Introduction

Abstract

People have been investigating and trying to explore the world around them since the most ancient times. In their early attempts to understand and to systematise the phenomena they encountered every day, the observations they made, together with their reflections on those observations led centuries ago to the emergence of philosophy in ancient Greece. This was a spark igniting human minds that later, over the upcoming centuries, had disciplined the human way of thinking and reasoning forging that which nowadays we call science.

People have been investigating and trying to explore the world around them since the most ancient times. In their early attempts to understand and to systematise the phenomena they encountered every day, the observations they made, together with their reflections on those observations led centuries ago to the emergence of philosophy in ancient Greece. This was a spark igniting human minds that later, over the upcoming centuries, had disciplined the human way of thinking and reasoning forging that which nowadays we call science.

Ancient Greek philosophers suspected that music results from a mysterious connection between various waves and sounds, behind which lay hidden air vibrations or disturbances—see Fig. 1.1. It was Pythagoras, a great Greek philosopher and mathematician, who said that “... there is geometry in the humming of the strings, there is music in the spacing of the spheres ...”. Since music was a particularly beloved and omnipresent field of art for ancient Greeks, accompanying their marriages, funerals, religious ceremonies as well as poetry recitation in theatres, it simply had to have a great and prominent impact on their lives. Perhaps this was a direct reason why ancient Greeks started to study various phenomena, which nowadays we can so easily place within the fields of mathematics (harmonics), physics (sound propagation) or architecture (outdoor theatres), and which could be thought of as the foundations of modern acoustics.

With no doubt Pythagoras (c. 570–c. 495) should be mentioned as the first great Greek philosopher, whose efforts to understand the universe left deep imprints on music and mathematics as well as astronomy. In the field of music Pythagoras, using the monochord, investigated vibrating strings producing harmonious tones in order to determine mathematical formulae describing relations between the lengths of these strings and the tones they produced. By dividing the strings into ratios of halves, thirds, quarters or fifths Pythagoras created music intervals of an octave, a perfect fifth, a second octave, and a major third, respectively. Thanks to these investigations he discovered what we now call the first five overtones, and which create the common intervals being the primary building blocks of modern musical harmony—see Fig. 1.2. For many centuries this idea of Pythagoras had been the foundation of all music theory, having its roots in antiquity but lasting up to modern times [1].

Fig. 1.1

Waves are inherently a part of the surrounding world as: a sea waves resulting from the interaction of the wind blowing over the free surface of the sea [2], b atmospheric gravity (internal) waves resulting from the interaction of the rapidly rising air through convection and the wind blowing over the ocean [3], c shock waves in the atmosphere resulting from the eruptions of volcanoes [4], d wave structure in Saturn’s rings known as the Janus 2:1 spiral density wave, resulting from the same process as this responsible for the creation of spiral galaxies [5]

Another great Greek philosopher Aristotle (384–322 BC), whose interests spanned nearly all fields of ancient science, also studied sounds and investigated the phenomenon of their propagation. He rightly found out that sound propagation in the air results from compressive air movements, but he was wrong to think that the speed of sound is proportional to the sound pitch, greater for sounds of higher frequencies and smaller for sounds of lower frequencies.

It should be noted that in the Middle Ages and the Renaissance it was music which directly stimulated investigations in the field of acoustics as well as other fields of science. An English philosopher John Blund (c. 1175–1248), who studied Aristotle’s works, established frameworks for future acoustics theories by studying generation of sound and its reception, its medium, etc. Another English music theorist Walter Odington (c. 1260–1346) gathered together all knowledge of music at that time, which he supplemented with his own theoretical considerations about the consonance of the minor and major thirds. A music theorist, Iacobus de Ispania (d.c. 1330), was the author of the longest work on music of the Middle Ages. In contrast to that Thomas Bradwardine (c. 1300–1349), an English scholar, mathematician and physicist together with Johannes Boen (d.c. 1367), a Dutch music theorist, held the position that separation between the theory of music and acoustics was at least questionable [6]. However, over the forthcoming centuries the leading role of music would be gradually fading. Music evolved to become a separate field of art, while those branches of science that were initially serving to music as tools to explain music rules and complexity, now released form this dependence, evolved to become individual science disciplines such as: acoustics, astronomy, chemistry, mathematics, mechanics, physics, etc. However, musical sounds continued to be a very important subject of then research.

