한국계산과학공학회 기념워크샵 고전역학에 기초한 전산역학 - CST 와 CFD 관점에서

1. 한국계산과학공학회기념워크샵고전역학에 기초한 전산역학 - CST와 CFD 관점에서 김 승 조* Professor, Seoul National University 김 민 기 Seoul National University 문 종 근 Seoul National University 2009. 10. 12, 코엑스인터콘티넨탈 호텔

2. Contents 1 2 3 4

3. Mechanics in Physics Fundamentals of Physics by David Halliday, Robert Resnick, Jearl Walker

4. Mechanics in Physics

5. Mechanics in Physics • Mechanics Ch1 Measurement Ch2 Motion Along a Straight Line Ch3 Vectors Ch4 Motion in Two and Three Dimensions Ch5 Force and Motion I Ch6 Force and Motion II Ch7 Kinetic Energy and Work Ch8 Potential Energy and Conservation of Energy Ch9 Center of Mass and Linear Momentum Ch10 Rotation Ch11 Rolling Torque, and Angular Momentum Ch13 Gravitation

6. Mechanics in Physics • Properties of Matter Ch12 Equilibrium and Elasticity Ch14 Fluids Ch19 The Kinetic Theory of Gases • Heat Ch18 Temperature, Heat, and the First Law of Thermodynamics Ch20 Entropy and the Second Law of Thermodynamics • Sound Ch15 Oscillations Ch16 Waves I Ch17 Waves II

7. Mechanics in Physics • Electricity and Magnetism Ch21 Electric Charge Ch22 Electric Fields Ch23 Gauss' Law Ch24 Electric Potential Ch25 Capacitance Ch26 Current and Resistance Ch27 Circuits Ch28 Magnetic Fields Ch29 Magnetic Fields Due to Currents Ch30 Induction and Inductance Ch31 Electromagnetic Oscillations and Alternating Current Ch32 Maxwell's Equations; Magnetism of Matter Ch33 Electromagnetic Waves

8. Mechanics in Physics • Light Ch34 Images Ch35 Interference Ch36 Diffraction • Atomic and Nuclear Physics Ch38 Photons and Matter Waves Ch39 More About Matter Waves Ch40 All About Atoms Ch41 Conduction of Electricity in Solids Ch42 Nuclear Physics Ch43 Energy from the Nucleus Ch44 Quarks, Leptons, and the Big Bang

9. Mechanics in Physics • Relativity Ch37 Relativity

10. Classical Mechanics ? • Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science, engineering and technology. • Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light; general relativity is employed to handle gravitation at a deeper level; and quantum mechanics handles the wave-particle duality of atoms and molecules. • In physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies. The other sub-field is quantum mechanics.

11. Classical Mechanics ? • The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century workers, building upon the earlier astronomical theories of Johannes Kepler, the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "relativistic physics" from that category. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form. • The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. More abstract and general methods include Lagrangian mechanics and Hamiltonian mechanics. Much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton.

12. Classical Mechanics ? – Leonardo da Vinci •Leonardo di ser Pieroda Vinci (April 15, 1452 – May 2, 1519) was an Italianpolymath, being a scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, architect, botanist, musician and writer. Leonardo has often been described as the archetype of the renaissance man, a man whose unquenchable curiosity was equaled only by his powers of invention. He is widely considered to be one of the greatest painters of all time and perhaps the most diversely talentedperson ever to have lived. Rhombicuboctahedron ClosLucé in France, where Leonardo died in 1519

13. Classical Mechanics ? – Leonardo da Vinci • Leonardo as observer, scientist and inventor design for a flying machine flight of a bird

14. Classical Mechanics ? – Leonardo da Vinci • Leonardo as observer, scientist and inventor flying machine helicopter

15. Classical Mechanics ? – Leonardo da Vinci • Leonardo as observer, scientist and inventor grinding machine various hydraulic machines

16. Classical Mechanics ? – Leonardo da Vinci • Leonardo as observer, scientist and inventor tank Arsenal

17. Classical Mechanics ? – Copernicus • Nicolaus Copernicus (February 19, 1473 – May 24, 1543) was the first astronomer to formulate a scientifically-based heliocentriccosmology that displaced the Earth from the center of the universe. His epochal book, De revolutionibusorbiumcoelestium (On the Revolutions of the Celestial Spheres), is often regarded as the starting point of modern astronomy and the defining epiphany that began the Scientific Revolution.

