한국계산과학공학회 기념워크샵 고전역학에 기초한 전산역학 - CST 와 CFD 관점에서

한국계산과학공학회 기념워크샵

고전역학에 기초한 전산역학 - CST 와 CFD 관점에서

김 승 조 *Professor, Seoul National University

김 민 기Seoul National University

문 종 근Seoul National University

2009. 10. 12, 코엑스 인터콘티넨탈 호텔

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Contents

고전 역학적 관점에서 구조역학과 유체역학구조역학과 유체역학의 수치기법 소개

유한요소 구조해석 기술 소개범용 구조해석 프로그램 및 DIAMOND/

IPSAP

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SolidMechanics

Mechanics in Physics

Fundamentals of Physics byDavid Halliday,Robert Resnick,Jearl Walker

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Mechanics in PhysicsTopics Contents

Mechanics Ch1 ~ Ch11, Ch13

Properties of Matter Ch12, Ch14, Ch19

Heat Ch18, Ch20

Sound Ch15 ~ Ch17

Electricity and Magnetism Ch21 ~ Ch33

Light Ch34 ~ Ch36

Atomic and Nuclear Physics Ch38 ~ Ch44

Relativity Ch37

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Mechanics in Physics• Mechanics

Ch1 MeasurementCh2 Motion Along a Straight LineCh3 VectorsCh4 Motion in Two and Three DimensionsCh5 Force and Motion ICh6 Force and Motion IICh7 Kinetic Energy and WorkCh8 Potential Energy and Conservation of EnergyCh9 Center of Mass and Linear MomentumCh10 RotationCh11 Rolling Torque, and Angular MomentumCh13 Gravitation

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Mechanics in Physics• Properties of Matter

Ch12 Equilibrium and ElasticityCh14 FluidsCh19 The Kinetic Theory of Gases

• HeatCh18 Temperature, Heat, and the First Law of Thermody-namicsCh20 Entropy and the Second Law of Thermodynamics

• SoundCh15 OscillationsCh16 Waves ICh17 Waves II

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Mechanics in Physics• Electricity and Magnetism

Ch21 Electric ChargeCh22 Electric FieldsCh23 Gauss' LawCh24 Electric PotentialCh25 CapacitanceCh26 Current and ResistanceCh27 CircuitsCh28 Magnetic FieldsCh29 Magnetic Fields Due to CurrentsCh30 Induction and InductanceCh31 Electromagnetic Oscillations and Alternating CurrentCh32 Maxwell's Equations; Magnetism of MatterCh33 Electromagnetic Waves

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Mechanics in Physics• Light

Ch34 ImagesCh35 InterferenceCh36 Diffraction

• Atomic and Nuclear PhysicsCh38 Photons and Matter WavesCh39 More About Matter WavesCh40 All About AtomsCh41 Conduction of Electricity in SolidsCh42 Nuclear PhysicsCh43 Energy from the NucleusCh44 Quarks, Leptons, and the Big Bang

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Mechanics in Physics• Relativity

Ch37 Relativity

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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.

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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.

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Classical Mechanics ? – Leonardo da Vinci

•Leonardo di ser Piero da Vinci (April 15, 1452 – May 2, 1519) was an Italian polymath, 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 talented person ever to have lived.

Rhombicuboctahedron

Clos Lucé in France, where Leonardo died in 1519

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Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

flight of a bird design for a flying machine

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Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

helicopter flying machine

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Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

various hydraulic machines grinding machine

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Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

Arsenal

tank

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Classical Mechanics ? – Copernicus

• Nicolaus Copernicus (February 19, 1473 – May 24, 1543) was the first astronomer to formulate a scientifically-based heliocentric cosmology that displaced the Earth from the center of the universe. His epochal book, De revolutionibus orbium coelestium (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.

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Classical Mechanics ?

Galileo Galilei (15 February 1564 – 8 January 1642)

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Classical Mechanics ? – Galilei

• Galileo Galilei (15 February 1564 – 8 January 1642) was a Tuscan physicist, 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.

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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.

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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 construc-tions 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, re-futing 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 be-fore 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).

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Classical Mechanics ? – Galilei

• Improvement of Telescope and Astronomical Observation

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Classical Mechanics ? – Galilei

• Pendulum Motion

glT 2

'Galileo's lamp' in the cathedral of Pisa

Galileo also claimed (incorrectly) that a pendulum's swings always take the same amount of time, independently of the amplitude.

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Classical Mechanics ? – Newton

Sir Isaac New-ton

(1642-1727)

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Classical Mechanics ? – Newton

Sir Isaac Newton, (4 January 1643  – 31 March 1727) was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian and one of the most influential men in human history. His Philosophiæ Naturalis Principia Mathematica, published in 1687, is consid-ered 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.

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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.

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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

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Classical Mechanics ?– Bernoulli family

• Bernoulli family tree

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Classical Mechanics ?• Daniel Bernoulli (29 January 1700 – 27 July 1782) was a Dutch-Swiss

mathematician 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.

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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.

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Classical Mechanics ?• Euler was a pioneering Swiss mathematician 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]

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Classical Mechanics ? – d’Alembert

• J Jean le Rond d'Alembert (November 16, 1717 – October 29, 1783) was a French mathematician, 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 af-ter him.

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Classical Mechanics ? – Lagrange

• Joseph Louis La-grange

(1736-1813)

.

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Classical Mechanics ?• Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia (

25 January 1736 – 10 April 1813) was an Italian mathematician 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 (Mécanique Analytique, 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.