Fig. 1.2

A concept of Pythagorean music intervals based on the note C

From one angle the primary subject of investigation was the air or water and their properties as the media within which sounds can propagate. An Anglo-Irish philosopher, chemist and physicist Robert Boyle (1627–1691), investigating properties of various gasses, proved experimentally that sound cannot propagate through the vacuum. Sir Isaac Newton (1642–1726), an English mathematician, astronomer, theologian and physicist was the first who tried to establish a formula for the speed of sound in air. Assuming harmonic motion of adjacent air particles, Newton expected the speed of sound in air to be equal to the square root of the ratio of the air pressure and density. This problem was also studied by Joseph Louis Lagrange (1736–1813), an Italian astronomer and mathematician, who corrected Newton’s calculations by formulating general equations of air motion and by their subsequent integration along the direction of their propagation. However, it was Pierre Simon Laplace (1749–1827), a French mathematician, physicist and astronomer, who further corrected both previous formulae by pointing out that elastic properties of the air should take into account the air heat capacity due to adiabatic compression. In turn, a Swiss physicist Daniel Colladon (1802–1893) determined the speed of sound in water. He also measured the compressibility of other principal liquids, thereby winning the the prize of the Academy of Sciences in Paris. Another French physicist, astronomer and mathematician Jean Baptiste Biot (1774–1862), carried out measurements on the propagation of sounds in pipes noting that the speed of sound in the pipes themselves is much higher that in the air. Propagation of sound in gasses other than the air was the area of interest of a German physicist and musician Ernst Florence Friedrich Chladni (1756–1827).

Approaching from the opposite angle, the subjects of investigation were various instruments in the form of strings, plates, membranes, etc. that were responsible for sound generation. Many great names can be mentioned here as well. An ingenious Italian polymath, Galileo Galilei (1564–1642), who worked as an astronomer, physicist, philosopher and mathematician, tried to establish a link between the pitch and frequency of sounds produced by vibrating monochord strings of different lengths. This was also a research subject of a French mathematician and physicist Joseph Sauveur (1653–1716), who investigated this link in great detail, and who is also credited with coining the term acoustique [7].

An English mathematician Brook Taylor (1685–1731), thanks to the application of a mathematical tool sophisticated for its time, now known as the calculus of finite differences, determined the fundamental form of vibrating strings. In the following years Daniel Bernoulli (1700–1782), a famous Swiss mathematician and physicist, managed to formulate and solve partial differential equations describing the motion of vibrating strings. This was possible thanks to the application of d’Alembert’s formula, which nowadays is broadly used in mechanics. The solution to the equation of motion obtained by Bernoulli was interpreted by Jean le Rond d’Alembert (1717–1783), a French mathematician, mechanician, physicist, philosopher as well as music theorist, as two independent waves travelling in opposite directions along the strings [8]. Transverse vibrations of strings were also studied by Lagrange, who disputed in his works the lack of generality in earlier works of Brook Taylor, d’Alembert or Euler, suggesting a new, general solution form.

A Swiss mathematician, physicist, astronomer, logician and engineer Leonhard Euler (1707–1783) helped to formulate the Euler-Benoulli equations for transverse vibrations of beams, which today is one of the most common analytical tools of classical mechanics. His studies included fluid dynamics too. In this field Euler is responsible for the formulation of a very important set of equations of fluid dynamics for inviscid fluids, known as the Euler equations. The work of Jean-Baptiste Joseph Fourier (1768–1830), a French mathematician and physicist, well-known for the Fourier theorem, should also be mentioned here. The impact of the Fourier theorem turned out to have very deep implications also in the field of acoustics. His theorem about representing periodic functions by an infinite series of sines and cosines, proved in fact the principle of superposition, which is one of the fundamentals of every linear analysis, including the analysis of sound propagation.

Based on the research that had been carried out in the previous decades as well as many indisputable milestones in various field of science, also driven by the demand of the Industrial Revolution, the 18th and 19th century appear as the period of great progress in the field of theoretical and experimental acoustics.

