김 승 조*
Professor, Seoul National University
김민기
Seoul National University
문종근
Seoul National University
2009. 10. 12, 코엑스 인터콘티넨탈 호텔
Contents
‹#›
1
고전 역학적 관점에서 구조역학과 유체역학
2
구조역학과 유체역학의 수치기법 소개
3
유한요소 구조해석 기술 소개
4
범용 구조해석 프로그램 및 DIAMOND/IPSAP
서울대학교 항공우주구조연구실
Solid
Mechanics
Mechanics in Physics
Fundamentals of Physics by
David Halliday,
Robert Resnick,
Jearl Walker
Aerospace Structures Laboratory
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
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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
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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
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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
Ch33 Electromagnetic Waves
Aerospace Structures Laboratory
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
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Solid
Mechanics
•
Mechanics in Physics
Relativity
Ch37 Relativity
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Solid
Mechanics
•
•
•
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.
Aerospace Structures Laboratory
Solid
Mechanics
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 socalled "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 workStructures
of Newton.
Aerospace
Laboratory
•
Solid
Mechanics
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
Aerospace Structures Laboratory
Clos Lucé in France, where Leonardo died in 1519
Solid
Mechanics
Classical Mechanics ? –
Leonardo da Vinci
• Leonardo as observer, scientist and inventor
flight of a bird
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design for a flying machine
Solid
Mechanics
Classical Mechanics ? –
Leonardo da Vinci
• Leonardo as observer, scientist and inventor
helicopter
Aerospace Structures Laboratory
flying machine
Solid
Mechanics
Classical Mechanics ? –
Leonardo da Vinci
• Leonardo as observer, scientist and inventor
various hydraulic machines
Aerospace Structures Laboratory
grinding machine
Solid
Mechanics
Classical Mechanics ? –
Leonardo da Vinci
• Leonardo as observer, scientist and inventor
tank
Arsenal
Aerospace Structures Laboratory
Solid
Mechanics
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.
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ?
Galileo Galilei
(15 February 1564 – 8 January 1642)
Aerospace Structures Laboratory
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, 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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Solid
Mechanics
•
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.
Aerospace Structures Laboratory
Solid
Mechanics
•
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 n
egligible, or in the limiting case of its falling through a vacuum. He also derive
d the correct kinematical law for the distance travelled during a uniform accel
eration 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 entirel
y 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 16t
h, had suggested that bodies falling through a homogeneous medium would
be uniformly accelerated[ Galileo expressed the time-squared law using geome
trical constructions and mathematically-precise words, adhering to the standar
ds 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 frict
ion—acts upon them, refuting the generally accepted Aristotelian hypothesis t
hat objects "naturally" slow down and stop unless a force acts upon them (phi
losophical ideas relating to inertia had been proposed by Ibn al-Haytham cent
uries 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 th
at it had been mathematically expressed, verified experimentally, and introduc
ed the idea of frictional force, the key breakthrough in validating inertia). Galil
eo's Principle of Inertia stated: "A body moving on a level surface will continu
e in the same direction at constant speed unless disturbed." This principle was
incorporated into Newton's laws of motion (first law).
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
Galilei
• Improvement of Telescope and Astronomical Observation
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
Galilei
• Pendulum Motion
T  2
l
g