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Classical Mechanics ? – Cauchy

• Augustin Louis Cauchy (1789-1857)

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Classical Mechanics ?• Augustin Louis Cauchy (21 August 1789 – 23 May

1857) was a French mathematician. 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.

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Classical Mechanics ? – Navier

• Claude Louis Navier (1785-1836)

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Classical Mechanics ?• Claude-Louis Navier (10 February 1785 in Dijon – 21 August 1836 in

Paris) was a French engineer and physicist who specialized in mechanics.

• The Navier-Stokes equations are named after him and George Gabriel Stokes.

• In 1802, Navier enrolled at the École polytechnique, and in 1804 continued his studies at the École Nationale 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 École Nationale des Ponts et Chaussées, and in the following year succeeded exiled Augustin Louis Cauchy as professor of calculus and mechanics at the École polytechnique.

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Classical Mechanics ? – Stokes

• George Gabriel Stokes

(1819-1903)

.

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Classical Mechanics ?• Sir George Gabriel Stokes (13 August 1819–1 February 1903), 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.

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Classical Mechanics ? – Stokes

• Navier-Stokes equation

The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, de-scribe the motion of fluid substances. These equations arise from applying Newton's second law to fluid mo-tion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term.

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

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

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Classification of Classical Mechanics

1. Dynamics : Kinematics, Kinetics, Rigid Body Motion • Rigid/Deformable Body Dynamics – Vibration - Axioms 1. 2. 3.

2. Solid Mechanics : Stress, Strain, Constitutive Equation • Structural Mechanics : Bar, Truss, Beam, Column, Frame, Plate

• Deformable Body Dynamics – Vibration - Axioms 1. 2. 3., sometimes 5.

3. Fluid Mechanics : Stress, Velocity Gradient, Fluid & Gas state • Stokes Hypothesis – Navier-Stokes Equation - Axiom 1. 2. 3. 4. 5.

4. Thermodynamics : Temperature, Heat Flux, Fourier’s Law • Heat Conduction, Convection, Radiation - Axiom 4. 5.

고전역학의 원리

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서울대학교 항공우주구조연구실

고전역학의 분류

• Lagrangian 방식• 각 입자의 관점에서 물리량의 시간변화를 기술• 모든 물리량은 각 질점 위에서 시간에 의해 (t,x0) 결정됨

T=t0 T=t0+Dt

격 자 계 가 입 자 의 움직임과 함께 변화

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

고전역학의 분류

그림 출처 : http://efdl.as.ntu.edu.tw/research/islandwake/description.html

• Eulerian 방식• 고정된 좌표 ( 격자계 ) 상에 입자의 흐름을 기술• 모든 물리량은 2 개의 변수인 시간과 공간 (t,x) 에 의해 결정됨

T=t0 T=t0+Dt

격자계 불변

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 변형 중의 물체의 변형텐서 정의고전역학의 원리

dXdZ

dY

X

Y

Z

dxdzdy

x

y

z

i

jij X

xF

• 변형 텐서 (deformation gradient Tensor) :

iii uXx

• 미소 부피 변화량 = 변형텐서의 행렬식 : det(F)=J

Xdxd

• 미소 위치벡터 변화량 : XFdudXdxd

변형 전 변형 후

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 연속체 장 방정식▫ 고전역학의 5 대 공리를 연속체에 적용한 방정식

1. 연속방정식 :

2. 선운동량 방정식 :

3. 각운동량 방정식 :

4. 열역학 제 1 법칙 :

5. 열역학 제 2 법칙 :

고전역학의 연속체 장 방정식

0 J

ba

T

)( Dtrrqe

0

rq

σ : 응력 텐서a : 가속도 벡터b : 체적력J : 미소부피 변화량ρ : 밀도e : 내부에너지 / 질량q : 열유속 벡터r : 복사열D : 속도구배텐서θ : 절대온도η : 엔트로피 / 질량

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서울대학교 항공우주구조연구실

• 속도 , 가속도 및 시간미분 관계식▫ 가속도 - 속도–변위 관계식

▫ 임의의 물리량과 시간미분의 관계식

고전역학의 연속체 장 방정식

mm

fixedX

m

mfixedxfixedX

fixedX

xvv

tv

tx

xv

tv

tv

dtvda

tx

dtxdv

XxtXxx

0 ,,

pvp

xp

vtp

tx

xp

tp

tp

dtdp

t

mm

fixedX

m

mfixedxfixedX

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서울대학교 항공우주구조연구실

• 연속방정식▫ 미소 부피 변화량의 시간미분

▫ 연속방정식 양변 시간미분

고전역학의 연속체 장 방정식

vJDJJ

)tr(

00

vt

J

vvt

J

vJ

vJJ

JJJ

0

vt

Eulerian 기반 연속방정식

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 운동량 방정식▫ 응력 텐서

▫ 점성응력 텐서의 특성 점성응력 텐서는 각운동량 보존 방정식에 의해 대칭텐서임 점성응력 텐서는 속도장과 점성계수 및 내부에너지 등의 변수로 결정됨

▫ Navier 운동방정식

ΤΤΤ

T

pI p : 압력T : 점성응력

고전역학의 연속체 장 방정식

bpvvtv

dtdva

Τ

운동량 대류 항압력 구배점성응력 구배

체적력 (body force)

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 운동량 방정식▫ 점성응력 텐서

뉴턴 유체 (Newtonian fluid) 의 구성방정식 (constitutive equa-tion) 뉴턴유체의 경우 점성계수 μ 라는 단일 물리량에 의해 점성응력이 결정