Fig. 1.3

First nine Chladni patterns of a free aluminium disc. Results of numerical computations by TD-SFEM

Simeon Denis Poisson (1781–1840), a French mathematician, engineer, and physicist was the first to presented a solution to the problem of vibrating circular and rectangular membranes, which was also studied by a German mathematician Rudolf Friedrich Alfred Clebsch (1833–1872). A simple method to visualise nodal lines of vibrating plates was proposed by Chladni, who for that purpose used sand sprinkled onto their surfaces—see Fig. 1.3. After his name the patterns produced by this technique are known as Chladni figures. In the opinion of Michael Faraday (1791–1867), a famous British scientist, it was the process of acoustic streaming which was responsible for the appearance of Chladni figures. The relationship between vibration frequencies and corresponding modes of vibrations for flat circular surfaces was given the name of Chladni’s law by Lord Rayleigh. Lord Rayleigh, in full John William Strutt, 3rd Baron Rayleigh (1842–1919), a Nobel prize winning British physicist, in his textbook Theory of Sound [9] presented a holistic and scientific approach to the most urgent problems of acoustics at that time. A solution to the problem of vibrating plates was given by Marie Sophie Germain (1776–1831), a French mathematician, physicist, and philosopher, who received for her work a prestigious grand prize from the Paris Academy of Sciences. However, it was Gustav Robert Kirchhoff (1824–1887), a German physicist, who corrected her results by presenting a more accurate approach to treat plate boundary conditions. John Tyndall (1820–1893) was a prominent Irish physicist. The results of his scientific investigations on vibrations of rods, plates and bells were published in his book on sound [10]. Tyndall was also an experimentalist, whose works in the field of acoustics focused on sound transmission in the air, especially on differences in sound propagation at particular locations, resulting from temperature differences of air masses. Hermann Ludwig Ferdinand von Helmholtz (1821–1894) was a German physician and physicist, not only to be remembered for his famous theorem, named after him as Helmholtz’s theorem. His works in the field of acoustics were mainly focused on sound perception. As he was also educated in medicine, in his book [11] he combined his knowledge of physics, physiology and music to show that the human sense of hearing is able to differentiate even very complex tones.

The following decades are a time of a great progress and rapid development primarily in mathematics, and consequently in other fields of science, including acoustics and mechanics, which very eagerly took advantage of the achievements of contemporary mathematics. Wave motion stayed at the very centre of scientific interest of then research. A simple classification in the case of various types of waves, as well as in the case of mechanical waves, are presented in Figs. 1.4 and 1.5.

In the field of acoustics the classical theory of longitudinal behaviour of rods is well established, which can be attributed together to d’Alembert, Bernoulli, Euler and Lagrange. The same can be said about the theory of flexural behaviour of beams, which we owe to the works of Bernoulli and Euler. Nowadays it is best known as the classical theory of flexural behaviour, or simply the classical beam theory. On the other hand the theory of torsional vibration can be attributed to Adhemar Jean Claude Barre de Saint-Venant (1797–1886), a French mechanician and mathematician, whose works concentrated on stress analysis and hydraulics. The equations developed by Saint-Venant for unsteady flows in shallow waters through open channels, also known as the Saint-Venant equations, are fundamental in modern hydraulic engineering. Two-dimensional structures such as membranes and plates were also a common subject of research.

Fig. 1.4

Simple classification of various types of waves

Fig. 1.5

Simple classification of mechanical waves in various types elastic media

In 1888 Augustus Edward Hough Love (1863–1940), an English mathematician famous for his works on mathematical theories of elasticity, developed a theory for flexural behaviour of plates. The theory was a two-dimensional extension of the classical theory of beams by Bernoulli and Euler, and was taking advantage of the same assumptions earlier formulated by Kirchhoff. Love is also known for his works on propagation of horizontally polarised seismic surface waves, which are called Love waves. It should be added that another type of seismic surface waves, which are polarised vertically, as opposed to horizontally polarised Love waves, takes its name from Lord Rayleigh, who predicted their existence. The problem of flexural behaviour of plates was also studied by Poisson and Cauchy, who based their approach on the general theory of elasticity. Baron Augustin-Louis Cauchy (1789–1857) was a famous French mathematician, engineer and physicist. His pioneering works contributed to several branches of mathematics, including mathematical analysis as well as continuum mechanics. The theory of bending was a field of specialisation of another great scientist at that time, Stephan Prokopovych Timoshenko (1878–1972). This Russian born, and since 1922 also an American engineer and academic, is recognised by many as the father of modern engineering mechanics.

