김 승 조 * Professor, Seoul National University

한국계산과학공학회 기념워크샵

고전역학에 기초한 전산역학 - CS T 와 CFD 관점에서

김 승 조 *

Professor, Seoul National University

김 민 기

Seoul National University

문 종 근

2

한국계산과학공학회 기념워크샵

Contents

1

2

3

4

Solid

Mechanics

Mechanics in Physics

Fundamentals of Physics by

David Halliday,

Robert Resnick,

Jearl Walker

Solid

Mechanics

Mechanics in Physics

Topics 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

Solid

Mechanics

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

Solid

Mechanics

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

Solid

Mechanics

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

Solid

Mechanics

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

Solid

Mechanics

Mechanics in Physics

• Relativity

Ch37 Relativity

Solid

Mechanics

Classical Mechanics ?

• Classical mechanics is used for describing the motion of macroscopic obj ects, 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, an d solids, and so on. Classical mechanics is enhanced by special relativity f or 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.

Solid

Mechanics

Classical Mechanics ?

• The term classical mechanics was coined in the early 20th century to des cribe the system of mathematical physics begun by Isaac Newton and ma ny contemporary 17th century workers, building upon the earlier astrono mical theories of Johannes Kepler, the studies of terrestrial

projectile motion of Galileo, but before the development of quantum ph ysics and relativity. Therefore, some sources exclude so-called "

relativistic physics" from that category. However, a number of modern so urces do include Einstein's mechanics, which in their view represents clas sical mechanics in its most developed and most accurate form.

• The initial stage in the development of classical mechanics is often referr ed to as Newtonian mechanics, and is associated with the physical conce pts employed by and the mathematical methods invented by Newton hi mself, in parallel with Leibniz, and others. More abstract and general me thods include Lagrangian mechanics and Hamiltonian mechanics. Much of the content of classical mechanics was created in the 18th and 19th c enturies and extends considerably beyond (particularly in its use of analyt ical mathematics) the work of Newton.

Solid Mechanics

Classical Mechanics ? – Leonardo da Vinci

•

painter, sculptor, architect, botanist, musician and writer. Leonardo has often b een described as the archetype of the renaissance man, a man whose unquenc hable curiosity was equaled only by his powers of invention.

He is widely considered to be one of the greatest painters of all time and perhap s the most diversely talented person ever to have lived.

Solid Mechanics

Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

Solid Mechanics

Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

Solid Mechanics

Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

Solid Mechanics

Classical Mechanics ? – Leonardo da Vinci

• Leonardo as observer, scientist and inventor

tank

Solid Mechanics

Classical Mechanics ? – ^Copernicus

•

astronomer to formulate a scientifically-based heliocentric cosmology that displa ced the Earth from the center of the universe. His epochal book, De

revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), i s often regarded as the starting point of modern astronomy and the defining epiphany that began the Scientific Revolution.

Solid

Mechanics

Classical Mechanics ?

Galileo

Galilei

Solid Mechanics

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, taug ht in nearly all high school and introductory college physics courses, was studi ed by Galileo as the subject of kinematics. His contributions to observational a stronomy include the telescopic confirmation of the phases of Venus, the disc overy 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 a pplied science and technology, improving compass design.

Solid

Mechanics

Classical Mechanics ?

• Galileo is perhaps the first to clearly state that the laws of nature ar

e mathematical. In The Assayer he wrote "Philosophy is written in t

his grand book, the universe ... It is written in the language of mat

hematics, and its characters are triangles, circles, and other geome

tric figures; ...". His mathematical analyses are a further developme

nt of a tradition employed by late scholastic natural philosophers,

which Galileo learned when he studied philosophy. Although he tri

ed to remain loyal to the Catholic Church, his adherence to experi

mental results, and their most honest interpretation, led to a rejecti

on of blind allegiance to authority, both philosophical and religiou

s, in matters of science. In broader terms, this aided to separate sci

ence from both philosophy and religion; a major development in h

uman thought.

Solid

Mechanics

Classical Mechanics ?