속도구배 텐서

비압축성 뉴턴 유체의 운동방정식 (Navier-Stokes equation)

DD 2)tr(32

Τ

bvpvvtv

2

j

i

i

jij x

vxv

D21

고전역학의 연속체 장 방정식

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 에너지 방정식▫ Fourier 의 전도법칙

▫ 단위질량 내부에너지

▫ 물성을 적용한 에너지 보존방정식 형태

고전역학의 연속체 장 방정식

Tkq

pTCphe p

)(2 DtrrTkTvtTC p Τ

열에너지 대류 항열 전도

복사 열 전달점성 소산 (viscous dissipation)

• k : 열전달계수• T : 온도• e : 내부에너지• h : 엔탈피• Cp : 정압비열

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 구성방정식과 상태방정식을 결합한 유동장 방정식▫ 응력텐서 구성방정식

▫ 열유속벡터 구성방정식

▫ 물성의 상태방정식

유동장 방정식

,...),,( eD

pIT

ΤΤΤΤ

Τ

// pTCpheTkq

p

....,...),(

,...),(,...),(

eekk

epp

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 구성방정식과 상태방정식을 결합한 유동장 방정식▫ 연속방정식 :

▫ 선운동량 방정식 :

▫ 각운동량 방정식 :

▫ 에너지 방정식 :

▫ 상태방정식 :

유동장 방정식

0 vt

bpvvtv

Τ

ΤΤ T

)(2 DtrrTkTvtTC p T

....,...),(

,...),(,...),(

eekk

epp

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 비압축성 뉴턴유체의 유동장 방정식▫ 연속방정식 :

▫ 운동량 방정식 :

▫ 에너지 방정식 :

▫ 상태방정식 : ρ, μ, k 등 모든 물성치는 불변 압력은 밀도 및 온도 ( 내부에너지 ) 와 무관함 수학적으로 압력은 질량보존 제약조건을 만족시키는 Lagrange 승수로

이해할 수 있음

유동장 방정식

0 v

bvpvvtv

2

)tr(2 22 DrTkTvtTC p

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 고체역학 방정식▫ Lagrangian 기반 동역학 방정식의 경우 격자계가 곧 질점의 위치 상에 있기 때문에 질량보존의 법칙은 자동으로 만족됨▫ 물체의 변형상태를 보는 관점에 따라 세 가지 종류의 응력텐서 정의 가능함

1. Cauchy Stress, true stress : 변형 후 형상에서 정의되는 응력2. 1st Piola-Kirchhoff stress : Cauchy 응력을 변형 전 초기형상으로 치환한 응력3. 2nd Piola-Kirchhoff stress : 1st PK 응력을 대칭화한 응력

▫ 실질적으로 비선형 구조해석용도로는 1 과 3 의 정의가 많이 사용됨 ( 대칭이므로 )

고체역학 방정식

Cauchy stress σ

1st Piola-Kirchhoff stress JσF-T

2nd Piola-Kirchhoff stress JF-1σF-T

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서울대학교 항공우주구조연구실

• 고체역학 방정식▫ 유동장과 다르게 고체의 응력은 주로 변위장에 지배됨▫ 응력텐서와 마찬가지로 변형률 텐서 역시 두 가지의 정의가 있음

1. Green-Lagrangian 변형률 텐서 : 변형 전 형상 기반2. Almansi-Hamel 변형률 텐서 : 변형 후 형상 기반

3. 이후의 논의를 단순화하기 위해 선형화된 계로 가정함4. 응력과 변형률의 정의를 변위가 작고 재질이 불변하다는 가정하에 아래와 같이 단순화시킬 수 있음

Green-Lagrangian ½(FTF-I)

Almansi-Hamel ½(I-F-TF-1)

klijklij

i

j

j

iij

E

xu

xu

,21 선형 변형률 텐서

응력 텐서와 탄성계수 텐서

고체역학 방정식

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서울대학교 항공우주구조연구실

• 일반적으로 고체역학 방정식은 아래의 미지수와 방정식으로 구성됨

• 이 중 위의 두 개의 방정식은 재질과 무관함▫ 첫째는 운동량 방정식이고 둘째는 변형률텐서의 정의임

• 마지막 응력 - 변형률 관계식은 재질의 특성에 의존적임▫ 일반화된 훅의 법칙 : Generalized Hook’s Law

고체역학 구성방정식

15 미지수 15 방정식• 6 strains• 6 stress• 3 displacement

• 3 equilibrium• 6 strain-displacement rela-tions• 6 stress-strain relations

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서울대학교 항공우주구조연구실

• 일반적인 선형 재질의 응력 - 변형률 관계식▫탄성계수 텐서 Eijkl

i, j, k, l = 1, 2, 3 3x3x3x3=81 components

▫대칭 조건 응력 대칭 : σij=σji

Eijkl = Eijlk

변형률 대칭 : εkl= εlk

Eijkl = Ejikl

열역학적 보존법칙으로부터 Eijkl = Eklij

고체역학 구성방정식

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

21 unknowns

앞 장의 대칭조건을 정리하면 아래 식처럼 21 개의 독립적인 미지수를 얻을 수 있다 .

고체역학 구성방정식

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

고체역학 구성방정식

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서울대학교 항공우주구조연구실

• 선형 등방성 재질의 응력 - 변형률 관계식▫ 선형 재질의 경우 탄성계수 텐서는 두 개의 물리량으로 정리할 수 있다 .