As shown above the tremendous progress in science during these times included not only physics and mathematics, but also acoustics and mechanics. Very often theoretical investigations were driven directly by luck and experiments. Based on the foundations laid by many great scientists and researchers from the previous centuries many new or refined theories, explaining better the observed phenomena or offering a deeper insight into the physics behind them were proposed during this Golden Age of acoustics and mechanics. Many well-known names contributing to these fields can be listed here including Pochhammer, Chree, Lamb, Reisner, Mindlin, Herrmann, Reddy, etc. to mention only a few.

Leo August Pochhammer (1841–1920) was a Prussian mathematician, who is mostly known for his works on special functions, but who had a great interest in the theory of elasticity [12]. At the same time as Pochhammer [13], Charles Chree (1860–1928), a Scottish born British physicist, also worked on elasticity problems, which concerned the longitudinal dynamic behaviour of elastic bars [14]. The result of their parallel work was greatly recognised by the scientific community, and nowadays this result is known as the Pochhammer-Chree equation. The phenomenon of sound propagation in the form of multi-mode flexural and in-plane elastic waves in infinite plates was studied by Sir Horace Lamb (1849–1934) [15]. Lamb was an English applied mathematician, who authored several influential textbooks on classical physics, mechanics and acoustics [16], as well as who contributed to hydrodynamics. Lamb coined the term vorticity, which since 1916 remains in use in fluid-dynamics.

Eric Reissner (1913–1996) was a German born, American civil engineer and mathematician, whos works in applied mechanics is fundamental to the theoretical understanding of the behaviour of elastic solids. Reissner [17] is recognised as a co-author of the most commonly used plate theory, which is known to the scientific community as the Mindlin-Reissner theory of plates, since the theory was developed in parallel and independently by Raymond David Mindlin (1906–1987), an American mechanical engineer [18]. George Herrmann (1921–2007) was a Russia born American scientist working in the field of mechanical and civil engineering. Together with Mindlin he co-authored a theory describing longitudinal behaviour of rods, know in the literature as the Mindlin-Herrman theory of rods [19]. Finally, Junuthula Narasimha Reddy (1945–present) is an Indian born, American civil engineer, scientist and researcher, who significantly contributed to the field of solid and fracture mechanics as well as mechanics of composite materials, who is also responsible for the development of a higher-order theory of plate flexural behaviour, known as the Reddy plate theory [20].

It may be interesting to note that the tremendous progress in many branches of science sometimes led to unusual situations. New theories were formulated, but the ability of the mathematics of the time to show practical solutions for these theories was insufficient. For example, the characteristic equation derived by Pochhammer and Chree had to wait nearly 65 years to get solved. However, its solution was obtained not thanks to the traditional analytical approach of mathematics, but thanks to numerical computations. A new era of numerical computations was about to come and brought to existence a powerful computational tool, which nowadays we call the computer. Together with computers new algorithms and methods were developed to offer computational abilities, which we know from the present day.

The first steps on the path leading to the emergence of modern computers can be attributed to Jospeh Marie Jacuquard (1752–1834), a French weaver and merchant, who is the inventor of the earliest programmable machine known as the Jacquard loom. The Jacquard loom invented in 1801 used punched wooden cards to weave fabric designs in exactly the same manner as an early version of IBM digital compiler. Based on the idea of Jacuquard, an American engineer and inventor Herman Hollerith (1860–1929) developed in 1890 an electromechanical punched card tabulator for the purpose of the 1890 census in the United States. His invention led to the beginning of the Tabulating Machine Company, which in 1911 joined with three other companies to become the Computing-Tabulating-Recording Company, which since 1924 has been known just as IBM. In 1936 Alan Mathison Turing (1912–1954), an English mathematician, computer scientist, logician and cryptanalyst came up with the idea of a universal machine, better known as the Turing machine, being a true and real ancestor of modern computers. Turing is considered as the father of theoretical computer science and artificial intelligence, the originator of the so-called Turing test from 1950 measuring the ability of a computing machine to mimic the behaviour of a human being in an intelligent and indistinguishable manner. In 1948 Turing developed and presented the algorithm of the LU decomposition method, which since that time remains in use for solving matrix equations.