Galileo also claimed (incorrectly) that
a pendulum's swings always take the
same amount of time, independently
'Galileo's lamp' in the cathedral of Pisa
Aerospace Structures Laboratory
of the amplitude.
Solid
Mechanics
Classical Mechanics ? – Newton
Sir Isaac Newton
(1642-1727)
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
Newton
Sir Isaac Newton, (4 January 1643 – 31 March 1727) was an English physic
ist, mathematician, astronomer, natural philosopher, alchemist, and theolog
ian and one of the most influential men in human history. His Philosophiæ
Naturalis 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 gravi
tation 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 sa
me set of natural laws by demonstrating the consistency between Kepler's
laws of planetary motion and his theory of gravitation, thus removing the l
ast doubts about heliocentrism and advancing the scientific revolution.
Aerospace Structures Laboratory
Solid
Mechanics
•
•
•
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.
Aerospace Structures Laboratory
Solid
Mechanics
•
•
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 socalled "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
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ?–
• Bernoulli family tree
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Bernoulli family
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
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.
Aerospace Structures Laboratory
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 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.
Aerospace Structures Laboratory
Solid
Mechanics
•
•
•
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]
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
• J Jean le Rond d'Alembert (November 16
, 1717 – October 29, 1783) was a French
mathematician, mechanician, physicist an
d philosopher. He was also co-editor with
Denis Diderot of the Encyclopédie. D'Ale
mbert's method for the wave equation is
named after him.
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d’Alembert
Solid
Mechanics
•
Classical Mechanics ? – Lagrange
Joseph Louis Lagrange
(1736-1813)
.
Aerospace Structures Laboratory
Solid
Mechanics
•
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.
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
• Augustin Louis Cauchy
(1789-1857)
Aerospace Structures Laboratory
Cauchy
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 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.
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
• Claude Louis Navier
(1785-1836)
Aerospace Structures Laboratory
Navier
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
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.
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
• George Gabriel Stokes
(1819-1903)
.
Aerospace Structures Laboratory
Stokes
Solid
Mechanics
•
•
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.
Aerospace Structures Laboratory
Solid
Mechanics
Classical Mechanics ? –
Stokes
• Navier-Stokes equation
The Navier–Stokes equations, named after Claude-Louis Navier a
nd George Gabriel Stokes, describe the motion of fluid substance
s. These equations arise from applying Newton's second law to fl
uid motion, 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.
Aerospace Structures Laboratory
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
서울대학교 항공우주구조연구실
고전역학의 원리
‹#›
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.
서울대학교 항공우주구조연구실
고전역학의 분류
‹#›
• Lagrangian 방식
• 각 입자의 관점에서 물리량의 시간변화를 기술
• 모든 물리량은 각 질점 위에서 시간에 의해(t,x0) 결정됨
격자계가 입
자의 움직임
과 함께 변화
T=t0
T=t0+Dt
서울대학교 항공우주구조연구실
고전역학의 분류
‹#›
• Eulerian 방식
• 고정된 좌표(격자계) 상에 입자의 흐름을 기술
• 모든 물리량은 2개의 변수인 시간과 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
서울대학교 항공우주구조연구실
고전역학의 원리
‹#›
• 변형 중의 물체의 변형텐서 정의
Y
y
dx
xi  X i  ui
dX
dy
dz
dY
dZ
dX
Z
변형 전
dx
X
변형 후
z
• 변형 텐서 (deformation gradient Tensor) : Fij 
• 미소 위치벡터 변화량 : d x  d X  d u  Fd X
x
x j
X i
• 미소 부피 변화량 = 변형텐서의 행렬식 : det(F)=J
서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 연속체 장 방정식
▫ 고전역학의 5대 공리를 연속체에 적용한 방정식
1. 연속방정식 : J   0
σ : 응력 텐서
a : 가속도 벡터
b : 체적력
J : 미소부피 변화량
ρ : 밀도
2. 선운동량 방정식 : a      b
3. 각운동량 방정식 :  T  