)]([133221111

E

)]([133112222

E

)]([122113333

E

13131 G

12121 G

23231 G

klijklij E

고체역학 구성방정식

)1(2

EG

Young’s modulus : EPoisson’s ratio : γ두 개의 물리량으로 정의

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 비선형 해석▫ 재질 비선형

hyper-elastic 등의 선형탄성 관계식이 아닌 경우 : ex) 고무 소성 / 항복 변형 : 재질이 탄성한계나 항복응력 (yield stress) 을 초과한 하중이 가해질 경우 탄 - 소성 (elastic-plastic), 탄 - 점성 (visco-elastic), 탄 - 점소성

▫ 기하학적 비선형 변형률 텐서의 비선형 항을 고려함 대변형이 가해질 경우 적용됨

▫ 경계 비선형 접촉 (contact) 비선형이 대표적인 예

▫ 비선형성을 모사하기 위한 수치기법 Implicit 법 : Newton-Rhapson 법 , Riks 방법 Explicit 법 : pure Lagrangian, 2 step Lagrangian – Eulerian 기법

구조역학 해석 종류

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 선형 정적해석▫ 재질이 선형 탄성관계식을 따르고 그 값이 불변이라고 가정함▫ 탄성한계 이내 하중이고 변위가 작을 경우 적합함▫ 운동방정식의 시간미분 항을 제거한 힘평형 방정식을 풀이▫ 구조물의 안전성을 평가하는 데에 가장 널리 사용되는 기법임▫ 산업체 / 연구소에서 수행하는 구조해석은 대부분 선형 정적해석임▫ 고정밀 / 최적설계와 관련하여 선형 정적해석 기법의 중요성은 예나 지금이나 무척 중요함

구조역학 해석 종류

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 고유치 해석▫ 주기적인 하중이 가해질 때 그에 맞는 구조물의 거대한 진동이 발생할 경우 , 이 주파수를 고유주파수 , 진동의 형상을 고유 모드라고 한다 .

▫ 수학적으로 선형시스템의 고유해를 구하는 문제로 설명할 수 있다 .

▫ 고유치 해석 역시 선형 정적해석과 더불어 널리 사용되는 구조해석 기법 중의 하나임 .

▫ 수학적인 의미의 구조물 고유치 해석으로 좌굴 해석이 있음

구조역학 해석 종류

0

0 ,

ew

eiwt

e

uKMw

ueuu

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

수치해석 기법 소개

• 유한차분법▫ 가장 기본적이고 이해하기 쉬운 수치기법▫ 수치적인 정확도가 낮고 보존식을 정확히 만족하기 힘들다는 단점이 있으나 현재도 단순한 격자계의 유동장에서는 사용되는 수치기법임▫ 전체 해를 격자계의 노드에 분포한 이산화된 해로 설정

u’’ (ui+1-ui+ui-1)/h2

u’ (forward difference) (ui+1-ui)/h

u’ (backward difference) (ui-ui-1)/h

u’ (central difference) (ui+1-ui-1)/2h

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

수치해석 기법 소개

• 유한체적법▫ 보존식을 만족시키기 위해 유한한 크기의 제어체적 개념을 도입▫ 전체 해는 각 제어체적의 보존식의 적분을 만족하는 이산화된 해▫ 유동장 해석에서 가장 널리 사용되는 기법임▫ 일반적인 유동장의 보존방정식을 제어체적으로 적분하면

▫ 대류항의 제어체적 사이의 물리량을 계산하는 방법 중앙차분법 (central difference)

수치적 정확성이 높지만 불안정성을 내포하고 있음 풍상차분법 (upwind difference)

수치적으로 안정하지만 가상점성 (false diffusion) 이라는 오차가 생김 이외에 MUSCL, TVD, ENO 등 여러 가지 차분법이 존재함

VVVV

dVfdSndSnvdVt )(

대류 항(convec-

tion)

확산 항(diffu-sion)

소스 항(sourc

e)

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 유한요소법▫ 구조물 해석을 위해 고안된 방법▫ 현재 유동장 해석 등 여러 분야의 편미분방정식 해법에 널리 사용됨▫ 원래 방정식과 임의의 테스트 함수와의 곱의 적분을 취한 범함수를 최소화하는 함수가 곧바로 방정식의 해가 됨▫ 타원형 방정식의 예

▫ 임의의 테스트 함수를 곱하고 적분을 취한 뒤 발산정리 적용

▫ 원래 방정식에 비해 생성된 범함수 J 는 1 차 공간미분항의 제곱이 적분가능한 함수 범위에서 해를 검색할 수 있음 미분가능성 제약조건이 약해짐 2 차 미분가능 -> 1 차 미분의 제곱이 적분 가능 범함수 J 가 0 이 되도록 하는 미지함수 u 가 곧 원래 방정식의 해가 됨

수치해석 기법 소개

fbuuk

BCsfvdVdVbuvvukvuJV V

)(),(

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

수치해석 기법 소개

• 유한요소법▫ 유한차원에서 해를 찾기 위해 미지함수 u 와 테스트함수 v 를 동일한 기저 (basis) 를 사용해서 이산화하면 범함수 J 는 다음과 같은 N 차원 함수로 표현된다 .

▫ 임의의 테스트 함수에 대해 J=0 이 되도록 하는 해 이산화 과정을 거쳐서 아래의 대수방정식으로 치환할 수 있다 .