A breakthrough was made by two American engineers John Presper Eckert (1919–1995) and John William Mauchly (1907–1980), who built between 1943–1945 the earliest and the most famous electronic computer named ENIAC (Electronic Numerical Integrator and Computer). However, it should be said that the title of the first computer is also claimed by Colossus, built in 1943 under the supervision of a British mathematician and codebreaker Maxwell Herman Alexander Newman (1897–1984) and an English engineer Tommy Flowers (1905–1998), as well as Alan Turing himself. Additionally, a computational machine Z3 designed in 1943 by a German civil engineer, inventor and computer pioneer Konrad Zuse (1910–1995) must be mentioned here together with the ABC computer (Atanasoft-Berry Computer) built between 1937–1942 by two American engineers John Vincent Atanasoff (1903–1995) and Clifford Edward Berry (1918–1963).

A great leap forward in the development of computers, as we understand this term today, was made in 1947 by William Bradford Shockley (1910–1989), an American physicist and inventor, John Bardeen (1908–1991), an American physicist and electrical engineer, as well as Walter Houser Brattain (1902–1987), an American physicist, who in Bell Laboratories invented the transistor, which today is the most essential and fundamental building block of modern electronic devices. For their joined invention they were awarded the Nobel Prize in Physics in 1956. However, the concept of the transistor undoubtedly belongs to Julius Edgar Lilienfeld (1882–1963), an American physicist and electronic engineer, who presented it more than two decades earlier earlier, in 1926. In 1958 the first integrated circuit (microchip) was built, which is considered the key element of modern computers, and which spurred the revolution in the field of personal computers. The invention of the integrated circuit should be attributed independently to an American inventor and engineer Jack St. Clair Kilby (1923–2005), who for his invention was awarded the Nobel Prize in physics in 2000, and an American physicist Robert Noyce (1927–1990), a co-founder of Fairchild Semiconductor in 1957 and Intel Corporation in 1968.

The inventions of the transistor and the integrated circuit denote a symbolic beginning of the era of personal computers. Altair 8800 is considered by many as the first personal computer. It was built in 1974 by Micro Instrumentation and Telemetry Systems (MITS) founded by an American engineer Ed Roberts (1941–2010), who is recognised as the father of the personal computer. In 1976 Apple Computer, Inc. was established by Stephen Gary Wozniak (1950–present), an American inventor, electronics engineer, programmer, and Steven Paul Jobs (1955–2011), which produced one of the first commercially successful and mass-produced home computers, Apple II. Apple II was the successor to a short-series of 200 hand-built Apple I computers. The first personal computer produced by Hewlett-Packard Company HP-85 enters the market in 1980. Only one year later, in 1981 the first personal computer produced by IBM named Acorn appears. The number of computers used for various purposes grows tremendously and new companies producing computers for everyday home use appear, including such brands as: ZX series, Commodore, Amiga or Atari. Today the total number of personal computers sold worldwide every year exceeds 250 million. Personal computers, as well as computers in general, assist every day life in every possible aspect of human activity. This also includes science and research.

From the very beginning of their history computers were found by scientists and researchers to be very helpful tools offering enormous speed of computations, which otherwise had to be carried out by many people over long hours or even days, and being so much prone to human error or mistake. Computer programming became practically a separate branch of science giving birth to new, more efficient and more intuitive programming languages, programming packages or environments such as: computer aided designing (CAD) or computer aided engineering (CAE). Among them, in the realm of scientific research and its needs, numerical computational methods and tools played the most important role. However, before this could have happened it is necessary to travel back in time to much earlier years and such great names as: Navier, Stokes, Ritz or Galerkin.