4. 열역학 제 1법칙 :  e    q  r  tr(D)
e : 내부에너지/질량
q : 열유속 벡터
r : 복사열
D : 속도구배텐서
θ : 절대온도
η : 엔트로피/질량
q
r
5. 열역학 제 2법칙 :          0

  
서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 속도, 가속도 및 시간미분 관계식
▫ 가속도-속도–변위 관계식
 
x  x X , t , x0  X
v
d x x

dt t
d v v
a

dt t
X  fixed
X  fixed
v

t
x  fixed
 v xm

xm t
X  fixed
v
v

 vm
t
xm
▫ 임의의 물리량과 시간미분의 관계식
dp p

dt t

X  fixed
p
t

x  fixed
p xm
xm t

X  fixed
p
p
 vm
t
xm
  t p  v  p
서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 연속방정식
▫ 미소 부피 변화량의 시간미분

J  J tr( D)  J  v
▫ 연속방정식 양변 시간미분



J    J   J

  J  J  v


 J      v 


 

 J
 v      v 
 t

 


   v  0
t
Eulerian 기반 연속방정식
 

 

 J
    v    0   0
 t

서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 운동량 방정식
▫ 응력 텐서
   pI  Τ
ΤT  Τ
p : 압력
T : 점성응력
▫ 점성응력 텐서의 특성
 점성응력 텐서는 각운동량 보존 방정식에 의해 대칭텐서임
 점성응력 텐서는 속도장과 점성계수 및 내부에너지 등의 변수로 결정됨
▫ Navier 운동방정식
a  
dv
 v

    v  v   p    Τ  b
dt
 t

체적력(body force)
운동량 대류 항
점성응력 구배
압력 구배
서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 운동량 방정식
▫ 점성응력 텐서
 뉴턴 유체(Newtonian fluid)의 구성방정식(constitutive equation)
 뉴턴유체의 경우 점성계수 μ라는 단일 물리량에 의해 점성응력이 결정
2
Τ    tr( D)  2D
3
 속도구배 텐서
1  v j vi
Dij  


2  xi x j




 비압축성 뉴턴 유체의 운동방정식(Navier-Stokes equation)
 v

 v  v   p   2 v  b
 t


서울대학교 항공우주구조연구실
고전역학의 연속체 장 방정식
‹#›
• 에너지 방정식
▫ Fourier 의 전도법칙
q  kT
▫ 단위질량 내부에너지
p
p
e  h   C pT 


• k : 열전달계수
• T : 온도
• e : 내부에너지
• h : 엔탈피
• Cp : 정압비열
▫ 물성을 적용한 에너지 보존방정식 형태
 T

 v  T   k 2T  r  tr(ΤD)
 t

C p 
점성 소산(viscous dissipation)
열에너지 대류 항
복사 열 전달
열 전도
서울대학교 항공우주구조연구실
유동장 방정식
‹#›
• 구성방정식과 상태방정식을 결합한 유동장 방정식
▫ 응력텐서 구성방정식
   pI  Τ
ΤT  Τ
Τ  Τ( D,  , e,...)
▫ 열유속벡터 구성방정식
q  kT
e  h  p /   C pT  p / 
▫ 물성의 상태방정식
p  p(  , e,...)
k  k (e,  ,...)
   (e,  ,...)
....
서울대학교 항공우주구조연구실
유동장 방정식
‹#›
• 구성방정식과 상태방정식을 결합한 유동장 방정식
▫ 연속방정식 :
 

   v  0
t
v

 v  v   p    Τ  b
 t

▫ 선운동량 방정식 :  
▫ 각운동량 방정식 :
▫ 에너지 방정식 :
▫ 상태방정식 :
ΤT  Τ
 T

 v  T   k 2T  r  tr(TD)
 t

C p 
p  p(  , e,...)
k  k (e,  ,...)
   (e,  ,...)
....
서울대학교 항공우주구조연구실
유동장 방정식
‹#›
• 비압축성 뉴턴유체의 유동장 방정식
▫ 연속방정식 :
v  0
▫ 운동량 방정식 :
 v

 v  v   p   2 v  b
 t


▫ 에너지 방정식 : C p  T  v  T   k 2T  r  2 tr( D 2 )
 t

▫ 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
서울대학교 항공우주구조연구실
고체역학 방정식
‹#›
• 고체역학 방정식
▫ Lagrangian 기반 동역학 방정식의 경우 격자계가 곧 질점의 위치 상
에 있기 때문에 질량보존의 법칙은 자동으로 만족됨
▫ 물체의 변형상태를 보는 관점에 따라 세 가지 종류의 응력텐서 정의
가능함
1.
2.
3.
Cauchy Stress, true stress : 변형 후 형상에서 정의되는 응력
1st Piola-Kirchhoff stress : Cauchy 응력을 변형 전 초기형상으로 치
환한 응력
2nd Piola-Kirchhoff stress : 1st PK 응력을 대칭화한 응력
▫ 실질적으로 비선형 구조해석용도로는 1과 3의 정의가 많이 사용됨
(대칭이므로)
Cauchy stress
σ
-T
1st Piola-Kirchhoff stress
JσF
2nd Piola-Kirchhoff stress
JF σF
-1
-T
서울대학교 항공우주구조연구실
고체역학 방정식
‹#›
• 고체역학 방정식
▫ 유동장과 다르게 고체의 응력은 주로 변위장에 지배됨
▫ 응력텐서와 마찬가지로 변형률 텐서 역시 두 가지의 정의가 있음
1. Green-Lagrangian 변형률 텐서
2. Almansi-Hamel 변형률 텐서
: 변형 전 형상 기반
: 변형 후 형상 기반
T
Green-Lagrangian
½(F F-I)
Almansi-Hamel
½(I-F F )
-T
-1
3. 이후의 논의를 단순화하기 위해 선형화된 계로 가정함
4. 응력과 변형률의 정의를 변위가 작고 재질이 불변하다는 가정하에 아래
와 같이 단순화시킬 수 있음
1  ui u j 
 ij  