N

iii

N

jjj

vv

uu

1

1

N

i

N

jijijiij FuKvvuJvuJ

1 1

),(),(

N

jijij FuK

1dVfF

dVbkK

iV

i

jiV

jiij

)(

)(

강성행렬 (stiffness ma-trix)하중벡터 (load vector)

• 기저가 되는 독립적인 함수들을 형상함수 (shape function) 라고 함

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

수치해석 기법 소개

• 유한요소법▫ 2 차원 평면응력 예제

EDuDD

EDuEDu

xy

y

xD

uu

u

TT

, ,0

0

, ,2

1

12

22

11

12

22

11

Vext

T

V

T

dVfF

dVDEDK

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 보존식에 적용된 유한요소법▫ 유동장의 수송방정식 (transport equation) 에 유한요소법을 적용

fvt

)()(

Vi

Vjic

Vji

Vji

c

dVfF

dVvK

dVK

dVC

FuKKCu

)('

용량 행렬 (capacity matrix)

강성 행렬 (stiffness matrix)

대류 행렬 (convection ma-trix)

하중 벡터 (load vector)

포물선형 방정식에서 대류항이 존재할 때 통상적인 유한요소 근사화를 적용할 경우 전체 강성행렬이 대류행렬의 존재로 인해 수치적 불안정성을 야기한다 .이를 극복하기 위해 다양한 유한요소 근사화가 개발되었다 .1. CV-FEM : Control volume based Finite Element Method2. Velocity-Pressure integrated Method3. PS/SU PG : Pressure Stabilized/Streamline Upwind Petrov Galerkin

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 1940 년대▫ Hrenikoff[1941] : Framework Method 선 요소 (1 차원 봉이나 보 )▫ Courant[1943] : Ritz Method 삼각형에서의 조각적 (piecewise) 보간함수 이용▫ Prager 와 Synge[1947] : 조각적 (piecewise) 보간함수 이용▫ Levy : 유연도법 ( 하중법 :flexibility matrix) 을 개발

• 1950 년대▫ propeller-->jet, flutter analysis▫ Turner :USA Boeing, seattle : Matrix Method▫ Argyris :London Univ. : Matrix Method▫ IBM 650 개발▫ Levy[5] ; 강성도법 ( 변위법 :stiffness matrix) 을 제안

초고속 컴퓨터의 발전과 더불어 그의 방법은 점점 각광받게 되었다 .▫ Argyris 와 Kelsey[1954] 는 에너지 원리를 이용한 행렬구조 해석법을 개발▫ Turner, Clough, Martin, Topp[1956] ; Plane Stress 2 차원 요소

유한 요소 구조해석 기술의 발전

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• 1960 년대▫ Progress in Aerospace Engineering : 1969 Apollo 11▫ Application in civil engineering▫ Clough : Univ. of California , Berkely▫ Martin : Univ. of Washington▫ Turner : Boeing▫ Zienkiewicz : Wales Univ. in UK▫ Clough, Topp : FEM - 유한요소란 용어를 처음 사용▫ Melosh, Grafton 과 Strome, Martin, Gallagher, Padlog, Bi-

jlaard, Melosh, Argyris, Clough 와 Rashid, Wilson ,Turner, Dill, Martin, Melosh, Gallagher, Padlog, Bijlaard, Gallagher and Padlog

▫ Zienkiewicz, Watson, King, Archer, Melosh▫ 장 (場 ) 문제 : Zienkiewicz 와 Cheung, Martin, Wilson and

Nickel▫ Oden : Nonlinear FEM▫ Zienkiewicz : Fluid, Heat,Piezo,Plasma, Chemical reaction▫ Brebbia : BEM▫ Pian : Hybrid and Mixed FEM

유한 요소 구조해석 기술의 발전

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

MP SCALAR

VECTOR Deg

ree

of P

aral

lelis

m

0.1 1 10 100 1000

FLUENT

ABAQUS

PAM-CRASHLS-DYNA

MSC.Nastran (101)

ADINAANSYS

Cache-friendlyMemory BW

Low

High STAR-CD

RADIOSSMARC

MSC.Nastran (108)

MSC.Nastran (103 and 111)

OVERFLOWCFD

Explicit FEA

Implicit FEA(Statics)

Implicit FEA(Direct Freq)

Implicit FEA(Modal Freq)

Compute Intensity Flops/word of memory traffic

Characterization of CAE Applica-tions

대표적인 상용 프로그램

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

대표적인 상용 프로그램 I

• MSC NASTRAN▫ Developed by NASA as analysis tools

for the structural analysis of space-craft. (1963) and managed by MSC

▫ Through 40 years of R&D, MSC/NASTRAN has been regarded as a standard analysis system in most area of industry.

▫ Capable of linear static analysis, buckling analysis, vibration and thermal analysis.

▫ Sparse matrix solver, Automated Component Modal Synthesis

▫ Analysis results of aerospace struc-tural parts are used as the certifica-tion of quality. Certificated by FAA (USA)

< Structural analysis of VAN >

Turbine blade thermal stress analysis

Stress analysis of Car Bumper

FE model vibration analysis

stress analysis

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

MSC NASTRAN- Parallel Performance

www.mscsoftware.com , Ver. : 2007

대표적인 상용 프로그램 I

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• ABAQUS▫ Developed by Hibbitt, Karlsson &

Sorrensen in 1978▫ In 2005, Dassault Systems(CATIA)

acquired ABAQUS : SIMULIA▫ Linear and nonlinear structural

analysis▫ Multifrontal solver, Block Lanczos

eigen solver▫ Vectorized Explicit Time Integra-

tion for the dynamic analysis▫ Conduction, convection and heat

transfer problem▫ Analysis of offshore structure

wave-induced inertial force, buoy-ant force and drag of fluid

Stress analysis of airplane engine

대표적인 상용 프로그램 II

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

• ABAQUS - Parallel Performance

E1: Car crash(274,632 elements)

