale is key to understanding a wide range of mechanical phenomena and has classically been dealt with by simplifying energy relaxation theory or by expensive finite element calculations. Here, we discuss a stabilized Fourier spectral technique for the homogenized response at the level of a representative volume element (RVE). We show that the FFT-based method admits sufficiently high resolution suitable to predict the emergence of energy-minimizing microstructures and the resulting effective response by computing the approximated quasiconvex energy hull. We test the method in the classical single-slip problem in single-and bicrystals. Especially the latter goes beyond the scope of traditional finite element and analytical relaxation treatments and hints at mechanisms of pattern formation in polycrystals. We also demonstrate that the chosen spectral finite-difference approximation, important for removing ringing artifacts in the presence of high contrasts, adds a natural regularization to the non-convex minimization. Finally, the technique is applied to polycrystalline pure magnesium, where we account for the competition between dislocation-mediated plasticity and deformation twinning. These inelastic deformation mechanisms result in complex texture evolution paths at the polycrystalline mesoscale and are simulated within RVEs of varying grain size and texture by a constitutive crystal plasticity model with an effective, volume fraction-based description of twinning.

show abstract

The Gibbs-Wilbraham phenomenon: An episode in fourier analysis

Cited by 300 publications

References 31 publications

Jay Wright Forrester (1918–2016): His Contribution to the Concept of Overshoot in Socioeconomic Systems

Jay Wright Forrester (1918–2016): His Contribution to the Concept of Overshoot in Socioeconomic Systems

On the interplay between cosmological shock waves and their environment

Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium

Contact Info

Product

Resources

About