• Galileo proposed that a falling body would fall with a uniform acceleration, as long as the r esistance of the medium through which it was falling remained negligible, or in the limitin g case of its falling through a vacuum. He also derived the correct kinematical law for the d istance travelled during a uniform acceleration starting from rest—namely, that it is proport ional to the square of the elapsed time ( d ∝ t ² ). However, in neither case were these disc overies entirely original. The time-squared law for uniformly accelerated change was alread y known to Nicole Oresme in the 14th century, and Domingo de Soto, in the 16th, had sug gested that bodies falling through a homogeneous medium would be uniformly accelerate d^[ Galileo expressed the time-squared law using geometrical constructions and mathemati cally-precise words, adhering to the standards of the day. (It remained for others to re-expr ess 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 hypothe sis that objects "naturally" slow down and stop unless a force acts upon them (philosophic al 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, verifi ed experimentally, and introduced the idea of frictional force, the key breakthrough in vali dating inertia). Galileo's Principle of Inertia stated: "A body moving on a level surface will c ontinue in the same direction at constant speed unless disturbed." This principle was incor porated into Newton's laws of motion (first law).

Solid Mechanics

Classical Mechanics ? – ^Galilei

• Improvement of Telescope and Astronomical Observation

Solid Mechanics

Classical Mechanics ? – ^Galilei

• Pendulum Motion

g T  2  l

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

Solid Mechanics

Classical Mechanics ? – ^Newton

Sir Isaac New- ton

(1642-1727)

Solid Mechanics

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

Solid

Mechanics

Classical Mechanics ?

• In mechanics, Newton enunciated the principles of conservation of b oth momentum and angular momentum. In optics, he built the first practical reflecting telescope^[5] and developed a theory of colour bas ed on the observation that a prism decomposes white light into the many colours which form the visible spectrum. He also formulated a n 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 de monstrated the generalised binomial theorem, developed the so-call ed "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 d emonstrated 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.

Solid

Mechanics

Classical Mechanics ?

• In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrate d the generalised binomial theorem, developed the so-called "

Newton's method" for approximating the zeroes of a function, and contri buted to the study of power series.

• Most modern historians believe that Newton and Leibniz developed

infinitesimal calculus independently, using their own unique notations. Ac cording to Newton's inner circle, Newton had worked out his method year s before Leibniz, yet he published almost nothing about it until 1693, and did not give a full account until 1704. Meanwhile, Leibniz began publishin g a full account of his methods in 1684. Moreover, Leibniz's notation an d "differential Method" were universally adopted on the Continent, and af ter 1820 or so, in the British Empire. Whereas Leibniz's notebooks show th e advancement of the ideas from early stages until maturity, there is only t he end product in Newton's known notes. Newton claimed that he had be en reluctant to publish his calculus because he feared being mocked for it

Solid Mechanics

Classical Mechanics ? – Bernoulli family

• Bernoulli family tree

Solid

Mechanics

Classical Mechanics ?

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

mathematician and was one of the many prominent mathematicians i n the Bernoulli family. He is particularly remembered for his applicatio ns of mathematics to mechanics, especially fluid mechanics, and for hi s pioneering work in probability and statistics.

• Born in Groningen, in the Netherlands, the son of Johann Bernoulli, n

ephew of Jacob Bernoulli, younger brother of Nicolaus II Bernoulli, an

d older brother of Johann II, Daniel Bernoulli has been described as "b

y far the ablest of the younger Bernoullis". He is said to have had a ba

d relationship with his father. Upon both of them entering and tying f

or first place in a scientific contest at the University of Paris, Johann, u

nable to bear the "shame" of being compared to his offspring, banne

d Daniel from his house. Johann Bernoulli also tried to steal Daniel's b

ook Hydrodynamica and rename it Hydraulica. Despite Daniel's attem

pts at reconciliation, his father carried the grudge until his death.