,

2  x j xi 
 ij  Eijkl kl
선형 변형률 텐서
응력 텐서와 탄성계수 텐서
서울대학교 항공우주구조연구실
고체역학 구성방정식
• 일반적으로 고체역학 방정식은 아래의 미지수와 방정식
으로 구성됨
15 미지수
15 방정식
• 6 strains
• 6 stress
• 3 displacement
• 3 equilibrium
• 6 strain-displacement relations
• 6 stress-strain relations
• 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
서울대학교 항공우주구조연구실
‹#›
고체역학 구성방정식
‹#›
• 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
서울대학교 항공우주구조연구실
고체역학 구성방정식
‹#›
앞 장의 대칭조건을 정리하면 아래 식처럼
21개의 독립적인 미지수를 얻을 수 있다.
21 unknowns
서울대학교 항공우주구조연구실
고체역학 구성방정식
‹#›
서울대학교 항공우주구조연구실
고체역학 구성방정식
‹#›
• 선형 등방성 재질의 응력-변형률 관계식
▫ 선형 재질의 경우 탄성계수 텐서는 두 개의 물리량으로 정리할 수
있다.
 ij  Eijkl kl
 11 
1
[ 11   ( 22   33 )]
E
 22 
1
[ 22   ( 11   33 )]
E
 33 
1
[ 33   ( 11   22 )]
E
 13 
1
 13
G
 12 
1
 12
G
 23 
1
 23
G
G
E
2(1   )
Young’s modulus : E
Poisson’s ratio : γ
두 개의 물리량으로 정의
서울대학교 항공우주구조연구실
구조역학 해석 종류
‹#›
• 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
서울대학교 항공우주구조연구실
구조역학 해석 종류
‹#›
• 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
서울대학교 항공우주구조연구실
구조역학 해석 종류
‹#›
• 고유치 해석
▫ 주기적인 하중이 가해질 때 그에 맞는 구조물의 거대한 진동이 발생
할 경우, 이 주파수를 고유주파수, 진동의 형상을 고유 모드라고 한
다.
▫ 수학적으로 선형시스템의 고유해를 구하는 문제로 설명할 수 있다.
u  ue eiwt ,
ue  0
 w M  K u
w
e
0
▫ 고유치 해석 역시 선형 정적해석과 더불어 널리 사용되는 구조해석
기법 중의 하나임.
▫ 수학적인 의미의 구조물 고유치 해석으로 좌굴 해석이 있음
서울대학교 항공우주구조연구실
수치해석 기법 소개
‹#›
• 유한차분법
▫ 가장 기본적이고 이해하기 쉬운 수치기법
▫ 수치적인 정확도가 낮고 보존식을 정확히 만족하기 힘들다는 단점
이 있으나 현재도 단순한 격자계의 유동장에서는 사용되는 수치기
법임
▫ 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
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

V t (  )dV  V  v  ndS  V   ndS  V f dV
대류 항
(convection)
확산 항 소스 항
(diffusion) (source)
▫ 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
서울대학교 항공우주구조연구실
수치해석 기법 소개
‹#›
• 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 최
소화하는 함수가 곧바로 방정식의 해가 됨
▫ 타원형 방정식의 예
   ku   bu  f
▫ 임의의 테스트 함수를 곱하고 적분을 취한 뒤 발산정리 적용
J (u, v)   (ku  v  buv)dV   fvdV  BCs
V
V
▫ 원래 방정식에 비해 생성된 범함수 J는 1차 공간미분항의 제곱이 적분
가능한 함수 범위에서 해를 검색할 수 있음
 미분가능성 제약조건이 약해짐
 2차 미분가능 -> 1차 미분의 제곱이 적분 가능
 범함수 J가 0이 되도록 하는 미지함수 u가 곧 원래 방정식의 해가 됨
서울대학교 항공우주구조연구실
수치해석 기법 소개
‹#›
• 유한요소법
▫ 유한차원에서 해를 찾기 위해 미지함수 u와 테스트함수 v를 동일한
기저(basis)를 사용해서 이산화하면 범함수 J는 다음과 같은 N차원
함수로 표현된다.
N
N
N