E2: Cell phone drop (45,785 elements)

E3: Sheet forming (34,540 elements)

E4: Projectile penetration (237,100 elements)

대표적인 상용 프로그램 II

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

▫ Developed by John Swanson in

1970▫ Utilized in conceptual design of

the product and the manufactur-ing process

▫ Provides general graphic utilities▫ Various analysis utilities

Basic structural analysis, CFD, Electro-magnetic analysis

Thermal stress, Acoustic analysis, Piezoelectric analysis

Multi-physics▫ AI*NASTRAN solver

Wavefrontal solver based on sparse matrix solver,

Substructuring analysis option for large structures

▫ Block Lanczos eigen solver▫ Distributed Pre-conditioned Con-

jugate Gradient (DPCG) Distrib-uted Jacobi Conjugate Gradient (DJCG)

Engine analysis

Infrared camera analysis

대표적인 상용 프로그램 III

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

- Parallel Performance▫ The solvers are:

Distributed Pre-conditioned Conjugate Gradient (D-PCG)

Distributed Jacobi Conjugate Gradient (DJCG) Distributed Domain Solver (DDS) Algebraic Multigrid (AMG)

출처 : www.ansys.com , ANSYS ver. 10.0

대표적인 상용 프로그램 III

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

세계적 전산 해석 소프트웨어 개발 현황

EU• Dassault, Falcon

7x• AirBus, VDD

Japan•Adventure Project •GeoFEM

USA• Famous Commercial Soft-wares• SALINAS Project

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

SALINAS project – USA Part of the ASCI Project of The U.S. Energy Department Developed by Sandia National Laboratory in 1999 Provision of Scalable Calculation Tool

such as Stress, Vibration, and Transient Response Analysis for Very Complex Structures

ASCI System composed of Several Thousand Processors Implicit Solver, DDM-based FETI-DP Algorithm

대형 병렬 소프트웨어 개발 연구 I

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

ADVENTURE Project - JAPAN ADVanced ENgineering analysis Tool for Ultra large REal world Development of Computational Mechanics System for Large

Scale Analysis and Design 1997 ~ 2002 Goal : Compute the 10~100million size model in 1 hour~1 day Composed with twenty pre-processing, post-processing modules

Pantheon Model(1.5M DOF) Solid Analysis Fluid Anal-ysis

대형 병렬 소프트웨어 개발 연구 II

84

한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

GeoFEM - JAPAN Parallel FE Solid Earth Simulator 1997 ~ 2003 Localized operation & optimum data structures

for massively parallel computation Pluggable design Platform : linear solver, I/O, visualization

GeoFEM Platform

A test dataset on the ES with 5,886,640 unstructured ele-

ments

Geodynamo process and fluid dy-namics in the Earth’s outer core

Modeling of Philippine Sea plate boundary

대형 병렬 소프트웨어 개발 연구 III

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

대형 병렬 소프트웨어 개발 연구 IV

High Per-formance Parallel

Computing

General Purpose &

User Friendly Software

Challenge Application

to Large Scale

AnalysisI P S A P

High Performance Parallel Finite Element Analysis Software based on Parallel Multi-Frontal Algorithm

High-Performance Parallel Software

Grand Challenge Applications (Large Scale)

High-Performance Hardware (Supercomputer, Clusters, GRID)

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Parallel Structural Analysis Software, IPSAP • IPSAP : Internet Parallel Structural Analysis Program

IPSAP• General Purpose FEA Program• Generality, Single & Parallel, Written C & C++• Libraries : BLAS, LAPACK, METIS

FEM Module• FE Model : Solid, Plate, Beam, Spring, Rigid Body Element,Concentrated Mass• Nodal force, Pressure, Acceleration, Temperature load

• Thermal Module• GUI Interface

Solver ModuleLinear Solver

• Multifrontal Solver• Hybrid Solver

Eigen Solver• Block Lanczos Solver

Lagrangian Eulerian

IPSAP/Standard

• Explicit Time Integration, Auto Time Step Control• Elastic, Orthotropic, Elastoplastic, Johnson-Cook• EOS (Equation of State) : Polynomial Model, JWL, Grüneisen• Artificial Bulk Viscosity• Contact Treatment : Contact Search : Bucket Sorting Master-Slave Algorithm, Penalty Method Single Surface Contact

IPSAP/Explicit

Time Transient Solver• Generalized Trapezoidal Solver for Thermal Problems

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Sponsored Research by Microsoft

• Porting and Managing IPSAP to Windows OS• Improving of IPSAP GUI with Microsoft,DIAMOND/IPSAP• Documentation of IPSAP• Construction of DB for Application Problems of IPSAP

• Linear Static Solver with Hybrid Solution Method• Block-Lanczos Eigen Solver with Hybrid Solution Method• Thermal Analysis•High-Level Contact and Crash Analysis

Focus on Serial and parallel Performance on the MPP Environment• Direct Solution Method – Developed Multifrontal Solver• Eigen Solution Method – Lanczos Eigen Solver• Build Up Cluster Super Computer – PEGASUS (Microsoft, Samsung, Intel Korea)• Applications – Cyclocopter, Cycloidal Windturbine, Composite Materials, etc.

IPSAP Ver. 1.0

IPSA

P Ve

r. 1.