18. Classical Mechanics ? Galileo Galilei(15 February 1564 – 8 January 1642)

19. Classical Mechanics ? – Galilei • Galileo Galilei (15 February 1564 – 8 January 1642) was a Tuscanphysicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations, and support for Copernicanism. Galileo has been called the "father of modern observational astronomy", the "father of modern physics", the "father of science", and "the Father of Modern Science." The motion of uniformly accelerated objects, taught in nearly all high school and introductory college physics courses, was studied by Galileo as the subject of kinematics. His contributions to observational astronomy include the telescopic confirmation of the phases of Venus, the discovery of the four largest satellites of Jupiter, named the Galilean moons in his honor, and the observation and analysis of sunspots. Galileo also worked in applied science and technology, improving compass design.

20. Classical Mechanics ? • Galileo is perhaps the first to clearly state that the laws of nature are mathematical. In The Assayer he wrote "Philosophy is written in this grand book, the universe ... It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures; ...". His mathematical analyses are a further development of a tradition employed by late scholastic natural philosophers, which Galileo learned when he studied philosophy. Although he tried to remain loyal to the Catholic Church, his adherence to experimental results, and their most honest interpretation, led to a rejection of blind allegiance to authority, both philosophical and religious, in matters of science. In broader terms, this aided to separate science from both philosophy and religion; a major development in human thought.

21. Classical Mechanics ? • Galileo proposed that a falling body would fall with a uniform acceleration, as long as the resistance of the medium through which it was falling remained negligible, or in the limiting case of its falling through a vacuum. He also derived the correct kinematical law for the distance travelled during a uniform acceleration starting from rest—namely, that it is proportional to the square of the elapsed time ( d ∝ t 2 ). However, in neither case were these discoveries entirely original. The time-squared law for uniformly accelerated change was already known to Nicole Oresme in the 14th century, and Domingo de Soto, in the 16th, had suggested that bodies falling through a homogeneous medium would be uniformly accelerated[ Galileo expressed the time-squared law using geometrical constructions and mathematically-precise words, adhering to the standards of the day. (It remained for others to re-express the law in algebraic terms). He also concluded that objects retain their velocity unless a force—often friction—acts upon them, refuting the generally accepted Aristotelian hypothesis that objects "naturally" slow down and stop unless a force acts upon them (philosophical ideas relating to inertia had been proposed by Ibn al-Haytham centuries earlier, as had Jean Buridan, and according to Joseph Needham, Mo Tzu had proposed it centuries before either of them, but this was the first time that it had been mathematically expressed, verified experimentally, and introduced the idea of frictional force, the key breakthrough in validating inertia). Galileo's Principle of Inertia stated: "A body moving on a level surface will continue in the same direction at constant speed unless disturbed." This principle was incorporated into Newton's laws of motion (first law).

22. Classical Mechanics ? – Galilei • Improvement of Telescope and Astronomical Observation

23. Classical Mechanics ? – Galilei • Pendulum Motion Galileo also claimed (incorrectly) that a pendulum's swings always take the same amount of time, independently of the amplitude. 'Galileo's lamp' in the cathedral of Pisa

24. Classical Mechanics ? – Newton Sir Isaac Newton (1642-1727)

25. Classical Mechanics ? – Newton Sir Isaac Newton, (4 January 1643  – 31 March 1727) was an Englishphysicist, mathematician, astronomer, natural philosopher, alchemist, and theologian and one of the most influential men in human history. His PhilosophiæNaturalis Principia Mathematica, published in 1687, is considered to be the most influential book in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientific revolution.

26. Classical Mechanics ? • In mechanics, Newton enunciated the principles of conservation of both momentum and angular momentum. In optics, he built the first practical reflecting telescope[5] and developed a theory of colour based on the observation that a prism decomposes white light into the many colours which form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. • In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function, and contributed to the study of power series. • Newton's stature among scientists remains at the very top rank, as demonstrated by a 2005 survey of scientists in Britain's Royal Society asking who had the greater effect on the history of science, Newton or Albert Einstein. Newton was deemed the more influential.