u   u j j
J (u, v)  J (u j , vi )   vi  Kiju j  Fi 
j 1
i 1
 j 1

N
v   vi i
• 기저가 되는 독립적인 함수들을 형상함수(shape function)라고 함
i 1
▫ 임의의 테스트 함수에 대해 J=0 이 되도록 하는 해
 이산화 과정을 거쳐서 아래의 대수방정식으로 치환할 수 있다.
N
K u
j 1
ij
j
 Fi
Kij   (ki   j  bi j )dV 강성행렬 (stiffness matrix)
V
Fi   ( fi )dV
하중벡터 (load vector)
V
서울대학교 항공우주구조연구실
수치해석 기법 소개
‹#›
• 유한요소법
▫ 2차원 평면응력 예제
 11 
   22 ,
 12 


 x
D 0



 y
 11 
   22 ,
 12 
 u1 
u 
u2 

0


,   Du ,   E  EDu
y 


x 
K   D  E D dV
T
V
F   T f ext dV
V
    DT   DT EDu
서울대학교 항공우주구조연구실
‹#›
• 보존식에 적용된 유한요소법
▫ 유동장의 수송방정식(transport equation)에 유한요소법을 적용
Cu '  ( K  K c )u  F
C    i j dV
(

 v   )    (   )  f 
t
용량 행렬(capacity matrix)
V
K    i   j dV
강성 행렬(stiffness matrix)
V
K c   v   i  j dV
대류 행렬(convection matrix)
V
F    i f dV
하중 벡터(load vector)
V
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 Method
2. Velocity-Pressure integrated Method
3. PS/SU PG : Pressure Stabilized/Streamline Upwind Petrov Galerkin
서울대학교 항공우주구조연구실
유한 요소 구조해석 기술의 발전
‹#›
• 1940년대
▫ Hrenikoff[1941] : Framework Method 선 요소 (1차원 봉이나 보)
▫ Courant[1943] : Ritz Method 삼각형에서의 조각적(piecewise)
보간함수 이용
▫ Prager와 Synge[1947] : 조각적(piecewise) 보간함수 이용
▫ Levy : 유연도법(하중법:flexibility matrix)을 개발
• 1950년대
▫
▫
▫
▫
▫
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.
▫ Argyris와 Kelsey[1954]는 에너지 원리를 이용한 행렬구조 해석법을
개발
▫ Turner, Clough, Martin, Topp[1956] ; Plane Stress 2차원 요소
서울대학교 항공우주구조연구실
유한 요소 구조해석 기술의 발전
‹#›
• 1960년대
▫
▫
▫
▫
▫
▫
▫
▫
▫
▫
▫
▫
▫
▫
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, Grafton과 Strome, Martin, Gallagher, Padlog, Bijlaard,
Melosh, Argyris, Clough와 Rashid, Wilson ,Turner, Dill, Martin,
Melosh, Gallagher, Padlog, Bijlaard, Gallagher and Padlog
Zienkiewicz, Watson, King, Archer, Melosh
장(場) 문제 : Zienkiewicz와 Cheung, Martin, Wilson and Nickel
Oden : Nonlinear FEM
Zienkiewicz : Fluid, Heat,Piezo,Plasma, Chemical reaction
Brebbia : BEM
Pian : Hybrid and Mixed FEM
서울대학교 항공우주구조연구실
대표적인 상용 프로그램
‹#›
Degree of Parallelism
Characterization of CAE Applications
FLUENT
CFD
OVERFLOW
STAR-CD
High
MP SCALAR
LS-DYNA Explicit FEA
Implicit FEA
(Direct Freq)
PAM-CRASH
MSC.Nastran (108)
RADIOSS
VECTOR
Low
ADINA MARC
Implicit FEA
(Statics)
ANSYS ABAQUS
Implicit FEA
(Modal Freq) MSC.Nastran
MSC.Nastran (101)
(103 and 111)
0.1
Memory BW
1
10
100
Compute Intensity
Flops/word of memory traffic
1000
Cache-friendly
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 I
‹#›
• MSC
NASTRAN
FE model
▫ Developed by NASA as analysis
tools for the structural analysis of
spacecraft. (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
structural parts are used as the
certification of quality.
 Certificated by FAA (USA)
vibration analysis
stress analysis
< Structural analysis of VAN >
Turbine blade thermal stress
analysis
Stress analysis of
Car Bumper
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 I
‹#›
 MSC
NASTRAN - Parallel Performance
www.mscsoftware.com , Ver. : 2007
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 II
‹#›
•
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 Integration
for the dynamic analysis
▫ Conduction, convection and heat