0 –

Rele

ase

to P

ublic

for

Free

IPSAP specialized in Windows OS

Improving IPSAP Usability

IPSAP Ver. 1.1 to Ver. 3.0 Microsoft• IPSAP specified for Windows• Inclusion of IPSAP to Windows• Revitalization of Windows HPC

NRL• Produce better research results• Magnify of IPSAP user• Increase Manufacture efficiency

2007.01.15Time Line

“Porting to Windows OS and Management of high performance FE software, IPSAP (Internet Parallel Structural Analysis Program) &Improvement of IPSAP GUI for convenient and smooth execution in Windows”

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Free Release by Website

• Homepage : http://ipsap.snu.ac.kr– Modules included : Stress analysis, vibration analysis– Elements : solid, shell, beam– Downloadable IPSAP executables

• Windows, Linux, OS-X• Serial, parallel version

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

IPSAP/Standard – Multi-Frontal Solver

Illustration of multi-frontal method

Step1. Domain partitioning

Step2.Symbolic factorization

Step3. Numerical factorization

Step4. Triangular solve

Procedure of multi-frontal method

Optimized for finite element

method !!

• Concept of Multi-Frontal solver▫ Utilization of multiple elimination fronts instead of single front▫ Domain-wise Approach for efficient elimination procedure

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

IPSAP/Standard – Parallel Multi-Frontal Solver

Parallel Stage : Distributed Memory Parallelization Merging makes the distributed frontal matrix Factorization is performed with distributed matrix

Proc 0 Proc 1

Proc 2 Proc 3 Proc 2,3

Proc 0,1

Factorization

Proc 0,1,2,3

Factorization

Stage 1 Stage 2

Stage 3 Stage 4Parallel extend-add Operation

2-D processor map is used

New parallel linear algebra subroutines are required which allow flexible block size

Parallel Linear Algebra Subroutine in C PLASC is developed

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서울대학교 항공우주구조연구실

IPSAP/Explicit

• Parallelization

1 2 3

4 5 6

7 8 9

FE Calculation Parallelization

• Compute at each processor independently.

• Interface values are swapped and added.

Contact Parallelization

1. 3D box define along with Master Segment2. Slave node information communication3. Contact force calculation independently4. Contact force vector communication

1 2 3 4

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서울대학교 항공우주구조연구실

Serial Performance• Comparison with Commercial Software

▫ 32x32x32 hexagonal elements (DOFs = 107,811)

2.1 Gflops

Windows Compute Cluster Server 2003, Intel Core2 Quad 2.66GHz, 8GB Memory

6.52 Gflops

Mesh Model

IPSAP CS1 CS20

100

200

300

400

500

600

700

44.7650000000003

602.625

88.143.063

552.327999999998

85.2

Elapsed TimeCPU Time

Alpha EV67 (667MHz)

IBM Power4 1.3GHz

Intel Xeon 2.4GHz (Linux)

0

200

400

600

800

1000

1200

1400

1600

305

112 93

331147 203

1345IPSAPCS1CS2

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서울대학교 항공우주구조연구실

Parallel Performance

• IPSAP/Standard – Static Analysis- Intel Quad Core 2.66GHz- Infiniband Network

1 4 160

2

4

6

8

10

12

14

16

1.00

4.16

12.87

SCALABILITY TEST(2D Topology)

IPSAPIdeal

Number of CPU

Scal

abil-

ity

– Vibration Analysis- Intel Quad Core 2.66GHz- Infiniband Network

1 4 160

2

4

6

8

10

12

14

16

1.00

5.11

14.07

SCALABILITY TEST (2D Topology)

IPSAPIdeal

Number of CPU

Scal

abili

ty

4 Node Shell Element

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서울대학교 항공우주구조연구실

Parallel Performance

• Performance of extremely large scale prob-lems▫ 100 million DOFs Problem

▫ FEA Model- 4096*4096*1 - 8 node hexagon element- DOFs : ~ 100 million

▫ Result of in Xeon 256 CPUs- Factorization : 7284.6 sec- Elimination & Substitution : 1387.6 sec

1st success of a 100 million FEA in 32-bit cluster system with direct solver !!

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서울대학교 항공우주구조연구실

Parallel Performance• IPSAP/Standard

– Vibration Analysis▫ comparison with Commercial

Software- PEGASUS Cluster– Distributed memory parallel

Parallel Performance Test on Windows Cluster

Intel Quad Core 2.66GHz Infiniband Network

CS1

One Lanczos iteration time ( summing up three above

routines ) - Less than 1 sec up to 64

CPUs

No. of CPU

meshDOF

1 N=10061206

2 N=141120984

4 N=200242406

8 N=282480534

16 N=400964806

32N=565

1922136

64N=800

3849606

128 N=11327702134

1 4 160

2

4

6

8

10

12

14

16

1.00

5.11

14.07

SCALABILITY TEST (2D Topology)

IPSAPIdeal

Number of CPU

Scal

abilit

y

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서울대학교 항공우주구조연구실

Parallel Performance

0 2 4 6 8 10 12 14 16 1802468

1012141618

IdealIPSAP(Time Step=1)IPSAP(Time Step=10)

number of CPUs

Spee

d-U

p

IPSAP/Standard – Heat Transfer Analysis- PEGASUS Cluster- OS : Linux Redhat 9.0 kernel 2.4.26- CPU : Xeon 2.2GHz 1 ~ 16

Comparison of Computing Time w.r.t . the number of Time Step Speed-Up w.r.t the number of CPU

Time = MFS Time + Transient Time

0 5 10 15 20 25 30 350

100200300400500600700

Transient TimeMFS Time

The number of Time Steps

Tim

e(se

c)