Claude-Louis Navier (1785–1836), was a famous French engineer and physicist, who is mostly known for his contribution to continuum mechanics. In 1821 Navier presented, formulated by himself, the general theory of elasticity using the language of contemporary mathematics. The input of Navier into the field of continuum mechanics allows him to be considered as the founder of modern structural analysis. The system of partial differential equations describing the behaviour of the elastic continuum under the influence of forces is known in mechanics as the Navier or Navier-Cauchy equations. However, the major contribution of Navier stays at the centre of fluid dynamics. Together with Sir George Gabriel Stokes (1819–1903), an Anglo-Irish physicist and mathematician, Navier is recognised as a co-author of the famous Navier-Stokes equations. Walther Ritz (1878–1909) was a Swiss theoretical physicist, famous for the formulation of a general method for finding approximate solutions of partial differential equations accompanied with sets of boundary conditions, known as boundary value problems. A special variant of the method proposed by Ritz was developed by Boris Grigoryevich Galerkin (1871–1945), who was a Russian mathematician and engineer. In his article from 1915 Galerkin proposed an idea for a new approach that could be effectively used for approximate solutions to partial differential equations. Nowadays the Ritz and Galerkin methods are considered as the foundation of many effective solution algorithms in the fields of mechanics, thermodynamics, electromagnetism, hydrodynamics and many others. One such a method is the Finite Element Method (FEM).

It should be said, however, that the mathematical origins of FEM were not only firmly embedded in the earlier works of Bernoulli, Euler, Lagrange, Legendre, Gauss, Cauchy and many more, considered as the fathers of the calculus of variations, but also in the contemporary works of Rayleigh, Ritz and Galerkin. Its principal idea, as well as the most important feature and advantage, is the subdivision of the computational domain into smaller sub-domains of simpler geometry, which are called finite elements (FEs). Based on numerical properties of FEs the solution of a given problem, through a simple element aggregation procedure, can be presented at the level of the entire domain and solved algebraically.

No precise date can be proposed as the date of birth of FEM, which undoubtedly is one of the most popular and efficient computational tools available these days. However, the beginnings of FEM can be dated back to the 1940 s the and names of Hrennikoff and Courant, when the idea of FEs was crystallised [21,22,23,24]. Alexander Pavlovich Hrennikoff (1896–1984) was a Russian born, Canadian structural engineer, who is considered as the originator of FEM. The unique approach of Hrennikoff employed to solve a boundary value problem was based on a lattice analogy [21, 22] used for subdivision of the computational domain. The approach presented by Richard Courant (1888–1972), a German born, American mathematician, was different. Courant suggested the subdivision of the computational domain into a regular mesh of triangular sub-domains [23, 24], in a manner more typical of the current approach of FEM. At first FEM was mostly used as a numerical technique by mechanical engineers in order to solve boundary value problems associated with various types of partial differential equations. In 1973 a fundamental book dealing with mathematical aspects of FEM was published [25]. Since then, FEM has been gradually reinforcing its mathematical foundations gaining its current strength as a tool used for numerical modelling of physical phenomena in a wide range of engineering disciplines.

However, the method had to wait until the 1960 and 1970 s, when it achieved its current, great level of interest among engineers and researchers all around the world thanks to the works of such pioneers of FEM as: John Hadji Argyris (1913–2004) [26,27,28], a Greek born engineer, academic and professor of aerospace engineering at the University of Stuttgart; Ray William Clough (1920–2016) [29,30,31], an American engineer, academic and professor of structural engineering at the University of California, Berkeley; Olgierd Cecil Zienkiewicz (1921–2009) [32,33,34], a British engineer and academic of Polish descent, professor at Swansea University; Philippe Gaston Ciarlet (1938–present) [35,36,37], a French mathematician, professor at Pierre and Marie Curie University in Paris as well as Richard Hugo Gallagher (1927–1997) [38,39,40], an American engineer and academic, a professor of civil engineering at Cornell University, to name only a few.

Nowadays, FEM is very well-established as a numerical tool as well as a mathematical computational method, which is still improving its capabilities and efficiency. A strong proof of this is perhaps the total number of scientific and research papers published every year, which directly refer to FEM in their titles. Figure 1.6 suggest that not only is the total number of papers published every year vast, but that this number is constantly growing. Despite the fact that the results presented in Fig. 1.6 only concern the Web of Science database, which is one out of many similar databases providing this kind of information, it should be emphasised that the expected number of papers, which will be published within the years 2021–2025 may far exceed 8,000, which is more than 1,600 papers per year. At the same time the total number of papers related to wave propagation problems, so relevant to this monograph, also stays on a high level and within the same years 2021–2025 should reach 5,000, which is 1,000 papers per year.