Solid

Mechanics

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 eve ntually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his mat ernal grandmother. At the age of thirteen he matriculated at the

University of Basel, and in 1723, received his M.Phil with a dissertation t hat compared the philosophies of Descartes and Newton. At this time, h e was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Ber noulli convinced Paul Euler that Leonhard was destined to become a gre at mathematician. In 1726, Euler completed his Ph.D. dissertation on the propagation of sound and in 1727, he entered the Paris Academy Prize P roblem competition, where the problem that year was to find the best w ay to place the masts on a ship. He won second place in the first competi tion but Euler subsequently won this coveted annual prize twelve times i n his career.

Solid

Mechanics

Classical Mechanics ?

• Euler was a pioneering Swiss mathematician and physicist who spent m ost 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 ter minology and notation, particularly for mathematical analysis, such as th e 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 analy tic tools to problems in classical mechanics, Euler also applied these tech niques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accom plishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and cal culating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.^[33]

Solid Mechanics

Classical Mechanics ? – ^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.

Solid Mechanics

Classical Mechanics ? – ^Lagrange

• Joseph Louis La- grange

(1736-1813)

.

Solid

Mechanics

Classical Mechanics ?

• Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia (

25 January 1736 – 10 April 1813) was an Italian mathematician a nd 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 r ecommendation of Euler and D'Alembert, in 1766 Lagrange succ eeded 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 severa l prizes of the French Academy of Sciences. Lagrange's treatise o n analytical mechanics ( Mécanique Analytique , 4. ed., 2 vols. Pari s: Gauthier-Villars et fils, 1888-89), written in Berlin and first publ ished in 1788, offered the most comprehensive treatment of

classical mechanics since Newton and formed a basis for the deve

lopment of mathematical physics in the nineteenth century.

Solid Mechanics

Classical Mechanics ? – ^Cauchy

• Augustin Louis Cauchy

(1789-1857)

Solid

Mechanics

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 mann

er 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 per

spicuous and rigorous methods Cauchy exercised a great influenc

e over his contemporaries and successors. His writings cover the

entire range of mathematics and mathematical physics.

Solid Mechanics

Classical Mechanics ? – ^Navier

• Claude Louis Navier

(1785-1836)

Solid

Mechanics

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 continue d his studies at the École Nationale des Ponts et Chaussées, from which h e graduated in 1806. He eventually succeeded his uncle as Inspecteur gen eral at the Corps des Ponts et Chaussées.

• He directed the construction of bridges at Choisy, Asnières and Argenteui l 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 183 0, he took up a professorship at the École Nationale des Ponts et Chaussé es, and in the following year succeeded exiled Augustin Louis Cauchy as p rofessor of calculus and mechanics at the École polytechnique.

Solid Mechanics

Classical Mechanics ? – ^Stokes

• George Gabriel Stokes

(1819-1903)

.

Solid

Mechanics

Classical Mechanics ?

• Sir George Gabriel Stokes (13 August 1819–1 February 1903), was a mathematician and physicist, who at Cambridge made important contr ibutions to fluid dynamics (including the Navier–Stokes equations),

optics, and mathematical physics (including Stokes' theorem). He was s ecretary, 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 moti on. These were followed in 1845 by one on the friction of fluids in moti on and the equilibrium and motion of elastic solids, and in 1850 by ano ther 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 m any natural phenomena, such as the suspension of clouds in air, and th e 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, th e skin resistance of ships and aerodynamics for airplane design.

Solid Mechanics

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

42

한국계산과학공학회 기념워크샵

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

43

한국계산과학공학회 기념워크샵

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

고전역학의 원리

44

한국계산과학공학회 기념워크샵

고전역학의 분류

• Lagrangian 방식

• 각 입자의 관점에서 물리량의 시간변화를 기술

• 모든 물리량은 각 질점 위에서 시간에 의해 (t,x ₀) 결정됨

T=t T=t +Dt

격 자 계 가 입 자 의 움 직 임 과 함께 변화

45

한국계산과학공학회 기념워크샵

고전역학의 분류

• Eulerian 방식

• 고정된 좌표 ( 격자계 ) 상에 입자의 흐름을 기술

• 모든 물리량은 2 개의 변수인 시간과 공간 (t,x) 에 의해 결정됨

T=t T=t +Dt

격자계 불변

46

한국계산과학공학회 기념워크샵

• 변형 중의 물체의 변형텐서 정의

고전역학의 원리

dX

dZ dY

X Y

Z

dx dy dz

x y

z

i j

ij X

F x



 

