P. J. Masson, P. M. Silveira, C. Duque, and P. F. Ribeiro

Chapter 5 - Fourier Series: Visualizing Joseph Fourier’s Imaginative Discovery via FEA and Time-Frequency Decomposition

6.1 Introduction

Joseph Fourier was with no doubt one of the greatest minds whose work changed the face of engineering. His contributions to physics and engineering are numerous; however, he is mostly remembered for his work on heat transfer that leads to the widely used transform named after him. Through a simple heat transfer experiment, Fourier noticed that the shape of temperature distribution in a ring was varying with time to become eventually a sinusoidal distribution. He then had the idea of representing the periodic temperature distribution by a sum of sinusoids giving birth to a novel signal analysis method nowadays used in all disciplines of engineering.

Fourier analysis is one of the most utilized mathematical techniques across a wide range of disciplines that covers from physics and engineering to biology, economics, oceanography, and other areas. However, the geniality of Joseph Fourier in the process of his discovery is not well known and recognized. This chapter attempts to reproduce some of the original experiments carried out that Joseph Fourier conducted via the use of finite element analysis (FEA). The authors expect that the results will bring more insight into the understanding of this revolutionary technique as well as illustrate the fundamental need for an integration of physical intuition and mathematics for the progress of scientific developments.

This chapter presents Fourier’s heat transfer experiment through the use of finite element analysis. An iron ring is modeled and transient thermal analysis is performed to reproduce the data Fourier obtained experimentally.

6.2 The historical context

The life of Jean Joseph Fourier was an exciting and controversial one, to say the least. Among the several facets of his life one can list: teacher, secrete policeman, political prisoner, governor of Egypt, mayor, friend of Napoleon and finally secretary of the Academy of Sciences in Paris.

Joseph Fourier was born in Auxerre, France, as the son of a tailor. He was recommended to the Bishop of Auxerre, and was educated by the Benvenistes of the Convent of St. Mark. He accepted a military lectureship on mathematics, took a prominent part in promoting the French Revolution, and was rewarded by an appointment in 1795 in the École Normale Supérieure, and by a chair at the École Polytechnique. A few years later, he traveled to North Africa with Napoleon Bonaparte and became Governor of Lower Egypt. After he came back to France to become prefect of Isere, he resumed his work on heat propagation. In 1822, he became permanent secretary of the French Academy of Sciences. In this year, Fourier published his “Theorie Analytique de la Chaleur”; in this work he claimed that any function of a variable can be decomposed in a sum of cosines of a multiple of that variable. Even though this statement was not entirely correct, the breakthrough was there. Dirichlet later determined the restrictions to Fourier’s statement. The bases left by Fourier were reworked by numerous mathematicians until 1829 when a final demonstration was proposed by Sturm.

Fourier’s work was not limited to heat transfer and numerous discoveries have been attributed to his genius such as greenhouse effect gases and our planet’s energy balance. [1-5]

6.3 The Experiment

Joseph Fourier was working on heat transfer / propagation when he started experimenting and observing the heat transfer and temperature distribution along metallic materials. Having used a thin bar first, he later on used a polished iron ring with a diameter of approximately 30 cm held in place by wooden supports. Holes were drilled into the ring for thermometers. A time dependent experiment was carried out where the ring was placed halfway into a furnace and then removed to an insulating bath of sand. The initial distribution of temperature was of uniformly hot around one half and cold around the other as shown in Fig. 2. Figure 1 was constructed using FEA and the characteristics of the ring used by Joseph Fourier. The temperature then started to change as the heat flowed from hot to cold and illustrated in Fig. 2 to Fig. 7. Fourier observed the nature of the distribution of the variation and associated it with simple sinusoidal patterns. Despite the imprecision of the instruments and tools used, Joseph Fourier’s genial mind was able to think of the initial distribution of temperature, and shape as time went on, as a superposition of simple sinusoids that varied from peak to trough to peak an integer number of times along the circumference of the ring. As a result, Fourier could propose his creative and revolutionary equation.

A summary of the idea that prompted the concept is depicted in Fig. 1a and Fig. 1b, where the behavior of the temperature distribution along the spatial discontinuity versus time (1a) and as a function of the number of Fourier series coefficients (1b) are respectively illustrated.

Figure 1 – (a) Temperature distribution around the discontinuity obtained by FEA for different times.

Figure 1 – (b) Temperature distribution around the discontinuity as a function of the number of Fourier coefficients .

6.4 The finite element analysis modeling simulation

As in Fourier’s experiment, an iron ring has been modeled in Finite Element Analysis software COMSOL Multiphysics. The ring shown in figure 1 is composed of iron with a thermal conductivity and heat capacity depending on temperature. The boundary is assumed adiabatic. Due to its large thermal diffusion time, iron can accommodate a pretty steep temperature gradient which is set to maximum at initial time. The initial temperature conditions are 600 K in half of the ring and 300 K for the second half. At t=0, heat transfer begins as represented in figures 2 to 7. Of course, as no external heat transfer exists, the equilibrium temperature reached is the average 450 K. Fig. 8a shows the evolution of temperature along the mean radius of the ring; as time passes, the temperature profile shows less harmonics content to end up being a pure sinusoid.

Fig. 2 – Geometry modeled in COMSOL Multiphysics.

Fig. 3 – Initial temperature distribution

Fig. 4 – Time step after initial distribution – variation seems to take a simple sinusoidal pattern

Fig.5 – Additional time step

Fig. 6 – Additional time step

Fig. 7 – Additional time step

Fig. 8 – Final temperature distribution for the entire ring.