transfer problem
▫ Analysis of offshore structure
Stress analysis of
airplane engine
 wave-induced inertial force, buoyant
force and drag of fluid
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 II
‹#›
•
ABAQUS - Parallel Performance
E1: Car crash
E2: Cell phone drop
E3: Sheet forming
E4: Projectile penetration
(274,632 elements)
(45,785 elements)
(34,540 elements)
(237,100 elements)
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 III
‹#›
▫ Developed by John Swanson in 1970
▫ Utilized in conceptual design of the
product and the manufacturing
process
▫ Provides general graphic utilities
▫ Various analysis utilities
 Basic structural analysis, CFD, Electromagnetic analysis
 Thermal stress, Acoustic analysis,
Piezoelectric analysis
 Multi-physics
Engine analysis
▫ AI*NASTRAN solver
 Wavefrontal solver based on sparse
matrix solver,
 Substructuring analysis option for large
structures
▫ Block Lanczos eigen solver
▫ Distributed Pre-conditioned Conjugate
Gradient (DPCG) Distributed Jacobi
Conjugate Gradient (DJCG)
Infrared camera analysis
서울대학교 항공우주구조연구실
대표적인 상용 프로그램 III
‹#›
- Parallel Performance
▫ The solvers are:
 Distributed Pre-conditioned Conjugate Gradient (DPCG)
 Distributed Jacobi Conjugate Gradient (DJCG)
 Distributed Domain Solver (DDS)
 Algebraic Multigrid (AMG)
출처 : www.ansys.com , ANSYS ver. 10.0
서울대학교 항공우주구조연구실
세계적 전산 해석 소프트웨어 개발 현황
‹#›
EU
• Dassault, Falcon 7x
• AirBus, VDD
USA
Japan
• Famous Commercial Softwares
• SALINAS Project
•Adventure Project
•GeoFEM
서울대학교 항공우주구조연구실
대형 병렬 소프트웨어 개발 연구 I
‹#›
 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
서울대학교 항공우주구조연구실
대형 병렬 소프트웨어 개발 연구 II
‹#›
 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 Analysis
서울대학교 항공우주구조연구실
대형 병렬 소프트웨어 개발 연구 III
‹#›
 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
Geodynamo process and fluid dynamics in
the Earth’s outer core
A test dataset on the ES with
5,886,640 unstructured elements
Modeling of Philippine Sea
plate boundary
서울대학교 항공우주구조연구실
대형 병렬 소프트웨어 개발 연구 IV
‹#›
IPSAP
High Performance Parallel Finite Element Analysis Software based on Parallel MultiFrontal Algorithm
High-Performance Hardware
(Supercomputer, Clusters, GRID)
Grand Challenge Applications (Large Scale)
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
IPSAP/Standard
FEM Module
• FE Model
: Solid, Plate, Beam, Spring,
Rigid Body Element,
Concentrated Mass
• Nodal force, Pressure,
Acceleration, Temperature
load
• Thermal Module
• GUI Interface
Solver Module
Linear Solver
• Multifrontal Solver
• Hybrid Solver
Eigen Solver
• Block Lanczos Solver
Time Transient Solver
• Generalized Trapezoidal
Solver for Thermal Problems
IPSAP/Explicit
Lagrangian
Eulerian
• 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
서울대학교 항공우주구조연구실
Sponsored Research by Microsoft
‹#›
“Porting to Windows OS and Management of high performance FE software,
IPSAP (Internet Parallel Structural Analysis Program)
&
IPSAP Ver. 1.0
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.
Time Line
IPSAP Ver. 1.0 – Release to Public for Free
Improvement of IPSAP GUI for convenient and smooth execution in Windows”