• 변형 텐서 (deformation gradient Tensor) :

i i

i

X u

x  

X d

x d

• 미소 위치벡터 변화량 d x  d X  du  Fd X

변형 전 변형 후

47

한국계산과학공학회 기념워크샵

• 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



 J 

b

a  

    







) ( D tr

r q

e  



    

 0

 



 



 









 

 ^{q r}

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

한국계산과학공학회 기념워크샵

• 속도 , 가속도 및 시간미분 관계식

▫ 가속도 - 속도 변위 관계식 –

고전역학의 연속체 장 방정식

 

m m

fixed X

m fixed m

x fixed

X

fixed X

x v v

t v t

x x

v t

v t

v dt

v a d

t x dt

x v d

X x

t X x x



 



 







 



 



 





 







 





, ,

x v p t p t

x x

p t

p t

p dt

dp

m m

fixed X

m m

fixed x fixed

X



 



 







 



 



 

 



49

한국계산과학공학회 기념워크샵

• 연속방정식

▫ 미소 부피 변화량의 시간미분

▫ 연속방정식 양변 시간미분

고전역학의 연속체 장 방정식

v J

D J

J

 tr( )   

 

 

^{ }

 



 



 



    



 



 



  





























 



 















v J

v t v

J

v J

v J J

J J

J

  ⁰

   



  v

t 



Eulerian 기반 연속방정 식

50

한국계산과학공학회 기념워크샵

• 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

Τ Τ

Τ









T

 pI

T : 점성응력

고전역학의 연속체 장 방정식

b p

v t v

v dt

a  dv  

     



 



  



 

 Τ

운동량 대류 항 점성응력 구배

체적력 (body force)

51

한국계산과학공학회 기념워크샵

• 운동량 방정식

▫ 점성응력 텐서



뉴턴 유체

의 구성방정식

 뉴턴유체의 경우 점성계수 μ 라는 단일 물리량에 의해 점성응력이 결정

 속도구배 텐서

 비압축성 뉴턴 유체의 운동방정식 (Navier-Stokes equation) D

D 

 tr( ) 2 3

2 



 Τ

b v

p v

t v

v

 





 



  



 ₂















 



 

j i i

j

ij x

v x

D v

2 1

고전역학의 연속체 장 방정식

52

한국계산과학공학회 기념워크샵

• 에너지 방정식

▫ Fourier 의 전도법칙

▫ 단위질량 내부에너지

▫ 물성을 적용한 에너지 보존방정식 형태

고전역학의 연속체 장 방정식

T k

q   





T p p C

h

e   



)

2

T r tr ( D

k T

t v

C

T      Τ



 



   



 



열에너지 대류 항 복사 열 전달

점성 소산 (viscous dissipa- tion)

• k : 열전달계수

• T : 온도

• e : 내부에너지

• h : 엔탈피

• C_p : 정압비열

53

한국계산과학공학회 기념워크샵

• 구성방정식과 상태방정식을 결합한 유동장 방정식

▫ 응력텐서 구성방정식

▫ 열유속벡터 구성방정식

▫ 물성의 상태방정식

유동장 방정식

,...) ,

, (D e

pI

T



 Τ Τ

Τ Τ

Τ













 /

/ C T p p

h e

T k q

p















,...) ,

(

,...) ,

(

,...) ,

(









 e e k k

e p

p







54

한국계산과학공학회 기념워크샵

• 구성방정식과 상태방정식을 결합한 유동장 방정식

▫ 연속방정식 :

▫ 선운동량 방정식 :

▫ 각운동량 방정식 :

▫ 에너지 방정식 :

▫ 상태방정식 :

유동장 방정식

 





 

 v

t 



b p

v t v

v 

     



 



  



 Τ

Τ Τ



)

2T r tr( D

k T

t v

C_p T      T



 