6.5 Time-frequency Analysis could have been discovered

Fourier reasoned that the higher frequency sinusoids would damp out rapidly. For example, a sinusoid with twice the frequency would imply that the distance between hot peak and cold trough was halved and on top of that the temperature gradient would be doubled. As a result, a sinusoidal distribution with twice the frequency would dampen at four times the rate. Based on this observation, the Fourier series was developed for steady-state signals which assume that all the sinusoidal components would be present all the time. But in reality Joseph Fourier’s experiment was a time-dependent one and time-varying components could have been included.

In order to exemplify to analyze the experiment with time-frequency resolution, the behavior of the temperature distribution can be approximated by a quadratic waveform, whose initial discontinuity changes and smoothes along the time as the temperature reaches stabilization, as illustrated in Fig. 9.

Fig. 9 – Representation of temperature variation versus time

The time-varying components can be prompted and observed if Fourier had more advanced mathematical tools (time-frequency analysis). Figure 10, for example, illustrates the solution of the time-dependent experiment using a methodology developed [6] in which the behavior time-varying harmonic components are observed as the temperature varies. This methodology based on Fourier’s analysis can be accomplished by several approaches which uses a sliding window to overcome the steady-state requirement of the Fourier series.

Fig. 10 – Decomposition of temperature variation by time-varying harmonics

Considering that during the first cycles of the temperature behavior is like a square waveform, it is difficult to recompose the original signal using only sinusoidal components, even using a high order of components (N). In fact, since the square wave satisfies the Dirichlet conditions, the limit of f(t) at the discontinuities, as Nà¥, is the average value of the discontinuity. According to [7], when a signal has a discontinuity of unity height, the partial sum of the components results in a maximum value of 1.09, i.e., there will be an overshoot of 9% of the height of the discontinuity, independently of how large N is (see Fig. 2 and Fig. 8a). This behavior has been known as the Gibbs phenomenon, since it was the famous mathematical physicist Josiah Gibbs, who investigated it and presented his explanation in 1899.

With his experience, Joseph Fourier had already inferred two important behaviors of the harmonic components: (i) a fast decay of the higher frequency components along the time, as we can see in Fig. 10 and, (ii) a fast decay of the higher order components at the discontinuities of the signal.

For this second behavior, although intuitive, it is possible to observe it easily by using another methodology of decomposition, instead of the Fourier technique. In this case, for example, the application of Wavelet Transform provides different components, each one localized inside a different bandwidth. The higher frequency components, which are present at the discontinuities of the signal, can be extracted and analyzed. Considering the signal shown in Fig. 9, one can perform its decomposition using a wavelet based filter bank with six decomposition levels, whose results are shown in Figure 11. In this case the Meyer wavelet has been used; taking into account that it is a very suitable wavelet to extract signals with smoothed features. The Joseph Fourier’s observations are clearly depicted in this figure. First, the higher the frequency component, the faster the decay along the time, which can be seen by comparing the output signals (d1 to a6). Second, considering the zoomed details of the Fig. 11, one can observe the existence of high frequency components in each discontinuity of the signal. These characteristics can be observed at the different detailed levels shown in Figure 11 where the decomposition of temperature variation using a wavelet based filter bank method has been used. Figure 11 also shows that the temperature discontinuity can be adequately represented by only six components using a wavelet based filter bank decomposition, whereas the traditional harmonic decomposition would require many times that number of components to achieve similar resolution since they are all of steady-state nature. Direct physical interpretation of temperature variation from the higher detail coefficients levels would, however, be meaningless.

Fig. 11 – Decomposition of temperature variation using a wavelet based filter bank method

6.6 Conclusions

Joseph Fourier's many contributions to modern engineering science are so critically important and so pervasive that he is rightly regarded as the father of modern engineering. Great discoveries such as Fourier’s transform can be found through basic physics experiments coupled with mathematics. Fourier’s physical intuition lead to one of the most used analysis methods that changed the face of engineering. Finite Element Analysis allowed for a simple reproduction of Fourier’s experiment of heat propagation / temperature variation through a metallic ring. Simulated data gave a clear view of how Fourier first thought of representing temperature distribution in a ring as a combination of sinusoidal functions and how this experiment gave information about how harmonics content is modified in time. The use of new signal processing methods, based on time-frequency decomposition, further illustrates Joseph Fourier’s physical intuition to visualize the time varying components long before the mathematical foundation was developed.

6.7 References

[1] Joseph Fourier – Politician & Scientist, David A. Keston, Today in Science - - http://www.todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm

[2] Grattam-Guiness, Ivor: Joseph Fourier (1768-1830): a survey of his life and work, The MIT Press, 1972.

[3] Herivel, John: Joseph Fourier : The Man and the Physicist, Clarendon Press, 1975.

[4] Fourier, J.-B.-J. Mémoires de l'Académie Royale des Sciences de l'Institut de France VII. 570-604 (1827) (greenhouse effect essay)

[5] The Project Gutenberg EBook of Biographies of Distinguished Scientific Men by François Arago

[6] Duque, C., P.M. Silveira, T. Baldwin, P. F. Ribeiro, "Novel Method for Tracking Time-Varying Power Harmonic Distortions without Frequency Spillover," submitted to the PES GM2008, Pittsburgh, 20 - 24 July, 2008.

[7] Oppenheim, A. V., Willsky A. S., Signals and Systems, Prentice-Hall, 1983.