Fig. 1.6

Number of publications published according to the Web of Science database related to two different topic search keywords: A—finite and element and method, B—wave and propagation

The growing popularity of FEM is no coincidence. The strength of the method comes from its constant theoretical development, behind which many names of great scientists and researchers can be found. Among many such names, the following may be mentioned: Oden, Babuška, Doyle, Gopa-lakrishnan or Patera.

Ivo Milan Babuška (1926–present) [41,42,43] is a Czech born, American mathematician, well-known for his studies of FEM as well as the error estimation associated with this numerical technique. John Tinsley Oden (1936–present) [44,45,46], is an American mathematician and academic, whose works in the field of FEM not only concern problems related to error estimation, but also non-linear mechanics and computation mechanics in general. James Francis Doyle (1951–present) [47,48,49] is an Irish born, American mechanical engineer and academic, and an expert in the field of computational mechanics. He is one of the fathers of the Frequency-domain Spectral Finite Element Method (FD-SFEM), an extremely efficient and powerful FEM clone, which is based on the application of the fast Fourier transform (FFT) together with analytical shape functions to built finite elements. An adopter of the idea of FD-SFEM and its further development is Srinivasan Gopalakrishnan (1960–present) [49,50,51], an Indian aerospace engineer and academic. Another clone of FEM is the Time-domain Spectral Finite Element Method (TD-SFEM), which has been originally proposed and developed by Anthony Tyr Patera (1959–present) [52,53,54] an American mathematician, academic and professor of mechanical and computational engineering. This specialised variant of FEM, based on the application of special, orthogonal approximation polynomials of higher orders in comparison with the classical FEM, is a particularity efficient computational tool in solving fluid-mechanics problems as well as wave propagation problems—see Fig. 1.7.

Fig. 1.7

Patterns of elastic waves propagating in a laminated wind turbine rotor blade in consecutive moments in time: a 112.5 $\mu$s, b 187.5 $\mu$s, c 375.0 $\mu$s, d 562.5 $\mu$s, e 750.0 $\mu$s and f the mesh of SFEs. Results of numerical computations by TD-SFEM

However, the popularity of FEM also results from its general availability as an easy-to-use computational tool, which is offered to and used worldwide by engineers, scientists and researchers. This is mostly thanks to many FEM software packages guiding their users through all required steps of FEM analysis including pre- and post-processing of all resulting computational data.

With no doubt NASTRAN [55] (from NASA¹ STRucture ANalysis) developed in late 1960 s, thanks to the financial support of the US government, was one of the first FEM packages used to help engineers in complex structural dynamic analysis. Its evolution over the following years changed its original purpose to a multi-physics computational tool that allows engineers to carry on structural analysis computations of any type. Since that time its source code has been integrated into many different FEM software packages that are available, including such ones as: MSC² Software (MSC NASTRAN), NEi³ Software (NEi NASTRAN) or Siemens PLM Software (NX NASTRAN).

On the same list of early FEM software packages available nowadays is ANSYS [56] (from ANalysis SYStems). ANSYS was originally developed in 1970 by John Arthur Swanson (1940–present), founder of Swanson Analysis Systems Inc. Swanson is an American engineer and entrepreneur who, according to many experts in computational methods, is regarded as a pioneer in the application of FEM in numerous fields of engineering. Since its début ANSYS has been gradually taking over the market of FEM software packages and nowadays the company is one of the biggest producer of computer simulation engineering software in the world. It is the holder of such brand names as FLUENT, ICEM CFD,⁴ Reaction Design and many more.

Finite element analysis software ABAQUS [57] (probably derived from the word abacus) was originated in 1978 as a computational product based on the application of FEM. It was developed by the team of three scientists Dr Hugh David Hibbitt (1944–present), Dr Bengt Karlsson (1944–present) and dr Paul Sorensen (1959–present), who established Hibbitt, Karlsonn and Sorensen, Inc. that later on changed its name to ABAQUS Inc. Since that time ABAQUS has established a firm position in the market of computational tools amongst engineers and researchers all over the world. ABAQUS offers a number of core products addressed to structural engineers, especially those seeking solutions to linear and non-linear structural dynamics problems, as well as problems of computational fluid dynamics and also electromagnetism. ABAQUS remains a highly regarded tool for its multi-physics computational capabilities including various problems of coupled fields, for example acoustic-structural, thermo- and electromechanical, and more.