  



 



,...) ,

(

,...) ,

(

,...) ,

(









 e e k k

e p

p







55

한국계산과학공학회 기념워크샵

• 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 이해

할 수 있음

유동장 방정식

 0



 v

b v

p v

t v

v

 





 



  



 ₂

) tr(

2 ²

2T r D

k T

t v

C_p T  

     



 



  





56

한국계산과학공학회 기념워크샵

• 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 가능 함

1. Cauchy Stress, true stress : 변형 후 형상에서 정의되는 응력

2. 1^st Piola-Kirchhoff stress : Cauchy 응력을 변형 전 초기형상으로 치환한 응력

3. 2^nd Piola-Kirchhoff stress : 1^st PK 응력을 대칭화한 응력

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

고체역학 방정식

Cauchy stress σ

1^st Piola-Kirchhoff stress JσF^-T

57

한국계산과학공학회 기념워크샵

• 고체역학 방정식

▫ 유동장과 다르게 고체의 응력은 주로 변위장 에 지배됨

▫ 응력텐서와 마찬가지로 변형률 텐서 역시 두 가지의 정의가 있음

1.

2.

3.

4.

이 단순화시킬 수 있음

Green-Lagrangian ½(F^TF-I)

Almansi-Hamel ½(I-F^-TF^-1)

i j j

ij i

x u x

 u _ ^





 







 



  ,

2

1

고체역학 방정식

58

한국계산과학공학회 기념워크샵

• 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

방정식

• 6 strains

• 6 stress

• 3 displacement

• 3 equilibrium

• 6 strain-displacement rela- tions

• 6 stress-strain relations

59

한국계산과학공학회 기념워크샵

• 일반적인 선형 재질의 응력 - 변형률 관계식

▫ 탄성계수 텐서 E

 i, j, k, l = 1, 2, 3

 3x3x3x3=81 components

▫ 대칭 조건

 응력 대칭 : σ

=σ

 E

= E

 변형률 대칭 : ε

= ε

 E

= E

 열역학적 보존법칙으로부터

고체역학 구성방정식

60

한국계산과학공학회 기념워크샵

21 unknowns

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

고체역학 구성방정식

61

한국계산과학공학회 기념워크샵

고체역학 구성방정식

62

한국계산과학공학회 기념워크샵

• 선형 등방성 재질의 응력 - 변형률 관계식

▫ 선형 재질의 경우 탄성계수 텐서는 두 개의 물리량으로 정리할 수 있다 .

)]

( 1 [

33 22

11

11    

   

E

)]

( 1 [

33 11

22

22    

   

E

)]

( 1 [

22 11

33

33    

   

E

13 13

1 

  G

12 12

1 

  G

23 23

1 

  kl

ijkl

ij

E 

 

고체역학 구성방정식

) 1 ( 2 

 E

G

Young’s modulus : E Poisson’s ratio : γ 두 개의 물리량으로 정

63

한국계산과학공학회 기념워크샵

• 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 : pure Lagrangian, 2 step Lagrangian – Eulerian 기법

구조역학 해석 종류

64

한국계산과학공학회 기념워크샵

• 선형 정적해석

▫

▫

▫

▫

▫ 산업체 /

▫ 고정밀 /

구조역학 해석 종류

65

한국계산과학공학회 기념워크샵

• 고유치 해석

▫ 주기적인 하중이 가해질 때 그에 맞는 구조물의 거대한 진동이 발생할 경 우 , 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 .