# of CPUs 1 2 4 8 16Time

(Time Step=1) 291.18 147.85 76.61 44.39 25.54 Time

(Time Step=10) 391.50 198.95 107.66 65.21 41.08

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서울대학교 항공우주구조연구실

Parallel Performance• IPSAP/Explicit

▫ comparison with Commer-cial Software

- PEGASUS Cluster– Distributed memory parallel– 127 Speed up / 128 CPUs

Parallel Performance Test on Windows Cluster

Intel Quad Core 2.66GHz Infiniband Network

1 4 160

2

4

6

8

10

12

14

16

1

4.01

15.13

SCALABILITY TEST (Taylor Impact Test)

IPSAPIdeal

Scal

abil-

ity

Number of CPU

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서울대학교 항공우주구조연구실

Inexpensive SuperComputing Re-sources

• New Type Super Computing System– Windows HPC Server 2008 – Infiniband Network

N001hpc.ne

t

N002hpc.ne

t

N003hpc.ne

t

N004hpc.ne

t

Infiniband Network Device(10 Gbps)

Public Network

Head Node & Compute Node

Compute Node

Unit Node Total System

CPU Intel Core 2 Quad 3.2 GHz (4 Core X 2) 32 Cores

RAMDELL : DDR2 ECC 64GB

(4GB X 16)Mac : DDR2 ECC 32GB

(2GB X 16)224 GB

HDD SATA 500 GB 2 TB

Network Infiniband (10Gbps) for MPI network

OS / Compiler

Windows HPC Server 2008 / Visual Studio 2008

MPI MS-MPI in HPC Pack

HPL Benchmark Results : 207 Gflops/409.6 Gflops Gflops

(Rmax/Rpeak)

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서울대학교 항공우주구조연구실

• Scalability & Speed-up test ▫ 2D Mesh topology

No. of Cores Mesh Number of

UnknownsPerfor-mance

(GFLOPS)

Scaled Speedu

p

8 1000_1000 6,012,006 20.282 1

16 1260_1260 9,540,726 46.121 2.274

32 1588_1588

15,149,526 74.734 3.685

Computing Performance Result

Speed-Up TestMesh : 2D Quad_1260_1260

Scalability Test

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서울대학교 항공우주구조연구실

• Scalability & Speed-up test▫ 3D Mesh topology

No. of Cores Mesh Number of

UnknownsPerfor-mance

(GFLOPS)

Scaled Speedu

p8 64_64_64 823,875 46.125 1

16 72_72_72 1,167,051 98.255 2.13032 80_80_80 1,594,323 137.443 2.980

Computing Performance Result

Speed-Up TestMesh : 3D Hexa_72_72_72

Scalability Test

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서울대학교 항공우주구조연구실

Computing Performance Result• Example : DNS (Direct Numerical Simulation)

▫Woven Composite Extension test in x-direction 3,713,328 DOFs Factorization time ( in 4 Nodes )

: 379.3 sec Factorization Performance ( in 4 Nodes ) : 164.6 Gflops

No. of Nodes 1,237,776

No. of Elements 1,152,000

No. of DOFs 3,713,328

Speed-Up Test

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서울대학교 항공우주구조연구실

Computing Performance Result• Example : Ship Hull Structure▫ 1,085,120 DOFs▫ Factorization time▫ : 45.2 sec (in 4 Nodes)▫ Factorization Performance

: 50.093 Gflops (in 4 Nodes)

No. of Nodes 180,855

No. of Elements 294,657

No. of DOFs 1,085,120

Speed-Up Test

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서울대학교 항공우주구조연구실

Necessity of Pre/Post GUI Software for IPSAP

IPSAP/EXPLICIT

VIBRATION ANALYSIS

STRESS ANAL-YSIS

Pre-Post GUI Tool

Analysis

Post-Pro-cessing

Pre-pro-

cessing DIAMOND/IPSAP

Recognizing the necessity of GUI-based Visualiza-tion Toolkit

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Development of User-Friendly Visualization Toolkit - DIAMOND/IPSAP

import

IPSAP input file

IPSAPFEM SOLVER

output file

V i s u a l i z e C o n t e n t sI n D i a m o n d

d o c u m e n t

Diamond document

meshproperty, material . . .

Geometry

Line, Surface, Solid

MeshBeam, Plate

Createinput file

Load, Boundary

Material, Property

DAISManager

View Control

Development of Pre-/Post-Visualization Toolkit for IPSAPDevelopment Environ-

ment Windows Visual Studio 2008 Graphic Library : Open CASCADE

6.2.0 Adoption of Ribbon UI

Realized Functions Pre-Processing Analysis of IPSAP Post-Processing Displacement/Stress View Eigen Mode View Parallel Analysis of IPSAP

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Main Frame of DIAMOND/IPSAP

File

Analysis

Geometry

Mesh

Post

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한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Specific Application Module of DIAMOND/IPSAP

Realization of Several Application Modules based on DIAMOND/IPSAPSatellite Bus

Design Optimization

Module

Virtual Simulation & Experiment

DIAMOND/

IPSAP

DIAMOND/SBD

DIAMOND/VE

Dynamics & Stability of Helicopter Rotor Blade System

Crash & Impact Simulation

Educational FE Code for Partial Differential Equation

Semi-Conductor & MEMS Packaging Simulation

DIAMOND/VTOL

DIAMOND/PDE

DIAMOND/PACK

DIAMOND/IE

Option PricingDIAMOND/Finance

107

한국계산과학공학회 기념워크샵

서울대학교 항공우주구조연구실

Thank you!