27. Classical Mechanics ? • In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function, and contributed to the study of power series. • Most modern historians believe that Newton and Leibniz developed infinitesimal calculus independently, using their own unique notations. According to Newton's inner circle, Newton had worked out his method years before Leibniz, yet he published almost nothing about it until 1693, and did not give a full account until 1704. Meanwhile, Leibniz began publishing a full account of his methods in 1684. Moreover, Leibniz's notation and "differential Method" were universally adopted on the Continent, and after 1820 or so, in the British Empire. Whereas Leibniz's notebooks show the advancement of the ideas from early stages until maturity, there is only the end product in Newton's known notes. Newton claimed that he had been reluctant to publish his calculus because he feared being mocked for it

28. Classical Mechanics ?– Bernoulli family • Bernoulli family tree

29. Classical Mechanics ? • Daniel Bernoulli (29 January 1700 – 27 July 1782) was a Dutch-Swissmathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. • Born in Groningen, in the Netherlands, the son of Johann Bernoulli, nephew of Jacob Bernoulli, younger brother of Nicolaus II Bernoulli, and older brother of Johann II, Daniel Bernoulli has been described as "by far the ablest of the younger Bernoullis". He is said to have had a bad relationship with his father. Upon both of them entering and tying for first place in a scientific contest at the University of Paris, Johann, unable to bear the "shame" of being compared to his offspring, banned Daniel from his house. Johann Bernoulli also tried to steal Daniel's book Hydrodynamica and rename it Hydraulica. Despite Daniel's attempts at reconciliation, his father carried the grudge until his death.

30. Classical Mechanics ? • Leonhard Paul Euler (15 April, 1707 – 18 September, 1783) was born in Basel . Paul Euler was a friend of the Bernoulli family—Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he matriculated at the University of Basel, and in 1723, received his M.Phil with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph.D. dissertation on the propagation of sound and in 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He won second place in the first competition but Euler subsequently won this coveted annual prize twelve times in his career.

31. Classical Mechanics ? • Euler was a pioneering Swissmathematician and physicist who spent most of his life in Russia and Germany. • Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[4] He is also renowned for his work in mechanics, optics, and astronomy. • Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.[33]

32. Classical Mechanics ? – d’Alembert • J Jean le Rondd'Alembert (November 16, 1717 – October 29, 1783) was a Frenchmathematician, mechanician, physicist and philosopher. He was also co-editor with Denis Diderot of the Encyclopédie. D'Alembert's method for the wave equation is named after him.

33. Classical Mechanics ? – Lagrange • Joseph Louis Lagrange (1736-1813) .

34. Classical Mechanics ? • Joseph-Louis Lagrange, born Giuseppe LodovicoLagrangia (25 January1736 – 10 April1813) was an Italianmathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. On the recommendation of Euler and D'Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing a large body of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (MécaniqueAnalytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888-89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.

35. Classical Mechanics ? – Cauchy • Augustin Louis Cauchy (1789-1857)

36. Classical Mechanics ? • Augustin Louis Cauchy (21 August 1789 – 23 May 1857) was a Frenchmathematician. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner and was thus an early pioneer of analysis. He also gave several important theorems in complex analysis and initiated the study of permutation groups. A profound mathematician, through his perspicuous and rigorous methods Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

37. Classical Mechanics ? – Navier • Claude Louis Navier (1785-1836)

38. Classical Mechanics ? • Claude-Louis Navier (10 February 1785 in Dijon – 21 August 1836 in Paris) was a Frenchengineer and physicist who specialized in mechanics. • The Navier-Stokes equations are named after him and George Gabriel Stokes. • In 1802, Navier enrolled at the Écolepolytechnique, and in 1804 continued his studies at the ÉcoleNationale des Ponts et Chaussées, from which he graduated in 1806. He eventually succeeded his uncle as Inspecteur general at the Corps des Ponts et Chaussées. • He directed the construction of bridges at Choisy, Asnières and Argenteuil in the Department of the Seine, and built a footbridge to the Île de la Cité in Paris. • In 1824, Navier was admitted into the French Academy of Science. In 1830, he took up a professorship at the ÉcoleNationale des Ponts et Chaussées, and in the following year succeeded exiled Augustin Louis Cauchy as professor of calculus and mechanics at the Écolepolytechnique.