IPSAP Ver. 1.1 to Ver. 3.0
IPSAP specialized in
Windows OS
• 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
Improving IPSAP
Usability
• Linear Static Solver
with Hybrid Solution
Method
• Block-Lanczos Eigen
Solver with Hybrid
Solution Method
• Thermal Analysis
•High-Level Contact
and Crash Analysis
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
2007.01.15
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
서울대학교 항공우주구조연구실
IPSAP/Standard – Multi-Frontal Solver
‹#›
• Concept of Multi-Frontal solver
▫ Utilization of multiple elimination fronts instead of single front
▫ Domain-wise Approach for efficient elimination procedure
Step1.
Step2.
Step3.
Step4.
Domain
partitioning
Symbolic
factorization
Numerical
factorization
Triangular
solve
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 0,1
Factorizatio
n
Proc 2
Proc 3
Proc 0,1,2,3
Factorizatio
n
Proc 2,3
Stage 1
Stage 2
Stage 3
Stage 4
New parallel linear algebra subroutines are
required which allow flexible block size
Parallel Linear Algebra Subroutine in C
PLASC is developed
서울대학교 항공우주구조연구실
IPSAP/Explicit
‹#›
• Parallelization
FE Calculation Parallelization
1
2
4
5
3
•
Compute at each
processor independently.
6
•
Interface values are
swapped and added.
7
8
9
Contact Parallelization
1
2
3
4
1. 3D box define along with Master Segment
2. Slave node information communication
3. Contact force calculation independently
4. Contact force vector communication
서울대학교 항공우주구조연구실
Serial Performance
‹#›
• Comparison with Commercial Software
▫ 32x32x32 hexagonal elements (DOFs = 107,811)
1500
1,345
Mesh Model
IPSAP
CS1
CS2
1000
700
602.625
552.328
600
500
305
Elapsed Time
CPU Time
500
331
112
147
93
203
0
400
300
Alpha EV67 (667MHz) IBM Power4 1.3GHz
Intel Xeon 2.4GHz
(Linux)
200
100
2.1 Gflops
88.1
44.765 43.063
85.2
0
IPSAP
6.52 Gflops
CS1
CS2
Windows Compute Cluster Server 2003, Intel
Core2 Quad 2.66GHz, 8GB Memory
서울대학교 항공우주구조연구실
Parallel Performance
‹#›
• IPSAP/Standard
– Static Analysis
- Intel Quad Core 2.66GHz
- Infiniband Network
4 Node Shell Element
– Vibration Analysis
- Intel Quad Core 2.66GHz
- Infiniband Network
SCALABILITY TEST(2D Topology)
16
14
14.07
14
IPSAP
12
12.87
IPSAP
12
Ideal
10
Ideal
10
Scalability
Scalability
SCALABILITY TEST (2D Topology)
16
8
6
4.16
4
8
6
5.11
4
2
2
1.00
1.00
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Number of CPU
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Number of CPU
서울대학교 항공우주구조연구실
Parallel Performance
‹#›
• Performance of extremely large scale problems
▫ 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
서울대학교 항공우주구조연구실
Parallel Performance
‹#›
• IPSAP/Standard
– Vibration Analysis
▫ comparison with Commercial Software
- PEGASUS Cluster
– Distributed memory parallel
1
2
4
8
16
32
64
128
mesh
DOF
N=100
61206
N=141
120984
N=200
242406
N=282
480534
N=400
964806
N=565
SCALABILITY TEST (2D Topology)
16
14.07
14
IPSAP
12
CS1
1922136
Ideal
10
8
6
5.11
4
N=800
2
3849606
N=1132
7702134
-
Parallel Performance Test on
Windows Cluster
Intel Quad Core 2.66GHz
Infiniband Network
Scalability
No. of CPU