구조역학 해석 종류

 ^{   ,} 

^{  0 0}

e iwt

e

u K M

w

u e

u

u

66

한국계산과학공학회 기념워크샵

수치해석 기법 소개

• 유한차분법

▫ 가장 기본적이고 이해하기 쉬운 수치기법

▫ 수치적인 정확도가 낮고 보존식을 정확히 만족하기 힘들다는 단점이 있 으나 현재도 단순한 격자계의 유동장에서는 사용되는 수치기법임

▫ 전체 해를 격자계의 노드에 분포한 이산화된 해로 설정

u’’ (u_i+1-u_i+u_i-1)/h² u’ (forward difference) (u_i+1-u_i)/h u’ (backward difference) (u_i-u_i-1)/h

u’ (central difference) (u -u )/2h

67

한국계산과학공학회 기념워크샵

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











V V V

V

dV f dS

n dS

n v

t (







(convec-대류 항 tion)

확산 항(diffu- sion)

소스 항(sourc e)

68

한국계산과학공학회 기념워크샵

• 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 곧바 로 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 범위에

서 해를 검색할 수 있음

 미분가능성 제약조건이 약해짐

수치해석 기법 소개

 ^{k  u}  ^{ bu  f}







BCs fvdV

dV buv

v u

k v

u J

V V













  ^{( )} 

)

,

(

69

한국계산과학공학회 기념워크샵

수치해석 기법 소개

• 유한요소법

▫ 유한차원에서 해를 찾기 위해 미지함수 u 와 테스트함수 v 를 동일한 기 저 (basis) 를 사용해서 이산화하면 범함수 J ^{는 다음과 같은} N ^{차원 함수} 로 표현된다 .

▫ 임의의 테스트 함수에 대해 J=0 이 되도록 하는 해



이산화 과정을 거쳐서 아래의 대수방정식으로 치환할 수 있다













N i

i i N j

j j

v v

u u

1 1





 

  



 



 



 ^N

i

N

j

i j

ij i

i

j v v K u F

u J v

u J

1 1

) , ( )

, (





j

i j

ij

u F

K

1

F f dV

dV b

k K

i i

j i V

j i

ij

) (

) (



























trix)

하중벡터 (load vector)

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

70

한국계산과학공학회 기념워크샵

수치해석 기법 소개

• 유한요소법

▫ 2 차원 평면응력 예제

EDu D

D

EDu E

Du x

y

y x

D

u u u

T

T 











































 

 







 



 































































,

, 0

0

,

,

2 1

12 22 11

12 22 11

   









V

ext T V

T

dV f

F

dV D

E D

K







71

한국계산과학공학회 기념워크샵

• 보존식에 적용된 유한요소법

▫ 유동장의 수송방정식 (transport equation) 에 유한요소법을 적용

 

 

 v f

t      



 ) ( )

( 

































V i V

j i c

V

j i

V

j i

c

dV f F

dV v

K

dV K

dV C

F u K K Cu

 















) (

'



용량 행렬 (capacity ma- trix)

강성 행렬 (stiffness ma- trix)

대류 행렬 (convection ma- trix)

하중 벡터 (load vector)

포물선형 방정식에서 대류항이 존재할 때 통상적인 유한요소 근사화를 적용할 경우 전 체 강성행렬이 대류행렬의 존재로 인해 수치적 불안정성을 야기한다 .

이를 극복하기 위해 다양한 유한요소 근사화가 개발되었다 .

1. CV-FEM : Control volume based Finite Element Method

72

한국계산과학공학회 기념워크샵

• 1940 년대

▫ Hrenikoff[1941] : Framework Method

선 요소

차원 봉이나 보

▫ Courant[1943] : Ritz Method

삼각형에서의 조각적

보간함수 이용

▫ Prager

와

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

는 에너지 원리를 이용한 행렬구조 해석법을 개발

유한 요소 구조해석 기술의 발전

73

한국계산과학공학회 기념워크샵

• 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 -

▫ Melosh, Grafton

Strome, Martin, Gallagher, Padlog, Bijlaard, Melosh, Argyris, Clough

Rashid, Wilson ,Turner, Dill, Martin, Melosh, Gallagher, Padlog, Bijlaa rd, Gallagher and Padlog

▫ Zienkiewicz, Watson, King, Archer, Melosh

▫ 장 ( 場 )

: Zienkiewicz

Cheung, Martin, Wilson and Nickel

▫ Oden : Nonlinear FEM

▫ Zienkiewicz : Fluid, Heat,Piezo,Plasma, Chemical reaction

▫ Brebbia : BEM

유한 요소 구조해석 기술의 발전