39. Classical Mechanics ? – Stokes • George Gabriel Stokes (1819-1903) .

40. Classical Mechanics ? • Sir George Gabriel Stokes (13 August1819–1 February1903), was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier–Stokes equations), optics, and mathematical physics (including Stokes' theorem). He was secretary, then president, of the Royal Society. • His first published papers, which appeared in 1842 and 1843, were on the steady motion of incompressible fluids and some cases of fluid motion. These were followed in 1845 by one on the friction of fluids in motion and the equilibrium and motion of elastic solids, and in 1850 by another on the effects of the internal friction of fluids on the motion of pendulums. These inquiries together put the science of fluid dynamics on a new footing, and provided a key not only to the explanation of many natural phenomena, such as the suspension of clouds in air, and the subsidence of ripples and waves in water, but also to the solution of practical problems, such as the flow of water in rivers and channels, the skin resistance of ships and aerodynamics for airplane design.

41. Classical Mechanics ? – Stokes • Navier-Stokes equation The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusingviscous term (proportional to the gradient of velocity), plus a pressure term.

42. 고전역학의 원리 Principles of Classical Mechanics (Axiomatic Approach, 공리적 접근) Axiom 1. Mass Conservation, - Continuity equation Axiom 2. Conservation of Linear Momentum - Force Equilibrium Equation Axiom 3. Conservation of Angular Momentum - Moment Equilibrium Equation Axiom 4. Conservation of Energy - The 1st Law of Thermodynamics Axiom 5. Entropy Inequality - The 2nd Law of Thermodynamics

43. 고전역학의 원리 Classification of Classical Mechanics Dynamics : Kinematics, Kinetics, Rigid Body Motion • Rigid/Deformable Body Dynamics – Vibration - Axioms 1. 2. 3. Solid Mechanics : Stress, Strain, Constitutive Equation • Structural Mechanics : Bar, Truss, Beam, Column, Frame, Plate • Deformable Body Dynamics – Vibration - Axioms 1. 2. 3., sometimes 5. Fluid Mechanics : Stress, Velocity Gradient, Fluid & Gas state • Stokes Hypothesis – Navier-Stokes Equation - Axiom 1. 2. 3. 4. 5. Thermodynamics : Temperature, Heat Flux, Fourier’s Law • Heat Conduction, Convection, Radiation - Axiom 4. 5.

44. 고전역학의 분류 • Lagrangian방식 • 각입자의 관점에서물리량의 시간변화를 기술 • 모든 물리량은 각 질점 위에서 시간에 의해(t,x0) 결정됨 격자계가 입자의 움직임과 함께 변화 T=t0 T=t0+Dt

45. 고전역학의 분류 • Eulerian방식 • 고정된 좌표(격자계) 상에 입자의 흐름을 기술 • 모든 물리량은2개의 변수인시간과 공간(t,x)에 의해 결정됨 격자계 불변 T=t0 T=t0+Dt 그림 출처 : http://efdl.as.ntu.edu.tw/research/islandwake/description.html

46. 고전역학의 원리 변형 중의 물체의 변형텐서 정의 Y y dy dz dY dx dZ X x dX Z 변형 전 변형 후 z • 변형 텐서(deformation gradient Tensor) : • 미소 위치벡터 변화량: • 미소 부피 변화량= 변형텐서의 행렬식 : det(F)=J

47. 고전역학의 연속체 장 방정식 • 연속체 장 방정식 • 고전역학의 5대 공리를 연속체에 적용한 방정식 • 연속방정식 : • 선운동량 방정식 : • 각운동량 방정식 : • 열역학 제 1법칙 : • 열역학 제 2법칙 : σ : 응력 텐서 a : 가속도 벡터 b : 체적력 J : 미소부피 변화량 ρ: 밀도 e : 내부에너지/질량 q : 열유속 벡터 r : 복사열 D : 속도구배텐서 θ : 절대온도 η : 엔트로피/질량

48. 고전역학의 연속체 장 방정식 • 속도, 가속도 및 시간미분 관계식 • 가속도-속도–변위 관계식 • 임의의 물리량과 시간미분의 관계식

49. 고전역학의 연속체 장 방정식 • 연속방정식 • 미소 부피 변화량의 시간미분 • 연속방정식 양변 시간미분 Eulerian기반 연속방정식

50. 고전역학의 연속체 장 방정식 • 운동량 방정식 • 응력 텐서 • 점성응력 텐서의 특성 • 점성응력 텐서는 각운동량 보존 방정식에 의해 대칭텐서임 • 점성응력 텐서는속도장과점성계수및 내부에너지 등의 변수로 결정됨 • Navier운동방정식 p : 압력 T : 점성응력 체적력(body force) 운동량 대류 항 점성응력 구배 압력 구배