1.00
One Lanczos iteration time
( summing up three above routines )
- Less than 1 sec up to 64 CPUs
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Number of CPU
서울대학교 항공우주구조연구실
Parallel Performance
‹#›
 IPSAP/Standard
– Heat Transfer Analysis
Time = MFS Time + Transient Time
- PEGASUS Cluster
- OS : Linux Redhat 9.0 kernel 2.4.26
- CPU : Xeon 2.2GHz 1 ~ 16
# of CPUs
1
2
4
8
16
Time
(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
700
20
Time
500
15
400
Speed-Up
Time(sec)
Ideal
IPSAP(Time Step=1)
IPSAP(Time Step=10)
Transient
600
300
200
10
5
100
0
0
10
20
30
The number of Time Steps
40
0
0
5
10
15
20
number of CPUs
서울대학교 항공우주구조연구실
Parallel Performance
‹#›
• IPSAP/Explicit
Parallel Performance Test on
Windows Cluster
Intel Quad Core 2.66GHz
Infiniband Network

-
SCALABILITY TEST (Taylor Impact Test)
16
IPSAP
14
15.13
Ideal
12
10
Scalability
▫ comparison with Commercial
Software
- PEGASUS Cluster
– Distributed memory parallel
– 127 Speed up / 128 CPUs
8
6
4
4.01
2
0
1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Number of CPU
서울대학교 항공우주구조연구실
Inexpensive SuperComputing Resources
‹#›
• New Type Super Computing System
– Windows HPC Server 2008
– Infiniband Network
Compute Node
Head Node &
Compute Node
N001
hpc.net
Public
Network
Unit Node
N002
hpc.net
N003
hpc.net
Total System
CPU
Intel Core 2 Quad 3.2 GHz
(4 Core X 2)
32 Cores
RAM
DELL : DDR2 ECC 64GB
(4GB X 16)
Mac : DDR2 ECC 32GB
(2GB X 16)
224 GB
HDD
SATA 500 GB
2 TB
N004
hpc.net
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)
서울대학교 항공우주구조연구실
Computing Performance Result
‹#›
• Scalability & Speed-up test
▫ 2D Mesh topology
No. of
Cores
Mesh
Number of
Unknowns
Performance
(GFLOPS)
Scaled
Speedup
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
서울대학교 항공우주구조연구실
Computing Performance Result
‹#›
• Scalability & Speed-up test
▫ 3D Mesh topology
No. of
Cores
Mesh
Number of
Unknowns
Performance
(GFLOPS)
Scaled
Speedup
8
64_64_64
823,875
46.125
1
16
72_72_72
1,167,051
98.255
2.130
32
80_80_80
1,594,323
137.443
2.980
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
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
서울대학교 항공우주구조연구실
Necessity of Pre/Post GUI Software for IPSAP
‹#›
STRESS
ANALYSIS
Preprocessing
VIBRATION
ANALYSIS
Analysis
DIAMOND/IPSAP
Post-Processing
IPSAP/
EXPLICIT
서울대학교 항공우주구조연구실
Development of User-Friendly Visualization Toolkit
- DIAMOND/IPSAP
‹#›
 Development
of
Pre-/Post-Visualization
Toolkit for IPSAP
 Development



Windows Visual Studio 2008
Graphic Library :
Open CASCADE 6.2.0
Adoption of Ribbon UI
 Realized






Environment
Functions
Pre-Processing
Analysis of IPSAP
Post-Processing
Displacement/Stress View
Eigen Mode View
Parallel Analysis of IPSAP
IPSAP input file
import
DAIS
Manager
View
Control
Diamond document
mesh
property, material . . .
output file
input file
Create
Geometry
Line, Surface, Solid
Mesh
Beam, Plate
Load, Boundary
Material, Property
서울대학교 항공우주구조연구실
Main Frame of DIAMOND/IPSAP
‹#›
서울대학교 항공우주구조연구실
Specific Application Module of DIAMOND/IPSAP
Realization of Several Application Modules based on DIAMOND/IPSAP
‹#›
 Educational FE Code for
Partial Differential Equation
 Satellite Bus Design
Optimization Module
 Semi-Conductor
& MEMS Packaging
Simulation
 Virtual Simulation
& Experiment
 Dynamics & Stability
of Helicopter
Rotor Blade System
 Crash & Impact
Simulation
 Option Pricing
서울대학교 항공우주구조연구실
‹#›
Thank you!
서울대학교 항공우주구조연구실