In 1986 a company was founded by Klass-Jürgen Bathe (1943–present) under the name of ADINA [58] R &D Inc. It is the producer and developer of a FEM computational tool, called ADINA (from Automatic Dynamic Incremental Non-linear Analysis), which is addressed to academics as well as industry. In a similar way to other FEM commercial packages ADINA offers a complex computational environment to solve problems of fluid dynamics, heat transfer, electromagnetism and more, but primarily ADINA is highly regarded as a numerical computational tool for structural analysis, especially its non-linear analysis capabilities. As a result of this its non-linear solver is also employed by other FEM packages, such as NASTRAN, for example.

In the same year 1986 COMSOL [59] (probably from COMputer SOLution) was founded in Sweden, at the Royal Institute of Technology in Stockholm, by Svante Littmarck (1954–present) and Farhad Saeidi (1962–present). Since the time of its emergence COMSOL FEM software has developed its reputation as a multi-platform and multi-physics computational tool offering scientists and researchers an integrated environment to carry on numerical computations in the case of electrical, mechanical and chemical engineering problems as well as fluid dynamics.

It should be emphasised at this point that the list of FEM software that is available nowadays is much longer and the names already mentioned are meant to represent only historical beginnings of FEM software development. The rapid development of numerical computation techniques together with a tremendous increase in the computational power of modern computers is directly responsible for the boom that can be observed in the field of numerical computational tools available to both academia and industry. Another interesting aspect of it comes from the fact that now an increasing number of such computational tools is made available to scientists and researchers as free, very often customised and optimised, computational tools. It is worth mentioning here such FEM packages as: Z88, Code_Aster, Elmer, CalculiX or SimScale.

Z88 [60] is a free FEM software package, which was developed in 1985 at the University of Bayreuth, in Germany, by a team led by Frank Rieg (1955–present), who is a German academic and professor of mechanical engineering at the same university. Since its development Z88 has been adopted by many universities and small-sized enterprises as a tool offering a three-dimensional analysis of structural problems as well as topology optimisation.

Code_Aster [61] (from Analyses des Structures et Thermomécanique pour des Études et des Recherches in French, which can be translated to Structural and Thermomechanical Analysis for Study and Research in English) appeared in 1989 as a response to the demand of the French Department of Energy seeking a computational tool, which could allow for structural and thermal analysis of nuclear facilities in France. Since that time the program is maintained and developed to serve as a numerical solver for both academics and engineers. Code_Aster is freely distributed as open-source FEM software together with Salomé [62], which is a generic software platform for pre- and post-processing.

The origins of Elmer [63] date back to 1995. This FEM software package was developed under support of the Finnish Funding Agency for Technology and Innovation, named Tekes, with help from Finnish universities, research laboratories and industry. Its development was a part of a national CFD technology programme. Nowadays Elmer is a free and open-source FEM-based computational tool for multi-physics problems including fluid and structural mechanics, electromagnetism and acoustics.

CalculiX [64], as open-source and free FEM software, emerged in 1998. Since that time it has been developed by its authors Dr Guido Dhondt (1961–present), who is responsible for the development of the solver, as well as Klaus Wittig (1961–present), who is the author of the pre- and post-processing software. It is interesting to note that CalculiX is compatible, in terms of input data, with a number of commercial FEM software packages such as: NASTRAN, ANSYS or ABAQUS. Additionally, CalculiX offers its users generation of input data for other open-source solvers.

Another free FEM software package is SimScale [65]. It is interesting to note that SimScale is a fully web-based FEM solution for computer aided engineering (CAE) available as a free tool for non-commercial use. It is well integrated with another web-based solution for computer aided design (CAD) named Onshape [66]. Both of them, integrated together, represent a powerful cloud-based computation tool for scientists and engineers employing FEM for solving scientific and/or engineering problems from various fields.

It should be mentioned here once more that the constant development of FEM, as well as many similar numerical techniques based of FEM, comes from the fact that the number of software packages for more general purposes or very specialised ones, changes very rapidly due to the constant demand of engineers and academia. Many specialised and customised solutions are reported in the available literature devoted to the subject of FEM. What is more interesting, many such solutions are successfully adopted by commercial FEM software developers, which can be clearly seen over all these years that have passed since the beginning of FEM.